journal of mathematics teacher education_5

EDITORIAL

JMTE: A TIME TO REFLECT

The Journal of Mathematics Teacher Education (JMTE) has reachedVolume 5. This means it is in its fifth year, and is no longer ‘newlyborn’. In its four years of issues it has established a scholarly tradition inreporting research on the learning of teachers and development of teachingof mathematics. Fundamentally, of course, as with all academic writing inmathematics education, it is concerned with the enhancement of learningof students of mathematics. I shall say more of this shortly. My reason foremphasising these matters is that this issue of JMTE comes under a neweditorial team. JMTE’s originating editor, and visionary scholar in the fieldof mathematics teacher education, Tom Cooney, has decided it is time toretire from this editorship. It is therefore a time to reflect on JMTE, onwhat has been established, and to look ahead to the future.

Tom has shown a fine judgment and sensitive regard, both for the disci-pline and the authors, in editing JMTE. It is unsurprising, given his longrecord in this field, that he has brought knowledge, expertise and wisdom tothe role of editor. However, he has also brought a sincere desire to enhancethe processes and practices of mathematics teacher education through thecareful nurturing of academic writing and the respectful, caring, time-giving and painstaking way he has worked with contributors. We have alot to learn from Tom’s dedication and professionalism.

In taking over from Tom, I am delighted that three eminent scholars inmathematics teacher education have agreed to join me in editing JMTE.They are Konrad Krainer, of the University of Klagenfurt, Austria; PeterSullivan from La Trobe University, Melbourne, Australia; and Terry Wood,from Purdue University in the United States. We intend to work closelyas a team, as well as to continue Tom’s personal touch in working withauthors. We welcome also a new and distinguished Editorial Board. I amvery pleased that its members are enthusiastic about working with JMTEand supporting its development and scholarship.

In his very first editorial (JMTE 1) Tom expressed his intention that“JMTE will honour diversity”. He recognised that

Journal of Mathematics Teacher Education 5: 1–5, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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. . . the knowledge domains of mathematics teacher education have no sharp demarcationlines and often involve fields seemingly unrelated, at first glance, to the education ofmathematics teachers. This theoretical eclecticism is not unique to mathematics teachereducation, as it characterizes the field of mathematics education more generally (p. 1).

These observations remain central to our philosophy for the progress ofJMTE. We share Tom’s belief in “the integration of our study of teachereducation with our practice of teacher education”, and all areas of theoryand practice that impinge on this relationship are welcomed as focuses forJMTE.

So, we recognise and applaud Tom Cooney’s vision and its realisationin the early years of JMTE. We acknowledge him as Founding Editor;we are honoured to have him on the new Board, and hope that retirementwill not prevent him from contributing to the ongoing development of thisjournal. Thank you Tom, wholeheartedly!

TEACHING, LEARNING AND MATHEMATICS

Pearson (1989) tells us that the intention of teaching is ‘bringing aboutlearning’ (p. 64; my emphasis). Through a study of mathematics teaching,we therefore learn about the ways in which mathematics teaching bringsabout the learning of mathematics. Such study provides insights into howmathematics teaching develops or can develop for the enhancement ofmathematics learning of students. Of course, teaching does not developin the abstract, but through the growth of knowledge and experienceof mathematics teachers as they engage in the practices and processesof teaching and participate in activities related to their own learningas teachers. The perspectives and practices of educators, working withteachers to foster and facilitate teacher learning are as much objects ofstudy as those of teachers working with students. The research process atall these levels is itself a learning enterprise. Thus, teachers, educators andresearchers are all learners and our integration of research and practice isfundamentally a study of learning.

The three papers you will find in this issue all address learning ata multiplicity of levels. Janine Remillard and Pamela Geist discuss thelearning of teacher educators as they work to facilitate the learning ofelementary teachers. Of course the teachers are concerned with the mathe-matical learning of their students, and thus discussion between educatorsand teachers focuses critically on the learning of mathematics. The authors,skillfully, weave the two themes in addressing issues and dilemmas inteacher education and the learning of educators. Ruth Heaton and WilliamMickelson have engaged in research into the learning of statistics, by

EDITORIAL 3

student teachers and their pupils in classrooms, through a process ofconducting statistical investigations. We are introduced to the questionsand issues, both mathematical and pedagogical, that the student teachersencountered in conducting their own statistical investigations, and orches-trating those of their pupils. The authors, as educators, reflect on theirown learning experiences in researching these processes and practices. EricKnuth reports on a study of secondary mathematics teachers’ perceptionsof mathematical proof. His study explores how teachers’ mathematicalperceptions relate to their pedagogical perceptions, and how teachers’ ownunderstandings of proof develop. Through the teachers’ perceptions weglimpse some of the issues related to students’ development of knowledgeof proof in classroom settings.

MATHEMATICS

As editors, as we read papers from authors, we sometimes have to addressthe question of what distinguishes papers in mathematics teacher educa-tion from those in teacher education more widely. The answer is notsimply that mathematics figures in them somewhere. In the general fieldof teacher education, there is always something that is taught, and thisis sometimes mathematics. However, in mathematics teacher education itseems reasonable to expect that mathematics will be a major considera-tion in issues addressed. Such is the case with papers in this issue whichinclude significant discussion of the teaching and/or learning of proof,statistics, and decimal numbers, all important areas of mathematics thatfigure in curricula internationally. Unsurprisingly, within a discipline ofteacher education, the discussion of mathematics and its learning is woveninto the wider discussion of teacher learning and teaching development.However, it seems important to ask questions about how teacher learningand teaching development are distinctively related to mathematics as theirsubject.

One way in which this distinction can be made is in considerations ofteachers’ mathematical knowledge and its suitability or adequacy for thelevel of teaching in which the teachers are engaged. In the Heaton andMickelson paper, for example, we see teachers learning more about statis-tics as they work with their students on statistical investigations. To whatextent do we expect teachers’ mathematical knowledge to grow throughpedagogical engagement, and to what extent do teachers need a moredirect mode of being taught mathematics? What is the nature of such a‘more direct’ mode, and how can it relate to the way teachers work on

4 EDITORIAL

mathematics with their pupils? These are issues that we hope JMTE canaddress.

TEACHERS AND TEACHING

JMTE is interested in studies of mathematics teacher learning and teachingdevelopment at all levels of education, including elementary/primary,secondary and tertiary/university education. This includes teaching issuesrelated to the mathematical learning of pupils in classrooms, the learningof student teachers, and that of experienced teachers. It also includes thelearning of teacher educators. How these layers of learning are distinctor interwoven is a fascinating consideration in many papers. In the paperfrom Remillard and Geist, for example, we find discussion of the natureof ‘facilitation’ as educators work with teachers. In what ways is facili-tation the same as teaching, or something different? How does knowledgegrow in the various layers? What are the implications of seeing classroomteaching of mathematics as ‘facilitation’? And, importantly for this paper,who facilitates the facilitators? As educators, seeking to ‘bring aboutlearning’ for others, we learn ourselves: a form of self-regulation inlearning. To what extent is our self-regulation related to our engagementin research, and can such relationships be exploited fruitfully for learningat other levels? Again, these are issues that we hope JMTE can address.

EDUCATION AND RESEARCH

The questions articulated above just start to open up the field of mathe-matics teacher education, and to lead to the kinds of studies that willaddress such questions. In order to develop knowledge and theory in thefield we need not only good examples of programmes in teacher educa-tion and the issues they raise, but also a critical and cogent addressing ofthese issues through research. This is not to say that we do not welcomepapers that focus on particular programmes, but that we would look forinsights arising from such programmes to be related through research andthe wider literature to concerns that affect mathematics teacher educationmore generally.

JMTE is an international journal, and therefore we seek insightsinto ways in which practices and issues relate to national and interna-tional contexts, and vary according to political and cultural concerns.In addressing the literature related to issues, it seems important not tobe restricted just to national contexts but to relate our national settings

EDITORIAL 5

to knowledge and theory internationally. We should like to see JMTEfostering international awareness, and thus overtly encourage authors fromall countries to see JMTE as an outlet for their work.

A new direction in which JMTE could develop is to include shortcommentaries, for example, those which respond to a published paper, orthose which take up an issue and develop it theoretically. In addition to oursection of papers on Teacher Education Around the World, to which westill invite contributions, we forsee a section that devotes itself to ReaderCommentary. Of course, as with all papers submitted to JMTE, such shortcommentaries will go through the same rigorous review process with highexpectations of scholarship. We invite short papers of this nature.

In conclusion, as least where this editorial is concerned, I should liketo see us identifying the ‘big ideas’ in mathematics teacher education, and,from these, developing significant theoretical perspectives through whichthis new discipline of mathematics teacher education can gain substanceand through which our future practices can be enhanced. I leave the lastword with Tom Cooney, who wrote,

But if we are to move beyond collecting interesting stories, theoretical perspectives needto be developed that allow us to see how those stories begin to tell a larger story. That is weshould be interested in how local theories about teachers can contribute to a more generaltheory about teacher education. (Cooney, 1994, p. 627)

REFERENCES

Cooney, T. J. (1994). Research and teacher education: In search of common ground.Journal for Research in Mathematics Education, 25, 608–636.

Pearson, T. A. (1989). The teacher: Theory and practice in teacher education. London:Routledge.

JANINE T. REMILLARD and PAMELA KAYE GEIST

SUPPORTING TEACHERS’ PROFESSIONAL LEARNING BYNAVIGATING OPENINGS IN THE CURRICULUM �

ABSTRACT. Researchers agree that achieving the fundamental changes called forby current reforms in mathematics education requires new learning on the part ofteachers. Currently, across the United States there exists a tremendous variety of teacher-enhancement projects representing a range of perspectives and approaches to supportingteachers’ learning. This paper presents a comparative analysis of three teacher educatorsusing a curriculum designed for use with elementary teachers in an inquiry-group setting.The aim of the study was to examine the process and demands of supporting teachers’learning and teachers’ efforts to reform their practices. Analyses revealed that the centraldemand of supporting teachers’ learning through inquiry involved navigating through whatwe have called openings in the curriculum. These openings took the form of unanticipatedquestions, challenges, observations, or actions by participating teachers that required facili-tators to make on-the-spot judgments about how to guide the discourse. Examinations ofthe facilitators’ processes of navigating these openings revealed a set of three activitiesthey employed in determining how to respond. Analysis of the activities of facilitators inresponse to openings further illuminates the work involved in supporting teachers’ learningand has implications for the skills needed by teacher educators engaged in this work.

KEY WORDS: curriculum material use, mathematics education, professional develop-ment, teacher development, teacher inquiry, teacher learning

INTRODUCTION

The images of mathematics teaching and learning envisioned by thecurrent reform movement in the United States are foreign to most U.S.teachers. As a result, reforming mathematics education requires substan-tial new learning on the part of teachers (Ball, 1997; Simon, 1997).To encourage this learning, professional development opportunities forteachers also must change (Cohen & Barnes, 1993; Heaton, 2000; Sykes,1996). In recent years, the growing body of research on teacher learningand change has provided insights into the kinds of learning that are likelyto support significant shifts in mathematics teaching. Many researchersagree that teachers need opportunities to develop deep understandings

� This research was funded in part by the National Science Foundation (grant no. ESI-9553908). The views expressed in this paper are the authors’ and are not necessarily sharedby the grantors.


8 JANINE T. REMILLARD AND PAMELA KAYE GEIST

of mathematics and of students’ mathematical thinking and development(Ball, 1993; Schifter, 1998). We have learned that teachers’ pedago-gical decisions are closely connected to their beliefs about students,learning, and the aims of education (Fennema, Carpenter & Franke, 1996;Thompson, 1992). Finally, considerable evidence suggests that the kindof learning that supports fundamental change in teaching occurs over along period of time, with extensive support and multiple opportunities toexperiment and reflect (Loucks-Horsley, 1997; Nelson, 1997).

There is less agreement, however, about how to foster and support thiskind of learning (Ball, 1997). Currently a tremendous variety of teacher-enhancement projects exists across the world, representing a range ofapproaches to promoting teacher learning and change (Loucks-Horsley,1997). This study examines one such project. The authors studied threeteacher educators using an innovative teacher development curriculum inthe United States. The aim of our research was to study the process anddemands of fostering learning that supports teachers’ efforts to reform theirpractices.

THE CURRICULUM AND THE CHALLENGE IT OFFERS

Developing Mathematical Ideas (DMI), designed by Schifter, Bastableand Russell (1999), is a curriculum intended for use with elementaryteachers in an inquiry-group setting. Through cases of students’ mathe-matical thinking written by teachers, group discussions, and mathematicalinvestigations, the materials provide opportunities for teachers to simul-taneously examine central mathematical ideas and students’ thinking aboutthem. The curriculum rests on an assumption that, through examiningtheir own and children’s understandings of mathematical structures andrelationships underlying the elementary curriculum, teachers will learnmathematics in new ways and reconsider what it means to learn and knowmathematics. The DMI developers expect new insights teachers gain fromthese mathematical explorations to prompt rethinking of what it means toteach mathematics. This approach is in concert with Ball’s (1997) call forprofessional development to foster a stance of critique and inquiry, ratherthan one of answers.

At the time of the study, the DMI curriculum included two modules,“Building a System of Tens” and “Making Meaning of Operations.”Through a sequence of eight 3-hour sessions, each module chroniclesthe development of children’s mathematical understandings as they movefrom kindergarten into the middle grades. In preparation for each session,participants read teacher-narrated cases of classroom episodes that illus-

OPENINGS IN THE CURRICULUM 9

trate student thinking and work. In addition to reading and discussingcases, teachers explore mathematics for themselves, share and discusssamples of their own students’ work and understandings, view videotapesof mathematics classrooms, and write their own cases.

The DMI curriculum guides facilitators’ work by providing activities,readings, and a structure for each meeting. It also includes reflectivejournal entries of a fictitious DMI instructor as she guides a group ofteachers through the modules. Each entry provides an image of howthe facilitator interprets and reflects on the interactions of participants.Research on K-12 teachers using reform-oriented curricula, however,suggests that implementing an innovative curriculum is not simply a matterof picking it up and using it (Cohen, 1990; Heaton, 2000; Lloyd, 1999;Remillard, 1996, 1999). It involves interpreting new and unfamiliar ideasabout teaching and learning. Thus, using an innovative curriculum forteacher development is likely to involve at least two layers of complexityfor teacher educators. The first layer involves working with unfamiliarideas about children’s mathematical learning. The second layer involvesfinding one’s way through new approaches to teachers’ learning. Forthis reason the DMI curriculum provided a productive site to examinethe following question: What is involved for facilitators as they (a)use an innovative teacher-development curriculum and (b) support thekinds of teacher-learning opportunities compatible with reform ideas inmathematics education?

WHAT WE KNOW ABOUT CURRICULUM AND REFORM INMATHEMATICS EDUCATION

The preceding question is related to existing research on reform-inspiredteaching and teachers’ use of curriculum materials. Nevertheless, there islittle research that examines such questions when the teacher is a teachereducator and students are practicing teachers. In fact, research on howteachers interact with and use curriculum materials is relatively new. Previ-ously, textbooks and curricula were viewed as accurate representationsof classroom curriculum (Walker, 1976). Implicit in this perspective wasa view of the teacher as a conduit for curriculum, not a user or shaperof it. Observations of teachers using the “teacher-proof” materials ofthe 1950s and 1960s suggested that many teachers did not use the newcurriculum materials as the authors had intended. Stake and Easley (1978)described adaptations to inquiry-based curriculum that reflected teachers’notions about teaching and the nature of the subject matter. Sarason(1982) observed teachers’ struggles to understand the “New Mathematics”


materials, noting a clash between their beliefs about mathematics and theideals represented in the materials. Studies such as these illustrated thesubstantial role that teachers play in shaping the curriculum experiencedby students.

Researchers have since examined teaching and teachers’ use ofcurriculum guides, seeking further insight into teacher-text relationships.Scholars who have examined the beliefs underlying teachers’ use ofcurriculum materials have concluded that a variety of factors tend toinfluence teachers’ decisions, including their knowledge of and viewsabout mathematics (Graybeal & Stodolsky, 1987; Thompson, 1984), theirperceptions of the text (Bush, 1986; Remillard, 1991; Woodward & Elliot,1990), their perceptions of external pressures (Floden et al., 1980; Kuhs &Freeman, 1979), and their ideas about the purpose of school and the natureof learning (Donovan, 1983; Stephens, 1982).

From another perspective, researchers have argued that placing theteacher-text relationship at the center of analyses oversimplifies teachers’curricular decisions. In a study of elementary teachers, Sosniak andStodolsky (1993) found that teachers did not see textbooks and teachers’guides as “blueprints” or “driving forces,” but as “props in the service ofmanaging larger agendas” (p. 271). By capturing the role of the text inrelation to teachers’ varied responsibilities, these findings suggest a needto consider teachers’ larger curricular agendas and the role the curriculumguide plays in them.

Research on what Doyle (1993) called the “curriculum process”considers teachers’ larger agendas by focusing on how they enactcurriculum in their classrooms. This research focuses less on the teacher-text relationship and more on the teacher-curriculum relationship. It oftenincludes how teachers draw on resources such as curriculum guides, butassumes that this process necessarily involves interpreting the meaningsand intents of these resources (Doyle, 1993; Lemke, 1990; Snyder, Bolin& Zumwalt, 1992). Implicit in studies of teachers’ curriculum processesis a view that the enacted curriculum is more than what is capturedin official policy documents or textbooks. It is the events teachers andstudents experience in the classroom (Clandinin & Connelly, 1992). Fromthis perspective, studying teachers’ use of innovative curriculum resourcesinvolves trying to understand teachers’ processes of constructing theenacted curriculum and the role in it that resources play. With this in mind,our research examined the curriculum enacted by teacher educators inprofessional development settings and how they used the DMI curriculumin the process.


Our research questions also were influenced by recent research thatexamines the work of teaching in today’s reform context. As severalscholars have pointed out, the current calls for reform envision a modelof teaching that is significantly more complex than the traditional imageof the all-knowing guide who corrects students and monitors their practice(e.g., Ball, 1997; Remillard, 1999; Simon, 1997; Steffe, 1990). Reform-inspired goals for all students that include mathematical thinking, problemsolving, and communication require teachers to engage simultaneouslyin a number of inquiry-oriented activities. Through ongoing observationand analysis of students’ performances, teachers build models of students’mathematical understandings and generate hypotheses regarding how theirlearning might progress (Simon). They take actions based on these hypo-theses, but must continually modify their models and subsequent plans. Ina sense, they must both direct and follow the activities of students. Simonaptly characterizes the teachers as “function[ing] within the tension amonghis or her current goals for student learning and commitment to respond tothe mathematics of the student” (p. 80).

We assume that the work of teacher educators in this reform contextis equally complex. Not only must they help teachers engage in learningabout teaching that is unfamiliar and highly complex, they must takeinto account new ideas about how teachers are likely to learn. The DMIcurriculum proposes one hypothesis about teacher learning: By examiningtheir own and children’s understandings of mathematics, teachers willlearn mathematics in new ways and rethink their teaching of it. Our aimin this research was to examine facilitators’ work supporting this sort oflearning whether through use of the DMI curriculum or other means.

METHODS AND CONTEXT

To examine the work involved in supporting teachers’ learning, we usedqualitative, interpretive methods to study three teachers/teacher educatorsusing the DMI curriculum.1 The three teachers were among a group ofapproximately 10 facilitators involved in piloting the DMI materials priorto final publication. Our research represents part of the research undertakenduring this pilot year. Other concurrent studies examined participatingteachers’ learning and classroom practices. The three research sites variedacross several dimensions, which we describe below. We selected thesethree facilitators to study because they were from such varied settings andrepresent differing backgrounds. None of the facilitators received specialtraining in using the curriculum; however, most were involved in ongoing


conversations with the developers about its design and intent and about theprocesses of using the curriculum.

The Three Contexts

Marilyn,2 a middle school mathematics teacher, was new to teacher devel-opment work. She facilitated the DMI seminar through an agreementbetween her school district and a local college. She offered the seminarto elementary teachers in her district as a two-credit mathematics coursewhile continuing to teach middle school mathematics. Four participantsenrolled in the course: three veteran elementary school teachers and onemiddle school teacher with two years’ teaching experience. Participantsmet weekly for three hours over the course of the spring term.

Jennifer was a veteran teacher educator. A former elementary schoolteacher, she served as the curriculum specialist of her district for 13years and sponsored and facilitated a wide range of teacher-enhancementprojects. Jennifer offered the DMI seminar to a group of 30 teachers whohad been meeting monthly for professional development and discussionwhile piloting a new elementary mathematics curriculum. The experienceof teachers in this group ranged from 7 to 13 years of service. The groupmet once each month for six months, devoting mornings to the DMIcurriculum and afternoons to the pilot project.

Connie, a mathematics teacher educator in a university setting and anexperienced middle school teacher, used the DMI materials in a continuingeducation master’s degree course offered through her institution. Thecourse met once a week for one semester. Drawn from school districtssurrounding the university, the participants were practicing teachers withexperience ranging from 1 to 30 years. A few of the participants wereformer students in Connie’s mathematics methods course. The seminarbegan with 30 students, but enrollment dwindled to 15 by the end of thesemester. Connie brought to her use of the DMI materials an array ofexperiences facilitating professional development activities for practicingteachers.

Data Collection and Analysis

We collected data on each seminar through observations of the three-hoursessions and follow-up interviews with the facilitators, which were audio-taped and transcribed. In our observations, we paid particular attention tothe facilitator’s role in orchestrating the activities of the session, her useof the curriculum resources, and the way she responded to participants’ideas and questions. During the interviews, we asked the facilitators togive us their impressions of the session and to point out any instances that


concerned or excited them. We also asked each facilitator about specificevents from the session that stood out to us as possible decision-makingpoints and inquired into her use of the curriculum materials and her plan-ning for the sessions. As the investigation proceeded, we became partic-ularly interested in the facilitators’ responses to unanticipated participantcomments or actions.

We analyzed the data using within-case and cross-case inductivemethods of analysis (Patton, 1990). One researcher analyzed data for eachfacilitator with an eye toward characterizing the work involved in facili-tating the curriculum. Drawing on fieldnotes, we identified the kinds ofactivities the facilitators engaged in before, during, and after the sessionsand looked for themes across these activities. The cross-case analysisinvolved iterations of comparative examinations of each facilitator’s workby both researchers together, followed by additional checks with each caseto confirm validity and check for disconfirming evidence. As we exploredthe variety of challenges the facilitators faced, we developed the themeof openings to characterize the way their work was similar. Navigatingopenings, as described in our analysis, was a central piece of all threefacilitators’ ongoing work.

OPENINGS IN THE CURRICULUM

Despite the differences among facilitators and the seminar contexts, thethree facilitators confronted unanticipated and at times awkward pointsin the conversations through which they had to navigate. These instanceswere prompted most often by participants’ questions, observations, chal-lenges, or resistant stands on issues that were important to them. We havelabeled these instances openings in the curriculum because they requiredfacilitators to make judgments, often on-the-spot decisions, about how toguide the discourse. Here the curriculum refers to the enacted curriculum,the events teachers and students experience (Clandinin & Connelly, 1992).Initially we viewed these openings as interruptions in the natural flow ofthe sessions because they often felt clumsy or precarious to these first-timefacilitators of DMI. Through our analysis, we came to view these breaksas potentially rich spaces in the curriculum because they presented oppor-tunities for facilitators to foster learning by capitalizing on mathematicalor pedagogical issues as they arose. As we discuss later, openings reflecttensions inherent in the type of teacher development work envisionedby the DMI creators. They are the natural consequence of interactionsbetween what participants bring to the seminars and the kinds of learningopportunities proposed by the DMI curriculum.


In the sections that follow we describe three openings that stood outin the data. We selected these three examples because they challengedfacilitators in ways that made the process of navigating through themparticularly visible. Yet, the three openings discussed here are not theonly types of openings we observed. As we worked to understand whatopenings in the curriculum involved, we began to identify openings thatwere more transparent than the three examples that follow. The apparentinvisibility of these openings was due to the seamless way in which theywere navigated by facilitators.

Searching for Pedagogical Guidance

The first type of opening we describe occurred as participants in theseminars turned conversations away from the agenda at hand and towardquestions about mathematics pedagogy. Often the content of these conver-sations involved participants’ questions relating to a particular pedagogicalapproach, which, as the questions implied, they believed the facilitatorsadvocated. These queries forged openings in the curriculum that called onthe facilitator to respond.

Connie confronted this type of opening often. Participating teachersregularly sought specific advice from her about their teaching. These soli-citations occurred during class sessions, but also arose as conversations“on the side.” In fact, the first instance arose early in the first meeting.In preparation for the session, participants read an assigned set of cases onchild-derived algorithms for adding and subtracting two-digit numbers andbrought examples of three students’ work. Connie instructed the teachersto group themselves according to the grade they taught and discuss thework samples. After moving around the room and giving the groups a fewminutes to get started, Connie sat down with the five first-grade teachersand listened to their conversation.

Almost immediately Lucille, one of the teachers, began to solicitConnie’s advice about her teaching. Lucille’s face was strained and hervoice determined. She explained that she showed her students how touse counters to add together two numbers. “Am I overshadowing themby showing them how to do this?” she asked, pointing to the exampleof student work which showed number sentences like “2 + 3” andcorresponding drawings of 2 circles and 3 circles.

Connie paused for a moment and then asked several questions: “Whatkinds of responses have you had from your students when you give themproblems like these? How many problems do you give them? How muchdo you let them struggle to figure out their own strategies? What kinds ofstrategies do you see?”


Lucille’s response was puzzled and sincere. “Don’t you have to teachthem?” she asked. “We teach them the basic things and then I see them usestrategies.” She reminded Connie that her students had been in first gradefor just a few weeks. Connie listened and nodded as Lucille explained herconcerns. Connie then looked at the other four teachers in the group, whowere listening, and reminded them to be sure that everyone got a chance toshare the work they had brought. She then slipped to another group. Lucilleand other teachers attempted to draw Connie into similar conversationsthroughout the seminar (Observation, 9/10/96).

In a conversation after class, Connie referred to these interactions asmoments when her “biggest fears came true” (Interview, 9/10/96). “Iwould sit down, they would revert away from the discussion of what theywere sharing to talking about curriculum and hitting me with questions.”She found these questions frustrating because they seemed to be skep-tical responses to the pedagogy in the cases or attempts to sidestep themathematics or students’ engagement with the content.

Marilyn and Jennifer struggled with similar openings in the curriculum.Jennifer generally chose not to respond directly to questions aimed atprobing her views about good pedagogy. Instead, she often waited forparticipants to respond, which many were inclined to do. If no oneresponded, Jennifer directed the group on to the next focus question,suggesting that participants postpone their inquiries and focus first onlearning what they could about students’ engagement with the mathe-matical ideas.

In navigating openings created by participants’ pursuits of pedagogicalguidance, all three facilitators resisted making specific recommendationsor assertions about teaching. They did so for a variety of reasons. As aresult of their own beliefs and the inquiry orientation of the curriculum,they all shared the view that it was not the facilitator’s role to promoteparticular approaches to teaching; they believed the facilitator shouldprovide opportunities for participants to construct their own ideas aboutteaching. Connie explained, “I deliberately tried not to give advice.” Whenteachers persisted, she took a more explicit approach in defining her role,which she described in an interview:

I said, “You know, I want us to continue to ask this question every week, and talk delib-erately about what you are thinking. If I told you what I wanted you to do, that wouldn’treally make any impact, and what I’d really like you to do is see where your beliefs andconceptions are moving to. But, I do want us to continue to talk about it each week, andI’d be happy to be a sounding board” (Interview, 9/10/96).

The facilitators’ ideas about the central purpose of the seminar alsoinfluenced their decisions to avoid responding to these questions or chal-


lenges. Both Connie and Jennifer believed that the purpose of the DMIcurriculum was to engage teachers in inquiry about mathematics andstudents’ thinking. Jennifer explicitly suggested that participants postponetheir pedagogical questions and focus on the issues in the cases.

The facilitators also avoided challenging these solicitations for guid-ance – particularly the confrontational ones – because they hoped to avoidconflict. They worked hard to create a supportive and congenial atmo-sphere in which participants respected the views of others. Because theyfelt that confrontational questions or statements would threaten the atmo-sphere they had created, they chose to avoid them. For example, Marilynresisted taking a pedagogical stance when challenged by a participantbecause she considered that her role was to help others feel comfortablein the seminar. She believed that taking a stance counter to one voiced bya participant had the potential to foster disagreement within the group, andshe worried that this conflict would be counterproductive to the goal ofinquiry. Thus she responded by agreeing with the participants about thepreponderance of barriers to change in teaching.

This type of opening in the curriculum seemed to be motivated byparticipants’ searches for pedagogical suggestions and guidance. Theparticipants enrolled in what they understood to be a professional devel-opment seminar. The seminar, however, was unlike the workshops theyhad attended in the past. Traditionally, teacher development activities takea how-to approach, providing teachers with a selection of activities andlessons they can use in their classrooms (Little, 1993; Sparks & Loucks-Horsley, 1990). This approach to professional development is based on “adiscourse of answers” and “a confident stance of certainty” (Ball, 1997).In contrast, the DMI developers assumed that genuine and productiveteacher learning should begin with inquiry into mathematics and children’smathematical ideas (Schifter, 1998).

Taking a Prescriptive Stance

A second type of opening in the curriculum also occurred when thediscourse turned toward pedagogical practices. As we noted above, parti-cipants often expressed questions or ideas about what or how they wereteaching in their own classrooms. During some of these instances, otherparticipants offered prescriptive advice. Speaking with great authority,these participants told others what to do or gave advice about whatworked for them. The facilitators we studied were uncomfortable withthese instances because they did not want discussions of pedagogy to beshut down by simple prescriptions.


Jennifer found herself struggling with this type of opening when a parti-cipant offered advice that had the potential to undermine the focus of adiscussion of student-invented rules to compare decimal numbers. In aninterview, Jennifer confided that she “hoped that participants would workon the idea that comparing the value of decimal numbers and memorizingrules were not synonymous” (Interview, 6/8/97). She believed that the casethe participants had read, which described fifth graders struggling to artic-ulate generalizations of the strategies they developed to compare decimalnumbers, illustrated that, although the rules frequently taught could leadstudents to arrive at the correct answer, “they may not be the starting pointfor student learning.” At the same time she knew that most of the teachershad not thought about what it meant to understand decimal numbers.

Jennifer began the case investigation by asking participants whether thestudents in the case understood decimals. Several participants answeredthat they thought the students probably didn’t understand decimals verywell or they would have been able to state a rule for determining which waslarger. Another participant was unsure. She explained that she would haveliked to read the students’ notebooks in order to assess their understanding.She pointed out, “But the students seemed to think that they did understandhow to compare numbers smaller than one. Stating it in a rule was lessimportant to them.”

To push the participants to examine their ideas about rules more closely,Jennifer asked them to construct the rules they might use to comparedecimal numbers. As they proceeded, several participants began to voicequestions about learning rules and how they related to students’ under-standings of decimals. For example, one participant explained that shecould see the difficulty students might have in connecting numbers suchas 0.38 to the value of 38 hundredths or approximately 4/10 because theemphasis on rules led students to consider decimals digit by digit (e.g., 3tenths and 8 hundredths). Jennifer listened with interest; the participantsseemed to be circling around the issues she hoped would emerge. Then asingle comment derailed the conversation. Marvin, a fifth-grade teacher,stated very directly and with authority, “I have tried lots of things, but theonly thing that works with kids is having them memorize the rules, espe-cially when they have numbers with lots of zeros.” He explained furtherthat he had tried other approaches to teaching children to compare decimalsincluding using manipulatives, but had success only when he showedstudents how to “insert the correct number of zeros and compare thenumbers digit by digit.” He then challenged the group to show him anotherway that worked. The group grew quiet. No one responded (Observation,6/8/97).


Afterward Jennifer expressed frustration and disappointment with thisdiscussion. She admitted that she just didn’t know what to say. She feltthat both what Marvin said and how he said it shut down the discussionand potential learning opportunities for others. She noted, “You could evenfeel how the atmosphere changed in the room. It seemed that even thoughthere were probably those that disagreed, it would have been really hardto say so. I just didn’t know how or what to offer to the group or to thisperson. I just let it go” (Interview, 6/8/97).

Jennifer was particularly disappointed about this incident becauseshe had carefully planned goals for the session’s explorations. She hadwanted participants to explore the rules children might use to comparewhole numbers and then consider whether the rules would work acrossthe decimal point. She hoped this would prompt them to think aboutwhat understandings students might employ in comparing numbers withdecimals. At the same time Jennifer confided that, since her own under-standing of rational numbers was fragile, she was not sure about theanswers to these questions herself. She attributed her uneasiness about howto respond to Marvin to her lack of confidence in her own understandingof decimals.

Connie faced similar openings, although none that felt so confronta-tional. In her sessions, some teachers – usually those with only a few yearsof experience – openly expressed questions or doubts they had about theirteaching, and other, more experienced teachers offered specific advice.The responding teachers described approaches that worked best for themin terms of imperatives, while the younger teachers took copious notes.For example, during a discussion on place value, one teacher observedthat a number of her students struggled with place value. This commentprompted a flood of suggestions and advice. Teachers described specificactivities or mnemonics that worked for them. During these exchanges,Connie did not join the conversation, question the participants, paraphrase,or summarize what she heard. Although these strategies were typicallypart of her facilitating repertoire, she did not use them to extend theseconversations. Generally she allowed these exchanges to run their courseand then tried to move the group on to the next question or issue.

In Marilyn’s seminar one participant consistently challenged what heviewed as the favored pedagogical stance with claims such as these: “Iwould never do this in my class because I wouldn’t have time”; “Mystudents would not respond like that”; or “We have tests to get studentsready for.” These statements tended to halt the inquiry of the seminar.Marilyn appeared challenged by the comments and often responded byfocusing on the comments or obstacles raised by the participant.


All three facilitators found participants’ prescriptive offerings awkwardbecause they did not promote the kind of critique and inquiry about mathe-matics, teaching, or student learning that they hoped to cultivate in theseminar. Furthermore, they were concerned that by supporting commentssuch as these they would promote stratification within the group, implyingthat some participants knew more than others. At the same time thefacilitators hesitated to shut down the conversations too abruptly or tochallenge the specifics of the advice for fear that such moves wouldcommunicate disregard of these experienced teachers’ knowledge. So theytiptoed through these instances, neither supporting nor challenging theadvice offered.

This type of opening and the opening discussed previously are similarin that both result from mismatches between what teachers have cometo expect from professional development and what they encounter in theseminar. Familiar with the discourse of answers prevalent among profes-sional development opportunities (Ball, 1997), participants expected togive advice as well as receive it. Thus it is not surprising that the discus-sion occasionally lapsed into exchanges of advice. In these openings,facilitators struggled to find ways of acknowledging the expertise thatteachers brought with them while maintaining a stance of critique andinquiry. The facilitators were convinced that establishing a supportive andopen environment was key to fostering the level of inquiry sought by theDMI curriculum, yet they were unsure about how to maintain an openenvironment in these circumstances.

Invitations to Explore Mathematical Ideas

The third opening occurred frequently for all three facilitators. Thisopening involved the challenges arising from exploration of mathe-matical ideas. The prevalence of these openings is natural since the DMIcurriculum focuses on the conceptual underpinnings of the elementarymathematics curriculum. The mathematical ideas are richly complex andinterrelated, and yet unfamiliar to many elementary school teachers. Mostteachers learned mathematics as a set of rules to follow, not as ideasthat made sense, and their understandings were fragile. Even those whofelt familiar with the mathematics found themselves seeing new relation-ships and patterns. The explorations they engaged in took participatingteachers onto new mathematical ground. As a result, teachers frequentlyexpressed surprise, inspiration, insight, confusion, frustration, and curi-osity in their encounters with the mathematical ideas that undergird thefacts and procedures they had once memorized. Faced with these reactions,the facilitators needed to decide how to respond. At times the facilitators


Figure 1. The side-by-side drawings Marilyn showed the participants of array models of16 × 18 and (x + 6) (x + 8). The equations, which were not on Marilyn’s drawing, havebeen added for clarification.

themselves proceeded into new mathematical terrain even as they facedthese decisions. In the following example, Marilyn’s decision to push parti-cipants to examine a mathematical relationship more deeply was promptedby both new conceptual insights she gained and changes in her view of thefacilitator’s role.

During an activity focusing on two-digit multiplication, Marilyn askedher group to create representations of 16 × 18 with diagrams usingbase-ten materials. As Marilyn worked on the problem, she began tonotice a mathematical connection she had never made before. In a some-what surprised tone, she exclaimed to the others, “Multiplying two-digitnumbers in the form of an array looks a whole lot like multiplying binomialfactors (x + a)(x + b). This is the first time I have actually put multiplicationand algebra together.” The room fell silent for a few moments, after whichMarilyn commented:

You know, it seems that somehow, somewhere along the way, I have overlooked theidea that simple two-digit multiplication underlies the more complicated idea of binomialexpansion. It occurs to me now that I have been working with middle school children forall these years and never really made that connection.

The participants in the group stopped to listen to Marilyn. She continued,“It may have been useful to have students use arrays to multiply two-digitnumbers in eighth grade and then draw on this experience to learn aboutmultiplying binomials.”


While talking to the group about her insights, Marilyn studied theirfaces to assess whether they saw the same connection and whether itwas important to them. She asked, “Is anyone else seeing this?” Noone acknowledged that they were. Marilyn hesitated for a moment andthen began to show the other participants her diagram and explain theconnection she had made. She illustrated the rectangular array she hadconstructed for multiplying 16 × 18 and beside it another array illustratingthe multiplication of (x + 6)(x + 8). After several minutes of explanationthe participants remained puzzled.

As Marilyn pressed on, several related questions and observationsarose from the group. Looking carefully at Marilyn’s drawing of 16 ×18, one participant noticed that the arrangement of rods represented thepartial products of the conventional multiplication algorithm. The drawingincluded a 10 × 10 square representing 100, a set of 6 and a set of 8 longrods representing 6 tens and 8 tens, and 48 small squares representing 48.Other participants’ questions focused on the associative and distributiveproperties. As Marilyn waited and watched the conversation evolve, sheseemed pleased that participants were asking questions, but decided not topush the relationship to algebra any further (Observation, 4/15/97).

In an interview that followed, Marilyn explained her decision to openthe discussion about what she noticed. “I really saw for the first time whykids have so much trouble understanding algebra. In that moment I knewthat I had taken my own understanding of algebra for granted. I wantedothers to see this.” This inspiration, however, was not her only reason. Shewas also reassessing her role as facilitator. She explained, “In the begin-ning of the seminar, I relied on the materials to stimulate the conversations.I saw my role as the organizer. I also wanted to make everyone comfort-able. But, over time, my views began to change.” Marilyn explained thatshe began to realize that “in walking the cake walk where you try to beso careful that no one’s feelings get hurt, the agenda often got lost. Really,nothing got challenged, no one was willing to risk putting themselves outthere” (Interview, 4/15/97).

For Marilyn, this new perspective on her role as facilitator representeda significant shift. Earlier in the seminar she was less likely to initiatea discussion about a complex mathematical idea. Instead, she avoidedpressing participants to explore mathematical ideas so that no one wouldfeel put on the spot or forced to risk revealing what they did not understand.Over time, Marilyn had grown increasingly dissatisfied with the conversa-tions in the seminar. Inspired by the mathematical connection she made,she saw an opening that she thought was worth the risk and decided topush participants in that direction. Although she knew that participants


did not fully grasp the connection she made, she thought the discussionwas significantly more ambitious than those the group had earlier in theseminar. She was impressed with both the observations participants madeand the connections they drew between the base-ten model and the partialproducts of the multiplication algorithm. “In the case of one participant,”Marilyn speculated, “I think this may have been the first time to ever seea physical model of the traditional [multiplication] algorithm” (Interview,4/15/97).

Each facilitator we observed confronted questions about whether andhow to support participants’ mathematical explorations and learning. Inmany of these instances, the facilitator made the decision to pursue theparticular mathematical idea, as Marilyn did. Connie was particularlyinclined to follow up on mathematical questions and observations thatarose. Jennifer tended to probe many of the mathematical discussions withgeneral statements, such as “I know there is something more to this” and“Let’s keep looking.”

We also observed a number of instances in which the facilitators chosenot to follow mathematical leads. Connie made that choice during a discus-sion of children’s work with base-ten materials. Several participants raisedquestions about students’ understanding of place value and the role playedby the study of other bases. In response, other participants offered observa-tions and perspectives about the base-ten system with little or no evidenceto support their claims. Some of the claims seemed to be based on partialknowledge. For example, one teacher claimed that base 10 was chosen forthe conventional system because of the patterns inherent in a system basedon 10. Connie listened and nodded. At that moment she did not interjectspecific questions or challenges, and the conversation moved swiftly to anew issue (Observation, 9/24/96).

Later Connie described her thoughts during this discussion. She notedthat the conversation was moving quickly, which made interjecting diffi-cult. She recalled being troubled by many of the assumptions reflectedin participants’ comments, explaining, “I thought, ‘They have so manyvaried ideas of what base 10 is.’ ” She offered two reasons for not pursingthat issue at that moment: “I didn’t have a ready-available task to poseto them, nor did I think that was the place to do that. It wouldn’t havehelped me in the goals I had for the class.” She explained that in othersituations she might have provided a task in a later session to encourageparticipants to examine their ideas about place value. She knew she wouldnot do this, however, because she wanted to follow the agenda offeredin the curriculum. Given the focus of the curriculum, she was confidentthat the same questions would reemerge and “We can say, remember this


conversation.” While Connie seemed comfortable with the decision shehad made during the session, she also recalled feeling uncertain about it inthe moment (Interview, 9/24/96).

Similar invitations to explore mathematical ideas emerged frequentlyin the seminars and all three facilitators struggled with whether and how torespond. In some instances the facilitator backed off completely to avoidputting participants on the spot. At other times she chose to move forwardwith the agenda of the curriculum rather than pursue an emerging mathe-matical issue. Still, at other times, the facilitator chose to veer from theplans to explore the particular idea.

THE TENSIONS UNDERLYING THE OPENINGS

Our examination of the openings just described revealed that a set oftensions underlay the discourse in the professional development sessionsand that these tensions often figured into the facilitators’ decision making.We do not view these tensions as “problems” that were avoidable or easyto fix. Instead, they are similar to the dilemmas of teaching identified byLampert (1985). She argues that because the work of teaching involvesattending to multiple, often competing agendas, teachers constantlyconfront dilemmas of practice. The work of teaching involves managingthese dilemmas, rather than seeking to eliminate them. As we describebelow, the tensions DMI facilitators faced involved competing goals,expectations, and approaches to teaching that facilitators need to navigate.Examining these tensions can provide insights into the work of facilitatingteacher learning.

The first opening involved participants’ assumptions that facilitatorsadvocated a particular approach to teaching mathematics. Although eachfacilitator responded to questions and challenges differently, all threewrestled with tensions between what participants wanted from profes-sional development and the assumptions underlying the curriculum. Inthe second opening, we identified the tendency for participants to insertprescriptive advice about how to teach. The tension facilitators faced inthese instances between the participants’ images of professional learningand that envisioned in the curriculum is similar to those in the firstopening. As Ball (1997) reminds us, the professional development thatmost teachers are familiar with focuses on solutions and answers. A secondtension associated with this opening was between the facilitators’ desiresto maintain a stance of inquiry in the seminars and at the same timecreate a safe environment in which all participants’ ideas were valued andwelcomed. This tension is parallel to Simon’s (1997) characterization of


tensions faced by teachers of mathematics who must take their lead fromstudents, but work toward particular mathematical goals. Wilson and Berne(1999) remind us that maintaining this balance in professional develop-ment settings is particularly challenging, since inquiry and subjecting one’spractice to examination are risky for teachers.

In the third opening, we describe the multiple and varied opportunitiesthat arose in each session around explorations of mathematical ideas. Inopenings of this type, facilitators had to decide what they thought wasimportant mathematically for participants to explore or learn, how theymight invite participants to do so, how much to push the explorations,and whether the timing seemed right. In some cases, facilitators wereexploring the particular mathematical ideas themselves for the first time.One tension, in this case, was between mathematical goals and the imme-diate interests or questions of the participants. Another was between thedesigned trajectory of the curriculum and the opportunities offered in theimmediate moment of the seminar.

NAVIGATING OPENINGS IN THE CURRICULUM TOSUPPORT TEACHERS’ PROFESSIONAL LEARNING

A close look at the three openings and the tensions that underlie themreveals conflicts between and within the goals and commitments of thefacilitators, the expectations of the participants, and the agenda of theDMI curriculum. Our analysis of the openings sheds light on the ques-tions guiding this study: What is involved in using an innovative teacherdevelopment curriculum and supporting the kinds of teacher learningcompatible with current reforms in mathematics education? We see open-ings, because they signal places of conflict or discontinuity between thegoals of participants and the facilitator or the curriculum, as holding signi-ficant potential for inquiry and learning. Often initiated by the concernsand observations of participants, including the facilitator, these openingsinvite opportunities for facilitators to structure conversations and explora-tions that can extend or challenge participants’ knowledge and beliefs. Ouranalysis of the three facilitators’ efforts to navigate openings in this studyilluminates the nature of the work of supporting teacher learning.

While the idea of openings in the curriculum grew out of our analysisof facilitators using the DMI curriculum, it is reasonable to suggest that allteacher educators engaging teachers in reexamining mathematics teachingand learning are likely to confront similar openings – unanticipated andat times awkward points in the conversations – through which they had tonavigate. Thus, the following discussion of the work involved in navigating


openings has relevance to the work of teacher educators in general whoseek to support reform in mathematics education through fostering teacherlearning. Later we discuss issues directly related to the use of innovativecurricula.

Our observations revealed that facilitators’ responses to openings variedtremendously. On some occasions their responses seemed spontaneous,even automatic. At other times facilitators seemed entirely cognizant ofthe competing goals at play and of the trade-offs that a particular decisionwas likely to entail. Regardless of the extent to which the facilitators’decisions were explicit or tacit, we found that all three facilitators engagedin a set of three activities central to the navigating process: (a) readingthe participants and the discourse, (b) considering responses and possibleconsequences, and (c) taking responsive action. This cycle of activities,which includes analysis and consideration of goals and taking action, issimilar to a model proposed by Remillard (1999) of the teacher’s role inshaping the enacted curriculum. Similar to Remillard’s findings, we foundthat this process was not linear, but fluid and interactive, often appearing tobe spontaneous. In the following sections, we draw on our earlier examplesof openings to illustrate these activities.

Reading Participants and the Discourse

Throughout the openings we saw examples of facilitators reading bothparticipants and the discourse, to take stock of the general group, andindividual participants’ understandings, interests, intentions, and comfortlevels. For example, when the teacher in Connie’s seminar asked her foradvice about teaching students computational strategies, Connie respondedby asking a series of questions to learn more about the teacher’s views andapproaches. Similarly, when a participant in Jennifer’s seminar interjectedan imperative related to teaching decimals, Jennifer noted the authoritywith which he spoke and the awkward silence that fell over the group.Marilyn also read participants in her seminar as they explored the arraymodel of two-digit multiplication. She listened to them discuss the model,assessing the insights they were gaining.

Reading both individual students and the class as a whole is viewed asa critical component of teaching (Remillard, 1999; Ball, 1997; Sullivan,2000). Such reading in reform-oriented professional development settingsinvolves reading more than the learners’ understanding of the content. Italso involves assessing participants’ willingness to learn in nonconven-tional ways.


Considering Responses and Possible Consequences

As facilitators read participants and the discourse, they considered possibleresponses the participants might make with respect to their goals for theseminar. Sometimes this process was tacit occurring rapidly. Reminiscentof Lampert’s (1985) dilemmas of teaching, the goals against which thefacilitators weighed options were multiple and sometimes conflicted withone another. For example, in considering whether to question Marvin’spedagogical claim, Jennifer weighed several goals. She believed thatneither the content of his advice nor the way in which he stated it supportedher goal of promoting an inquiry-oriented stance toward learning. Onthe other hand, she was concerned that challenging Marvin would flyin the face of another goal – creating a safe and congenial environmentfor learning. Similarly, when considering how to respond to participants’pedagogical challenges and questions, each facilitator weighed the goalof supporting and acknowledging participants’ developing insights aboutteaching against the goal of encouraging inquiry into mathematical ideas.

In several instances the process of weighing responses againstcompeting goals involved balancing mathematical goals against thecurriculum agenda. This was the case for Connie when she consideredwhether to pursue claims that participants made about place value or tostick with the suggested plans offered in the curriculum. She believedthat mathematical questions were a productive way to encourage teachersto examine their assumptions; but she also valued the insights of thecurriculum developers and felt reluctant to move the conversation too faroff track. Understanding that unpacking the structure of the number systemwas an underlying goal of the complete module, she was confident that theissues raised in the moment would reemerge, so she chose not to pursuethe topic of place value at that time. Yinger (1988) referred to the kind ofdeliberation teachers do in the midst of teaching as a “three-way conver-sation between teachers, students, and the problem” (p. 86). He used theterm “problem” to refer to the actual task being taken up by the teacherand students. Our analysis suggests that in many cases the conversationmight include a fourth member – the curriculum guide. Considering howto respond involved all three facilitators in weighing the interests of thesefour constituencies.

Taking Responsive Action

Through the process of considering responses and goals, the facilitatorsdecided how to respond to the opening. Sometimes their actions favoredone goal over another. When Jennifer decided not to respond to Marvin’spedagogical interjection in order to keep the discourse smooth, she was


choosing what she believed would foster a safe learning environment. Atother times facilitator actions reflected attempts to navigate two or moreconflicting goals. Connie’s response to teachers who continually askedfor pedagogical advice exemplifies this approach. In deciding to explainher perspective on pedagogical development, and by agreeing to act as asounding board in these conversations, Connie acknowledged participants’yearning for pedagogical advice while maintaining her commitment toprioritizing mathematical explorations.

The responsive actions described here are similar to what Gay (1995)identified as “considered” action. According to Gay, considered actions arebased on self-reflection and analysis and are open to ongoing adjustment.The extent to which the facilitators’ actions were deliberate varied acrossthe three facilitators and changed over time. As we argue below, awarenessof the navigational process is critical to the work of supporting teachers’professional development.

Awareness of the Navigational Process

Looking comparatively at the three facilitators, we found that the moreoptions and possible consequences facilitators were able to consider inlight of competing goals, the more likely they were to take deliberateaction. Connie, who had experience working with practicing teachersaround reform-related ideas, brought to her use of the curriculum arepertoire of options she might take in responding to the teachers’ ques-tions and concerns. She also had a deeper understanding than the othertwo facilitators of the mathematical and pedagogical goals of the DMIcurriculum. This knowledge and experience supported her in navigatingopenings and taking on the tensions they represented. In Marilyn, wealso saw differences over time in the choices she made regarding howhard to press participants to examine mathematical ideas; these differencesseemed to indicate she had gained knowledge of possible consequencesof her choices. Initially Marilyn tended to avoid these opportunities. Sheexpected that the curriculum itself would support participants’ mathe-matics learning. Her primary concern was to create a safe learning envi-ronment for the participants. Over time, as her understanding of competinggoals increased, she recognized that the seminar contained limited oppor-tunities for deep mathematical learning. She began to weigh her concernsfor a comfortable learning environment against the goal of extendingmathematical learning opportunities. Her choice to encourage participantsto explore the mathematics reflected both her growing understanding ofthe goal to foster mathematical learning and her sense that a comfortablelearning environment may involve risk taking.


We believe that as facilitators gain experience working with curriculumlike DMI, they are increasingly likely to recognize places where openingsin the curriculum may occur. Marilyn, for example, represents a case of afacilitator whose tendency to recognize and deliberately navigate open-ings increased with her use of the curriculum. Initially she viewed herrole as primarily organizational. As the sessions proceeded, she began totake a more deliberate stance in facilitating participants’ learning. Throughgaining awareness of openings and competing goals associated with them,facilitators may become more likely to take explicit action in response toopenings in the curriculum. The relationship between navigating openingsand growth in facilitators’ awareness of this process is reminiscent ofRemillard’s (1996) observation regarding teacher learning and curriculumuse:

The most fruitful sites for learning occurred when the teachers had to read the text, theirstudents, or situations in their teaching with an eye toward designing or constructingcurriculum. This process of reading and decision making caused the teachers to reex-amine their beliefs and understandings that, in turn, influenced the curriculum they enacted(p. 256).

Reconsidering Curriculum Resources

The view of openings we propose has implications for how scholars andpractitioners think about the role of curriculum material in the work ofteachers and teacher educators. We believe that well-navigated openingsallow facilitators to take deliberate action to foster the kind of learningintended by DMI developers even when doing so involves “veering” fromthe plans suggested in the curriculum. In a sense, openings may be signalsthat the curriculum is working.

This stance assumes a novel perspective on curriculum materials andtheir use that is critical to the successful implementation of reform-inspiredprograms. Traditional views of curriculum materials hold that they arefixed guides to be either followed or veered from. A view of curriculumuse as embracing and exploiting openings, as we propose, assumes aninteractive relationship between written and enacted curriculum. As Lloyd(1999) suggests, “Curriculum implementation consists of a dynamic rela-tion between teachers and particular curricular features” (p. 244). In otherwords, implementing an inquiry-oriented curriculum demands that facili-tators/teachers take advantage of openings as they emerge and that theseopenings are likely to vary from context to context.

This view of the role of curriculum resources is also different fromthat suggested by Sosniak and Stodolsky (1993) who argued that teachersview curriculum materials as “props in the service of managing larger


agendas” (p. 271). We believe that well-designed curriculum resources cancontribute to teachers’ or facilitators’ larger agendas without constrainingthem. It is our view that learning to use innovative curriculum resourcesincludes learning to assume an interactive relationship with these materialsand their developers. Implicit in this view is the assumption that curriculumdevelopers redefine their role in guiding the work of teachers and teachereducators.

CONCLUSION

In this final section, we consider what our analytic frame of openings in thecurriculum suggests for improving professional learning opportunities forteachers of mathematics. Our analysis highlighted the critical role teachereducators play in fostering inquiry and exploration within teachers’ profes-sional learning opportunities regardless of whether curriculum materialsare involved. It follows that assisting teacher educators in learning tonavigate openings can contribute substantially to learning opportunities forteachers. We suggest that productive learning opportunities for facilitatorsmust help them learn to (a) recognize openings, (b) identify and unpackthe tensions that underlie them, and (c) understand processes of navigatingthem. We examine each of these suggestions and speculate about how thislearning could support teacher educators.3

We identified openings initially by focusing on places in the curriculumthat seemed difficult or uncomfortable for facilitators or participants.Through our analysis we came to view openings as opportunities forfacilitators to pursue critical learning goals. Facilitators of professionaldevelopment would benefit from learning to recognize and expect open-ings in the curriculum and frame them as points where participants’learning can be supported. Such a view of openings would cast a differentlight on the facilitator’s role. Although facilitators may continue to findopenings difficult to navigate, they would recognize them as the naturalconsequences of how participants engage with a curriculum intended tosupport reform in mathematics education. In short, openings could beunderstood as an indication that a curriculum is working because theyreveal places where some of the most critical and deliberative workof the facilitator can occur. Still, we do not see teacher learning as astraightforward and uncomplicated process.

To support participants’ learning within openings, facilitators wouldneed to learn to unpack them. We argued earlier that each opening involvedtensions among competing goals. Further, we saw important differencesin each facilitator’s awareness and understanding of these tensions. We


believe that facilitators need to learn to uncover and understand thetensions underlying openings. At the same time they need to have oppor-tunities to examine their own goals and learn about the goals of thecurriculum. This learning would involve coming to understand the rangeof tensions at play in any one opening. It would also involve helpingfacilitators expand their repertoires of responses within an opening. Weobserved, for example, the change in options available to Marilyn asshe began to consider a wider array of alternative goals. As a result,she deliberately chose to push participants to explore mathematical ideasmore deeply. We can speculate that, through learning to recognize andunpack competing goals underlying openings, Marilyn would be increas-ingly likely to take deliberate action to foster the learning agendas of thecurriculum.

As facilitators learn to unpack openings, they also need to considerpossible consequences of actions they might take. This learning, we specu-late, would promote a clearer sense of the connections between facilitators’decisions and participants’ learning. This understanding is perhaps themost critical of the navigational process. A facilitator’s choosing, forexample, to act as a sounding board in conversations that move quicklyaway from mathematical learning and into pedagogical approaches hasdifferent consequences for participant learning than a facilitator’s askingparticipants to refocus their conversation on the mathematics. We are notsuggesting that one approach is preferable. Instead, we argue that eachapproach has consequences for the interaction and how it develops, aswell as for what and how participants learn. We believe that, as they learnto examine various responses in light of possible consequences, facilitatorswill grow increasingly aware of the range of navigational choices availableto them and of the connection between those choices and what participantsmay be learning over time. This awareness likely will result in facilitatorstaking increasingly deliberate stances to foster learning that the curriculumwas designed to initiate.

It is important to recognize that relationships between facilitators’choices, or any professional development activity, and what teachers learnare not well established. In fact, as Wilson and Berne (1999) point outin their review of contemporary professional development, “The ‘what’of teacher learning needs to be identified, conceptualized, and assessed”(p. 203). We concur that the field would benefit from more clarity on whatteachers learn from inquiry-oriented professional development. Findingsfrom such research would inform the work of facilitators as they considerthe possible consequences and aims of their pedagogical choices.


Earlier we claimed that in some instances facilitators began to recog-nize openings in the curriculum and considered various ways to navigatethem. However, we argue that relying on experience alone to fosterfacilitators’ abilities to identify openings and to use them deliberately topromote participant learning leaves to chance much of what is needed tosupport teachers’ professional learning. The need for classroom teachersto receive professional support when using new curriculum materials orexperimenting with new practices is well established (Remillard, 1996;Ball, 1997; Cohen & Barnes, 1993; Heaton, 2000; Lloyd, 1999). Teachereducators using an innovative curriculum for teachers require similarsupport. Just as facilitators need to take deliberate action to support parti-cipants’ learning, those supporting facilitators must intentionally structureopportunities for facilitators to learn to recognize, unpack, and navigateopenings in the curriculum.

ACKNOWLEDGEMENT

The authors are grateful for the suggestions and insights offered by TomCooney, Barbara Jaworski and anonymous reviewers. Our analysis hasbenefited from conversations with Deborah Schifter, Virginia Bastable,Annette Sassi and Linda Ruiz Davenport.

NOTES

1 By using the DMI curriculum with teachers, the three participants were taking on the roleof teacher educator regardless of their formal titles (teacher educator, teacher, curriculumspecialist). For clarity, we use the term facilitator to refer to the three participants workingwith the curriculum because it is the term used by the DMI developers. However, whenreferring to the work of supporting teachers or guiding teacher learning more generally, weuse the term teacher educator.2 The facilitators’ names are pseudonyms.3 The developers of the DMI curriculum have continued to examine this issue. Besides ajournal written by a fictitious facilitator in the materials to support facilitators’ thinking,they offer a teacher educator institute and an electronic discussion forum for facilitatorsusing the curriculum. These forms of support provide additional sites for future research.

REFERENCES

Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representationalcontexts in teaching fractions. In T. Carpenter & E. Fennema (Eds.), Rational numbers(157–195). Hillsdale, NJ: Lawrence Erlbaum Associates.


Ball, D. L. (1997). Developing mathematics reform: What don’t we know about teacherlearning – but would make good hypotheses? In Susan N. Friel & G. W. Bright(Eds.), Reflecting on our work: NSF teacher enhancement in K-6 mathematics (77–112).Lanham, MD: University Press of America.

Bush, W. S. (1986). Preservice teachers’ sources of decisions in teaching secondarymathematics. Journal for Research in Mathematics Education, 17(1), 21–30.

Clandinin, D. J. & Connelly, F. M. (1992). Teacher as curriculum maker. In P. W. Jackson(Ed.), Handbook of research on curriculum (363–401). New York: Macmillan.

Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. EducationalEvaluation and Policy Analysis, 12(3), 327–345.

Cohen, D. K. & Barnes, C. (1993). Pedagogy and policy. In D. K. Cohen, M. W.McLaughlin & J. E. Talbert (Eds.), Teaching for understanding: Challenges for practice,research, and policy (207–239). San Francisco: Jossey-Bass.

Donovan, B. F. (1983). Power and curriculum in implementation: A case study ofan innovative mathematics program. Unpublished doctoral dissertation, University ofWisconsin, Madison.

Doyle, W. (1993). Constructing curriculum in the classroom. In F. K. Oser, A. Dick & J.Patry (Eds.), Effective and responsible teaching (66–79). San Francisco: Jossey-Bass.

Fennema, E., Carpenter, T. P. & Franke, M. L. (1996). A longitudinal study of learning touse children’s thinking in mathematics instruction. Journal for Research in MathematicsEducation, 27(4), 403–434.

Floden, R. E., Porter, A. C., Schmidt, W. H., Freeman, D. J. & Schwille, J. R. (1980).Responses to curriculum pressures: A policy capturing study of teacher decisions aboutcontent. Journal of Educational Psychology, 73, 129–141.

Gay, G. (1995). Modeling and mentoring in urban teacher preparation. Education andUrban Society, 28(1), 103–118.

Graybeal, S. S. & Stodolsky, S. S. (1987, April). Where’s all the “good stuff”?: An analysisof fifth-grade math and social studies teachers’ guides. Paper presented at the annualmeeting of the American Educational Research Association, Washington, DC.

Heaton, R. M. (2000). Teaching mathematics to the new standards: Relearning the dance.New York: Teachers College.

Kuhs, T. M. & Freeman, D. J. (1979). The potential influence of textbooks on teachers’selection of content for elementary school mathematics (Research Series No. 48). EastLansing, MI: Institute for Research on Teaching.

Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems inpractice. Harvard Educational Review, 55(2), 178–194.

Lemke, J. L. (1990). Talking science: Language, learning and values. Norwood, NJ: Ablex.Little, J. W. (1993). Teachers’ professional development in a climate of educational reform.

Educational Evaluation and Policy Analysis, 15(2), 129–151.Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implica-

tions for mathematics teacher development. Journal of Mathematics Teacher Education,2(3), 227–252.

Loucks-Horsley, S. (1997). Teacher change, staff development, and systemic change:Reflections from the eye of the paradigm. In S. N. Friel & G. W. Bright (Eds.), Reflectingon our work: NSF teacher enhancement in K-6 mathematics (133–150). Lanham, MD:University Press of America.

Nelson, B. (1997). Learning about teacher change in the context of mathematics reform:Where have we come from? In E. Fennema & B. S. Nelson (Eds.), Mathematics teachersin transition (3–15). Mahwah, NJ: Lawrence Erlbaum Associates.


Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed.). ThousandOaks, CA: Sage Publications.

Remillard, J. (1991). Abdicating authority for knowing: A teacher’s use of an innovativemathematics curriculum (Elementary Subjects Center Series No. 42). East Central, MI:Michigan State University, Institute for Research on Teaching, Center for the Learningand Teaching of Elementary Subjects.

Remillard, J. T. (1996). Changing texts, teachers, and teaching: The role of curriculummaterials in mathematics education reform. Unpublished Doctoral Dissertation,Michigan State University, East Lansing, MI.

Remillard, J. T. (1999). Curriculum materials in mathematics education reform: A frame-work for examining teachers’ curriculum development. Curriculum Inquiry, 29(3),315–342.

Sarason, S. (1982). The culture of the school and the problem of change (2nd ed.). Boston:Allyn and Bacon.

Schifter, D. (1998). Learning mathematics for teaching: From a teachers’ seminar to theclassroom. Journal of Mathematics Teacher Education, 1(1), 55–87.

Schifter, D., Bastable, V. & Russell, S. J. (1999). Developing mathematical ideas.Parsippany, NJ: Dale Seymour Publications.

Simon, M. (1997). Developing new models of mathematics teaching: An imperativefor research on mathematics teacher development. In E. Fennema & B. S. Nelson(Eds.), Mathematics teachers in transition (55–86). Mahwah, NJ: Lawrence ErlbaumAssociates.

Snyder, J., Bolin, F. & Zumwalt, K. (1992). Curriculum implementation. In P. W. Jackson(Ed.), Handbook of research on curriculum (402–435). New York: Macmillan.

Sosniak, L. A. & Stodolsky, S. S. (1993). Teachers and textbooks: Materials use in fourfourth-grade classrooms. The Elementary School Journal, 93(3), 249–275.

Sparks, D. & Loucks-Horsley, S. (1990). Models of staff development. In W. R. Houston(Ed.), Handbook of research on teacher education (234–250). New York: Macmillan.

Stake, R. E. & Easley, J. (1978). Case studies in science education. Urbana: University ofIllinois.

Steffe, L. (1990). Mathematics curriculum design: A constructivist perspective. In L.Steffe & T. Wood (Eds.), Transforming children’s mathematics education: Internationalperspectives (389–398). Hillsdale, NJ: Lawrence Erlbaum Associates.

Stephens, W. M. (1982). Mathematical knowledge and school work: A case study ofthe teaching of developing mathematical processes. Unpublished doctoral dissertation,University of Wisconsin, Madison.

Sullivan, A. M. (2000). Notes from a marine biologist’s daughter: On the art and scienceof attention. Harvard Educational Review, 70(2), 211–227.

Sykes, G. (1996). Reform of and as professional development. Phi Delta Kappan, 77(7),465–467.

Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics andmathematics teaching to instructional practice. Educational Studies in Mathematics, 15,105–127.

Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. InD. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (127–146). New York: Macmillan.

Walker, B. F. (1976). Curriculum evolution as portrayed through old textbooks. TerreHaute: Indiana State University, School of Education.


Wilson, S. M. & Berne, J. (1999). Teacher learning and the acquisition of professionalknowledge: An examination of research on contemporary professional development. InA. Iran-Nejad & P. D. Pearson (Eds.), Review of research in education, 24 (173–210).

Woodward A. & Elliot, D. L. (1990). Textbook use and teacher professionalism. In D.Elliot & A. Woodward (Eds.), Textbooks and schooling in the United States (178–193).Chicago: University of Chicago Press.

Yinger, R. (1998, May). The conversation of teaching: Patterns of explanation in mathe-matics lessons. Paper presented at the meeting of the International Study Association onTeacher Thinking, Nottingham, England.

Graduate School of Education3700 Walnut St.Philadelphia, PA 19104USAE-mail: [email protected]

RUTH M. HEATON and WILLIAM T. MICKELSON

THE LEARNING AND TEACHING OF STATISTICALINVESTIGATION IN TEACHING AND TEACHER EDUCATION

ABSTRACT. Current national standards in mathematics education in the United Statescall for revision of content and pedagogy, including the addition of statistics as a new topicin the elementary curriculum. The purpose of this study is to examine the collaborativeefforts of a mathematics educator and statistician to help prospective elementary teachersdevelop statistical knowledge and experience through merging statistical investigationinto existing elementary curricula. Findings from this study offer insight into preserviceteachers’ statistical and pedagogical content knowledge based on their application of theprocess of statistical investigation themselves and with children. Findings also indicatethat integrating statistics into the elementary curriculum is a more difficult and complexteaching and learning problem than expected.

KEY WORDS: elementary mathematics education, inquiry, integrated curriculum, preser-vice teacher education, statistical content knowledge, statistical investigation, statisticalpedagogical content knowledge, statistics education

INTRODUCTION

Statistics was formally introduced in the United States as a topic forelementary children in the Curriculum Standards for School Mathematics(National Council of Teachers of Mathematics, 1989) and remains high-lighted in the revised version of these standards (NCTM, 2000). Thestatistical emphasis is on helping children learn to formulate questions,gather data, and use data wisely in solving problems. In spite of its decadelong presence in curricular reform in elementary mathematics educationin the United States, statistics is an area still in its infancy (Lajoie &Romberg, 1998). There is little precedent regarding what aspects of statis-tics and probability to teach, much less how children can or should applystatistical reasoning in the context of solving real problems.

The statistics education literature has focused predominantly on theadult learner with some movement toward considering the teaching ofstatistics to children. One main pedagogical perspective involves firststudying learners’ informal knowledge about statistics and probability(Ainley, Nardi & Pratt, 2000; Fischbein, 1975; Fong, Krantz & Nisbett,1986; Horobin & Acredolo, 1989; Shaughnessy, 1992) with researchers


36 RUTH M. HEATON AND WILLIAM T. MICKELSON

paying attention to particular features of statistical knowledge. Forexample, Hancock, Kaput and Goldsmith (1992) & Lehrer and Romberg(1996) attend to children’s construction and modeling of data. Jacobs(1993) and Konold (1991, 1989) attend to the language used by chil-dren to talk about statistical ideas. Once an understanding of the learners’informal knowledge is gained, the goal is then to create instructional inter-ventions to target correct statistical understanding (Garfield, 1995). Thereare many examples to illustrate this point. Specifically, Beres (1988) andSchaeffer (1988) created and studied appropriate sequences in learningstatistical ideas. Russell and Friel (1989) created and studied system-atic sets of probes to facilitate children’s communication about statisticalideas. Ainley, Nardi and Pratt (2000) have addressed graphing throughthe use of spreadsheets in elementary classrooms in the United Kingdom.Cobb (1999) used computer minitools with elementary children andstudied their individual and social mathematical development in statistics.Lehrer and Schauble (2000) worked with children to design classificationmodels around the drawings of elementary age children. Feldman, Konold,Coulter, with Conroy, Hutchison and London (2000) designed classroominstruction in statistics around electronic shared databases accessible tochildren through network science curricula.

Existing literature supports two general conclusions about statisticseducation. First, instructional interventions do improve the learners’ statis-tical reasoning ability, and second, statistics instruction needs to be activitybased in order to confront misconceptions and develop correct statisticalunderstanding. Furthermore, some research suggests the improvement inchildren’s understanding of statistics might be even more dramatic if statis-tics instruction is situated in more real or authentic contexts (Fong et al.,1986; Lajoie, Jacobs & Lavigne, 1995; Moore, 1997). Where and how thisis going to fit in an already full elementary curriculum remains unclear.One option is to use activities designed to address the development ofparticular aspects of statistical understanding (Riddiough & McCall, 1998;Edwards, 1995). However, as with all mathematical activities availableto elementary teachers, without teachers’ own deep understanding of themathematical content, activities can be done with children but the mean-ingfulness and significance of the mathematics may be overlooked (e.g.,Heaton, 1992; Putnam, 1992; Cohen, 1990). Another option is to do whatFriel and Bright (1998) suggest. They argue, based on experiences withveteran teachers, that statistics can easily be extended and integrated withother subjects like social studies or science. That statistics can be integratedinto elementary classrooms so that “consistency and connection across

THE LEARNING AND TEACHING OF STATISTICAL INVESTIGATION 37

subjects” is promoted seems sensible and possibly more feasible thantrying to add it on to an already full curriculum.

What novice teachers need to learn to actually implement such integra-tion in an elementary classroom is unknown. Teacher education programsin the US do not require a course in statistics for elementary educationmajors. The need for finding a forum for elementary teachers, novice andexperienced, to acquire statistical content and pedagogical content know-ledge (Shulman, 1986), including a familiarity with children’s ways ofunderstanding (Shaughnessy, 1992), is glaring. There is general consensusin the mathematics education community that teachers need a deepand meaningful understanding of any mathematical content they teach(Ma, 1999). Recently, mathematicians and mathematics educators repre-senting the Conference Board of Mathematical Sciences (CBMS) maderecommendations (CBMS, 2001) regarding the mathematical knowledgeneeded for teaching. With regard to statistics and elementary education,paraphrased from Chapter 8 of CBMS (2001), this knowledge includes:

• Designing data investigations

– understanding the kinds of questions that can be addressed– creating data sets– moving back and forth between the question (purpose of the study)

and the design (data)

• Describing data

– using graphical methods– using summary statistics– understanding variability– comparing sets of data

• Drawing conclusions

– understanding sampling and inference– communicating results relative to the purpose of the study

• Developing notions of probability

– becoming familiar with the idea of randomness– making judgements under uncertainty– assigning numbers to likelihood

This recommendation is completely consistent with the process of statis-tical investigation (Friel & Bright, 1997; Graham, 1987; Kader & Perry,1994). In addition to the basics of data collection, summarization, andinterpretation, it is particularly noteworthy that CBMS (2001) systemat-ically links purpose of investigation and question formulation with data


collection and analysis, then reconnects interpretation in the context of theoriginal purpose.

Both our teaching practice and research took seriously the notion ofpurpose of statistical investigation. In this study, we created opportunitiesfor preservice teachers to learn the process of statistical investigation in thecontext of inquiry about elementary mathematics teaching and to learn toteach the process of statistical investigation in the context of merging thisprocess with elementary curriculum topics. We recognized that the preser-vice teachers entered our mathematics methods course with basic graphingskills and some previous teaching experience from earlier practicumopportunities associated with the teacher education program. As such,our purpose was not explicitly to teach statistical knowledge to teachers.Rather, we took an open-ended approach to obtain a baseline on preserviceteachers’ statistical and pedagogical content knowledge as they appliedthe process of statistical investigation themselves and with children. Wewanted to learn what the preservice teachers knew and could do with statis-tical investigation in these contexts with minimal statistics instruction.Our practices as teacher educators and researchers were consistent withJaworski’s (1994) idea of investigative teaching and investigative research(Jaworski, 1998). In this article, we describe our intentions as teachereducators and what we learned about learning, teaching, and learning toteach statistical investigation.

METHODOLOGY

Context of Study

The research setting was an undergraduate mathematics education course,situated in the final semester of coursework of a field-based teacher educa-tion program for prospective elementary teachers, at a large midwesternuniversity. For part of the course, our intent was for preservice teachers(referred to as students) to study and practice both elementary mathe-matics pedagogy and the process of statistical investigation while westudied the process and products of their efforts. Our goal was to constructand study authentic learning and teaching opportunities (Newmann &Wehlage, 1997) surrounding the process of statistical investigation.

We designed and implemented two assignments. In the first assign-ment, Inquiry Project I, students learned to apply the process of statisticalinvestigation to address a question they posed regarding some aspect ofmathematics teaching and learning at a practicum site. In this assign-ment, students were required to pose questions, identify variables, plan and


carry out data collection, summarize data, report findings, and recommendchanges in teaching practice. The products of Inquiry Project I were thestudents’ project reports and written reflections on their experiences. Inthe second assignment, Inquiry Project II, students developed and taughta statistical investigation unit with children in a practicum site, where theinvestigation was situated in the study of an elementary curriculum topic.As part of this assignment, students identified a curriculum topic with theircooperating teachers and developed possible investigation questions. Theythen helped children pose investigation questions, identify variables, planand carry out data collection, summarize data, and report findings. Theprojects of Inquiry Project II were the students’ written project reportsand reflections on their experiences. In both assignments, our aim wasfor learners, whether they were our students or children, to be engaged inauthentic learning of statistical process and content through investigation.

We recognized it was unreasonable to believe that prospective teacherswould develop into excellent quantitative inquirers in the span of onesemester through these two assignments. Realistically, our aim was to helpprospective teachers develop perspective, insight, some technical skill, andenthusiasm for statistical investigation. Our hope was that they wouldwant to continue to learn about statistics and teach reasoning with databeyond the single semester in which we worked with them. This goal wasconsistent with Heaton’s (2000) overall philosophy of teacher education,namely to launch prospective teachers as learners of what they need toknow rather than trying to teach them all they need to learn in the contextof a single course or program. This view is consistent with Dewey’s (1938)view of successful teachers as learners.

Participants

Our teaching occurred during the spring and fall semesters of 1999, withtwo different cohorts of 22 students each. The students in the course werein their last academic year of study as undergraduates in our elementary(K-6) teacher education program. At this point in the program, studentshad had four extensive, field-based experiences in K-6 classrooms aspart of their coursework and, finally, it was their last semester priorto student teaching. Demographically, students were primarily white,female, and middle class from the local urban and surrounding rural areas.Students worked as pairs with a cooperating teacher in a classroom ofelementary children two full days per week. Students in both semesterswere required to complete the two assignments described above. Thecooperating teachers at the practicum site had been working with Heatonin a school-university partnership (Goodlad, 1994) for three years prior to


this study and had classroom teaching experience ranging from three to30 years. The practices of the cooperating teachers represented a mixedacceptance of reform ideas and offered a full range of teaching styles andbeliefs about how children learn. Levels of cooperating teachers’ supportspanned a spectrum from simply “making room” for students to practiceteaching, to deep involvement with students’ professional development.

Heaton, a mathematics educator, was the main instructor for thecourse and supervised the practicum experience. Mickelson, a statisti-cian/research methodologist in a Department of Educational Psychology,took leadership regarding statistical content. He taught several classeson the process of statistical investigation, observed and offered feedbackin the practicum setting, and provided general consultation to Heaton,students, and their cooperating teachers.

Data Collection

Data include artifacts from classroom teaching, along with productsof Inquiry Project I and Inquiry Project II, as described above, inter-views with students and cooperating teachers, and our observation notes.Mickelson conducted interviews with pairs of students and separate inter-views with individual cooperating teachers. Open-ended questions wereused to elicit information about prior experiences with inquiry, reflectionon the two assignments, and views on the value of statistical investigationin the elementary classroom. In quoting from the data, all preservice andcooperating teachers’ names are pseudonyms.

Data Analysis

This is a qualitative case study (Creswell, 1998) of efforts to learn andteach statistical investigation. First, we examined all data pertaining toInquiry Project I. We looked for general themes in what students werelearning about the process of statistical investigation and the teaching andlearning of mathematics, what they found challenging about the assign-ment, evidence of what they were understanding and misunderstandingabout statistics, and their general response to the assignment and theprocess of statistical investigation. Second, we analyzed all data pertainingto Inquiry Project II in a similar manner. We looked for general themesin what students were learning about the process of statistical investiga-tion and merging statistical investigation into topics within the elementarycurriculum, what they found challenging about the assignment, evidenceof their understanding and misunderstanding about statistics, and theirgeneral response to the assignment. We added an analysis of our obser-vation notes. Finally, we looked for general themes relative to learning to


teach by examining all data, looking for common themes between InquiryProject I and Inquiry Project II and across semesters. Whenever we lookedfor evidence of understanding or misunderstanding, we drew directly fromevidence we saw in the data and inferred evidence from what appeared tobe lacking.

RESULTS

We present results consistent with our analysis in the same order as we didthe analysis. We present insights gained pertaining to the learning processfrom an analysis of data from Inquiry Project I; to the teaching processfrom an analysis of data from Project II; and to the process of learning toteach from an analysis of data between projects and across semesters.

From Inquiry Project I – On Learning Statistical Investigation

The AssignmentInquiry Project I was an assignment in which students were asked to inves-tigate the teaching and learning environment of the mathematics class intheir practicum setting through a statistical investigation. The assignmentcame early in the semester and coincided with an introduction to the stateand national reforms in mathematics education. Instead of lecturing on thesalient features of the environment of elementary mathematics classroomsconducive to teaching mathematics for understanding, we designed anassignment in which students were required to conduct their own inves-tigations. These were similar to the investigations of classroom culturedescribed by Lampert and Ball (1999). A difference here is that ourstudents were doing inquiry themselves in different classrooms in the prac-ticum setting, while Lampert and Ball’s artifacts from their own teachingoffer a shared context for inquiry for their students. In our study, preserviceteachers studied the teaching practice at their practicum site and came upwith their own research questions, subsequently collecting, analyzing, andinterpreting their own data.

To facilitate this project, we outlined the steps of the statistical inves-tigation process while we encouraged students to figure things out forthemselves, and offered guidance. For example, Mickelson led discus-sions on developing investigation questions, constructing data tables, andgraphing data. Heaton led discussions on the findings of students’ inves-tigations and provided an opportunity to look for patterns across students’graphs representing features of mathematics instruction across individualclassrooms in the practicum setting. A final class period was spent with


students in small groups to discuss their own efforts to alter a feature oftheir present math class environment. Our aim was to guide everyone inthe inquiry process while providing support tailored to individual projects.

Students’ Investigation Questions

Students were required to formulate three inquiry questions around whichthey gathered data over six math class periods. They were expected tocollect quantitative data on two of the three questions; to collect qualita-tive data on the third; and to analyze, represent, and interpret what theyfound across all three. We expected students to develop their own inves-tigation questions about teaching and learning issues of interest to them,which they did. Their quantitative questions included ones such as: Howmuch time does the teacher spend helping individual children during mathclass? During interactive instruction, where did questions come from –the teacher, the children, the previous day? How often did the teacher usethe whiteboard or the chalkboard? What did it get used for? What wasthe average wait time in math class between teacher questions and whenwas a student acknowledged to answer it? What was the average wait timebetween children’s answers and a teacher’s response to them? How manyhands were raised by the time a child was called on to answer? How manygirls or boys raised their hands in a math class? How many girls werecalled on? How many boys were called on? In reflecting on our fieldnotesassociated with the preservice teachers’ transition from initial questions toplans for data collection, a number of concerns emerged.

First, when we looked at the students’ investigation questions, the ques-tions were of the simplest type, namely asking how many or how much. Wewondered what useful knowledge would come from understanding howoften the teacher used the whiteboard as opposed to chalkboards, or howmuch time a teacher spends with one child as opposed to another. In theseexamples, there is no clear purpose to the investigation other than obtaininghow many or how much. From a closer look at the investigation questions,some students had a real concern about an issue in teaching and learningeven though their investigation question was of this simple “how many” or“how much” form. For example, comparing the number of boys that werecalled on relative to the number of girls could be connected to a largerissue of gender equity in mathematics education. The “so what” questionto the overall investigation was very important in the interpretation of theresults of student projects. Those students with a larger purpose to theirinvestigations were able to make meaningful recommendations for change.

Secondly, in conjunction with the first concern, formulating an inves-tigation question that can be addressed quantitatively was problematic. A


number of preservice teachers started with questions like: Why are partic-ular tasks or activities chosen to do with children? Or, what is the environ-ment of the mathematics class like? Or, who is more active in mathematicsclass? None of which are easily quantifiable questions. With these projects,the problem was one of negotiating that which students wanted to pursueand was doable statistically. This afforded opportunities to work closelywith those students to envision a statistical investigation from beginning toend consistent with the CBMS (2001) recommendations. Several preser-vice teachers, however, came to appreciate the challenge and importanceof formulating good questions. For example, Greg described the challenge,“It’s so hard to come up with questions and then come up with a way ofcollecting data that is going to answer that exact question.” Emma alsogrew to appreciate the value of coming up with a good question,

The most important thing I learned is to have a good question. It’s really important to have agood question that is precise, that makes sense. Because if you don’t have a good question,the data collection may or may not answer the question . . . so, it’s really important to havea good question, to have a good base.

Ultimately what is at issue is helping students to develop skills informulating pedagogically meaningful questions that can be addressedstatistically.

The third concern arose regarding operationalizing the variables onwhich data would be collected. On the one hand, even with a good ques-tion, some preservice teachers noticed that the physical process of datacollection could be challenging given the fast pace at which teaching andlearning happened in real classrooms. Sheila described data collectionabout gender and participation in her kindergarten classroom, “It’s hardto count fast. It’s all over, hands up and down, before you can count them.”Sheila had identified a salient feature of field-based research and a realityof teaching. On the other hand, she recognized the complexity of her datacollection task,

I think more boys raise their hand. But I think, then you have to look at how many kids arein the class because are there more boys than girls? And then you have got to look at didmore boys raise their hands? So it’s like you have to look at a number of things to get thecorrect data.

Sheila identified the need for a standardized unit of measure, like a ratio,for comparative purposes. While she did not articulate it in such technicalterms, this illustrates a teachable moment, where she had a personallymeaningful context to understand an abstract statistical concept. Unfortu-nately, Sheila was an isolated case, as most students merely reported totalcounts, overlooking or not recognizing the importance of a standard unitof measure.


Learning Outcomes for StudentsAs a final phase of the project, we expected students to recommendchanges in mathematics instruction within their practicum classroom basedon the results of their investigations. Regardless of the state of theiranalysis or interpretation, all made recommendations about teaching prac-tice based on data. They all made links between data and teaching practice.They were unable or unwilling to say data was insufficient to address theirquestions. There was no evidence they had critically evaluated the datathey had collected.

When we asked preservice teachers what they learned from InquiryProject I, their answers varied tremendously. Responses ranged fromseeing no connection to our goals to exceeding our expectations. Thatseveral students saw nothing worthwhile about this assignment was disap-pointing. In separate interviews, Beth and Renee, two preservice teachers,who happened to be close friends, both said the assignment was a wasteof time. Beth noted, “I would have come to the same conclusions withoutactually keeping track of the data as I did.” Renee also saw limited valueto the assignment. When asked about it in an interview, she said, “I thinkit’s important to be aware of what’s going on in our classroom. As far asan assignment, I don’t know if we needed a separate assignment, to carryit to that extent.” While these two students’ remarks were disappointing,their comments helped us consider the complexity of our expectations.While the first project had shown students that one could gather data aboutmathematics teaching and learning to confirm one’s informal observationsabout the environment, we were also urging them to consider what elsecould happen in the process, or as a result, of systematically collectingdata on mathematics teaching and learning.

In reflecting on the project, some students mentioned only learningstatistical content and process from the assignment. Maggie reflected,“I had a clearer idea after doing Inquiry Project I of what it meant totake tallies from a tally sheet and place it into a graph . . . and how toread a graph . . . so that’s the clearer understanding I got, how you takeinformation you’ve gathered and throw it on a graph.”

Others de-emphasized the process of statistical investigation and sawInquiry Project I as being only about mathematics teaching and learning.Winnie said, “It just got us familiar with the classroom environment, mathclass in general.” JoAnne noted, “I thought of it as, ‘I call on girls way toomany times than I call on boys’ . . . it was more of a gender thing.”

Some students saw Inquiry Project I as a genuine process from whichto gain insight into pedagogy. Monica wrote in her written reflections onInquiry Project I,


I found collecting the data to be very interesting. It helped me learn about the studentsand get to know details about their behavior better. This is part of education that is veryinteresting to me, observing and thinking about the students and why they act or thinkthe way they do. Moving the information into a data table allowed me to begin thinkingabout the big picture of what I learned from my questions. Instead of having six pages ofnotes, the data table allowed me to actually see patterns and draw conclusions. It was thebeginning stage of knowing what I had collected.

Tara offered similar insights in her interview with Mickelson. She said, “Iwas able to draw conclusions from it and then change my own strategiesto try to be better. It kind of opened my eyes to show me that I can recordinformation of what goes on in the classroom to better my own teachingskills. I don’t know.” Linda’s awareness of the usefulness of data washeightened. In her written reflections in Inquiry Project I, she stated,

I had never heard of reasoning with data so it was all new to me. This inquiry made methink in a way I do not usually think. I rarely use numbers to analyze what is happening in aclassroom. It was very interesting to see how numbers can be helpful and what informationthey show you that you might not notice with written or mental reflections. I learned themethod well enough that I think I could even do it again on my own in my classroom if Ifound myself with a question that numbers would solve.

Kathy, another preservice teacher, connected what she learned from theproject to her future development as a teacher. She said,

The most important thing . . . I think it probably was being able to learn how to do itmyself in my own classroom when I have my own classroom. You know it’s a way for meto evaluate what I’m doing and how well the kids are responding to it . . . it allows you todevelop changes . . . I think it helps you evaluate your teaching.

That students were seeing the process of reasoning with data as a toolrelevant to an ongoing understanding and revision of one’s own teachingpractice beyond the methods course far exceeded our expectations.

From Inquiry Project II – On Teaching Statistical Investigation

The purpose of Inquiry Project II was for students to plan, teach, andreflect on a statistical investigation with children in the practicum setting.Specifically, the students were to identify curriculum topics and investiga-tion questions, then help children articulate questions, collect, organize,and summarize data, and draw conclusions from their data. Our intent wasthat preservice teachers would take what they learned from Inquiry ProjectI as learners and apply it in the context of Inquiry Project II as teachers.We hoped all students would see the connections one of our students, Greg,saw between the two assignments, “It’s kind of a progression of doing itourselves and then teaching others to do it.”


The First AttemptIn the first semester, we did not intervene in the planning of Inquiry ProjectII. As we had done for Inquiry Project I, we created general guidelinesand then stepped back and watched what happened, acting as consultantsas needed. Our roles as teacher educators were to help problem solve asissues in planning and implementation arose. The children’s investigationswere situated in existing curriculum units in science, social studies, andlanguage arts, and centered around topics such as recycling, light, temper-ature, discrimination, health and hygiene, pets, communities, and authorstudies. For example, one third grade class investigated the food consump-tion of children relative to different food groups. Kindergarten childrenpolled families and friends to gather data on what pets they owned, andone fourth grade class developed and implemented a survey on recyclingpractices in their neighborhoods.

Although the preservice teachers gained teaching experience with thisassignment, the learning outcomes for children were of limited value bothin terms of knowledge about statistics and the curriculum topic underinvestigation. While children’s learning was important to us, our researchremained solely focused on the preservice teachers’ learning and teaching.In analyzing students’ written projects and reflecting on our observationfield notes, one source of difficulty appeared, in large part, traceable to theinvestigation questions and, in a larger sense, to the preservice teachers’conceptions of teaching and learning. In retrospect, it seemed obvious thatthe choice of question impacted the data collection and subsequent dataanalysis, summarization, and interpretation.

The Second AttemptWhen we revisited the initial questions posed by students for InquiryProject II, we could see that their questions were all very one-dimensionalwith one right answer. The synthesis of their data about food, temperatures,pets, or recycling became a descriptive summary of the facts generated.The purpose of the questions was to gain factual knowledge and the datacollected illustrated or elaborated the “fact”; for example, questions like“What is your favorite meat?” or “How many pages are in an Eric Carlebook?” or “How many teeth do you have?” were common. Rarely wererelational questions with a broader purpose, other than ascertaining factualinformation, posed. We were beginning to understand that there were sometopics and ways of asking questions that were more conducive to investiga-tion than others. Furthermore, we recognized that lessons learned duringProject I did not necessarily transfer into teaching insights in Project II.Not all questions led to interesting, intellectually worthwhile investiga-


tions. The practical issue became helping children and our students learnto distinguish limiting questions and to pose statistically and intellectu-ally interesting questions. Additionally, we needed to learn how to supportthe risk-taking required to pursue more open-ended, statistically rich, andcontent-appropriate questions with children.

When we repeated Inquiry Project II in the second semester, we decidedto take a more active role in the development of investigation questions,offering substantive, not just logistical or practical, help. We decided tointervene at the point of planning, specifically, the formulation of inves-tigation questions. We met with pairs of students who worked togetherin the same field placement. We tried to focus the investigation questionson broader purposes than what we observed in the first iteration. In ourmeetings with students, we tried to articulate the purpose behind teachinga specific topic in the curriculum, then tried to envision meaningful andimportant learning outcomes for children in terms of subject content. Wethen helped students envision the process of reasoning with data, includingthe data collection, summarization and representation, and interpretationconsistent with the desired outcomes.

In ten out of eleven conversations, we felt we had made progress inhelping students move in the direction of doing meaningful investigationaround topics and questions relevant to some aspect of the elementarycurriculum. As an illustrative example, a pair of students working withfourth grade students was scheduled to be involved with an author study onEd Young writings. Typically, an author study consisted of reading a largenumber of books written by the author and doing writing or art activitiesrelated to the author. The initial inquiry questions the preservice teachersbrought to our meeting were: “What is your favorite Ed Young book? Whatis your least favorite Ed Young Book? What year were Ed Young’s bookswritten? How many pages are in Ed Young’s books?” As our discussionon the purpose of the curriculum unfolded, it was mentioned that authorshave certain tendencies in the types of setting they use, the charactersthey create, as well as the plots, conflicts, and story resolutions they use.Ultimately, we examined the possibility of using Setting, Character Types,Plot, and Resolution as variables about which data could be collected andsummarized in an attempt to characterize Ed Young as an author and tolearn about components of literary composition.

As we observed these two students working with children we weresurprised to see their Ed Young author study and statistical investigationorganized around how many awards Ed Young received as an author andartist, how many books he wrote, illustrated, or did both, and whether thebook was fiction or non-fiction. The variables of Setting, Plot, Characters,


and Resolution were absent even though as instructors we had workedextensively with students to reorient and enrich their investigations. Thiswas difficult for us to understand until we read the preservice teachers’analyses of their own projects. Greg wrote,

The hardest part about this assignment was getting started with the children. They werehaving a hard time forming questions that qualified as investigation questions. We told thechildren that we wanted them to find answers to questions that required us to read a book.What we were getting at was that we did not want to answer questions like ‘How manykids does Ed Young have?’ or ‘Does Ed Young have any pets?’ or ‘Where does Ed Younglive?’ We were interested in finding out when Ed Young did most of his work, what typeof stories Ed Young was involved in, whether he wrote or illustrated more books, etc. . . .

the children had a time distinguishing between the two types of questions.

It is interesting to note the parallel between telling preservice teachers todo something, which does not happen, and the preservice teachers tellingchildren to do essentially the same task, which also does not happen.

The questions our students were encouraging children to ask werea move from personal biographical information to more professionalbiographical information. Linda, Greg’s practicum partner, wrote,

Our goal was to come up with questions the children could answer by reading Ed Young’sbooks because we wanted the children to learn about Ed Young as an author and illustrator,not just personal facts about him. This concept was really hard for children to grasp.

The biographical questions were interesting but tended not to lend them-selves to the “study” of an author. The answers to these personal andprofessional questions could be found, without ever reading one of theauthor’s books, in resources like book jackets, title pages, encyclope-dias, or authors’ web sites. On the surface, it may appear that there isa profoundly difficult task at hand, developing pupils’ understanding ofliterary appreciation and using statistical investigation. These two topicshave little in common; however, merging the content of literary appre-ciation with the process of statistical investigation is no different thanwith any other curriculum topic. Both content learning goals and processknowledge about the investigation must be kept in mind simultaneously forthe investigation to be successful. It merely appears that statistics is moreclosely related to a science investigation, for example, because the processis consistent with the scientific method, as opposed to more traditionalliterary methods of investigation.

From our perspective, actually reading an author’s texts in the contextof an author study is a highly desirable activity. Greg and Linda agreed andaimed to do this in their author study. Greg noted,

I really liked this method of doing an author study. I felt it was more worthwhile than justreading a bunch of Ed Young’s books and talking about what pets he has, for example.


Each child studied one specific book and we gave them some time to read whichever otherEd Young books they would like. I think all of them read at least two different books and Iknow many read more than that.

So, in fact, Greg and Linda may have felt like they had made major changesin the nature of author studies through their investigations with childreneven though from our perspective their changes were minor and illustrateda limited understanding of the potential of author studies.

Demands on Teachers and LearnersDeveloping substantive questions represented only part of the problem.Across both semesters, we observed our students and the children withwhom they worked become focused on merely the technical componentsof graphing data and lose any larger sense of purpose for the investiga-tion. Specifically, we observed in classroom after classroom, the making ofgraphs become the end point of data collection and inquiry. Making graphs,for the sake of making a graph, became the learning objective. Absent wasany meaningful connection back to the investigation questions through adeeper interpretation of the data or any recognition the connection waslacking. The factual questions the students insisted on working with andthe deliberate and controlled manner in which they created graphs withchildren seemed likely to be related to students’ own uncertainty aboutthe components and complexity of reasoning with data. Making graphswas a familiar and concrete task to pursue with children. While the formof “direct instruction” varied, across students’ practices, the task withchildren remained the same – collect data and make a graph, period.

We could argue that to the extent that the process of statistical inves-tigation, namely, posing questions, designing a study, collecting data,summarizing/analyzing the data, and drawing conclusions, was sufficientlynovel for the students, their teaching was limited. Even with classroominstruction and experience, moving between the process and the inquiryquestion was challenging for the students. So, when they were placedin the elementary classroom to teach this process, they resorted to thosecomponent parts of the process that they knew well and understood,namely data collection and graphing. Furthermore, this lack of flexibilitymade it difficult for them to recognize teachable moments and decidewhat to do with them when they arose. Nancy’s comments indicated theneed to understand something about statistics as well as the topic beinginvestigated – the challenge of integration of subject areas.

The conclusion about the pattern was hard to discuss. The discussion was difficult becauseI didn’t know what I wanted the children to get out of the inquiry. I didn’t know enoughabout reasoning of the inquiry myself so it was hard to make a discussion about something


I wasn’t too knowledgeable about myself. This part of the inquiry showed me I needed tobe more prepared about the inquiry. I need to know what I want the children to understandand make sense out of the inquiry.

These comments illustrated the complex intellectual demands placed onthe teacher. Tami’s following comments provided further evidence of weakknowledge on the part of the teacher.

The children did a better job at organizing the data than interpreting it. I don’t know ifthat’s because analyzing data is beyond a first grader’s developmental skills or if I’m justunderestimating these children or if it’s hard for me to teach those skills because I havelittle practice with them myself.

Her last explanation deals with statistical content knowledge. In spiteof everything we did, our students may not have been given adequateopportunities to construct sufficient statistical content knowledge. Withoutdeeper statistical content knowledge, the only analytic tool at their disposalwas simple graphs. Therefore, they taught that which they knew well.

Potential of Statistical InvestigationThere were a few bright spots and glimpses of the kind of learning we envi-sioned for children and our students. In one such project, a pair of studentsworked with sixth graders on an investigation about discrimination. Thestudents noted that the initial investigation questions, “Have you beendiscriminated against?” and “Have you discriminated against others?”in the context of a predominantly white and middle class school wouldnot generate data with sufficient variability for children to learn aboutdiscrimination. During our question formulation discussion, we helped thestudents consider how to work with children to obtain a diverse set ofresponses and how to form questions that illustrated what discriminationlooked and felt like for white middle class children who were not directlyexposed to this issue. The preservice teachers leading this investigationtook the notion of obtaining a diverse set of responses seriously and madecontact with sixth grade teachers at a high minority, low income, inner-city elementary school to initiate a joint investigation. They jointly helpedchildren develop a discrimination questionnaire, gather survey data, andanalyze the data to make comparisons between the schools.

Interestingly enough, the results from their survey data did not turn outexactly the way the sixth grade children at our practicum site expected.This motivated a serious and animated discussion amongst children,wholly unprompted by our students, on the topic of whether or not theirquestions truly measured discrimination, what discrimination actually was,and whether the data collected was right. As Mickelson wrote in his fieldnotes, “. . . they are having a graduate level discussion of construct validity


and external validity, its in the language of sixth graders, but I wish someof my graduate students would have this type of discussion.” Connie, oneof the students who led this project, wrote in her final reflections on theproject,

I believe that they learned more about discrimination from this than they would have fromreading articles on discrimination or from just reading the book on their own. It reallyinvests a teacher and the children in the topic and because of that, the learning is moreauthentic and natural than otherwise.

Tom, her practicum partner, saw the usefulness of the process of statisticalreasoning as a tool to learn about something. The project on discriminationgrew out of work with a literature group. He saw what it meant to integratesubject areas meaningfully. He wrote:

This inquiry and summary type project would be a great addition to any science or socialstudies or literature unit. As we have learned in science this semester and throughoutclasses in the Elementary Teacher Education Program, integrating areas of the curriculumis a great way to help children make connections between the different things they arelearning and get a clearer image for the big picture. For example, had I done the inquiryabout flying planes, the conclusions we could draw would be about concepts of physics,as it relates to flying. In either case, we took one area of the curriculum and managed toinclude examinations of related concepts in the realm of social studies or science.

The project on discrimination and the outcomes for the preservice teachersexceeded our expectations in terms of learning about statistics and inves-tigative teaching. It illustrates the great potential for merging statisticalinvestigation with the elementary curriculum.

Between Projects, Across Semesters – On Teacher Learning

We took a step back with our analysis and looked for general themes thatspanned projects and semesters related to teacher learning and change.This type of analysis helped us to see in new ways the systemic context ofschooling and the powerful force of tradition in teaching. Inquiry Project Iand Inquiry Project II, to a certain extent, challenged preservice teachers’conceptions of knowledge and their images of the roles and responsi-bilities of teachers and learners. By and large the preservice teachers inthis study have experienced in their own education the transmission modelof teaching and learning (Jackson, 1986). As Lortie (1975) reminded us,teachers teach the way they were taught. And as Cuban (1993) found,changing that is hard.

Our two inquiry projects challenged this conception in multiple ways.On one level, Inquiry Project I challenged preservice teachers as learnersof mathematical pedagogy by requiring them to infer an answer aboutteaching or learning based on data to another question they posed that did


not have a right or wrong solution. Furthermore, the process of pooling anddiscussing the results of these investigations afforded students the oppor-tunity to construct their own understandings of the culture and environmentof elementary mathematics classrooms. On a second level, Inquiry ProjectII challenged the students as teachers to design and implement their ownopen-ended investigations with children, with the processes of inquirybeing as important as the product. Finally, on a third level, these projectsafforded preservice teachers the opportunity to see a different role forlearners in their teaching.

This was related to Cohen’s (1988) notion of adventurous teaching.Repeatedly our students reminded us that this switch in perspective onteaching and learning was risky, not without cost, and led to teacher andlearner angst. For example, Beth, a preservice teacher, reflected on theassignments, “But it’s hard. It’s frustrating when you’re trying to figureout what to do if you’ve never done it. I was like, ‘I have no clue whatthey want . . .’.” Kathy, another preservice teacher, talked to Mickelson inan interview about her struggle with expectations, “And I know for a lotof us it was hard for us to understand . . . I didn’t understand what youwere asking of us . . . and you just keep telling us, ‘You just have to go outthere and try it and you’ll be surprised at what you know how to do’.” It’snot unlike the experiences children face when put in similar situations aslearners. Paul, a preservice teacher working with second graders, remindedus of what this kind of teaching demands of learners, “Some kids are goingto say, ‘you know this is too hard. I am not going to do it. I just want theanswer’.” It required a new way of working for everyone. Renee notedhow unfamiliar this way of working was for the fifth graders with whomshe worked, “the students were just somewhat shaken by being asked to bea part of the development of it. They’re so used to being just told what todo . . . they just kind of froze up and were like, ‘tell us what to do, please’.”It required giving up some control as a teacher and taking on ownershipas a learner. This kind of ownership did develop with opportunities andexperiences for Tawanda. She told Mickelson,

I had to come up with a graph, look at it and try to figure out what it meant to me . . .

how to make a graph on my computer . . . . I was quite proud of myself. I think the mostimportant thing is that as much as I complain about being like pushed and challenged,it’s a good way to kind of extend yourself in different areas that you’re not comfortablewith. That was probably the biggest lesson. That it’s okay to try something that you’renot accustomed to . . . . I would say that taking risks is what I learned from both of theseprojects.

Along with being asked to change perspective on teaching and learningthere is an embedded issue of understanding what inquiry or investigationactually entails.


As academics we understood investigation and inquiry. We lackedexperience trying to help others learn to teach in these ways in the contextof the realities of classroom life. Even though investigation and inquirywere stances toward teaching in other areas of our teacher educationprogram, the meaning of these words remained unclear to students. Whenasked whether they had experienced inquiry prior to this semester, studentsmentioned three different places the idea of inquiry had surfaced in theirexperiences in the teacher education program. One place was in a socialstudies methods class. In that experience, they were familiarized with theinquiry process. In describing a requirement for that course, Judy noted,“I did an inquiry on inquiry.” In reflecting on a reading and languagearts methods class, some students mentioned doing inquiry in the contextof a child study, where they were required to develop a case study of achild. Monica described the requirements as, “We did a lot with readinginquiries, a log of group studies, collecting data about a child, and puttingit together.” The third place students mentioned hearing about inquirywas in science methods class. Several made connections to what theywere learning about science experiments. JoAnne recollected, “I thinkwe covered it in science before we talked about it in math.” On the onehand, clearly inquiry was a familiar word to preservice teachers and theyhad experience implementing the general process. On the other hand, theyhad had limited experiences with it in the context of their role as teachershelping others to learn the process. As Maggie noted in an interview, “I’vedone inquiry projects before but it’s been more me doing it, not the chil-dren.” Her partner, Tara, agreed, “It hasn’t been in the classroom, it’s beenjust me.” The specific notion of conducting a statistical investigation wasuniformly new and was considered to be substantively different than theirother experiences with inquiry. From JoAnne’s perspective, we assumedstudents knew more than they did. JoAnne said, “So maybe she [Heaton]should have elaborated more. She thought we knew what we were doing.”

Even though we were disappointed in terms of what we observed inour students’ actions and learning, looking across the two assignments, welearned some things about the contextual and situational issues affectingstatistics as an element of the elementary curriculum, as an element ofthe knowledge of elementary teachers, and as an element of a teachereducation program. We learned a great deal about both processes of inquiryand the contribution of inquiry to learning at a multiplicity of levels, aswell as the shortcomings of inquiry in terms of curricular needs and theestablishment of traditional knowledge. Educators have to grapple withsuch issues, there are no easy answers, and the process of inquiry makesvivid the issues and needs for solutions.


DISCUSSION

Friel and Bright (1998) noted in the study of their own efforts at teachereducation, “much may be ascertained about teachers’ understanding ofcontent by watching them teach” (p. 106). We were venturing into newterritory as we contemplated the learning of teachers in statistics educationby trying to acquire an understanding of what teachers know and need tolearn about statistics. Within this new territory, we had the opportunity toconsider learning at three levels: children, preservice teachers, and teachereducators. In this study of preservice teachers and their work with children,our open-ended approach was in stark contrast to much of the statis-tics education research focused on children’s understanding of particularstatistical ideas. In contrast to our work, for example, Cobb (1999), noted,

[o]ur decision to focus on the competencies of developing and critiquing data-basedargument did lead us to make an important design decision when planning the teachingexperiment. In particular, we ruled out an open-ended project approach in which studentsinvestigate issues of personal interest by generating data and instead developed instruc-tional activities in which the students analyzed data sets created by others (p. 38).

We intentionally used an open-ended approach as a means to understandbeginning teachers’ knowledge and ability to implement the process ofstatistical investigation as it relates to teaching. Through our investiga-tive approach to our own teaching and research, we found that muchrelated to the process and content of statistical investigation, as a toolfor learning other content, was not obvious to beginning teachers. Whatwe wanted required not only an investigative approach to teaching mathe-matical content (Jaworski, 1994), in this case, statistical content, but aninvestigative approach to teaching the content of other subject areas, anequally difficult pedagogical challenge.

It did not appear that Inquiry Project I, alone, as initially implemented,provided sufficient opportunity for preservice teachers to develop thedepth of statistical content knowledge or experience with the process ofreasoning with data needed to teach children the process of statisticalinvestigation. What was clear was that Inquiry Project I, with its focuson the mathematical classroom environment, did provide a context forlearning this process that most preservice teachers found interesting andrelevant. A vested interest in the outcome of statistical investigation poten-tially opens a window for preservice teachers to learn and retain moretechnical statistical content. Our findings were consistent with the find-ings of Bradstreet (1996) and Garfield (1995). If this is the case, the nextproblem becomes, How can we help preservice teachers transfer theirunderstanding of statistics to the elementary classroom and help children


develop statistical skills as they study other curriculum topics? The placeand purpose in teacher education for statistical classroom activities, likeGummy Bears in Space (Schaeffer, Gnanadesikan, Watkins & Witmer,1996) and The Inclined Plane Activity (Edwards, 1995) are becoming moreapparent but the connections need to be made explicit as transfer was notseen as obvious, even with basic graphing. Our findings are consistentwith those of Project Teach-Stat (Friel & Bright, 1998) which found it isimportant to offer teachers repeated opportunities to experience statisticsas learners prior to teaching.

In our work as teacher educators and researchers, we were trying toaddress and assess both statistical content and process as well as the mean-ingfulness of the subject matter content learned through the investigationprocess. This attention to both statistics and subject matter content extendsthe work of Gal and Garfield (1997), whose primary attention was onassessment of statistical content knowledge with minor attention given tothe knowledge acquired in the process about anything other than statistics.If we are going to take seriously an effort to integrate statistical inves-tigation into the elementary curriculum and help preservice teachers andchildren appreciate and use statistical investigation as a tool for learningsomething meaningful about content, then we must assess and try toenhance both what is gained statistically as well as content-wise in theprocess.

For the teacher, this implies acquiring a challenging combination ofskills and knowledge. In addition to a deeper understanding of statis-tics content, the teacher also needs knowledge about doing investigationswith children, a quantitative research perspective on how to formulateinvestigation questions, an understanding of how to carry out the processof quantitative investigation, pedagogical knowledge on how to facili-tate the reasoning with data process with children, and a dispositiontoward teaching and learning that values knowledge obtained throughinvestigation.

CONCLUSIONS

We see problems that need to be addressed in implementing our vision ofmerging statistical investigation with the elementary curriculum, but wealso see tremendous possibilities. We must continually remind ourselvesthat, while we were somewhat disappointed with the teaching practicesthat resulted, repeatedly our students found this to be a good experienceand one from which they reported learning a great deal. As is often thecase, the changes and progress seem larger from the perspective and the


work of the insider than from the observations of the outsider (Wilson &Goldenberg, 1998; Cohen, 1991).

We continue to believe that it is important that our students get exper-iences with statistical investigation first as learners and then as teachers.We are finding the design to yield benefits just as Borasi et al. (1999)found in their design of professional development opportunities for exper-ienced teachers in mathematics. But from this work we would ideally needmultiple opportunities for students to be learners of the statistical investiga-tion process, rather than just one. Given that we want students to integratethis into other subject areas, it would be ideal to offer those opportunities inthe context of other methods courses within a teacher education program.This necessitates seeing and valuing statistical investigation as one of manytools for learning subject matter content in meaningful ways. Teaching thestatistical reasoning process with real and relevant data repeatedly acrossa teacher education program would be consistent with Bradstreet’s (1996)view on teaching statistics to nonstatisticians and Chance’s (1997) desireto have learners pick their own topics as a means of getting them interestedin statistics. Our aim is to help teachers see that statistical investigation canbe used as a tool to study their own teaching or as one of many means ofteaching content to children.

The NCTM standards (2000) have gone a long way in pushing onthe nature of knowledge and purpose of curriculum within the confinesof mathematics. As one tries to merge the notion of statistical investiga-tion into the elementary curriculum, one sees statistical investigation asa tool for learning about statistical content in meaningful ways. Moreimportantly, this approach appears to hold potential for pushing on thenature of knowledge and the purpose of curriculum in other subject areasacross the elementary curriculum, and could be seen, for example, as a toolfor implementing and fulfilling national standards in science and socialstudies. Statistical investigation could potentially lead to authentic inquiryor learning of subject matter (Newman & Wehlage, 1997). If that is one’sgoal as a teacher or teacher educator, statistical investigation is one meansto that end.

If statistical education is to be addressed seriously in elementary educa-tion, in the ways this study advocates, specific focus needs to be placedon the learning of teachers, with extension to the education of teachereducators. We cannot attend to children’s understanding of statisticswithout simultaneously attending to teachers’ understandings. We need totry to understand what teachers need and then offer them the intellectualsupport and encouragement necessary to understand and teach the processof statistical investigation in meaningful contexts.


Collaborative work between statisticians and mathematics teachereducators is a beginning step toward understanding the development ofreform-minded practices of elementary mathematics education (Borko etal., 1992) that would meet the content and pedagogical needs of begin-ning teachers (CBMS, 2001; Ma, 1999), challenge their beliefs aboutteaching and learning, and promote an understanding of statistics throughthe merging of statistics into the elementary curriculum.

REFERENCES

Ainley, J., Nardi, E. & Pratt, D. (2000). Towards the construction of meaning for trendin active graphing. The International Journal of Computers for Mathematical Learning,5(2), 85–114.

Beres, R. J. (1988). Statistics for college-bound students: Are the secondary schoolsresponding? School Science and Mathematics, 88(3), 200–209.

Borasi, R., Fonzi, J., Smith, C. F. & Rose, B. J. (1999). Beginning the process of rethinkingmathematics instruction: A professional development program. Journal of MathematicsTeacher Education, 2(1), 49–78.

Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D. & Agard, P. C. (1992).Learning to teach hard mathematics: Do novice teachers and their instructors give up tooeasily? Journal for Research in Mathematics Education, 23(2), 194–222.

Bradstreet, T. E. (1996). Teaching introductory statistics so that nonstatisticians experiencestatistical reasoning. The American Statistician, 50(1), 69–78.

Chance, B. L. (1997). Experiences with authentic assessment techniques in an introductorystatistics course. Journal of Statistics Education, 5(3), 1–15.

Cobb, P. (1999). Individual and collective mathematical development: The case ofstatistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43.

Cohen, D. K. (1991). Revolution in one classroom (or then again was it?). AmericanEducator, 15(2), 16–23, 44–48.

Cohen, D. K. (1990). A revolution in one classroom: The case of Mrs. Oublier. EducationalEvaluation and Policy Analysis, 12(3), 311–329.

Cohen, D. K. (1988). Teaching practice: plus que ca change. In P. W. Jackson (Ed.),Contributing to educational change: Perspective on research and practice (27–84).Berkeley, CA: McCutchen Publishing Corporation.

Conference Board of Mathematical Sciences (2001). The mathematical education ofteachers. Washington, D.C.: Author.

Creswell, J. W. (1998). Qualitative inquiry and research design: Choosing among fivetraditions. Thousand Oaks, CA: Sage Publications.

Cuban, L. (1993). How teachers taught: Constancy and change in American classrooms1890–1990. New York: Teachers College Press.

Dewey, J. (1938). Experience and education. New York: Collier Books.Edwards, T. (1995). Students as researchers: An inclined-plane activity. Mathematics

Teaching in the Middle Schools, 1(7), 532–535.Feldman, A., Konold, C., Coultier, B. with Conroy, B., Hutchinson, C. & London, N.

(2000). Network science, a decade later. Mahwah, NJ: Lawrence Erlbaum Associates.Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Boston: D.

Reidel.


Fong, G. T., Krantz, D. H. & Nisbett, R. E. (1986). The effects of statistical training onthinking about everyday problems. Cognitive Psychology, 18, 253–292.

Friel, S. N. & Bright, G. W. (1998). Teach-Stat: A model for professional development indata analysis and statistics for teachers K-6. In S. P. Lajoie (Ed.), Reflections on statistics:Learning, teaching, and assessment in grades K-12. Mahwah, NJ: Lawrence ErlbaumAssociates.

Friel, S. N. & Bright, G. W. (1997). A framework for assessing knowledge and learning instatistics (K-8). In J. Gal and J. B. Garfield (Eds.), The assessment challenge in statisticseducation (55–63). Amsterdam: IOS Press.

Gal I. & Garfield, J. B. (Eds.) (1997). The assessment challenge in statistics education.Amsterdam: IOS Press.

Garfield, J. (1995). How students learn statistics. International Statistical Review, 63, 25–43.

Goodlad, J. I. (1994). Educational renewal: Better teachers, better schools. San Francisco,CA: Jossey Bass.

Graham, A. (1987). Statistical investigations in the secondary school. Cambridge:Cambridge University Press.

Hancock, C., Kaput, J. J. & Goldsmith, L. T. (1992). Authentic inquiry with data: Criticalbarriers to classroom implementation. Educational Psychologist, 27, 337–364.

Heaton, R. M. (2000). Teaching mathematics to the new standards: Relearning the dance.New York: Teachers College Press.

Heaton, R. M. (1992). Who is minding the mathematics content? A case study of a fifthgrade teacher. Elementary School Journal, 93, 153–162.

Horobin, K. & Acredolo, C. (1989). The impact of probability judgments on reasoningabout multiple possibilities. Child Development, 60, 183–200.

Jacobs, V. (1993). Stochastics in middle school: An exploration of students’ informalknowledge. Unpublished master’s thesis, University of Wisconsin, Madison.

Jackson, P. (1986). The practice of teaching. New York: Teachers College Press.Jaworski, B. (1998). The centrality of the researcher: Rigor in a constructivist inquiry into

mathematics teaching. In A. Teppo (Ed.), Qualitative research methods in mathematicseducation (112–127). Reston, VA: National Council of Teachers of Mathematics.

Jaworski, B. (1994). Investigating mathematics teaching: A constructivist enquiry.London: Falmer Press.

Kader, G. & Perry, M. (1994). Learning statistics with technology. Mathematics Teachingin the Middle Schools, 1, 130–136.

Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 61(1),59–98.

Konold, C. (1991). Understanding students’ beliefs about probability. In E. von Glasersfeld(Ed.), Radical constructivism in mathematics education (139–156). The Netherlands:Kluwer.

Lajoie, S. P., Jacobs, V. R. & Lavigne, N. C. (1995). Empowering children in the use ofstatistics. Journal of Mathematical Behavior, 14, 401–425.

Lajoie, S. P. & Romberg, T. A. (1998). Identifying an agenda for statistics instruction andassessment in K-12. In S. P. Lajoie (Ed.), Reflections on statistics: Learning, teaching,and assessment in grades K-12. Mahwah, NJ: Lawrence Erlbaum Associates.

Lampert, M. & Ball, D. L. (1999). A design for a pedagogy of teacher education. Teaching,multimedia, and mathematics: Investigations of real practice (109–158). NY: TeachersCollege Press.


Lehrer, R. & Romberg, T. (1996). Exploring children’s data modeling. Cognition andInstruction, 14(1), 69–108.

Lehrer R. & Schauble, L. (2000). Inventing data structures for representational purposes:Elementary grade students’ classification models, Mathematical Thinking and Learning,2(1 & 2), 51–74.

Lortie, D. (1975). Schoolteacher. Chicago: University of Chicago Press.Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding

of fundamental mathematics in China and the United States. Mahwah, NJ: LawrenceErlbaum Associates.

Moore, D. S. (1997). New pedagogy and new content: The case of statistics. InternationalStatistical Review, 65(2), 123–137.

National Council of Teachers of Mathematics (2000). Principles & standards for schoolmathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.

Newmann, F. M. & Wehlage (1997). Successful school restructuring: A report to thepublic and educators. Center on Organization and Restructuring of Schools, Universityof Wisconsin, Madison.

Nisbett, R. E., Krantz, D. H., Jepson, C. & Kunda, Z. (1983). The use of statisticalheuristics in everyday inductive reasoning. Psychological Review, 90(4), 339–363.

Putnam, R. T. (1992). Teaching the ‘how’ of mathematics for everyday life: A case studyof a fifth grade teacher. Elementary School Journal, 93, 163–177.

Riddiough, R. & McCall, J. (1998). How long is a piece of string. Teaching Statistics,20(1), 13–16.

Russell, S. J. & Friel, S. N. (1989). Collecting and analyzing real data in the elementaryschool classroom. In P. R. Trafton & A. P. Shulte (Eds.), New directions for elementaryschool mathematics (134–148). Reston, VA: National Council of Teachers of Mathe-matics.

Schaeffer, R. L., Gnanadesikan, M., Watkins, A. & Witmer, J. A. (1996). Activity-basedstatistics. New York: Springer-Verlag.

Schaeffer, R. L. (1988). Statistics in schools: The past, present and future of the quantitativeliteracy project. Proceedings of the American Statistical Association from the Section onStatistics Education (71–78).

Shaughnessy, M. (1992). Research in probability and statistics: Reflections and directions.In D. Grouws (Ed.), Handbook for research in mathematics teaching and learning (465-494). New York: McMillan.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. EducationalResearcher, 15(2), 4–14.

Wilson, M. & Goldenberg, M. P. (1998). Some conceptions are difficult to change:One middle school mathematics teacher’s struggle. Journal of Mathematics TeacherEducation, 1(3), 269–293.

University of Nebraska-Lincoln118 Henzlik HallLincoln, NE 68588-0355USAE-mail: [email protected]

ERIC J. KNUTH

TEACHERS’ CONCEPTIONS OF PROOF IN THE CONTEXT OFSECONDARY SCHOOL MATHEMATICS

ABSTRACT. Current reform efforts in the United States are calling for substantial changesin the nature and role of proof in secondary school mathematics – changes designed toprovide all students with rich opportunities and experiences with proof throughout theentire secondary school mathematics curriculum. This study examined 17 experiencedsecondary school mathematics teachers’ conceptions of proof from their perspectives asteachers of school mathematics. The results suggest that implementing “proof for all”may be difficult for teachers; teachers viewed proof as appropriate for the mathematicseducation of a minority of students. The results further suggest that teachers tended toview proof in a pedagogically limited way, namely, as a topic of study rather than as atool for communicating and studying mathematics. Implications for mathematics teachereducation are discussed in light of these findings.

KEY WORDS: proof, reform, secondary mathematics, teacher conceptions

INTRODUCTION

Many consider proof to be central to the discipline of mathematics andto the practice of mathematicians;1 yet surprisingly, the role of proof inschool mathematics in the United States has been peripheral at best. Prooftraditionally has been expected to play a role only in the mathematicseducation of college-intending students and, even in this capacity, its rolehas been even further constrained – the only substantial treatment of proofhas been limited to the domain of Euclidean geometry. This absence ofproof in school mathematics has not gone unnoticed and, in fact, has beena target of criticism. Wu (1996) argued, for example, that the scarcity ofproof outside of geometry is a misrepresentation of the nature of proof inmathematics. He stated that this absence is

a glaring defect in the present-day mathematics education in high school, namely, the factthat outside geometry there are essentially no proofs. Even as anomalies in education go,this is certainly more anomalous than others inasmuch as it presents a totally falsifiedpicture of mathematics itself (p. 228).

Similarly, Schoenfeld (1994) suggested that “proof is not a thing separablefrom mathematics as it appears to be in our curricula; it is an essentialcomponent of doing, communicating, and recording mathematics. And I


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believe it can be embedded in our curricula, at all levels” (p. 76). Sowderand Harel (1998) also argued against limiting students’ experiences withproof to geometry, but more from an educational rather than mathematicalperspective: “It seems clear that to delay exposure to reason-giving untilthe secondary-school geometry course and to expect at that point an instantappreciation for the more sophisticated mathematical justifications is anunreasonable expectation” (p. 674).

Reflecting an awareness of such criticism, as well as embracing theimportant role of proof in mathematical practice, recent reform effortsin the United States are calling for substantial changes in both schoolmathematics curricula and teachers’ instructional practices with respect toproof. In contrast to the status of proof in the previous national standardsdocument (National Council of Teachers of Mathematics [NCTM], 1989),its position has been significantly elevated in the most recent document(NCTM, 2000). Not only has proof been upgraded to an actual standardin this latter document, but it has also received a much more prominentrole throughout the entire school mathematics curriculum and is expectedto be a part of the mathematics education of all students. More specifi-cally, the Principles and Standards for School Mathematics (NCTM, 2000)recommends that the mathematics education of pre-kindergarten throughgrade 12 students enable all students “to recognize reasoning and proof asfundamental aspects of mathematics, make and investigate mathematicalconjectures, develop and evaluate mathematical arguments and proofs, andselect and use various types of reasoning and methods of proof” (p. 56).

Enacting these recommendations, however, places significant demandson school mathematics teachers as approaches designed to enhance therole of proof in the classroom require a tremendous amount of a teacher,particularly in terms of teachers’ understanding of the nature and roleof proof (Chazan, 1990; Jones, 1997). The challenge of meeting thesedemands is particularly daunting given that many school mathematicsstudents have found the study of proof difficult (e.g., Balacheff, 1991;Bell, 1976; Chazan, 1993; Coe & Ruthven, 1994; Healy & Hoyles, 2000;Porteous, 1990; Senk, 1985). Further exacerbating these demands is thefact that mathematics teacher education and professional developmentprograms typically have not prepared teachers adequately to enact success-fully the lofty expectations set forth in reform documents (Ross, 1998).Consequently, to prepare adequately and support teachers to meet thesedemands successfully, it is necessary to understand the complex array offactors influencing teachers’ interpretations and enactment of such reformrecommendations.

TEACHERS’ CONCEPTIONS OF PROOF 63

One such set of factors, teachers’ knowledge and beliefs, have beenidentified as important determinants of teachers’ classroom practices and,consequently, have major implications for the extent to which teachersimplement reform recommendations (Borko & Putnam, 1996). Accord-ingly, the success of current reform efforts with respect to proof dependsin large part on the nature of teachers’ knowledge and beliefs about proof.Although researchers have focused on teachers’ conceptions2 of proof(e.g., Goetting, 1995; Jones, 1997; Harel & Sowder, 1998; Knuth, In press;Martin & Harel, 1989; Simon & Blume, 1996), this research typicallyhas not focused on teachers as individuals who are teachers of schoolmathematics; rather, such research has focused primarily on teachers asindividuals who are knowledgeable about mathematics. In highlighting theparticular focus of this previous research, I am not suggesting that teachers’conceptions as “knowers” of a discipline do not influence their teaching ofthe discipline. Indeed, in mathematics, for example, there is an extensivebody of literature that suggests teachers’ subject matter conceptions havea significant impact on their instructional practices (e.g., Fennema &Franke, 1992; Thompson, 1992). Research on teachers’ conceptions ofproof, however, has tended to focus exclusively on teachers as “knowers”of mathematics rather than as teachers of mathematics. Consequently,research that examines teachers’ conceptions of proof in the context ofsecondary school mathematics is greatly needed.

In this article, I describe the results from a study designed both toaddress this void and to identify areas of need for preparing teachers toenact the recommendations of reform successfully with respect to proof.Prior to presenting and discussing the results of this study, however, I firstpresent a framework for thinking about proof in school mathematics. Inaddition, I discuss briefly proving practices in school mathematics in termsof this framework.

A FRAMEWORK FOR CONSIDERING PROOFIN SCHOOL MATHEMATICS

Authors have suggested various roles that proof plays in mathematics:

• to verify that a statement is true,• to explain why a statement is true,• to communicate mathematical knowledge,• to discover or create new mathematics, or• to systematize statements into an axiomatic system (e.g., Bell, 1976;

de Villiers, 1999; Hanna, 1983, 1990; Schoenfeld, 1994).

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Although these particular roles were proposed in terms of proof in thediscipline of mathematics, I have found them to be useful for thinkingabout proof in school mathematics as well. Accordingly, I have used thesefive roles as a framework for considering proof in school mathematics inthis paper. I elaborate briefly on these roles below.

The role of proof in verifying that a statement is true requires littleelaboration. Indeed, few would question that a main role of proof in mathe-matics is to verify the correctness of a result or truth of a statement (Hanna,1983). Not surprisingly, this is typically the role most students encounterduring their school mathematics experiences. Students’ experiences withproof, however, often are limited to verifying the truth of statements thatthey know have been proven before and, in many cases, are intuitivelyobvious to them. Such experiences often lead students to view proof asa procedure for confirming what is already known to be true (Schoen-feld, 1994); as a consequence, proof reduces to “just a game because youalready know what the result is” (Wheeler, 1990, p. 3).

Mathematicians, however, expect the role of proof to include more thansimply the verification of results: “mathematicians routinely distinguishproofs that merely demonstrate from proofs which explain” (Steiner, 1978,p. 135). Making a similar distinction regarding this role of proof, that is,its explanatory role, Hersh (1993) contended that mathematicians are inter-ested in “more than whether a conjecture is correct, mathematicians wantto know why it is correct” (p. 390). Others also have echoed comparablesentiments: “[the] status of a proof will be enhanced if it gives insightas to why the proposition is true as opposed to just confirming that itis true” (Bell, 1976, p. 6) and “the best proof is one which also helpsmathematicians understand the meaning of the theorem being proved: tosee not only that it is true, but also why it is true” (Hanna, 1995, p. 47).In contrast, proof in school mathematics traditionally has been perceivedby students as a formal and, often meaningless, exercise to be done forthe teacher (Alibert, 1988). In fact, as Harel and Sowder (1998) suggested,“we impose on them [i.e., students] proof methods and implication rulesthat in many cases are utterly extraneous to what convinces them” (p. 237).Consequently, as Schoenfeld (1994) concluded, “in most instructionalcontexts proof has no personal meaning or explanatory power for students”(p. 75).

Many within the mathematics community also view proof as a socialconstruct and product of mathematical discourse (e.g., Davis, 1986;Hanna, 1983; Hersh, 1993; Richards, 1991). As Manin (1977) stated, “aproof becomes a proof after the social act of ‘accepting it as a proof”’(p. 48). Similarly, Hanna (1989) noted that “the acceptance of a theorem


by practising mathematicians is a social process” (p. 21). Consonant withthis view of proof is the approach to mathematical growth and discoveryoutlined in Lakatos’ (1976) seminal book, Proofs and Refutations.3 Inaddition to the social nature embodied in the process of accepting anargument as a proof, the “product” of such a process (i.e., a proof itself)also provides a means for communicating mathematical knowledge withothers (Alibert & Thomas, 1991; Schoenfeld, 1994). Yet, the social natureof proof traditionally has not been reflected in the proving practices ofschool mathematics. Chazan (1990) suggested that geometry instruction,for example, “downplays any social role in the determination of the validityof a proof; the teacher and the textbook are the arbiters of validity”(p. 20). Balacheff (1991) also noted the limited attention given to the socialnature of proof: “What does not appear in the school context is that amathematical proof is a tool for mathematicians for both establishing thevalidity of some statement, as well as a tool for communication with othermathematicians” (p. 178).

Proof also plays an important role in the discovery or creation of newmathematics. As de Villiers (1999) noted, “there are numerous examples inthe history of mathematics where new results were discovered or inventedin a purely deductive manner [e.g., non-Euclidean geometries]” (p. 5). Asdiscussed above, the role of proof in school mathematics typically hasbeen to verify previously known results. The role of proof in creatingnew mathematics, however, is beginning to play a larger part in manysecondary school geometry classrooms, particularly those classroomsin which students are utilizing dynamic geometry software (Chazan &Yerushalmy, 1998). Through their explorations, students generate conjec-tures and then attempt to verify the truth of the conjectures by producingdeductive proofs. In this case, students are using proof as a means ofcreating new results.

Finally, the role of proof that is the “most characteristically mathe-matical” (Bell, 1976, p. 24) is its role in the systematization of resultsinto a deductive system of definitions, axioms, and theorems. Althoughsecondary school geometry courses focus typically on a particular axio-matic system (i.e., Euclidean geometry), it is questionable whetherstudents are cognizant of the underlying axiomatic structure. In otherwords, I surmise (based on my experience both as a former high schoolteacher and as a teacher educator) that many students view the manytheorems that they are asked to prove as essentially independent of oneanother rather than as related by the underlying axiomatic structure.Geometry instruction typically does not include opportunities for studentsto reflect on the course from a “meta-level.”

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In sum, an informed conception of proof in school mathematics, onethat reflects the essence of proving in mathematical practice, must includea consideration of proof in each of these roles. There is, however, a longdistance between these roles of proof and their manifestation in schoolmathematics practices (Balacheff, 1991).4 As a result of such inconsis-tency, as well as students’ inadequate conceptions of proof, current reformefforts are calling for changes in the nature and role of proof in schoolmathematics (NCTM, 2000; Ross, 1998). My goal in this study was toexamine the extent to which teachers are prepared to enact these newrecommendations for proof in school mathematics. Specifically, this studyexamined teachers’ conceptions of proof in the context of secondaryschool mathematics. The study was guided by the following research ques-tions: (1) What constitutes proof in school mathematics? and (2) Whatare teachers’ conceptions about the nature and role of proof in schoolmathematics?

METHODS

Participants

Seventeen secondary school mathematics teachers (2 middle school and15 high school teachers) participated in this study.5 Their years of teachingexperience varied from three to twenty years, and the courses they taughtvaried from 7th grade mathematics to Advanced Placement Calculus.Eleven of the teachers taught either all lower-level mathematics courses(i.e., courses prior to geometry) or mixed-level mathematics courses (e.g.,first-year algebra and precalculus), while seven of the teachers taught onlyhigher-level courses (i.e., geometry and above). In addition, the teachersutilized various curricular programs in their classrooms; some of theschools in which the teachers teach have adopted reform-based programs,others utilized more traditional programs. I regarded both course level andcurricular program to be possible dimensions of contrast, that is, I hypo-thesized that teachers may have different conceptions of proof in schoolmathematics depending on the level of mathematics courses taught or onthe curricular program utilized. For example, many reform curricula placean emphasis on open-ended problems for which students are expected toprovide justification for their solutions; as a result, teachers may perceiveproof as appropriate in courses other than geometry.

The teachers were selected based on their willingness to participate inthe study; they were selected from among participants in two ongoingprofessional development programs. Although one might question how


representative the participating teachers are to the larger populationof secondary school mathematics teachers, it is worth noting that theparticipating teachers are committed to reform in mathematics educa-tion (as evidenced in their seeking professional development opportunitiesfocusing on reform). Consequently, it is likely that these teachers notonly are familiar with the most recent reform documents (e.g., NCTM,2000) and the corresponding recommendations, but also are interested inchanging their instructional practices to reflect more closely the vision ofpractice set forth in such documents (of which proof is to play a significantrole).

Data Collection

The primary sources of data were two semi-structured interviews. Eachinterview lasted approximately an hour and a half and was audiotaped andlater transcribed. The data were collected in two distinct stages, each witha different primary focus. The first stage focused on teachers’ conceptionsof proof in the discipline of mathematics (i.e., teachers’ conceptions asindividuals who are knowledgeable about mathematics), while the secondstage focused primarily on their conceptions of proof in the context ofsecondary school mathematics (i.e., teachers’ conceptions as individualswho are teachers of secondary school mathematics). At times this separa-tion into two stages seemed somewhat artificial as the teachers often hadtrouble removing their “teacher hats.” Yet, I tried to remain faithful to thisseparation throughout the data collection stages, often reminding teachersto think about a question or task as someone who is knowledgeable aboutmathematics rather than as someone who teaches mathematics. Becausethe focus of this article is on teachers’ conceptions of proof in the contextof school mathematics, the results presented and subsequent discussionfocus primarily on data from the second stage of data collection; however,data from the first stage of data collection are presented as applicable.6

The stage two interview questions focused on teachers’ conceptionsabout the nature and role of proof in the context of secondary schoolmathematics and their expectations of proof for students. Typical ques-tions included: What does the notion of proof mean to you (A questionrepeated from the first stage)? What constitutes proof in secondary schoolmathematics? Why teach proof in secondary school mathematics? Whenshould students encounter proof? What do the authors of the NCTM Prin-ciples and Standards for School Mathematics mean by proof?, and Whatdo you think about the recommendations for proof set forth in the NCTMPrinciples and Standards for School Mathematics?

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During the interview, teachers also were presented with different setsof researcher-constructed arguments for several mathematics statementsand were then asked to evaluate the arguments in terms of each argu-ment’s instructional appropriateness (i.e., would teachers use an argumentto convince students of a statement’s truth) and provide a rationale fortheir evaluation. My rationale for including this evaluation as a componentof the interview was that I expected the teachers’ responses might provideadditional insight into their views of proof in school mathematics. Thearguments presented were chosen to be appropriate mathematically forsecondary school mathematics students. Further, the arguments varied interms of their validity7 as proofs as well as the degree to which theywere explanatory (see Hanna, 1990, for further elaboration on explana-tory proofs). As an example, Figure 1 displays three arguments (froma set of 5) justifying a given statement and which differ in terms ofthese two variations. The argument presented in (a) is not a proof, whilethe arguments presented in (b) and (c) are valid proofs. With respect tothe arguments’ explanatory qualities, argument (a) provides little insightinto why the statement is true, while (b) and (c), to varying degrees, doprovide insight, that is, they provide “a set of reasons that derive from thephenomenon itself” (Hanna, 1990, p. 9). Although constructing argumentsin each set that varied in terms of their explanatory nature required an apriori categorization, I hypothesized that the rationale teachers providedregarding their responses would provide an indication of the degree towhich they found particular arguments more or less explanatory than otherarguments in a set.

Data Analysis

The data analysis was grounded in an analytical-inductive method inwhich teacher responses were coded using external and internal codes andthen classified according to relevant themes. Coding of the data beganusing a set of researcher-generated (external) codes that were identifiedprior to the data collection and that corresponded to, and were derivedfrom, my conceptual framework (i.e., the 5 roles of proof). The deductiveapproach utilized in producing the external codes was then supplementedwith a more inductive approach (Spradley, 1979). As the data were beingexamined, emerging themes required the proposal of several new codes(e.g., displaying student thinking as a role of proof). After proposingthese data-grounded (internal) codes, the data for each individual teacherwere then re-examined and re-coded incorporating these new codes. Asa means of checking the reliability of the coding and appropriatenessof the coding scheme, a second researcher read and coded samples of


(a) I tore up the angles of the obtuse triangle and put them together (as shown below).

The angles came together as a straight line, which is 180◦. I also tried it for an acutetriangle as well as a right triangle and the same thing happened. Therefore, the sum of themeasures of the interior angles of a triangle is equal to 180◦.

(b) I drew a line parallel to the base of the triangle.

I know n = a because alternate interior angles between two parallel lines are congruent.For the same reason, I also know that m = b. Since the angle measure of a straightline is 180◦, I know n + c + m = 180◦. Substituting a for n and b for m, gives a + b +c = 180◦. Thus, the sum of the measures of the interior angles of a triangle is equal to 180◦.

(c) Using the diagram below, imagine moving BA and CA to the perpendicularpositions BA’ and CA”, thus forming the second figure. In reversing this procedure (i.e.,moving BA’ to BA), the amount of the right angle, A’BC, that is lost is x. However, thislost amount is gained with angle y (since BA’ and DA are parallel, and x and y are alternateinterior angles). A similar argument can be made for the other case. Thus, the sum ofthe measures of the interior angles of any triangle is equal to 180◦ (Harel & Sowder, 1998).

Figure 1. Arguments demonstrating the sum of the angles in any triangle is 180◦.

70 ERIC J. KNUTH

the interview transcripts. The coded samples from both researchers werethen compared and differences were discussed until they were resolved.Data were then re-coded taking into account any changes made to thecoding scheme. Data for an individual teacher were also examined forconsistencies and/or inconsistencies in the nature of their responses; suchconsistencies/inconsistencies for individual teachers were then examinedacross data sets for all of the teachers with a focus on themes in theconsistencies/inconsistencies noted.

Upon completion of the coding of the data, a domain analysis ofthe data sets was conducted as a means for identifying, organizing,and understanding the relationships between the primary themes thatemerged through the coding process (Spradley, 1979). According toSpradley, domains are categories of meanings that are comprised ofsmaller categories, the smaller categories being linked to the correspondingdomain by a single semantic relationship. Domains selected for this stageof the analysis were informed by the research questions, that is, the issuesthat were deemed important for this study provided a backdrop againstwhich specific domains were proposed as the data sets were examined.As an example, I used domain analysis techniques to identify the natureof what the teachers seemed to believe constitutes proof in school mathe-matics – the result was the identification of three levels of proofs (discussedshortly). In this case, the domain chosen was “proof” and the smallercategories, the three levels of proof, were identified as kinds of proof(“kinds of” being the semantic relationship linking the domain to thesmaller categories). Similar to the approach taken in coding the data, amore inductive approach supplemented this deductive approach and led toadditional domains being proposed.

RESULTS AND DISCUSSION

This section reports and discusses the results of the study and is organizedaround the two aforementioned research questions: (1) What constitutesproof in school mathematics? and (2) What are teachers’ conceptionsabout the nature and role of proof in school mathematics? Included in thepresentation of the results are frequency counts for the relevant themesnoted during the data analysis; the counts allow for comparison of thesignificance of the different themes. In addition, interview excerpts thatare representative of particular themes are also provided. Due to spacelimitations, only themes evident in the responses from at least four teachersare presented (unless a theme is particularly interesting).


What Constitutes Proof in School Mathematics?

In describing what meaning they ascribed to the notion of proof8 ingeneral (responses were compared from Stages 1 and 2), the majority ofthe teachers (11) stated, to varying degrees, that a proof is a logical ordeductive argument that demonstrates the truth of a premise. The followingare representative of the teachers’ definitions:

I think it means to show logically that a certain statement or certain conjecture is trueusing theorems, logic, and going step by step (KK).9

I see it as a logical argument that proves the conclusion. You’re given a statement,and the logical argument has this statement as its conclusion (SP).

It’s the process of justifying a series of steps . . . . The justification of the steps isbased on already existing mathematical truths (PB).

Other teachers (6) ascribed a slightly more general meaning to proof,that of proof as a convincing argument. For example, one teacher statedthat proof is “a convincing argument showing that something that is saidto be true is actually true” (KA). Overall, whether defining proof as adeductive argument or as a convincing argument, teachers viewed proofas an argument that conclusively demonstrates the truth of a statement.

Turning now to the meaning of proof in the context of secondaryschool mathematics, the teachers’ descriptions could be categorized usingthree different degrees of formality: formal proofs, less formal proofs, andinformal proofs.10

Formal proofs. By many teachers (9), a clear distinction was madebetween what they considered to be formal proofs and what theyconsidered to be either less formal proofs or informal proofs. Theteachers’ descriptions of formal proofs were very ritualistic in nature, tiedheavily to prescribed formats and/or the use of particular language (cf.Martin & Harel, 1989). For example, one teacher focused on the formatrequired, “You have a prescribed set of rules that you have to follow anda prescribed format” (KA), while another alluded to the particulars of thelanguage used in formal proofs, “Is there a difference between us sayingthat the angles are equal as opposed to the angles are congruent? . . . Theyneed to have before the formal, the informal, where we’ll accept eitherone for now” (FF). Also included in this group of nine teachers werethose teachers (4) for whom two-column proofs (i.e., proofs in whichstatements are written in one column and the corresponding justificationsin a second column) are the epitome of formal proofs. “When I think offormal proof, I usually think of the two-column formal proof in geometry”

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(NA). Similarly, another teacher commented “When I think of a formalproof, I think of proofs where you have your little ‘T’ [i.e., a spatialdescription for the organizational structure of a two-column proof]” (DF).

Less formal proofs. Teachers (10) also talked about less formal proofs,proofs which do not necessarily have a rigidly defined structure or are notperceived as being “mathematically rigorous,” but were considered by theteachers to be valid proofs nonetheless (see Figure 1c for an example of arepresentative argument. These teachers defined less formal proofs morein terms of whether the argument established the truth of its premise for allrelevant cases rather than in terms of the rigor involved in the presentationof an argument. Typical definitions included:

Being able to come up with a general statement that always holds (FF).

It’s a convincing argument but it’s generalized . . . . It has to, in some way, begeneralized so it’s true for all cases (DL).

It’s a way to decide whether something is true in all situations, or not, based onmathematical justification (PB).

It’s a general argument why something mathematical is true (KB).

In short, the important quality common to arguments of this nature wasthat they were sound mathematically and proved the general case.

Informal proofs. Finally, all of the teachers considered explanationsand empirically-based arguments as representative of informal proofs –arguments not considered to be valid proofs because they are not proofsof the general case (see Figure 1a for an example). Proofs of this naturemight best be described as arguments in which one provides reasonsto justify one’s mathematical actions or presents examples to supportone’s claims (in either case, not arguments one would consider to bevalid proofs). In the case of viewing explanations as a type of informalproof, one teacher commented, “They [i.e., students] are always askedto justify their thinking. It seems like proof is everywhere” (SP). Inthis particular teacher’s case, she is utilizing reform-based curricularmaterials – materials that frequently ask students to justify the thinkingunderlying their solutions to presented tasks – thus, her statement that(informal) “proof is everywhere” is not too surprising. In the case ofviewing empirically-based arguments as a type of informal proof, anotherteacher stated, “One of the first [‘proofs’] they do is just prove by a millionexamples. They can use a bunch of examples and say that’s a proof” (QK).


The Nature of Proof in School Mathematics

As teachers talked about the nature of proof in secondary schoolmathematics, several themes emerged from the analysis of their responses:the centrality of proof in school mathematics, reform and proof, andstudents’ experiences with proof.

The centrality of proof. In response to being asked if they thoughtproof should play a central role in secondary school mathematicscurricula, teachers expressed varying perspectives depending upon themeaning they associated with proof. The majority of teachers (14) didnot consider proof (i.e., formal and less formal proof) to be a central ideathroughout secondary school mathematics, questioning its appropriatenessfor all students. As one teacher stated, “I’m not so sure that we ought todo a lot of teaching of proof” (DL). In general, teacher comments rangedfrom those that emphasized the types of courses in secondary mathematicsfor which proof is perceived as appropriate (or inappropriate), for example,

I think that [i.e., proof] is kind of one of those ivory tower ideas, unless you’re teaching inhonors pre-calc or honors calc. Actually, that’s not true. Any honors class you’re going toget into it a little more (FF);

I think if you’re asking kids to do them [i.e., proofs], the kids that are going to beable to do them are in the higher level mathematics classes (PB);

In secondary school mathematics proof is not a big part of algebra or analysis [i.e.,precalculus] courses (KB);

to those comments that emphasized the type of students who should beprovided experiences with proof, such as,

I think any student going into upper mathematics has to have a strong understanding ofproof (KU);

Using 10th grade as a boundary, as opposed to 11th and 12th for kids who aregoing to be going into mathematics and probably studying mathematics in college, 10thgrade and under I’m not convinced that proof has a real role with them (KD).

One teacher even commented that if she were to reduce the amount ofmaterial included in secondary school mathematics curricula, proof wouldbe her choice to go. “If you’re trying to get through curriculum, then that[i.e., proof] is what I would drop out” (LV). For this latter teacher, provingseemed to be a topic of study rather than a means of coming to understandmathematics.

Thus, for all of these teachers, proof seemed to be an appropriate ideaonly for those students enrolled in advanced mathematics classes andfor those students who will most likely be pursuing mathematics-related

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majors in college. These views are clearly inconsistent with those repre-sented in current reform documents in which proof is seen as playinga more central role for all students: “reasoning and proof [italics added]should be a consistent part of students’ mathematical experiences in pre-kindergarten through grade 12” (NCTM, 2000, p. 56). In addition, suchviews also are inconsistent with the views of mathematics educatorswho see the importance of proof in a fashion similar to Hanna (1983):“The axiomatic method and the concept of rigorous proof are among themost valuable assets of modern mathematics and should be among theintellectual acquisitions of any high-school student” (p. 4).

In contrast, all of the teachers considered informal proof to be a centralidea throughout secondary school mathematics, an idea that was viewedas appropriate for all students and one that should be integrated intoevery class. One teacher’s comment captured this view: “I think informalproof should play a big role. I think we should really work with kids onunderstanding how the mathematics developed or the justification for it.And pushing them in their work to be able to justify. I think informal proofis really important” (DL). To some extent this view is not surprising; inmany respects, the teachers’ views are consistent with the messages ofearlier reform efforts. The 1989 Curriculum and Evaluation Standards forSchool Mathematics (NCTM) emphasized reasoning and more informalmethods of proof as appropriate in the mathematics education of allstudents – an emphasis that is reflected in the teachers’ conceptions ofproof in secondary school mathematics. Hanna (1995), although agreeingwith the importance of informal proofs in school mathematics, objectedto limiting students’ experiences with proof to informal methods: “Thosewho would insist upon the total exclusion of formal methods, however,run the risk of creating a curriculum unreflective of the richness of currentmathematical practice. In doing so, they would also deny to teachers andstudents accepted methods of justification” (p. 46). Similarly, Wu (1996)noted that this emphasis on informal proof, even for students in lowerlevel mathematics classes is “a move in the right direction only if it isa supplement to, rather than a replacement of, the teaching of correctmathematical reasoning; that is, proofs” (p. 226). Yet, as will be discussedshortly, many teachers’ conceptions of proof do in fact limit their students’experiences with proof to informal methods.

Reform and proof. Given the disparity in the teachers’ views between proofand informal proof as central ideas in secondary school mathematics,the results regarding the teachers’ interpretations of the particularrecommendations set forth in the Principles and Standards for School


Mathematics (NCTM, 2000) with regard to proof are not particularlysurprising. All of the teachers expressed one of two opinions, dependingon how they interpreted the authors’ use of the word proof in theserecommendations. On the one hand, for those teachers (6) who interpretedthe authors’ use of proof as mathematical proof (i.e., formal or less formalproof), the recommendations were found to be appropriate only for higherlevel students (and thus, inappropriate for all students). Typical commentsfrom these teachers included, “I think it depends on the level of the class.Some students are not able to do that [i.e., develop and evaluate proofs]”(NA) and “I think the part about reasoning is okay, but constructing proofsis I think a little much to ask” (DF). One teacher spoke more adamantlyabout the reality of all students partaking in such recommendations:“I think they’re [authors of the Principles and Standards for SchoolMathematics] smoking crack [a drug]. I’d like to see how that wouldhappen, what that looks like in a classroom” (PB).

On the other hand, those teachers (11) who interpreted the authors’use of proof more broadly (i.e., includes formal, less formal, and/orinformal proofs), the recommendations were seen as more compatiblewith their own views regarding the centrality of proof in secondary schoolmathematics curricula. Several of these teachers differentiated their inter-pretations of the authors’ use of proof by student ability level; as one ofthese teachers stated:

Sounds like they want them to be doing proofs throughout 6–12. I think that in itselfindicates that they’re not expecting rigorous proof in grade 6. They’re wanting students torecognize relationships on their own, investigate patterns, use inductive reasoning. At thehigher levels learn more rigorous approaches to proving different relationships (KU).11

Others interpreted the authors’ use of proof as describing different aspectsof the entire proving process; thus, “proof” is appropriate for the mathe-matics experiences of all students. For example, one teacher noted, “Theyare saying that proof is an integral part of mathematics and it has todeal with reasoning and it has to do with making investigations . . . that’sall part of proving” (MQ). Finally, two teachers interpreted the authors’use of proof as being primarily informal in nature. As one of the twoteachers commented, “They’re still emphasizing, even though they’re stillusing proof a lot in here, it’s more the informal way of doing it” (LV).It is apparent from the comments of these eleven teachers that what theyconsider as mathematical proof (i.e., formal and less formal proofs) is stillperceived as appropriate for upper level mathematics students. In effect,the teachers have adopted a pragmatic stance regarding the Standards’ useof proof, that is, a particular meaning of proof is utilized depending on thestudents whom one teaches.

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Students’ experiences with proof. In response to being asked when studentsshould be introduced to the notion of proof, several teachers (5) suggestedthat proof in secondary school mathematics is primarily relegated to thedomain of Euclidean geometry, and it is in this domain that students actu-ally encounter more formal methods of proof. As one teacher recalled,“I’m not teaching geometry any more, but when I did teach geometry,that [i.e., formal methods of proof] was the focus” (NB). Another teacherprovided his rationale for why geometry was the domain of choice: “Ilike geometry because the medium is a little more concrete. It’s a subjectmatter that you can grasp this whole argument in terms of a formal proof”(CA). Although other teachers did not specifically mention geometry asthe “home” of proof in school mathematics, nine additional teachers didview upper level mathematics (including geometry) as appropriate coursesfor engaging students with proof. Five of these nine teachers, however,viewed proof as being implicit in upper level mathematics courses otherthan geometry and thus, by default, might be considered to view proof ashaving an explicit focus only in geometry.

The following is an example of what it means to treat proof as implicitwithin a non-geometry course. Teachers accepted as valid proofs variousalgebraic arguments (e.g., a derivation of the quadratic formula), yet, thesesame teachers stated that if they were to use the given arguments in theirown classrooms, they would not discuss them in terms of proof with theirstudents. For the teachers, the proofs would be discussed more as deri-vations, “I talked about it [i.e., quadratic formula] as a derivation. Here’swhere the formula comes from” (SP), or as rules, “This is just a rule. Wego from here to there” (CC), rather than as proofs in and of themselves.Three of the teachers, however, provided more pragmatic (or pedago-gical) reasons for not discussing the arguments as proofs; the followingis representative, “I probably would have avoided using proof because . . .

my experience with the kids is that they would shut down when you usethe word proof. They’re gone. Shades down” (KD).

Whereas the majority of teachers seemed to view proof as inappro-priate for students in lower level classes, all of the teachers reported thatthey would accept informal proofs (i.e., empirically-based arguments) asproof from their students in lower level mathematics classes. The followingteacher’s comment is representative: “When they say I noticed this patternand I tested it out for quite a few cases; you tell them good job. For them,that’s a proof. You don’t bother them with these general cases” (SP). Anunfortunate consequence of such instruction, however, is that students maydevelop the belief that the verification of several examples constitutesproof (Harel & Sowder, 1998). Wu (1996), recognizing the prevalence


of experimentation as a means of establishing truth in secondary schoolmathematics (and, in particular, in reform-based curricula), warned:

Now this is not to belittle the importance of experimentation, because experimentation isessential in mathematics. What I am trying to do is point out the folly of educating studentsto rely solely on experimentation as a way of doing mathematics. Mathematics is concernedwith statements that are true, forever and without exceptions, and there is no other way ofarriving at such statements except through the construction of proofs (pp. 223–224).

Only two of the teachers, however, heeded Wu’s warning and mentionedthat they would explicitly discuss the limitations of accepting such argu-ments as proof with their students. One of these two teachers stated thatstudents need to understand that “demonstrating it [with a few examples]doesn’t mean your proof is going to hold true for all cases” (KJ).

In further explaining the role informal arguments play in theirclassrooms, 11 teachers stated that they would use informal arguments asprecursors to the development of more formal arguments in their upperlevel classes. As one teacher commented: “It’s good for students to justifytheir answers . . . . That’s a step into developing proofs, for them to beable to justify their thinking” (MQ). A second teacher described how thisprocess unfolds in her geometry classes: “This [i.e., testing examples toinformally prove a statement] students do very early on to show that itworks. Then when we introduce other geometry concepts, we come back tothis and prove it formally” (SR). Whereas the preceding teacher’s commentsuggests that her students revisit the initially “proved” statement afteracquiring the needed tools to formally prove the statement, another teacherassumed a slightly different perspective regarding the need for generatinginformal proofs prior to formal proofs. In this case, the teacher viewed thegeneration of examples as essential to the proving process:

In order to develop a proof, first off you have to have the insight to say this appears to behappening over here. Why? Looking at what’s going on, seeing some interrelationships,there is this idea of using induction and saying it appears that these two, three, or fourthings are interrelated and they appear to be interrelated in this manner. It appears if I dothis, this other thing happens and this is related to this. It could be very simple. It couldbe very complex in nature. For a proof to really manifest, one needs to have that inductiveinsight (CA).

Such experiences with more informal methods of proof can providestudents with opportunities to formulate and investigate conjectures – bothimportant aspects of mathematical practice – and may help “studentsdevelop an inner compulsion to understand why a conjecture is true”(Hoyles, 1997, p. 8). Such practices also are reflective of the process ofexperimentation in mathematics: “Most mathematicians spend a lot oftime thinking about and analyzing particular examples. This motivatesfuture development of theory and gives a deeper understanding of existing

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theory” (Epstein & Levy, 1995, p. 670). Thus, for these teachers, informalproofs were viewed as often serving this very function (in higher levelclasses), namely, to the “development of theory.”

The Role of Proof in School Mathematics

Analysis of the teachers’ responses regarding the role of proof insecondary school mathematics revealed several categories – categoriesreflecting the study’s framework (presented at the beginning of the paper)as well as categories that emerged from the data. With respect to theformer, the role of proof in systematizing statements into an axiomaticsystem was the only role associated with the study’s framework notmentioned by the teachers.

Developing logical thinking skills. The majority of the teachers (13)identified the development of logical thinking or reasoning skills as aprimary role proof plays in secondary school mathematics. Includedwithin this category are teacher responses regarding a role of proof beingits applicability to the real world; the applicability role was subsumedby the logical thinking category because the teachers discussed logicalthinking skills in terms of the value outside the domain of mathematicsas well as inside. Typical comments in response to being asked what roleproof serves in secondary school mathematics included the following:

It develops that kind of thinking skill. We naturally use our intuition and we naturallythink inductively, but I think getting people to think deductively is not as easy. And that’sone thing I think proof causes kids to have to do (KB).

It’s not just in mathematics that you use logic. You use it in life problems too . . . .They just can’t say just because it is that way. They have to be able to support what they’rethnking (KU).

I would really say reasoning skills. Even if you become a carpenter, a businessman,understanding the limitations of your observations and trying to extrapolate them is onething that I think is really powerful. If you understand your proofs, I think that reallybuilds great reasoning (CA).

Interestingly, one of the teachers, although professing to believe that proofhelps develop students’ thinking skills, was unsure of its applicabilityoutside of mathematics; “other than the development of reasoningskills . . . I’ve never had to use proof outside of a math class. I don’t knowwhen they might use something like that” (KA).

Communicating mathematics. Ten teachers considered proof in secondaryschool mathematics to be a social construct. These teachers suggested


that in their classrooms, what is accepted as proof is the result ofan argument’s acceptance as such by the classroom community. Oneteacher’s statement captures this perspective: “They [students’ arguments]have to be convincing, accepted by all to be a proof. They [students] maybe convinced themselves, but unless they can convince other people, it’snot a proof” (EN). Another teacher provided details on how this socialprocess plays out in her classroom: “In class I have kids present theirwork, then they have a panel of critiquers, and so you can certainly putyour work out there for public inspection. The public can do it [i.e., acceptan argument as proof]” (SP). A similar process of “public inspection”takes place, reportedly, in another teacher’s classroom and serves not onlyas a means for accepting arguments as proof, but also as a means formaking distinctions between proofs and non-proofs.

I have students come up with different ways of proving something and then discuss whichof these really do prove it. They are able to see, able to compare one that does prove it andone that doesn’t, and can try to make the distinction between what a proof is and what it’snot (KA).

The social nature of proving as described by these teachers, to someextent, reflects the practice of proving in the discipline of mathematics;students present their arguments for public inspection and, as a result ofany ensuing deliberations, the arguments are either made more convincingand accurate or are found unacceptable as proofs. Such practices alsoclosely parallel the nature of the practices embodied in visions ofreform-based classrooms – classrooms in which students “should expectto explain and justify their conclusions” (NCTM, 2000, p. 342) and inwhich students “should understand that they have both the right and theresponsibility to develop and defend their own arguments” (p. 346).

Displaying thinking. Four teachers indicated that a role of proofin secondary school mathematics was to display students’ thinkingprocesses. In other words, a proof provides documentation (oral orwritten) of how a student arrived at a particular conclusion. Although thisrole could be perceived as a form of communication, I have chosen tocategorize it as a separate role because these teachers seemed to focusmore on the display as a means of assessing student understanding. Theteachers viewed the display of student thinking as beneficial to the studentpresenting a proof as well as to the audience reviewing a proof and, inparticular, to the teacher assessing the student’s level of understandingof the mathematics involved in producing a proof. For example, oneteacher commented that presenting a proof allows “students to be ableto demonstrate their understanding of why they’re able to do something.

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I think if they’re able to explain a process, their understanding is a bitmore solidified” (NA). Another teacher suggested that presenting a proofprovides “clarity to their audience or to their teacher that they understandthe mathematics they are dealing with” (EN).

Explaining why. The role of proof in explaining why a statement istrue surfaced (or failed to surface) in qualitatively different ways. Theresponses from seven teachers suggested they viewed a role of proof asenabling students to answer why a statement is true. In this case, studentslearn where statements come from or why they are true rather thanaccepting their truth as given (from some external source of authority).In this particular category, the focus is not so much on an argument’sillumination of the underlying mathematical concepts which determinewhy a statement is true as much as it is on showing how a statement cameto be true. For example, these teachers viewed a proof of the quadraticformula as an illustrative example of the role of proof in answering whysomething is true. A reader could follow the progression of steps in thederivation to understand how the general formula was derived (i.e., “why”it was true). As one teacher commented, “It gives a way [i.e., provides ameans] for kids to understand why things are the way they are. Some ofthe things we say, oh, that’s the way things are. Oh, that’s the formula.Instead of just accepting at face value, proofs give a way of justifying theformulas” (PB).

Noticeably missing in the teachers’ discussions was an explicit recog-nition of proof serving an explanatory capacity, that is, proof as a meansof promoting insight of the underlying mathematical relationships. Fiveteachers seemed to recognize this construct as evidenced in discussionsof their evaluations of the various arguments presented to them. As anillustration of this perspective, one teacher explained why she found theargument presented in Figure 1c to be particularly explanatory:

It gives you a picture of what’s going on and you can see that it’s going to be true.You can see how the amount you lose from one of the right angles is made up from thecorresponding part of the angle being formed. It is easy to see why the sum is 180◦ (KA).

None of the teachers, however, explicitly entioned this as a role proofshould play in school mathematics. It is possible that the explanatorynature of arguments is not something teachers consciously think aboutin designing their instruction (cf. Peled & Zaslavsky, 1998). In somerespects, it is not surprising that this role was not mentioned by any of theteachers; for many teachers, the focus of their previous experiences withproof as students themselves was primarily on the deductive mechanism oron the end result rather than on the underlying mathematical relationships


illuminated by a proof (e.g., Chazan, 1993; Goetting, 1995; Harel &Sowder, 1998). Nevertheless, of all the roles of proof, its role in promotingunderstanding is, perhaps, the most significant from an educationalperspective. In fact, the importance of this role of proof in secondaryschool is evident in a comment from the Mathematical Association ofAmerica’s Task Force on the NCTM Standards: “the emphasis on proofsshould be more on its educational value than on formal correctness. Timeneed not be wasted on the technical details of proofs, or even entire proofs,that do not lead to understanding or insight” (Ross, 1998, p. 3). Similarly,Hersh (1993) suggested, “at the high-school or undergraduate level, its[i.e., proof’s] primary role is explaining” (p. 398).

Creating mathematics knowledge. Four teachers viewed proof as anopportunity for students to become arbiters of mathematical truth ratherthan having to rely on their teacher or textbooks to perform this role.In order for students to be autonomous in mathematics classrooms,they must be able to create their own knowledge through validatingtheir own as well as their classmates’ knowledge claims. Consequently,this role of proof enables students to become producers of knowledgerather than consumers of other’s knowledge. Hanna (1995) saw this roleas an inherent characteristic of proof: “Proof conveys to students themessage that they can reason for themselves, that they do not need tobow down to authority. Thus the use of proof in the classroom is actuallyanti-authoritarian” (p. 46). Accordingly, for these teachers, proof providesstudents with an opportunity to become mathematically independentthinkers. The following two statements are characteristic of their views ofthis particular role:

It allows your students to be independent thinkers, instead of just robots who are told thisis the relationship, this is what works, use it to do these problems . . . . Students don’t haveto rely on a teacher or a book to give them information (KU).

It’s important that they can stand behind a statement or a solution, that they wouldbe able to have a discussion about that, other than saying the teacher told me I was right.That they themselves would have whatever they needed to explain it (EN).

Curricular Program and Course Level Influences

I hypothesized that the nature of the curricular programs used by theteachers and/or the level of the mathematics courses taught might influ-ence teachers’ conceptions regarding proof in school mathematics. Inthe former case, I thought that the conceptions of those teachers whowere implementing reform-based curricular programs might be different

82 ERIC J. KNUTH

from the conceptions of those teachers who were implementing tradi-tional curricular programs. Many of the tasks in reform-based curricularmaterials are open-ended and typically require students to provide justi-fication for their solutions. Such tasks – and the students’ correspondingjustifications – are very different from the tasks and expected justifi-cations in classrooms using traditional curricular materials (Schoenfeld,1994). Accordingly, it seemed reasonable to expect that teachers imple-menting reform-based curricular programs might develop conceptions thatdiffer from the conceptions of teachers implementing traditional curricularprograms. With few exceptions, however, the curricular programs fromwhich the teachers taught did not seem to have a significant influence ontheir conceptions. In other words, of teachers who held the conceptions ofproof in school mathematics discussed in the previous sections the numberwho taught from reform-based curricular programs were relatively equal tothe number who taught from traditional curricular programs. One possibleexplanation (among several possibilities) for the lack of influence mightbe the treatment of proof in the reform curricular programs, that is, reformcurricular programs may place a greater emphasis on informal reasoningthan formal reasoning (this conjecture requires further study).

In the latter case, I thought differences in teachers’ course loads mightinfluence their conceptions regarding proof in school mathematics. Forexample, teachers who teach lower-level classes may not see more formalmethods of proof as appropriate for their students and, thus, may havedeveloped different conceptions compared to their peers who teach thehigher-level classes (classes in which more formal methods of proof mightbe viewed as appropriate). Once again, however, no significant differenceswere noted in the conceptions of these two groups of teachers. As anexample, virtually all of the teachers agreed that more formal methodsof proof were most appropriate in higher-level mathematics courses, whilethe opposite was true in lower-level mathematics courses – informal proofdominated what the teachers considered to be appropriate experiences forstudents in these classes.

CONCLUDING REMARKS

As Edwards (1997) suggested, “the teaching of proof that takes placein many secondary level mathematics classrooms has often been incon-sistent with both the purpose and practice of proving as carried out byestablished mathematicians” (p. 187). Consequently, many students do notseem to understand why mathematicians place such a premium on proof(Chazan, 1993). In some sense the foregoing remarks are not surprising;


secondary school mathematics teachers – as well as their students – are,arguably, not mathematicians. Yet, the nature of classroom mathematicalpractices envisioned by recent mathematics education reform initiatives,and which teachers are expected to establish, reflects the essence of prac-tice in the discipline (Hoyles, 1997). There are examples in the literatureof elementary school students engaged in such proving practices (e.g.,Ball & Bass, 2000; Maher & Martino, 1996), yet such examples are morethe exception rather than the rule, and are rare at the secondary level. Atthe beginning of this paper, I stated that the purpose of this study was toexamine whether secondary school mathematics teachers are prepared toenact in their instructional practices the current reform recommendationsregarding proof. The findings of this study suggest that the successfulenactment of such practices may be difficult for teachers.

Although teachers tended to view proof as serving several importantfunctions in school mathematics, many of which reflect functions of proofin the discipline, they also tended to view proof as an appropriate goalfor the mathematics education of a minority of students. This latter view,however, is clearly inconsistent with the views of those who advocate amore central role for proof throughout school mathematics (e.g., Hanna,1995; NCTM, 2000; Schoenfeld, 1994). Thus, perhaps the greatest chal-lenge facing secondary school mathematics teachers is changing boththeir conceptions about the appropriateness of proof for all students andtheir enactment of corresponding proving practices in their classroominstruction. In turn, those parties chiefly responsible for the preservice andinservice education of teachers – mathematics education professionals anduniversity mathematics professors – face the challenge of better preparingand supporting teachers in their efforts to change. A starting point towardhelping teachers adopt and implement such a perspective may be to engageteachers in explicit discussions about proof.

In my work with teachers, for example, I have found discussions ofquestions pertaining to various aspects of proof to be particularly fruitfulin getting teachers to reconsider (or at least make explicit) their existingviews about proof. Particular questions have included: What is meant byproof? What purpose does proof serve in mathematics? What constitutesproof? Is a proof a proof or are there levels of proof? The latter twoquestions often have resulted in quite interesting (and, for teachers, oftenilluminating) discussions. More specifically, these two questions typicallyarise as teachers engage in the task of evaluating various sets of argumentsin terms of their validity as proofs (see Figure 1 for examples of typicalarguments). After having debated why particular arguments are or arenot valid proofs, many teachers express the realization that their views of

84 ERIC J. KNUTH

what constitutes proof may be too narrowly construed (e.g., proofs requireparticular format or language).

Relating their responses to the foregoing questions to the context ofschool mathematics also may serve to refine and extend teachers’ viewsabout proof. Other questions have focused specifically on proof in schoolmathematics and have included the following: Why include proof in schoolmathematics? Does what suffices as proof in the discipline differ fromwhat suffices as proof in school mathematics? Does what suffices as proofin one course differ from what suffices as proof in another course? Whattypes of experiences with proof should teachers provide students? Further,having teachers construct and present proofs of school mathematics tasks– tasks from various content areas and levels – provides a forum fordiscussing expectations of proof (e.g., what counts as proof) for studentsat differing levels of mathematical ability and in different mathematicscourses.

I would certainly be remiss not to mention the importance of under-graduate mathematics courses in shaping the conceptions of proof teachersdevelop. Many of the teachers in this study, for example, viewed proofas a object of study (i.e., a topic one teaches) rather than as an essen-tial tool for studying and communicating mathematics. As Schoenfeld(1994) suggested, “if students grew up in a mathematical culture wherediscourse, thinking things through, and convincing were important parts ofthe engagement with mathematics, then proofs would be seen as a naturalpart of their mathematics” (p. 76). Similarly, Harel and Sowder (1998)proposed that “for most university students, including even mathematicsmajors, university coursework must give conscious and perhaps overtattention to proof understanding, proof production, and proof appreciationas goals of instruction” (p. 275). In turn, such experiences and attentionto proof may influence the nature of the experiences with proof that theseteachers eventually provide their own students.

The success of teachers in establishing classroom mathematical prac-tices in which proof is an integral part may depend on their changing(or at the very least expanding) their current conceptions of proof in thecontext of secondary school mathematics. There are certainly conditionsother than those discussed in this article which contribute to the formationof teachers’ conceptions of proof and the manifestation of these concep-tions in their instructional practices, and for which further study is needed.Further, research is needed that examines how teachers’ conceptions playout in their day-to-day practices. It is my hope, however, that the findingsof this study challenge mathematics educators to recognize and to addressthe issues related to preparing and supporting secondary mathematics


teachers better to successfully enact the newest reform recommendationswith respect to proof.

ACKNOWLEDGMENTS

The study reported here is based on my dissertation research under thesupervision of Dominic Peressini and Hilda Borko. I would also like tothank Tom Cooney, Barbara Jaworski, and anonymous reviewers for theirhelpful comments on earlier versions of this paper.

NOTES

1 The centrality of proof in mathematics is not without controversy; in fact, the role ofproof in mathematics has received increased attention in recent years (e.g., Hanna, 1995;Horgan, 1993; Thurston, 1994). Although such literature contains some interesting discus-sions among mathematicians about directions related to the future of proof, discussion ofthis literature is beyond the scope of this paper. The position taken in this article is thatproof is, and will continue to be, an important part of mathematical practice.2 Rather than treating teachers’ knowledge and beliefs as separate domains, I use the term“conceptions” in order to represent the two domains in tandem. While separating teachers’knowledge and beliefs serves as a useful heuristic for thinking about and studying thefactors influencing teachers’ instructional practices, the separation is less distinct in realitythan it is in theory (Grossman, 1990).3 Although the social process of proofs and refutations Lakatos described has been criti-cized for its limited applicability in mathematics (Hanna, 1995), I think the process isworth noting as it undergirds the NCTM (1991) recommendation that teachers establishclassroom mathematical practices in which students: “make conjectures and present solu-tions; explore examples and counterexamples to investigate a conjecture; try to convincethemselves and one another of the validity of particular representations, solutions, conjec-tures, and answers; [and] rely on mathematical evidence and argument to determinevalidity” (p. 45).4 A variety of factors ranging from curricular emphases (e.g., Hoyles, 1997) to psycho-logical issues associated with learning to prove (e.g., Fischbein, 1982) to instructionalpractices with regard to proof (e.g., Alibert, 1988) have been suggested as contributingto the disparity between proving in mathematical practice and proving in school mathe-matics. The focus of this paper – teachers’ conceptions of proof – is another factor thatmight be seen to stand between the aforementioned roles of proof and their classroommanifestations.5 In the United States, secondary school typically refers to grades 7 to 12 (student agesvary from 12 years to 18 years). For most 7th and 8th grade students, the primary focusof their mathematics courses is on pre-algebra and informal geometry topics (a smallernumber of students enroll in a traditional algebra course in 8th grade). At the high schoollevel (grades 9–12), a typical 4-year course sequence is algebra (primarily the studyof linear functions), geometry (Euclidean geometry), intermediate algebra (primarily thestudy of non-linear functions), and pre-calculus (includes the study of trigonometric func-tions). In some cases, less well prepared students entering 9th grade might enroll in a

86 ERIC J. KNUTH

pre-algebra course while better prepared students might enroll in a geometry course (thusenabling them to enroll in a calculus course in 12th grade).6 See Knuth (In press) for a discussion of teachers’ conceptions of proof in the disciplineof mathematics.7 I recognize that there is not an absolute criterion for the degree of explicitness requiredin presenting a proof, nor for what mathematical results are acceptable to use; in bothcases, the conventions adopted in deciding what counts as a valid proof are those deemedappropriate in a secondary school context (from my perspective as a former secondaryschool teacher).8 Throughout the article, unless the context suggests otherwise, my use of the word proofrefers to a deductive argument that shows why a statement is true by utilizing other mathe-matical results and/or insight into the mathematical structure involved in the statement.When referring to non-proofs, I will use the term argument or informal proof.9 KK (a pseudonym) are the initials of teacher who was interviewed.10 These categories, formal, less formal, and informal, were researcher-generated basedon the teacher responses. It was unclear initially how teachers were using the word proof,whether in a mathematical sense or in an colloquial sense; as a result, I introduced the termsas I probed the teachers to further elaborate their usage of the term proof. I acknowledgethe possibility that by using the term formal, I may have influenced teachers to adopt aparticular perspective. In many cases, however, teachers actually used the term themselvesin making distinctions without my introducing the term.11 The data here seem to hold important findings for writers of the NCTM Standards. Acommon criticism of many reform documents is the level of ambiguity: in this case, theauthors never explicitly define what they mean by proof or what proof might look like atvarious grade levels [the one or two examples that are provided obviously leave teachersunsure how to interpret the recommendations put forth].

REFERENCES

Alibert, D. (1988). Towards new customs in the classroom. For the Learning of mathe-matics, 8(2), 31–35.

Alibert, D. & Thomas, M. (1991). Research on mathematical proof. In D. Tall (Ed.),Advanced mathematical thinking (215–230). The Netherlands: Kluwer AcademicPublishers.

Balacheff, N. (1991). The benefits and limits of social interaction: The case of mathe-matical proof. In A. Bishop, S. Mellin-Olsen & J. Van Dormolen (Eds.), Mathematicalknowledge: Its growth through teaching (175–192). The Netherlands: Kluwer AcademicPublishers.

Ball, D. & Bass, H. (2000). Making believe: The collective construction of public mathe-matical knowledge in the elementary classroom. In D. Phillips (Ed.), Constructivism ineducation. Chicago: University of Chicago Press.

Bell, A. (1976). A study of pupils’ proof – explanations in mathematical situations.Educational Studies in Mathematics, 7, 23–40.

Borko, H. & Putnam, R. (1996). Learning to teach. In R. Calfee & D. Berliner (Eds.),Handbook of educational psychology (673–725). New York: Macmillan.

Chazan, D. (1990). Quasi-empirical views of mathematics and mathematics teaching.Interchange, 21(1), 14–23.


Chazan, D. (1993). High school geometry students’ justification for their views ofempirical evidence and mathematical proof. Educational Studies in Mathematics, 24,359–387.

Chazan, D. & Yerushalmy, M. (1998). Charting a course for secondary geometry.In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developingunderstanding of geometry and space (67–90). Mahwah, NJ: Erlbaum.

Coe, R. & Ruthven, K. (1994). Proof practices and constructs of advanced mathematicsstudents. British Educational Research Journal, 20(1), 41–53.

Davis, P. (1986). The nature of proof. In M. Carss (Ed.), Proceedings of the Fifth Inter-national Congress on Mathematical Education (352–358). Adelaide, South Australia:Unesco.

de Villiers, M. (1999). Rethinking proof with the Geometer’s Sketchpad. Emeryville, CA:Key Curriculum Press.

Edwards, L. (1997). Exploring the territory before proof: Students’ generalizations in acomputer microworld for transformation geometry. International Journal of Computersfor Mathematical Learning, 2, 187–215.

Epstein, D. & Levy, S. (1995). Experimentation and proof in mathematics. Notices of theAmerican Mathematical Society, 42(6), 670–674.

Fennema, E. & Franke, M. (1992). Teachers’ knowledge and its impact. In D. Grouws(Ed.), Handbook of research on mathematics teaching and learning (147–164). NY:Macmillan.

Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–24.Goetting, M. (1995). The college students’ understanding of mathematical proof (Doctoral

dissertation, University of Maryland, 1995). Dissertations Abstracts International, 56,3016A.

Grossman, P. (1990). The making of a teacher: Teacher knowledge and teacher education.New York: Teachers College Press.

Hanna, G. (1983). Rigorous proof in mathematics education. Toronto, Ontario: OISE Press.Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20–23.Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6–13.Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics,

15(3), 42–49.Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies.

In A. Schoenfeld, J. Kaput & E. Dubinsky (Eds.), Research in collegiate mathematicseducation III (234–283). Washington, DC: Mathematical Association of America.

Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal forResearch in Mathematics Education, 31(4), 396–428.

Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathe-matics, 24, 389–399.

Horgan, J. (1993). The death of proof. Scientific American, 269(4), 93–103.Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the

Learning of Mathematics, 17(1), 7–16.Jones, K. (1997). Student-teachers’ conceptions of mathematical proof. Mathematics

Education Review, 9, 21–32.Knuth, E. (In press). Secondary school mathematics teachers’ conceptions of proof.

Journal for Research in Mathematics Education.Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.Maher, C. & Martino, A. (1996). The development of the idea of mathematical proof: A

5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.

88 ERIC J. KNUTH

Manin, Y. (1977). A course in mathematical logic. New York: Springer-Verlag.Martin, W. G. & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal

for Research in Mathematics Education, 20(1), 41–51.National Council of Teachers of Mathematics (1989). Curriculum and evaluation stan-

dards for school mathematics. Reston, VA: Author.National Council of Teachers of Mathematics (1991). Professional standards for teaching

mathematics. Reston, VA: Author.National Council of Teachers of Mathematics (2000). Principles and standards for school

mathematics. Reston, VA: Author.Peled, I. & Zaslavsky, O. (1998). Counter-examples that (only) prove and counter-

examples that (also) explain. Focus on Learning Problems in Mathematics, 19, 49–61.Porteous, K. (1990). What do children really believe? Educational Studies in Mathematics,

21, 589–598.Richards, J. (1991). Mathematical discussions. In E. von Glasersfeld (Ed.), Radical

constructivism in mathematics education (13–51). The Netherlands: Kluwer.Ross, K. (1998). Doing and proving: The place of algorithms and proof in school

mathematics. American Mathematical Monthly, 3, 252–255.Schoenfeld, A. (1994). What do we know about mathematics curricula? Journal of

Mathematical Behavior, 13(1), 55–80.Senk, S. (1985). How well do students write geometry proofs? Mathematics Teacher,

78(6), 448–456.Simon, M. & Blume, G. (1996). Justification in the mathematics classroom: A study of

prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31.Sowder, L. & Harel, G. (1998). Types of students’ justifications. Mathematics Teacher,

91(8), 670–675.Spradley, J. (1979). The ethnographic interview. New York: Holt, Rinehart and Winston.Steiner, M. (1978). Mathematical explanations. Philosophical Studies, 34, 135–151.Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In

D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (127–146). NY: Macmillan.

Thurston, W. (1995). On proof and progress in mathematics. For the Learning ofMathematics, 15(1), 29–37.

Wheeler, D. (1990). Aspects of mathematical proof. Interchange, 21(1), 1–5.Wu, H. (1996). The role of Euclidean geometry in high school. Journal of Mathematical

Behavior, 15, 221–237.

Teacher Education Building225 N. Mills StreetUniversity of WisconsinMadison, WI 537606E-mail: [email protected]

BARBARA JAWORSKI

EDITORIALLAYERS OF LEARNING IN INITIAL TEACHER EDUCATION

As a new editor I am finding it fascinating to encounter the variety ofthemes in the papers that are submitted to JMTE and appreciate the oppor-tunity to engage with a wide range of issues in teacher education andmathematics teaching development. It is interesting to see what themesemerge for a particular issue of the journal as papers reach the publicationstage. The editorial team will consider, in the future, the possibility ofspecial issues planned around particular themes, but for the moment anissue is constituted from the papers that are ready for publication at anappropriate time.

In this particular issue (JMTE 5.2) the three papers are all situated inthe area of initial, or pre-service, teacher education. The first paper (1),from João Pedro da Ponte, Hélia Oliveira and José Manuel Varandas,focuses on a one semester course in the fourth year of a five year under-graduate teacher preparation programme where student teachers engagewith computer technology and consider its roles in and contribution tomathematical learning for themselves and ultimately their future pupils.Central to this course are the ways in which students’ use of software suchas Geometer’s Sketchpad and Modellus stimulates thinking and contrib-utes to their mathematical learning. Publication of the outcomes of theirmathematical exploration on the Internet proves to be a demanding butmotivating experience through which students look critically at what theyhave achieved. The second paper (2), from Maria Blanton, focuses ondiscourse in the teaching of a geometry course to students in their final twoyears of an undergraduate teacher preparation programme. The course’sexplicit focus on discourse has a two-fold dimension, firstly it is seenas a means of enhancing students’ learning of geometry; secondly it isconsidered as a pedagogical device enabling learning. Student teachersrecognize both the value of engaging with alternative modes of discourseand the difficulty of teaching in a way that creates discourse oppor-tunities for pupils in a classroom. The third paper (3), from BarbaraKinach, focuses on prospective teachers’ conceptual learning of additionand subtraction of integers in a mathematics methods course. The course


90 EDITORIAL

exposes student teachers’ incomplete understandings based on rules orprocedures for which explanations are instrumental and sometimes irra-tional. It develops a pedagogy in which student teachers’ explanationshave to include justification, and shifts into a consideration of how suchpedagogy translates into classroom environments.

Each of the courses discussed in these papers focuses on mathematicsto differing degrees relative to other focuses. In (2), geometry is the mainfocus of the course; its purpose is that students learn geometry. However,discourse is the focus of research into the course, and is also a pedagogicfocus alongside the learning of geometry by the future teachers. In (3), themathematical focus is addition and subtraction of integers; however, thisis part of a methods course, and justified explanation of the mathematicalconcepts is the principal (pedagogic) focus; it is also the focus of research.In (1) the focus of research and of the course is the use of aspects ofcomputer technology to promote mathematical thinking and learning. Wesee students engaging with diverse, self-chosen aspects of mathematics asthey explore their use of software and plan web sites.

This variety of focuses linking mathematics and pedagogy raises ques-tions and issues for educators. Where and how do our student teacherslearn mathematics, and what is the nature of their mathematical know-ledge and understanding? How does pedagogical knowledge develop? Inwhat ways do prospective teachers link knowledge of mathematics andpedagogy to plan effectively for classroom teaching? At what points orstages do they learn to relate their knowledge in these areas to an aware-ness of students’ mathematical thinking and understanding, and the socialsituations and issues in and surrounding classrooms? As educators werecognize such questions and address them in a variety of ways in ourdifferent courses. As researchers we want to know more about relation-ships between the differing focuses, the emerging issues, and layers ofknowledge and understanding that guide initial teaching practices. We areaware that interpretation of theoretical ideas in a university course, evenwhen it is designed to focus student teachers’ awareness on elements ofpupils’ mathematical learning in school classrooms, cannot provide theimmediacy of experience through which issues in pupils’ learning can berecognized and addressed. The complexity of making such links meansthat papers in this field often deal with only a small portion of the multi-dimensional spectrum of educating new teachers. Yet, as practitioners(educators and researchers) we deal with, and have to make sense of amuch wider part of this spectrum in our ordinary professional lives.

The book review in this issue, written by Daniel Chazan, reflects onMagdalene Lampert’s Teaching Problems and the Problems of Teaching.

EDITORIAL 91

Here the author as teacher and researcher seeks to deal with a variety ofdimensions in engaging in the real sustained practice of teaching mathe-matics to a 5th grade class for a year. As the review shows, the book weavesthis variety through a range of focuses, allowing the reader access throughdiffering lenses to the range of issues that the teacher-researcher has to faceduring her year of teaching and in analysis of her data. It is significant ofcourse that the author is experienced in all three roles (teacher, educatorand researcher) which she plays during her year. Thus, she can speakwith authority from each of these positions, addressing her own learningdirectly and that of her pupils indirectly, to reach for the generic and moregeneral insights that the research reveals.

As teachers and educators we have responsibility to foster the learningof others; as researchers we often study the learning of others. In doingso, frequently, we reflect on and describe our own learning, and seekexplanations that move beyond the particular into the beginnings of generaltheories. There is a number of models around that start to capture thevarious layers of this learning process. Ron Tzur (2001) offers one modelthat captures and generalizes from his own experiences as a learner in fourmodes of practice: learning mathematics; learning mathematics teaching;learning educating teachers and learning mentoring educators. StephaniePrestage and Pat Perks (2001) offer a model that moves towards whatthey call, ‘A Pedagogy for Teacher Education’, in a model of ‘Learner-knowledge for the teacher educator’. This consists of a stack of tetra-hedra, each of which links four types of knowledge (learner knowledge,teacher knowledge, professional traditions and practical wisdom) withclassroom events in some domain of professional practice (such as mathe-matics, class management, or assessment). It will be interesting to seeresearchers exploring the value of such models to broaden understandingsof professional learning.

As Tom Cooney suggested in his final editorial (JMTE 4.4), we needto honour the practical arenas in which learning and teaching are rooted,while providing theoretical constructs that help our explanations andprovide a basis for rigorous analyses (both quantitative and qualitative) ofdata in learning and teaching. JMTE will continue to welcome papers thatprovide a fine focus on particular programmes, issues and phases of educa-tion. In addition we welcome papers, both research papers and theoreticalpapers, that seek to address the broader spectrum of levels, issues andrelationships fundamental to ongoing practice.

92 EDITORIAL

REFERENCES

Prestage, S. and Perks, P. (2001). Models and super models: Ways of thinking about profes-sional knowledge in mathematics teaching. In C. Morgan and K. Jones (Eds.), Researchin Mathematics Education: Papers of the British Society for Research into LearningMathematics. London: BSRLM.

Tzur, R. (2001). Becoming a mathematics teacher educator: Conceptualizing the terrainthrough self-reflective analysis. Journal of Mathematics Teacher Education, 4(4).

JOÃO PEDRO DA PONTE, HÉLIA OLIVEIRA and JOSÉ MANUEL VARANDAS

DEVELOPMENT OF PRE-SERVICE MATHEMATICS TEACHERS’PROFESSIONAL KNOWLEDGE AND IDENTITY IN WORKING

WITH INFORMATION AND COMMUNICATION TECHNOLOGY1

ABSTRACT. This paper describes the work undertaken in a course in communicationand information technology in a pre-service program for secondary school mathema-tics teachers. This course aimed to help pre-service teachers develop a positive attituderegarding ICT and use it confidently. It focused on the exploration of educational softwareand of the Internet’s potential as a means of research and production of web sites. Wediscuss how the pre-service mathematics teachers evaluate their work concerning theircommitment, difficulties they found, learning they identified, and personal relationship. Wealso analyse the effects of the course on the development of their professional knowledgeand identity.

KEY WORDS: information and communication technology, mathematics teacher educa-tion, pre-service teacher education, professional identity, professional knowledge

INTRODUCTION

Mathematics teachers need to know how to use the tools of information andcommunication technology (ICT), including subject-specific educationalsoftware and general software (NCTM, 1991). This technology enables thedevelopment of new perspectives about mathematics teaching, as it stressesthe role of graphic language and new forms of representation and putsless emphasis on computation and symbolic manipulation. It also allowsthe teacher to propose more activities and projects that include explora-tion, investigation, and modelling. As a result, ICT may enhance pupils’development of important competencies, foster better attitudes towardsmathematics, and stimulate a wider vision of the nature of this science(NCTM, 2000).

The Internet may be regarded as a “metatool” where one can findinformation about new developments in mathematics and mathematicseducation, software, sample tasks for pupils, ideas for the classroom,reports of experiences, and news about meetings and other events. Inaddition, the Internet allows for the dissemination of personal productionssuch as texts, images, video sequences, applets, and hypertext documents.Rendering synchronous and asynchronous communication possible, it is


94 JOÃO PEDRO DA PONTE, HELIA OLIVEIRA & JOSE MANUEL VARANDAS

a very useful tool for collaborative work. Facilitating and stimulatingpersonal interactions, the Internet supports human development in thepersonal, social, cultural, recreational, civic, and professional dimensions.It is an essential working tool in present days, playing an increasinglyimportant role in education.

Pre-service mathematics teachers need to be acquainted with the poten-tial of ICT for mathematics teaching and to develop confidence in usingit. In our university, this is problematic since most pre-service teachersarrive at this stage of their professional preparation with very little previouscontact with this technology. Many of these pre-service teachers are quitesuspicious of the use of ICT in education and are uncomfortable abouthandling it, even for their own personal use.

This paper is about the work carried out in a course that was createdas an attempt to change this situation, offering pre-service mathematicsteachers a positive working experience with ICT. Since this technologyinvolves many dimensions and is constantly evolving, careful choices haveto be made concerning the curriculum. It is also essential to pay great atten-tion to the aims and working modes involved in the use of ICT. We assumethat learning about ICT and its use in mathematics education should helppre-service mathematics teachers develop their professional knowledgeregarding this domain as well as their knowledge about learning andteaching mathematics, as both aspects are interrelated (Berger, 1999). Inaddition, working with ICT may help pre-service teachers to develop aprofessional identity by stimulating the adoption of a standpoint and valuesthat are appropriate to a mathematics teacher. So, in this paper we describethe activities carried out in this course, report on the evaluation done bythe participants, and discuss the role of such work in the preparation ofpre-service mathematics teachers.

ICT AND THE DEVELOPMENT OF PROFESSIONALKNOWLEDGE AND IDENTITY

ICT is an increasingly important tool in mathematics teachers’ activity thathas several dimensions. It may be regarded as: (i) an auxiliary educationalmeans to support pupils’ learning; (ii) a mathematics teacher’s tool ofpersonal productivity, to prepare material for classes, carry out managerialtasks, and search for information and materials, and (iii) a medium forinteracting and collaborating with other teachers and educational part-ners. Teachers need to know how to use ICT; evaluate its potential,strengths, and weaknesses; and develop an awareness of its social andethical implications.

DEVELOPMENT OF PRE-SERVICE MATHEMATICS TEACHERS 95

In pre-service mathematics teacher education, participants mustbecome acquainted with applications such as word processing, databasemanagement systems, image processing software, spreadsheets, statis-tics software, presentation software (like Powerpoint), electronic mail,educational software concerning the learning of specific topics, and theInternet in terms of information search and production. However, a recentstudy about the ICT preparation offered in teacher education programsin Portugal shows that the competencies and knowledge acquired by pre-service teachers are far from satisfactory (Ponte & Serrazina, 1998). Thesecompetencies are clearly insufficient regarding, for example, statisticssoftware, databases, Internet navigation, and use of electronic mail.

ICT can be used to reinforce teacher-centered practices as well as tofacilitate educational change. In fact, even today, many people regard theteacher’s role as one of providing pupils with information, controllingthe discourse and managing the class. But curriculum documents increa-singly advocate the teachers’ role as one of creating stimulating learningsituations, challenging pupils to think, supporting their work, and encour-aging diversification of learning routes. Therefore, pre-service teachersmust develop not only various technical competencies but also a soundeducational perspective regarding the use of ICT in the mathematicsclassroom.

The Internet also makes specific challenges regarding the teacher’s role.Its use requires that the teacher pay great attention to pupils’ developmentof a critical stance. In using the Internet, students may simply follow aconsumer orientation, looking for the information that is already available.Students using this resource also may be oriented towards production –creating new information, materials, and documents that may in turn beappropriated and changed by a whole community of users.

Thus, ICT poses specific challenges to the mathematics teachers’professional knowledge. Such knowledge may be regarded as a blend ofdeclarative, procedural, and strategic knowledge that is used in situationsof practice (Shulman, 1986). This knowledge has tacit and personal dimen-sions and develops through experience and personal reflection (Elbaz,1983; Schön, 1983). Among its structuring elements are the conceptionsthat frame the way a range of objects, processes, and problems are handled(Ponte, 1994; Thompson, 1984). Professional knowledge concerns not justthe teaching practice in the classroom but also other professional rolessuch as tutoring students, participating in school activities and projects,interacting with members of the community, and working in professionalgroups. For these roles, mathematics teachers need to (i) have knowledgeof educational theories and issues; (ii) be competent in their teaching


domain, and (iii) have a strong preparation in mathematics education,the specialized field that concerns their activity. This involves developingperspectives about curriculum, student learning, classroom instruction, andstudent evaluation (Boero, Dapueto & Parenti, 1996).

However, it is not enough for pre-service mathematics teachers to haveknowledge of mathematics, educational theories, and mathematics educa-tion. Experience with these matters established on a purely theoreticallevel, in terms of declarative knowledge, does not guarantee an effectiveacquisition of professional knowledge. The fact that this knowledge isdeeply personal and connected to action and to reflection upon experience(Fiorentini, Nacarato & Pinto, 1999), implies that for its development pre-service teachers need imaginative and varied working environments as wellas experience of situations as close as possible to real professional practice.

As ICT changes the environment in which teachers work and the waythey relate to other teachers, it has an important impact on the nature of theteachers’ work and therefore on their professional identity. The develop-ment of a professional identity involves assumption of the essential normsand values of a profession. Also related to a strong professional identity isan attitude of commitment to self-improvement as an educator and willing-ness to contribute towrads the development of the educational institutionswhere one works. A mathematics teacher should be able to carry out theproper professional activities of the teacher and identify personally withthe teaching profession. That means assuming a teacher’s point of view,internalizing the teacher’s roles and ways of dealing with professionalissues. For example, the ability to decide on the value of a variety ofresources available and learn to use them promptly is, increasingly, animportant part of the teacher’s work. It requires, for instance, knowinghow to explore software and web sites as well as flexibility and confidencein using computers (Berger, 1999).

Berger & Luckman (1966) regard the development of a professionalidentity as an aspect of the development of secondary socialization.According to these authors, primary socialization refers to a person beingintroduced to society, becoming a part of it. The child internalizes theroles, attitudes, and values of significant others, with scarce possibility forcritical distance. Secondary socialization comes later on, with the internali-zation of “institutional worlds”, involving the acquisition of specializedknowledge (including professional knowledge). This specialized know-ledge is constructed with reference to particular fields of activity that drawon specific symbolic universes.

In the view of Dubar (1997), the construction of social identitiesinvolves two complementary processes. One, the biographical process, is


the personal construction by individuals, throughout time, of social iden-tities, using the different categories offered by the institutions in theirenvironment. It involves a transaction between inherited identities anddesired identities. The other is the relational process that involves externaltransactions between individuals and significant others. It concerns theacknowledgment of identity at a given moment and the legitimization ofthe space of the identities related to knowledge, competencies, images, andvalues stated in various action systems.

Research carried out in the last few years shows that ICT may actu-ally play an important role in teacher education, thus contributing tothe professional and personal development of pre-service teachers. Forexample, a study by Robinson & Milligan (1997) aimed to investigatehow to influence pre-service teachers’ conceptions about mathematics,technology, instruction, and evaluation strategies in a pre-service teachereducation course. This course was designed to take place in a totallyelectronic environment. It was structured in modules with three types ofactivities: (i) developmental, to acquaint participants with technologicalresources; (ii) experimental, to expand participants’ mathematical know-ledge through technology, and (iii) instructional, to have participants applywhat they learned (namely using new software) in the development ofinstructional material. The results of this study show that the pre-serviceteachers changed their conceptions about the classroom environment, theteacher’s and pupils’ roles, and the learning strategies.

Another study, conducted by Yildirim & Kiraz (1999), aimed to analyzehow electronic mail can be used by the different actors in the pre-serviceteacher education process. The authors found that the participants viewedelectronic mail as an important communication tool, although their levelof use was quite variable. They also state that the participants revealed acertain degree of anxiety regarding computers but that trainees were moreat ease using ICT than their supervisors. The authors conclude that elec-tronic mail has several advantages, the main ones being to promote mutualdevelopment, overcome time and distance limitations, and encourage theexchange of ideas among teachers.

Finally, drawing on several principles of adult education, Rogan (1996)states that pre-service teachers are more likely to engage in learningsituations that involve participation where there is mutual respect amongparticipants (students and instructors). This author stresses the role ofcollaborative activity with distributed leadership, where both pre-serviceteachers and instructors learn, in a process geared towards personal libera-tion and action. He also highlights the need for critical reflection, that weanalyze the grounds of our conceptions regarding the learning that is taking


place. Finally, Rogan underlines the need to aim to develop self-oriented“apprentices” with great initiative and ease. In the development of thiscourse we were concerned about taking into account these experiences,as we strived to construct a learning environment that would foster thedevelopment of positive attitudes regarding ICT and its use in mathematicsteaching.

AN ICT INTRODUCTORY COURSE CENTERED ON THEINTERNET

The course discussed here is a one-semester course included in the fourthyear of the program for middle and secondary school pre-service mathema-tics teachers at the Faculty of Sciences, University of Lisbon. This programprepares teachers for middle (grades 7–9) and secondary school (grades10–12). The first three years of the program are dedicated to mathematicsitself, and the fourth year to educational preparation; the fifth year is a paidpracticum carried out at a middle or secondary school. In the fourth year,besides this course the pre-service teachers also take courses in the historyand philosophy of education, psychology of education, sociology of educa-tion, pedagogy, mathematics education, and an optional subject. They alsohave an introduction to professional practice based on the observation ofeducational situations in schools.

In 1998/99 and 1999/2000 this course, meeting four hours a week, wasoffered as an introduction to ICT. Its main objective was to help pre-serviceteachers to develop a positive attitude regarding ICT and use it compe-tently, from an educational perspective, focusing on the exploration ofspecific mathematics educational software and the potential of the produc-tion and publishing of web pages. In 1998/99, 66 pre-service mathematicsteachers attended the course and in 1999/2000 this number rose to 94.There were two instructors, each with two classes. The classroom had 6computers in the first year and 9 in the second, all connected to the Internet.

Pre-service teachers got to know a range of ICT tools, especially thosedirectly related to the Internet (browsers and html editors). They also hadthe opportunity to learn to work with educational software that is particu-larly important to mathematics, such as the Geometer’s Sketchpad (GSP)and Modellus,2 as well as with general tools like spreadsheets and wordprocessors. This course does not deal with predefined mathematics topics.Participants can choose to work on the topics they want and are encour-aged to look at them from the point of view of history, applications, andconnections with other topics.


The activities carried out in class with GSP and Modellus were intendedto give pre-service teachers a general view of the possibilities of thesoftware for mathematics teaching. Generally, classes started with aninvestigative task, to be carried out by the groups and ended with awhole-class discussion considering how ICT was used and how the taskcould be adapted to the mathematics classroom. Pre-service teachersgot acquainted with this software and its educational applications withinpresent mathematics curricula in Portugal.

Pre-service mathematics teachers used the software GSP to do geomet-rical constructions and to study invariant properties of different types oftransformations, as well as other mathematics topics. In this course, pre-service teachers explored GSP mainly from an investigative perspective,starting with simple mathematical questions about properties of trianglesand quadrilaterals, then moving on to some features of conics and, finally,invariant properties of certain geometrical transformations. Pre-serviceteachers also learned about Java Sketch, which was useful for includinganimations with GSP in their web pages. They also used Modellusfor constructing mathematical models of different kinds of phenomena.Such models were particularly useful in showing the dynamics of thephenomena over time, using a variety of representations. This softwarewas also used in some groups’ projects.

However, as already mentioned, the main activity proposed to pre-service teachers was to develop a project that involved the creation of agroup web page dealing with a mathematical theme taken from the middleor secondary school curricula that could be of interest to teachers or pre-service teachers. More specifically, we suggested that in this web page,besides developing the main theme, they should refer to other sites relatedto the theme, comment on the remaining activities carried out in the course,and also make a brief presentation of the group. In order to do this, thepre-service teachers were required to learn how to use the software Front-page, to do research in the Internet, and to pay special attention to thepage design. In their research, pre-service teachers also used traditionalsources of information such as scientific and professional journals, books,and textbooks.

During the whole semester, they usually worked in groups – this wasconsidered to be a desirable but also inevitable feature due to the limitednumber of computers available. Besides the work carried out during class,pre-service teachers could use the classroom in their own time, and in factmuch of their work was done autonomously and independently.

This course is eminently practical. Most of the time was spent on prac-tical activities carried out by the pre-service teachers while the instructor


supported each group, checking on their progress and trying to help themsolve their problems. The instructor focused on the introduction of newconcepts and explaining the basic workings of the various software.

The work in this course assumed that pre-service teachers could appro-priate a new tool and language – in this case ICT, stressing Internet asa means of production and expression – through a process based on twomain types of activities: exploring materials and resources and carrying outa project. These activities included (i) periods of practice, in which pre-service teachers worked on tasks proposed by the instructors or carried outspecific tasks of their own initiative; (ii) periods of discussion, whetherin group, between the group and the instructor or involving the wholeclass, and (iii) periods of creative activity, in which pre-service teacherswere designing and developing an educational project. This activity wasincluded in the specific preparation required for teaching their subject,since the task required the re-elaboration of mathematical issues from ateaching perspective.

The web pages produced by pre-service mathematics teachers duringthese two school years cover topics such as numbers, geometry, trigono-metry, history of mathematics, probability, logic, functions, derivatives,conic sections, sequences, and equations. Next we refer briefly to someof these web pages so as to show the work carried out by the courseparticipants.

One of these web pages, “The World of Fractals” (www.educ.fc.ul.pt/icm/icm99/icm14), is a resource for those intending to learn about someaspects of fractal geometry. These pre-service teachers make a referenceto Benoit Mandelbrot and briefly explain what a fractal is; in the “FractalGallery” they also present a chronology of the most representative fractals.In this gallery the user may also listen to fractal music and downloadcertain parts of this music. There also is reference to chaos theory and itsrelation to fractal geometry. Fractals also are connected to the secondaryschool curriculum. In the “Activities” item, a set of working proposals ispresented for pupils at this level. Among these the most interesting are theconstruction of a fractal from cuts on a sheet of paper and the constructionof the Koch curve with the GSP software.

The “Pascal Triangle” web page (www.educ.fc.ul.pt/icm/icm99/icm48)stands out for the pleasant and suggestive way in which the pre-serviceteachers organized its presentation. This page deals not only with thePascal triangle but also with some of its properties that are related toparticular sets of numbers. Therefore, detailed reference is made as tohow to find prime, figurative, Fibonacci, and Catalan numbers as well asthe powers of 2 and 11 in the Pascal triangle. The page also shows the


relation between the structure of the Sierpinski triangle and odd numbersin the Pascal triangle. In the “How to construct” item, we may learn howto construct the Pascal triangle. An application is also available to the userwho wants to visualize a triangle with a given number of lines. This webpage also includes an item on problems.

The web page “The Fibonacci World” (www.educ.fc.ul.pt/icm/icm99/icm31) has interesting information concerning sequences, with specialconsideration to the Fibonacci sequences. In the item “Applications of theFibonacci sequences”, besides the traditional problem of rabbit reproduc-tion, the pre-service teachers present some examples on the relation of thissequence to nature, to the Pascal triangle, and to the golden sections. Thisitem also presents a set of application proposals regarding the eighth andeleventh grade curricula. This web page includes another section called“Fibocuriosities”, where a trick with Fibonacci numbers is presented. Italso suggests further research into this theme through a number of linksrelated to the Fibonacci sequence.

Finally, we mention the web page “Decomposition of Figures andthe Pythagoras Theorem” (www.educ.fc.ul.pt/icm/icm99/icm25), designedmainly for eighth grade teachers and pupils. Besides brief referenceto basic geometrical notions, this web page illustrates the situations itpresents with a strong graphic component. It has a section dedicated topuzzles where the tangram and the pentominoes are explored. This is aweb page that fosters user interaction through animations produced withGSP. It presents a Java Aplet that allows users to work on the tangrampieces and construct the suggested shapes. Technically speaking, this pageis very good and reveals quite a sophisticated use of the software presentedin this course.

Pre-service teachers were especially interested in the GSP, whichseveral groups used in their web pages. Some even used Java Sketch toproduce animations. Due to time constraints, we did not cover all thesoftware that could be relevant. We felt it was better to study in depthsome good pieces of software, as pre-service teachers may explore othersoftware later on. In their pre-service education, the important thing is thatthey thoroughly appreciate the features of a few good examples of softwarethat may be used in mathematics teaching.

Working with ICT involves many unforeseen technical obstacles (faultycomputers, problems with the local server, network problems, and prob-lems with external communication). At times these problems disturbedthe class process and forced the instructors to change their plans. Produ-cing the web pages often took pre-service teachers much more time than


initially envisaged. These obstacles and problems show the need for carefulplanning and monitoring in this type of course.

During classes, the instructors tried to pay attention to the way pre-service mathematics teachers were involved in the different activities,joining in their projects, helping to solve problems, and making sugges-tions. At times the instructors would sense a strong concern among thepre-service teachers as to whether it would be possible for them to finishtheir projects within the established deadlines. In moments like these theinstructors’ role was to help them overcome their specific difficulties. Also,and probably most importantly, the instructors encouraged the pre-serviceteachers and made them believe in their capacity to deal with the problemsthey were facing. At the end of the semester, the work carried out in thecourse, especially the web page produced, was discussed during an hourwith each group.

STUDY METHODOLOGY

The observations carried out by the instructors during classes, the discus-sions with pre-service teachers at the end of the semester, and the reflectionregularly undertaken by the instructors provided a general evaluation ofthe course. We were concerned with what went well and what went badlywith our approach and with understanding the meaning of this educationalexperience for the participants.

In order to study in more depth the effect of the work carried out in thecourse on the pre-service teachers we also administered a questionnaireincluding the following items:

1. How do you define your current relationship with ICT? What evolutionoccurred in this regard during this semester?

2. Did this course provide you with the development of perspectivesabout the role of ICT in mathematics teaching? Specify.

3. How do you see the future of ICT in schools?4. How do you evaluate the work that you carried out in this course?5. Comment on the working methodologies used in this course.6. What suggestions can you give to improve this course?

This questionnaire was handed out in the last class and included sixfree-response questions with ample space to answer them. This question-naire addresses pre-service teachers’ relation to ICT, their views on the roleof ICT in mathematics teaching and in school, the evaluation of the workthat was carried out, and the working methodologies used in the course.The free-response format was chosen to capture pre-service teachers’ own


perspectives regarding their experiences in this course and, indeed, mostof them provided lengthy answers to the questions proposed.

A set of categories and sub-categories was developed in order tocode the answers. For example, pre-service teachers’ views encom-passed commitment, difficulties, identified learning, and personal rela-tions. Perspectives about these issues were gathered from the responsesto all six questions. The same was done for the categories of professionalknowledge and professional identity. We used the NUDIST (version 4.0)data analysis software to code the data and to provide the correspondingreports.

In the following sections, first, we seek to identify the way pre-serviceteachers view the work they carried out and their own evolution; next, wediscuss to what extent this work helped them develop their professionalknowledge and identity.

PRE-SERVICE TEACHERS’ VIEWS ON THE COURSE

At this point we show the assessment pre-service teachers make of thework they carried out. Namely, we refer to the commitment they putinto their work, the difficulties they encountered in fulfilling it, what theylearned and, finally, the evolution of their relation to the computer.

Commitment

The pre-service mathematics teachers were unanimous in stating that thissubject demanded a lot from them, not only due to the amount of hours theyspent constructing their web pages but also because of the obstacles theyhad to overcome, which often corresponded to intense learning moments.Generally speaking, they describe their involvement in this subject in termsof the need for a lot of commitment and effort. As one pre-service teacherstates:3

. . . I tried really hard in this subject for several reasons. First because some of the newtechnology we used was totally unknown to me and I had to learn (that took a lot of hours).Secondly, I felt motivated to learn, I thought it was great to do a web page for the Internet.

Their comments also show us that at the end they felt successful and that,looking back at the whole process, they feel personally fulfilled. As one ofthe pre-service teachers says:

The work involved a lot of effort, a lot of research, a lot of commitment and some tiredness,but I can say it was really gratifying. My “scope” of knowledge increased significantly withthis work.


Difficulties encountered

The drawbacks pre-service teachers encountered, mostly technical ones,were viewed as a major limitation because they used up too much time, alot more than seemed reasonable to the teachers. Despite this situation, ingeneral they praised this course, revealing that the positive aspects, namelyregarding what they learned, outweighed these obstacles:

Sometimes I actually felt an excessive and unpleasant pressure (lack of time, faultycomputers, software that didn’t respond as required). In any case, looking back, I thinkI learned so much, so much, that it was really useful.

The positive way in which they evaluate this experience is closely associ-ated to the degree of satisfaction with the final result of their work, whentheir web pages were published in the Internet.

Identified learning

Many of the pre-service teachers took a truly important qualitative leapregarding the use of the Internet, where many of them had never even doneany research. After considering all the effort involved in the process, oneof the pre-service teachers in these circumstances stated:

For me it was a great conquest never to have navigated on the Net and to wind up, afterthree months, editing a web page that now cruises the whole world. This reality seemed soremote to me at the start of the school year! . . .

I must say it was a challenge which, like all challenges, made me sweat, but fortunately allended well. Once again, I think the fact that I attended this course really thrilled me andtaught me things that will stay with me for the rest of my life.

Besides what they learned regarding the use of the Internet, many alsorefer to the opportunities for exploring educational software, which theyconsider to be quite relevant in the present mathematics teaching context.

Concerning the project they developed, the pre-service teachersconsidered that the most important aspects of their experience are thosedirectly related to the process of constructing a web page; however, theyalso stress their work on mathematical themes. The level of familiarity withthese themes varied: sometimes one theme was chosen because the pre-service teachers already were interested and felt at ease in that area; at othertimes the intention was to learn more about subjects that were practicallyor totally unknown to them. Some pre-service teachers stressed the factthat they could study topics that are not usually part of the curriculum andthey could study them in a different way, through explorations with ICT.They indicated that this work promoted their mathematical development:“it allowed me to expand my mathematical knowledge”. By exploring


these themes on their own, pre-service teachers developed a broader senseof internal and external connections, history and applications.

Evolution of the personal relationship

Finally, we analyze the way these pre-service teachers talk about theirrelationship to ICT and the evolution of this relationship. This is a themeabout which they talk a lot and willingly. Their responses show that themajority changed their attitude regarding ICT and established a betterrelationship with the computer. In many cases, this change occurred interms of their readiness to learn to work with the computer, enhancing theirself-confidence. An example that illustrates this quite well is the following:

If at first I was mainly scared (almost) to touch the computer, nowadays I am morecomfortable doing so, which doesn’t mean I don’t make mistakes, but it is an easierrelation.

When they attempt to describe their present attitude towards new tech-nology the pre-service teachers use terms like “more comfortable”, “lessformal”, “positive”, “closer” and “new”. In general they also point to avery positive evolution of this relationship. In some cases this evolution inthe way they view the computer is also apparent with the disappearanceof a negative metaphor – the “beast” (and in certain cases the “hideousbeast”).

Throughout the program two other courses have put me in touch with that “beast”, but onlythis year . . . did I lose my fear of computers and stopped viewing them as “beasts”.

Ignorance regarding the basic aspects of using a computer is one ofthe reasons pointed out by the pre-service teachers when explaining theirinsecurity and a certain attitude of rejection towards the computer fromthe start. Much of the computer’s important potential was also unknown tosome of them:

The start of the semester coincided with me buying a computer. At the beginning I justused it as a typewriter, whereas now I use it as a computer, capable of doing much morethan a simple typewriter.

A great many of the pre-service teachers declare that, prior to thiscourse, ICT did not arouse their interest in the least and sometimes theybegan with poor expectations related to work in this course: “When thesemester began my expectations of the work I would be doing were low.This is because my experience of new technology was frankly limited.”This is one of the aspects where an evolution was most felt. Interest in ICTgrew throughout the semester and at the end pre-service teachers wereeager to learn more in this field, though most consider what they havealready learned to be very positive:


So I think that this year, in this course, I learned a lot and I feel much more at ease. Idon’t consider myself an expert, far from it, I still have plenty to learn. But I’m much moremotivated to learn more.

The will that some pre-service teachers express to investigate new softwareis yet another indicator of having developed a stronger relationship to newtechnology:

I think I sort of trivialized my fear for computers. Working constantly with a computerallowed me to develop curiosity and pleasure in discovering new software. So I underwenta very positive evolution in relation to new technologies.

Another very important evolution is found in some pre-service teacherswho go from a bad relationship to frequently using ICT, even on a personallevel:

My relation has improved quite a lot, for before having [this course] I was suspicious ofcomputers and now I’m “almost” totally dependent. A day doesn’t go by without turningon the computer and seeing something on the Internet.

DEVELOPMENT OF PROFESSIONAL KNOWLEDGE ANDIDENTITY

Next, we discuss the role of this course in pre-service mathematicsteachers’ education. We analyze to what extent pre-service teachersdeveloped their professional knowledge in two fields – their conceptionsabout ICT in mathematics teaching and the impact of working methodolo-gies – and how they show certain aspects that refer to the development oftheir professional identity.

General perspectives about ICT in mathematics teaching

The first question we discuss is the contribution of the course to the devel-opment of a general perspective about the role of ICT in mathematicsteaching.

Pre-service teachers acknowledge that this course clearly made a differ-ence in their professional preparation through raising their awareness ofthe potential of ICT for mathematics teaching. Many of them probably hadheard talk in the media about the increasing importance of this technologyin society and in school, but they knew little else besides this. According totheir answers, it is possible to conclude that they evaluate the new perspec-tives the course provided very positively, especially because they feel theeducational system expects mathematics teachers to be well prepared inthis field:


In fact, since the beginning of the semester, the educational issues I learned enabled me tomake some progress regarding the use of new technology. So even though in the beginningI didn’t quite understand what the computer was for in the classroom, today my opinion hasnot only changed but it has really enriched, through discovering software and techniquesto apply in mathematics classes.

Most pre-service teachers refer to the Internet, GSP, and Modellus as faci-litators of the teacher’s role. Many of them regard these tools as sourcesof motivation: “ICT in the mathematics class is a must. This is the onlyway to make mathematics approachable and attractive to our pupils in thefuture.” Others feel that the software explored in the course will be usefulto support learning of specific topics, such as geometry. As one of the pre-service teachers puts it: “Using GSP showed me that when pupils use itthey understand geometry better, so I think it’s important to apply GSP inmiddle and secondary school classes”. In fact, many pre-service teachersbelieved that this software would offer great possibilities for mathematicsteaching.

Another aspect revealed by pre-service teachers’ answers is that manyof them feel that the use of ICT in mathematics teaching offers the possi-bility of promoting a new vision of mathematics, notably because thistechnology can make mathematics applications more visible. This goesalong with the development of a perspective of ICT use that values thepupils’ active role in learning and autonomous work, showing a high regardfor experimentation and exploration in the mathematics classroom. Twopre-service teachers commented in this respect:

By using new technology we can portray a smoother view of mathematics, so that pupilsfeel more motivated to “discover” mathematics, since nowadays any kid can have accessto a computer.We can use the computer, the Internet, GSP, to carry out different activities through whichwe allow pupils to explore mathematics by themselves, because when they themselvesmake a discovery, classes become active and autonomous and this is the only way for themto construct their own learning.

Another pre-service teacher pointed how computer representations canhelp to “visualize abstract concepts that are difficult to understand”. Othersacknowledged the technical possibilities the computer holds for mathema-tics teaching regarding the visualization, exploration and manipulation ofobjects:

The movement that can be created in a PC is impossible to illustrate in a book, such asgeometrical figures turning, or seeing three-dimensional graphs under other angles, and soon.New technology in school may facilitate the whole teaching-learning process in the sensethat they enable an enormous range of exploration, visualization, and experimentation thatwould otherwise be practically impossible.


Pre-service mathematics teachers also valued the work carried out onsearching for information and developing pages on the Internet. Theyconsider that this activity has great potential for both teachers and pupils.As one of them says: “using the Internet we can easily have access tocontents from all over the world and this expands our knowledge, includingmathematics”. As for the pupils, one pre-service teacher says they “canlearn a lot by searching in the Internet”. Pre-service teachers regard theresearch activity on the Internet as inquiry and point out the possibility ofdrawing a parallel with pupils’ learning processes. One comments that: “itwas also important to get to discover the Internet more ‘intimately’ becausethat enabled me to see how it may be applied in research projects that caneasily be developed in a mathematics classroom”. The uses of the Internetin the mathematics classroom foreseen by many of the pre-service teachersimply a different role for the teacher.

Pre-service teachers indicated that they would like to use ICT intheir teaching. They tend to agree that this technology will have a verystrong role in the school of the future. Regarding the current situation inschools, they expressed strong concerns, mentioning the limited number ofcomputers available and perceiving a dominant opinion in teachers againstthe use of technology. However, some visits to schools that they undertookin various field experiences of the teacher education program showed themthat the use of computers in schools is possible when there is a group ofteachers committed to put innovative activities into practice.

Impact of working methodologies

The second question that concerns us is the contribution of this courseto pre-service mathematics teachers’ development of an appreciationfor working methodologies that value the pupil’s active role, research,collaboration, and group work.

At the beginning of the semester most pre-service teachers found thetasks extremely challenging, mainly due to their lack of knowledge andfamiliarity with computers. As we have already mentioned, they felt thatin order to meet these challenges, they needed a lot of determination, bothin individual and group work. The pre-service teachers’ evaluation of thelevel of involvement required reveals that some of them finished the coursewith a sense of personal development, namely with a more positive attitudetowards new learning situations. As one of them tells us:

At first, and speaking for myself, it was a blend of fun (when we solved a problem) andoutrage (when the computer decided to be “mean” at inappropriate moments). But aboveall it was positive to “sweat” until we managed to get where we wanted. Once again I thinkhaving attended this course was something I really enjoyed and it taught me things thatwill stay with me for the rest of my life.


Pre-service teachers considered that the work carried out in this coursewas essentially practical and did not feel that the approach was directive ortransmissive. In some of them we also find a change in perspective abouthow to learn to work with ICT:

Throughout the whole semester I progressively lost my fear of the computer and abandoneda wrong idea I had. I thought that to work with a computer you needed a course, which istotally wrong, because it’s the computer that teaches you.I think that [the work] carried out was productive . . . When I say productive, I am especiallyreferring to the fact that I learned through my own experience, discovering with my grouppartners and/or by myself, and therefore what I learned will not “get lost” easily.

Their responses emphasise learning by doing and learning by discoverythrough their own experience, as the work with the computer involves thestudents’ active involvement in the learning process.

Several pre-service teachers explicitly recognize that the student-centered nature of the methodology used led them to participate activelyand promoted exploration and experimentation. Some thought of it asan example of “teaching through discovery” which sought to facilitatemeaningful learning. The following comment shows this:

I think the methodology that was used, which often or almost always led the pupil todiscover things for himself (or in a group) and to investigate, is very fruitful, for throughit we develop capacities that will be useful in new situations we will have to solve byourselves.Working with the Sketchpad was important because it made us think and discover geomet-rical properties about the matters we had to study. It is a good methodology that we, futureteachers, can adopt in mathematics teaching, if possible.

The projects and research carried out in this context stand out as themost relevant aspects of the course activity. Some pre-service teacherswere pleased with the opportunity they had to choose the theme fortheir project and learn more about it, emphasizing the inquiry process.For instance, one of the pre-service teachers evaluates group work in thefollowing way:

An interesting work on an interesting theme which is still not well known . . .. Researchwork was performed at several levels and after collecting information it was “filtered” andpresented in the form of a web page.

As a result of the research that was carried out within their projects, somepre-service teachers mentioned that they began to do Internet searchesmore often and that they developed a taste for investigating new softwareby themselves.

Many pre-service teachers feel that group work was a very positiveaspect of the course and had several explanations for this. For example,one of them considered that group work improved the quality of the final


product: “I think that with my colleagues’ collaboration, with the work wedid together, the result was a really successful web page.” In some casespositive reference is also made to the discussions within the group:

Group work is a fundamental working methodology. Of course there is or there may bea clash of opinions . . . But this clash leads to a “discussion” and intensive exchange ofopinions until we reach a consensus.

Besides this, some of the pre-service teachers see group work as apreparation for their professional activity in schools. For example, oneof them says that the experience acquired in this field will be extremelyimportant in the future “as collaboration among teachers is fundamentalfor the evolution . . . of mathematics teaching”.

Development of a professional identity

A third point of interest is the impact of the activity on the developmentof a professional identity. Pre-service teachers’ responses show aspectsof this process, especially as they assumed new perspectives and valuesthat they related to their future professional role. The development ofa professional identity as a mathematics teacher involves, among manyother things, the biographical process of establishing a personal relationwith ICT and developing perspectives about the mathematics teacher’s roleregarding this technology. The following statements illustrate this process:

Contrary to my initial opinion, I think that the way one acquires more knowledge is nothaving always someone helping us but discovering on our own. It was what happened withGSP and it worked perfectly with me.Right now, my relationship with technologies is very good, because I try to do as much as Ican of my work with the computer and I do not have problems investigating new softwareor new technologies.At the beginning I did not see the ICT course as a useful one . . .. Today, with the familiaritythat I acquired, all that I do involves the use of the computer and the Internet, where I findcountless things that I need and that have become indispensable for carrying out my “job”well as a student and that, for sure, I will use in the future in the same way as a teacher.As well as GSP, there is other software equally useful in mathematics teaching and I thinkthat teachers should learn how to use them to innovate their classes, make them less teacher-centered and lead pupils in exploring and discovering.

Explicitly or implicitly, these statements project future activities and rolesas well as assessments of past ideas and perspectives that pre-serviceteachers no longer value. There is evidence of a great change in their rela-tionship with ICT as well as in recognizing its importance in mathematicsteaching. Many of them claim that they developed new perspectives abouthow learning takes place, emphasizing discovery learning versus trans-missive teaching. These experiences led participants to anticipate their


future role as teachers and to relate this to what they think is happening inschools. These statements mark aspects of pre-service teachers’ biograp-hical identity defining processes, involving transactions between inheritedand envisioned identities as they reflect upon past ideas and concep-tions and start showing appreciation regarding what their future work asmathematics teachers will be.

In other responses we see influences of relational processes involvingpre-service teachers’ interactions with others, including their instructorsand other pre-service teachers:

The class went by so fast and . . . we were left to ourselves, given freedom to work and tosolve our own problems. When we needed the instructor, she was always there to help us.The instructor-student relation couldn’t have been better, whenever we needed help theinstructor would run to our rescue, but would always encourage us to try to solve theproblem we were facing by ourselves first. The work carried out with Frontpage developedus as researchers.Of course, we may have a big mismatch of opinions [within the group] but they’re easy tosolve. But this clash leads to a “discussion” and intensive exchange of opinions until wereach a consensus.And in the future, [the fact that we worked in groups] will be extremely important, ascollaboration among teachers is fundamental for the evolution . . . of mathematics teaching.

This relational process led pre-service teachers to appreciate the valueof group work, despite all its inherent difficulties, and to value theteacher-student relation as a complex interplay involving both support andchallenge. They acknowledge the need for negotiations involving differentpeople in order to reach some level of agreement. They also indicatetheir appreciation of collaboration, an important aspect of mathematicsteachers’ professional identity.

CONCLUSION

In this paper we present the work developed in a one-semester course onICT in a pre-service mathematics teacher education program. The orienta-tion we adopted led to successful changes for those pre-service teacherswhose initial attitudes were of fear and suspicion and who developed aremarkably positive relationship with this technology. This course led pre-service teachers to stop feeling menaced by ICT and to become confidentusers of the Internet.

Pre-service teachers take this course in the beginning of the fourthyear in the program. The course is concerned with developing ideasabout teaching and learning but it does not aim directly at a prepara-tion for using ICT in the classroom. That preparation will come later,


in the next semester, in a mathematics methods course that builds uponthe issues discussed here. The course aims to promote a positive rela-tionship with ICT in pre-service teachers who tend to know very littleabout this technology and are suspicious about its use in education. Itdeals with mathematics in a cultural and educational perspective but itis not a mathematics course that uses ICT to promote the learning ofsome mathematics topics. The work carried out helped pre-service teachersto grasp more connections among mathematics topics, their historicaldevelopment, applications, and aspects of classroom learning processes.

The technical quality of the pre-service teachers’ web pages exceededour expectations. In general, these pages have an excellent presentation andmany include interesting solutions in their effects and general structure.Content-wise, the production of these pages represents a very importantmoment in the process of assuming a professional perspective, as pre-service teachers increasingly seek to deal with mathematical themes froma teacher’s point of view. In this work, there was plenty of inquiry, reflec-tion, and discussion among pre-service teachers and between them and theinstructors. That was apparent in their conceptual evolution.

Pre-service teachers got a good idea of the multiple educational possibi-lities of this new tool. Some seemed to become somehow disturbed withthe fact that they did not thoroughly discuss the teaching activities that theyshould propose in the future to their pupils. This is a common concern thatthey carry to all their courses at this point of their preparation, regardingwhich it could be premature to provide the type of answers they are lookingfor. Even so, pre-service teachers developed new perspectives on the useof ICT in mathematics education and some appreciation for working meth-odologies that promote active learning. Both are important aspects of theprofessional knowledge necessary for teaching mathematics. Pre-serviceteachers also took important steps in assuming professional values andattitudes, such as the need to discover and investigate by themselves andthe important role of discussions and collaboration in carrying out profes-sional tasks. These are undoubtedly important aspects that characterizemathematics teachers’ professional identity.

The course described in this paper aimed to change the personal rela-tionship of pre-service teachers with respect to ICT and to provide themwith a general perspective about the uses of this technology in mathematicseducation. It used a rather open-ended approach, especially in the projectphase, assuming that pre-service teachers need to appreciate the value oflearning by exploring and carrying out supervised projects. We regard thisexperience as an important foundation to help participants reflect on therole of the teacher in a technologically rich classroom.


The pedagogic approach used in this course, which focused on pre-service teachers’ exploration and discovery and on group productionof a project, was effective in supporting the objectives that were setout. Specific educational software for mathematics was also dealt with.However, contrary to what often happens in introductory ICT courses (see,for example, Robinson & Milligan, 1997), we deliberately decided to workin-depth with a very small number of software (browsers, Frontpage, GSP,and Modellus). Our option allowed pre-service teachers to master the soft-ware they studied and, at the same time, to develop the capacity to exploreother software by themselves in the future.

The course’s aims, structure, and contents were targeted at the charac-teristics of the participants. The course design is not original or unique,but the evaluation carried out showed that it constituted an innovative andreasonable response to our particular working conditions, contributing tothe development of the professional knowledge and identity of the parti-cipants. Of course, as technology develops and its uses diversify, manyaspects will have to be reconsidered in terms of global planning, activitiesproposed, and discussions about the educational aspects of ICT.

As we mentioned previously, anxiety regarding technology is a signi-ficant issue for pre-service teachers (Yildirim & Kiraz, 1999). By gainingconfidence in the production of web pages, the participants in this coursebecame not only consumers but also potential producers of contents forthe Internet. This is quite an important nuance. The production of webpages about projects, studies, centers of interest, etc., is one of the mostpromising possibilities this resource provides both for the teachers’ workand for the pupils themselves. Here, pupils may find an important wayof expressing their activity, interacting with other pupils, teachers, andmembers of the educational and non-educational communities. Thus, theschool is provided with new possibilities, the development of which canbe facilitated through teachers’ pre-service (and inservice) education. Theresults achieved suggest that in the future we may expect many teachersto be not only consumers of Internet contents but also producers and co-producers of web pages with their pupils, sharing their explorations ofmathematics themes and their teaching-learning experiences in this course.

ICT is not just a simple auxiliary tool. It is an essential technolo-gical element that shapes the social environment, including mathematicsteaching. Therefore, it influences the mathematics teacher’s evolutionregarding professional knowledge and identity. Future teachers mustdevelop confidence in using ICT and a critical attitude towards it. Theyneed to be able to integrate ICT within the goals and objectives formathematics teaching. The task of pre-service teacher programmes is not


to help participants learn how to use this technology in an instrumentalway, but to consider how this technology fits into the development of theirprofessional knowledge and identity. The curriculum of this course wasintended to provide pre-service teachers with comprehensive experiencesof working in ICT projects, but other contexts must be created that take intoaccount other aspects of this quickly expanding technology, in particularits potential in terms of long distance interaction and collaborative work.

NOTES

1 Parts of this paper were presented at the II European Conference of Research inMathematics Education, held in February 2001, in Mariánské Lázne, The Czech Republic.The authors thank Sofia Coelho, of the Centro de Investigação en Educação, for her helpin correcting the English language.2 The Geometer’s Sketchpad is a dynamic geometry software that allows studying proper-ties of figures that remain invariant in different kinds of transformations. Modellus is asoftware that allows the construction of mathematical models using different representa-tions and the simulation of physical phenomena.3 Pre-service teachers’ quotations are translations from Portuguese. The expressions usedin the original words are often idiomatic and involve imprecise use of words (such asusing “perspectives” to mean “expectations”). We opted for an interpretive and not a literaltranslation to enable a better understanding by English speaking readers.

REFERENCES

Berger, P. (1999). Affective component of teachers’ computer beliefs: Role specificaspects. In K. Krainer & F. Goffree (Eds.), On research in teacher education: Froma study of teaching practices to issues in teacher education (63–78). Osnabrück:Forschungsintitut für Mathematikdidaktik.

Berger, P.I. & Luckmann, T. (1966). The social construction of reality. New York, NY:Doubleday.

Boero, P., Dapueto, C. & Parenti, L. (1996). Didactics of mathematics and the profes-sional knowledge of teachers. In A.J. Bishop, K. Clements, C. Keitel, J. Kilpatrick& C. Laborde (Eds.), International handbook of mathematics education (1097–1122).Dordrecht: Kluwer.

Dubar, C. (1997). A socialização: Construção das identidades sociais e profisssionais[Socialization: The construction of social and professional identities]. Porto: PortoEditora.

Elbaz, F. (1983). Teacher thinking: A study of practical knowledge. London: Croom Helm.Fiorentini, D., Nacarato, A.M. & Pinto, R.A. (1999). Saberes da experiência docente em

matemática e educação continuada [Knowledge of experience of teaching mathematicsand inservice education]. Quadrante, 8(1–2), 33–60.

NCTM (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.NCTM (2001). Principles and standards for school mathematics. Reston, VA: NCTM.


Ponte, J.P. (1994). Mathematics teachers’ professional knowledge. In J.P. Ponte & J.F.Matos (Eds.), Proceedings of PME XVIII, Vol. I (195–210). Lisboa, Portugal.

Ponte, J.P. & Serrazina, L. (1998). As novas tecnologias na formação inicial de professores[ICT in pre-service teacher education]. Lisboa: DAPP do ME.

Robinson, S. & Milligan, K. (1997). Technology in the mathematics classroom. Journal ofComputing in Teacher Education, 14(1), 11–15.

Rogan, J.M. (1996). Online mentoring: Reflections and suggestions. Journal of Computingin Teacher Education, 13(3), 5–13.

Schön, D.A. (1983). The reflective practitioner: How professionals think in action.Aldershot Hants: Avebury.

Shulman, L.S. (1986). Those who understand: Knowledge growth in teaching. EducationalResearcher, 15(2), 4–14.

Thompson, A.G. (1984). The relationship of teachers’ conceptions of mathematics andmathematics teaching to instructional practice. Educational Studies in Mathematics, 15,105–127.

Yildirim, S. & Kiraz, E. (1999). Obstacles in integrating online communications tools intopre-service teacher education. Journal of Computing in Teacher Education, 15(3), 23–28.

Departamento de Educação e Centro de Investigação em EducaçãoFaculdade de Ciências da Universidade de LisboaE-mail: [email protected]@[email protected]

MARIA L. BLANTON

USING AN UNDERGRADUATE GEOMETRY COURSE TOCHALLENGE PRE-SERVICE TEACHERS’ NOTIONS OF

DISCOURSE1

ABSTRACT. This investigation uses classroom discourse in an undergraduate mathe-matics course to challenge pre-service secondary mathematics teachers’ notions aboutmathematical discourse, what it might resemble in the classroom, and how its various formscan be cultivated by classroom teachers. The research setting was a required geometrycourse taught by the author. Eleven pre-service mathematics teachers in their junior orsenior year2 of an undergraduate program participated. Results indicated participantsmade a transition toward an image of discourse as an active process by which studentsuse the collective knowledge of their peers to build mathematical understanding anddeveloped in their ability to participate in such discourse. This awareness, along withparticipants’ analyses of their own habits of discourse as classroom teachers, promptedshifts in their projected image of the role of discourse in their future practices of teaching.Results suggested further that the undergraduate mathematics classroom (as opposedto the methods classroom) offers a powerful and unique forum in which pre-servicesecondary teachers can practice, articulate, and collectively reflect on reform-minded waysof teaching.

INTRODUCTION

The development of a cadre of classroom mathematics teachers whosepractices reflect current research on teaching mathematics rests in parton how pre-service teachers, as students, experience mathematics. Sucha claim is theoretically grounded in the sociocultural argument that thenature of intermental functioning is subsequently reflected in an indi-vidual’s intramental functioning (Vygotsky, 1978/1934; Wertsch & Toma,1995). That is, the way in which teachers and students interact with mathe-matical ideas in the social context of the classroom, whether passively oractively, structures students’ thinking about mathematics. From this, wecan infer that classrooms rooted in traditional models of teaching portendthe nature of future instruction. That is, the power of pre-service teachers’mathematical experiences extends to their pedagogical thinking as well.Lortie (1975) maintains that one’s internalized models of teaching area legacy of the ‘apprenticeship of observation’ realized through years


118 MARIA L. BLANTON

of schooling. In effect, as students of mathematics, pre-service teachersacquire what might be described as incidental pedagogies. By this, I meanmodels of teaching that are necessarily “intuitive and imitative ratherthan explicit and analytical” (Lortie, 1975, p. 62) because they derivefrom an orientation (the student’s) that is necessarily bound by his or herperspective as a student. Moreover, in the case of pre-service teachers,such models are likely to reflect the most recent (hence, more accessible)memories of the undergraduate classroom (Grossman, 1990).

EXPLORING AN ALTERNATIVE CONTEXT FORMATHEMATICS PRE-SERVICE TEACHER EDUCATION

This latter point concerning the models of teaching which pre-serviceteachers appropriate from their undergraduate experiences imposes anobvious dilemma for mathematics teacher educators. That is, as educatorsour goal is to craft undergraduate experiences that are organized aroundreform-minded ways of teaching, yet we sometimes have limited access toadvanced courses in mathematics or the faculty who teach them. However,within the undergraduate mathematics community an emerging awarenessof and commitment to systemic reforms on behalf of all students (Kaput& Dubinsky, 1994; Schoenfeld, 1990; Steen, 1992) could make it easierto carve a niche within mathematics that addresses the peculiar domainof pre-service secondary teachers. In this context, I would argue alongwith others (e.g., Zeichner, 1996) for subject-matter courses in whichpre-service secondary teachers seriously explore content in parallel withpedagogy.

My own thinking about the need for mathematics courses with apedagogical focus crystallized while teaching a one-semester under-graduate geometry course for pre-service secondary3 teachers. Prior tothis, I had decided to explore how to design a course that preserved itsintended mathematical integrity while challenging pre-service teachers’incipient notions of practice – in essence, a course that integrated contentand pedagogy. (I distinguish such a course from traditional mathematicsmethods courses because of the inherent emphasis on subject matter in acontent course.) The geometry course seemed an ideal research setting.Ten of the eleven students participating in the course were pre-servicesecondary mathematics (or science and mathematics) teachers in their finaltwo years of a four-year teacher preparation program; one student wasreturning for teacher certification.

CHALLENGING PRE-SERVICE TEACHERS’ DISCOURSE 119

DISCOURSE AS A PEDAGOGICAL FOCUS IN THEMATHEMATICS CLASSROOM

While ‘notions of practice’ might invoke an expanse of objects for study(e.g., technology-based instruction), I was interested in the particularexperience of classroom discourse4 as it contributes to one’s thinking aboutmathematics and, consequently, teaching mathematics. That is, how couldan undergraduate mathematics course be used to heighten students’ aware-ness of and ability to engage in and cultivate a discourse that promotesa conceptual understanding of mathematics? The earlier premise thatthe nature of pre-service teachers’ mathematical experiences shapes howthey ultimately teach mathematics brought such a question to the forein my own thinking. Thus, I targeted students’ thinking about classroomdiscourse, what it might resemble, and how its various forms could becultivated. To this end, the intent of this study was to extend currentresearch about the role of discourse in teaching and learning mathematicsto a setting that contributes to pre-service teachers’ notions of mathema-tical discourse (i.e., the undergraduate mathematics classroom), but is lessunderstood for this purpose.

Although there are variant terminologies about discourse in the litera-ture that could have served this focus, I found Lotman’s (1988) charac-terization of text succinctly appealing in framing our class discussionsabout mathematics, about discourse, and, consequently, about this studyon discourse. Lotman’s notion of text, which he defines inclusively asa “semiotic space in which languages interact, interfere, and organizethemselves hierarchically” (p. 37), includes verbal text such as classroomdiscourse (see e.g., Peressini & Knuth, 1998). Lotman argues that textfunctions dualistically as either a “passive link in conveying some constantinformation between input (sender) and output (receiver)” or as a “thinkingdevice” that generates new meaning when a participant actively interpretsthe text by questioning, validating, or even rejecting it (p. 36). In the formercase, text is viewed as information to be received, encoded, and stored,and its goal is the alignment of codes, or languages, between the speakerand listener. Furthermore, any discrepancy between what is transmittedby the speaker and received by the listener is attributed to a defect incommunication. In contrast, the latter case describes text which serves asa starting point for making sense of an idea or constructing new ideas (seealso Wertsch & Toma, 1995). (Hereafter, I will use the respective termsunivocal and dialogic, ascribed by Wertsch and Toma, when referring tothese functions.) Finally, Lotman argues that for clear communication to


take place, discourse (or text, more generally) should include both univocaland dialogic functioning.

Since the function of discourse (i.e., whether univocal or dialogic) isbased on both the speaker’s intent and the respondent’s passive or activeinterpretation of a speaker’s utterance, identifying the function requiresthe researcher to analyze each speaker’s utterance and the response to it.That is, it requires the analysis of utterances not in isolation but in thecontext of an interaction. Moreover, interpreting the function of discourseis a subjective process that ultimately requires the researcher to identify anutterance as being predominantly, not exclusively, univocal or dialogic. AsPeressini and Knuth (1998) describe, any social interaction requires eachparticipant to decipher text and generate his or her own meaning. Thus, alldiscourse contains a measure of both dialogic and univocal functioning.

The theoretical justification for a focus on discourse was further drawnfrom Vygotsky’s (1978/1934, 1986/1934) sociocultural perspective, whichespouses the primacy of language in an individual’s development. Inparticular, Vygotsky maintained that “higher voluntary forms of humanbehavior have their roots in social interaction, in the individual’s parti-cipation in social behaviors that are mediated by speech [italics added]”(Minick, 1996, p. 33). He believed that psychological tools, such aslanguage, serve to guide human behavior by “transforming the naturalhuman abilities and skills into higher mental functions” (1986, p. xxv). Inthis, he argued that language is the primary medium through which thoughtdevelops and that it is a “manifestation of the transition between socialspeech on the inter-psychological plane (between individuals) and innerspeech on the intra-psychological plane (within the individual)” (Wertsch,1988, p. 86).

Integrating Vygtosky’s perspectives on language with Lotman’s char-acterization of text, Wertsch and Toma (1995) argue not only for theexistence and necessity of classroom discourse, but that its very form(that is, whether it is univocal or dialogic in its function) will be intern-alized by and reflected in the individual’s inner speech. Thus, for instance,if the purpose of classroom discourse is to make sense of ideas andto use those ideas to generate new thinking, then it can reasonably beexpected that students will interpret utterances as thinking devices, “takingan active stance toward them by questioning and extending them [and]by incorporating them into their own external and internal utterances”(p. 171).

The ongoing emphasis on classroom discourse in mathematics educa-tion is rooted philosophically in extant reform agendas such as the NationalCouncil of Teachers of Mathematics (NCTM) Principles and Standards


for School Mathematics [PSSM] (2000) and Professional Standards forTeaching Mathematics (1991). Current research reported in these docu-ments argues that “learning with understanding can be further enhancedby classroom interactions, as students propose mathematical ideas andconjectures, learn to evaluate their own thinking and that of others, anddevelop mathematical reasoning skills” (Hanna and Yackel, in press, ascited in PSSM, 2000). Beyond this, classroom discourse draws its signi-ficance from research perspectives that explore linkages between socialinteractions and the development of teaching and learning mathematics.In an analysis of elementary students’ small group mathematical activity,Cobb, Yackel, and Wood (1992) found that students learned mathematicsas they “participated in the interactive constitution of the situations inwhich they learned” (p. 119). In this, they attributed the development ofgroup consensus to interactions within the group, in which individualsnegotiated incongruities between their own and others’ mathematicalactivity. This suggests that discourse among students can be seen as acentral characteristic of learning mathematics.

Research on teaching K-12 mathematics indicates that classroomdiscourse analyses help us understand what students are learning. Peressiniand Knuth (1998) used Lotman’s characterization of text as a frame-work for analyzing an experienced secondary mathematics teacher’s verbalparticipation in a discrete mathematics course for in-service teachersand the effect of the course on the teacher’s ability to cultivate dialogicdiscourse in his own practice. They found that modeling dialogic discoursein a professional development setting was not sufficient to produce changein the teacher’s ability to foster dialogic discourse. They concluded thatprofessional development should address univocal and dialogic discourseexplicitly and, particularly, instruct teachers in how to create dialogicdiscourse. Additionally, Blanton, Berenson, and Norwood (2001) arguedthat the influence discursive acts in the secondary classroom have onthe student teacher’s practice underscores the need explicitly to addressclassroom discourse in undergraduate settings prior to student teaching.These theoretical and empirical results inform us of (a) the necessity forcreating contexts in which students (including teachers as students) buildtheir mathematical thinking through social, discursive interactions; and(b) the value of classroom discourse both as an object of study for in-service and pre-service teachers and as a tool for understanding teacherlearning. As such, this research suggests that a course involving a focuson classroom discourse could be a valuable tool for building students’capacity as teachers to engage in and promote the kinds of discourse thatwill support their students’ mathematical thinking.


DESIGNING A MATHEMATICS COURSE TO CHALLENGEPRE-SERVICE TEACHERS’ NOTIONS OF DISCOURSE

The goals of the course were thus to (a) develop an understanding ofgeometries (emphasis on the plural); (b) develop logical thinking skillsthrough an emphasis on constructing proofs; and (c) develop an under-standing of meaningful mathematical discourse. These goals are elabor-ated in the remainder of this section.

The Pedagogical Focus of the Course

The geometry course was structured to challenge students’ thinking aboutmathematical discourse on three levels: (a) that of student, as a participantin discourse; (b) that of pre-service teacher, as a student in the pedagogyof discourse; and (c) that of teacher, as an architect of discourse. Althoughthe levels comprising this organizing triad are recorded here as disjointevents, they were in fact intricately connected within the classroom, wheremathematical discourse naturally merged into reflections about the natureof that discourse and its implications for future teaching practices. Buthow might this triad have challenged these pre-service teachers’ notionsabout discourse? First, it is in the mathematics classroom that pre-serviceteachers, as student participants, internalize culturally-sanctioned rulesof (mathematical) discourse. As such, this offers a compelling forum inwhich to engage students’ habits of discourse. Moreover, in this setting,pre-service teachers can dissect a discursive mathematical event as theycreate it, allowing them to apprentice powerful techniques of discourse insitu. By comparison, the same task in other undergraduate contexts (e.g.,methods courses) seems at best academic since mathematics is not theprimary focus. Finally, situating pre-service teachers as architects of anobject for critique (such as discourse) within the mathematics classroomshifts their challenge to an arena in which they can construct understandingabout teaching mathematics with the continuous guidance of the classroominstructor.

Students as participants in mathematical discourseAccording to the PSSM (2000), “the act of formulating ideas to shareinformation or arguments to convince others is an important part oflearning. When ideas are exchanged and subjected to thoughtful critiques,they are often refined and improved” (p. 348). From this perspective,challenging pre-service teachers at the level of student participant meantorchestrating conversation through which students could build their know-ledge as they “participated in the interactive constitution of the situations in


which they learned” (Cobb, et al., 1992, p. 119). That is, assuming studentsalready were well versed in the syntax of univocal discourse (the claim ismade by Wertsch and Toma (1995) that classroom discourse in Americanschools is 80 percent univocal), I set out to craft classroom experiences thatencouraged dialogic discourse. (Students later reported that they had nothad significant experience with dialogic discourse in their undergraduatemathematics courses.) This meant students led the conversation while asteacher my role was to occasionally pose questions intended to clarify orextend their thinking, with the intent of placing students (not the teacher)in a position of justification and argumentation.

Students in the pedagogy of discourseFrom the mathematical discourse, we discussed the nature of ourdialoguing in order to challenge pre-service teachers as students in thepedagogy of discourse. By the ‘pedagogy of discourse’, I mean the func-tion of discourse as described by Lotman (1988) and its implications forclassroom instruction. The characterization of text as univocal or dialogicwas integrated into the course through a progression of events. Initially,students were assigned readings from current literature that applied ordelineated this framework or some other aspect of discourse (Blanton,1998; Wertsch & Toma, 1995; Wood, 1995) and were asked to providewritten reflections on the nature of our discourse. From this there followedclass discussions about the meanings of this framework. We then used theframework as a tool for informal analysis of our mathematical classroomdiscourse as it occurred. Questions such as ‘What was the nature ofour discourse?’ and ‘What should it be?’ were considered. As my ownconflicts about balancing the teacher’s role in discourse became an arti-fact for discussion, pre-service teachers were given the opportunity tosee and analyze a mathematics teacher’s dilemmas as they occurred.Students subsequently used the framework to describe their own teachingexperience in the course.

Students as architects of discourseChallenging pre-service teachers as classroom teachers took the form ofa Discourse Analysis Project (DAP). As part of this project, pairs ofstudents selected one lesson in Euclidean geometry to teach the class. Afterteaching an approximately one-hour lesson (as a dyad or individually),each student completed as an out-of-class assignment the transcriptionof the videotaped lesson and an analysis of these data according to theunivocal or dialogic function of classroom discourse. As part of thisassignment, each student provided a written analysis to support a partic-


ular designation of the function of discourse as univocal or dialogic. Theanalysis also included reflections on perceived strengths and weaknessesin the discourse and the benefit of the DAP to the student’s professionaldevelopment. In the final stage of the DAP, I conducted a structured, 30-minute clinical interview with each student to discuss the results of his orher analysis.

I wish to differentiate teaching in a mathematics classroom as experi-enced by these pre-service teachers from micro-teaching in a methodscourse. In micro-teaching, the mathematics typically constitutes a reviewof mathematical concepts, which can detract from the feeling of an actualmathematics classroom. And while one could argue that teaching mathe-matics in an undergraduate classroom of pre-service teachers is itselfsomewhat contrived, it does have the advantage of being an authenticmathematical setting. I will conjecture here (and later revisit this conjec-ture with supportive data) that the DAP provided these pre-service teacherswith a substantive experience in their preparation for teaching mathe-matics.

The Mathematical Focus of the Course

The dominant mathematical themes of the course were proof and justi-fication, in particular, building formal and informal arguments for topicsin Euclidean, non-Euclidean, and finite geometries. While the courseaddressed traditional Euclidean topics, including congruency and simil-arity relationships in polygons, special properties of triangles, the ParallelPostulate and transversals, geometric properties of circles and Platonicsolids, we also explored Reimannian (spherical) geometry and finitegeometries. Our study of finite and non-Euclidean geometries, whichrepresented about twenty-five percent of the course material, began with anexploration of axioms as well as the development of models consistent withthose axioms. For finite geometries, we constructed proofs, both collec-tively (in class) and individually (out of class), for theorems that arosewithin these axiomatic systems.

Investigations using physical manipulatives (e.g., Lenart Spheres�) andGeometer’s Sketchpad � technology were also incorporated into in-classactivities to strengthen students’ mathematical understanding and exposethem to grade-appropriate ways for building their students’ mathematicalunderstanding. The inclusion of a pedagogical focus in the coursenecessarily precluded time for the study of some geometric concepts. Evenso, mathematics was the predominant focus of the course, requiring abouteighty-five percent of in-class activity and discussion (the remaining time


TABLE I

Sample Geometry Problems Assigned to Students

1. The six executives of Company X have a business charter that requires the following:

• every pair of executives is on exactly one committee together

• each committee consists of at least 3 executives

• there must be 3 executives who are not all on the same committee.

What do you think and why? (Hint: Use Fano’s geometry)

2. PROVE: In any right triangle, the altitude to the hypotenuse forms two right trianglesthat are similar to each other and to the original triangle.

3. PROVE: Interior angles formed by two lines and a transversal such that the anglesare on the same side of the transversal are supplementary if and only if the lines areparallel.

was used to explore issues about discourse and its relation to teachingmathematics).

In addition to in-class mathematical problems geared toward peer argu-mentation and the development of dialogic discourse, students were askedapproximately weekly to complete out-of-class geometry assignments (12total) that emphasized constructing proofs and solving non-routine prob-lems. None of the geometry problems assigned were rote or repetitivein nature. A typical assignment included three to five such geometryproblems (see Table I for sample problems).

Assignments also included alternative problems such as reflectivewritings (e.g., Use metaphors to describe your understanding of mathe-matics and teaching mathematics.) or designing investigative activitiesusing geometric concepts from class. The alternative assignments weregiven in addition to the geometry assignments, with geometry assign-ments comprising approximately seventy-five percent of the total problemsassigned. Additionally, a final exam comprised of 70 percent geometryconcepts and 30 percent pedagogy concepts was administered at the end ofthe semester. Each out-of-class assignment was graded and students wereprovided with detailed feedback of their work.

DATA FOR THE STUDY

The data corpus for this study consisted of video recordings of thegeometry class; students’ discourse analyses of their teaching (the


TABLE II

Axioms for Three-Point Geometry (Smart, 1994)

AXIOM 1: There exist exactly three distinct points in this geometry.

AXIOM 2: Two distinct points are on exactly one line.

AXIOM 3: Not all points of the geometry are on the same line.

AXIOM 4: Two distinct lines are on at least one point.

Discourse Analysis Project); a clinical interview with each student abouthis or her discourse analysis; students’ reflective writings about theirnotions of mathematical discourse and the pedagogical and mathematicalstructure of the geometry course; and selections of students’ in-class work.A description of how the data were analyzed is given in the next section.

FINDINGS AND INTERPRETATIONS

Emerging Mathematical Discourse

Changing students’ notions about discourse required more than their parti-cipation in dialogue or my efforts to model appropriate pedagogy (seealso Peressini & Knuth, 1998); it also required a focused attention onand analysis of the nature of our conversations. Eventually, the discourseseemed to mature towards that dualistic balance set forth by Lotman (1988)in which utterances are, as appropriate, questioned for the purpose ofgenerating new thinking (dialogic) or clarified in order for informationto be communicated adequately (univocal). I have selected the followingexcerpt, which occurred over halfway through the semester and afterstudents had completed the DAP, to convey the nature of dialoguing,whether univocal or dialogic in its function, that had emerged in our class.To characterize the function of discourse in this excerpt, sequences ofutterances were examined to determine how one person’s utterances wereinterpreted by a respondent. That is, given an utterance, I tried to infer ifthe respondent actively questioned it to build deeper knowledge, or if heasked the speaker for clarification. This involved reading an utterance andthe response to it and determining the intent of the response based on itscontent and its tone as conveyed in the video recordings.

A class discussion about three-point geometryIn this episode, the mathematical task was to develop a model, orrepresentation, for three-point geometry (see Table II).


Figure 1. Representation proposed for three-point geometry by Jia’s group.

One of the students, Jia, had drawn her group’s representation on theboard as a source for discussion (see Figure 1). As the excerpt opens,students are trying to verify that the axiom “two distinct points are onexactly one line” is true for this representation. In particular, Brad seems tobe confusing the task of showing that any two distinct points are on exactlyone line with showing that any line contains two points. The excerpt islengthy because it is difficult to convey the essence of the discussionotherwise. Even so, this represents only a portion of the class discussion,which extended for most of an hour (all names are pseudonyms).

1 Teacher: OK, find two distinct points on that model.2 Brad: Well they’re not distinct . . . because those are two points [on

this line] (he indicates line AB), but there are also two pointsfor that [line] (he indicates line BC), so they’re not distinct.


Brad seems to take the notion of “distinct” to mean that only one line canhave exactly 2 points. That is, it seems that he is interpreting the axiom“two distinct points are on exactly one line” as “exactly one line containstwo distinct points”. As he argues here, there is at least a second linethat also has exactly 2 points which, for him, contradicts the notion ofdistinctness. Alternatively, Brad might have been arguing that since B wascommon to both lines, it was not distinct to either line.

3 Teacher: Well, you circled 2 [points] over there. Are you saying they’renot distinct or they are distinct?

My goal here was not to explain my own thinking (and hence the errorin Brad’s), but to push Brad’s thinking further and get him to articulatehis position. My purpose in declaring this goal is not to justify it, but toexplain why I made certain pedagogical choices.

4 Brad: According, no, they’re not.5 Laura: They’re distinct, they’re distinct.

At this point, students begin, simultaneously, to argue their interpretationsof ‘distinct’. Brad’s question (6)5 overrides the other comments.

6 Brad: What is distinct?7 Teacher: What is distinct? Good question.

At this point, I made the choice to turn the question back to students,rather than giving my own clarification of ‘distinctness’. This decision, asI was aware at the time, was intended to confront their readiness to leanon the teacher’s knowledge instead of their own or that of their peers. Asit turned out, Laura and Jia were able to clarify this notion for Brad (8, 17).

8 Laura: Not the same points. They’re two different points.9 Brad: But on that line (he indicates line BC), well (pause) . . ..10 Teacher: Do we agree that what Jia has drawn up there (see Figure 1)

. . . [are] three distinct points, A, B, and C? (Students indicateagreement.) Now, I want you to show me that 2 distinct pointsare on exactly one line.

My purpose here was to see if, in the course of our dialogue, clear commu-nication had occurred and students understood the notion of distinct points.Additionally, it was to remind students of the focus of our dialogue,namely, to verify that 2 distinct points were on exactly one line.


11 Beth: Shouldn’t we define what a line is?12 Teacher: What is a line?

Again, the class spontaneously erupts in an intense (but untranscribable)discussion about what is meant by ‘line’, thus calling into question anotion they had heretofore taken for granted.

13 Laura: What I don’t understand, if it’s saying there are two distinctpoints on exactly one line, does that mean a point can only beon one line and not another line?

14 Brad: That’s what I thought . . .

15 Laura: That’s what I thought when it said exactly one line, that onlythese two points (she indicates points A and B) can make oneline and they can’t be on another one.

16 Brad: That means [this is] wrong (Brad indicates the representationon the board).

At this point, I asked students from Jia’s group, who had until now beensilent, to respond. The pedagogical issue here perhaps is that it seemsstudents had been enculturated into more traditional classroom cultures,in which they were not active verbal participants.

17 Jia: As far as the 2 distinct points are on exactly one line, so thatmeans A and B are on the line that connects them, and theyare 2 different points, and A and C are 2 different points thatare just on one line. Same for B and C.

18 Andrea: [Our group] looked at [the points] as a set: {A, B}, {A, C},and {B, C} (Andrea and Jia were two members of the triadwhose representation (see Figure 1) was being argued.)

19 Teacher: So let’s go back to what you were saying Jia. You pointed out{A, B}, for example, as being 2 distinct points that are on thisline connecting points A and B (indicating the respective linein the representation)

20 Jia: Right.21 Teacher: And are A and B, those two distinct points, on another line,

an additional line?22 Laura: Yes.23 Teacher: Which one?24 Beth: Wait, wait.25 Laura: Oh, they’re not on [another line] together.


26 Beth: Couldn’t you have a third [point] on that line though (sheindicates line AB)? It says two distinct points are on exactlyone line, so what about the other third point (referring to pointC)? Could that also be on the line?

27 Jia: But not all three points in that geometry are on the same line(Here, Jia is referring to Axiom 3. See Table II.)

28 Teacher: That’s good. Now, let’s . . . make sure everybody’s clear.We’ve got two distinct points A and B. They are on the lineconnecting A and B. Are they on a different line?

My response to Jia was intended to affirm her axiomatic argument toBeth’s question (26). Additionally, I wanted to describe the consensus thatseemed to be developing and to focus the discussion on what we needed todetermine, namely, if the two distinct points A and B were on a differentline. This kind of comment was important. As the ‘official’ more-knowingother in this discussion, I had to bring to bear on the conversation theexpertise that would help students to develop valid mathematical ideas, arole which sometimes took the form of assessment and affirmation.

29 Laura: All right, when you say those two distinct points (referringto points A and B), are you saying those two distinct pointstogether, or just those with another one?

30 Teacher: No, A and B, two distinct points collectively.31 Laura: No (indicating they are not on a different line). (Other

students register their agreement.)

At this point, we seemed to have established the idea that “two distinctpoints are on exactly one line” for the representation posed by Jia’s group.Later in the discussion, Beth proposed another model (see Figure 2) inwhich she questioned if lines had to be “straight”. As students began toconfront their long-held Euclidean notions of a line, a debate ensued overhow a line might be represented in three-point geometry.

32 Teacher: Is that a model (referring to Beth’s representation – seeFigure 2)?

33 Laura: Because on that one you don’t have . . . I think it would be.Because, you have to define a line . . . .

34 Beth: Yeah.35 Laura: . . . is a line straight or is it curved? Earlier we just defined a

line as connecting 2 distinct points.36 Teacher: Worth? (He goes to the board).


Figure 2. Beth’s proposed model.

37 Worth: I think this (he indicates the model drawn by Jia – seeFigure 1) is closer to correct where it (referring to line AB)has to terminate there (i.e., at point A) because what aboutthis example where this line (referring to line AB) can curve.It can do all sorts of things. What if it did that? (He extendsline AB in a curved manner so that it intersects the third point,C [see Figure 3]). In Euclidean geometry, it never would dothat. But if we allow it to go here (i.e., to the point C), andwe say it can do whatever it wants, and it hits that third line(referring to line BC), then it (the model) doesn’t pass thoseaxioms anymore.

38 Teacher: It picks up another point.39 Worth: It picks up another point. So I think [a line] has to terminate

at the [point].

Worth is arguing against a model for which the ‘lines’ extend beyonda point of intersection (as in lines BC and AC, which extend beyondpoint C in Figure 2). Although this occurrence does not contradict theaxioms of three-point geometry (i.e., ‘lines’ can extend beyond a point ofintersection), his argument is that if this is allowed, then the line might“eventually” intersect with a third point, thus contradicting the axioms.


Figure 3. Worth’s counterexample of a three-point geometry model.

40 Laura: But when it terminates, [the line] can still terminate at [a]point and not be straight.

41 Worth: In between the 2 points it can curve, but it can’t go throughanother point.

42 Teacher: Why can’t it go through another point?43 Laura: Because then it would be like . . .

44 Worth: [It would contradict] axiom 3.

As our discussion continued, we were able to establish that representationssuch as those given in Figure 2 were models for three-point geometry andthat Worth’s argument was legitimate and should be (and was) consideredin generating a model. That is, it was agreed that a line could be curvedand could extend beyond a point of intersection as long as it did notintersect the third point of the geometry. Meanwhile, Laura amendedWorth’s model to consist of two “straight” lines (AB and BC) and one“curved” line (AC) and proposed it as yet another representation (seeFigure 4).

45 Teacher: How is Laura’s model (see Figure 4) different from [themodel proposed by Jia’s group] (see Figure 1)?

46 Beth: It’s not.


Figure 4. Model exposed by Laura as an extension of Worth’s representation.

47 Brad: It sure looks different.48 Jia: It isn’t different though.49 Beth: Just like if you drew . . . a circle and [divided the circle into

three parts and] you considered each part as like one line.That’s why I was asking if you could curve [a line].

50 Katy: The model [drawn by Jia’s group] (see Figure 1) is exactlylike the model that Laura just drew (Figure 4) because youcould say that ABCA is a line, whereas the teepee (Figure 5)has 3 distinct lines.

Katy seems to be arguing that lines are distinct when their ‘endpoints’ donot adjoin another line. In this sense, she argues that Jia’s and Laura’smodels (see Figure 1 and Figure 4, respectively) are alike in that theyeach represent exactly one ‘line’ containing 3 points. Figure 5, which Katyreferred to as a ‘teepee’, was a representation that I had drawn on the boardsubsequent to and in contrast with the representation given by Figure 1. Inparticular, it was in response to a question by Laura (for constraints oflength, not included in the transcripts here) concerning whether ‘lines’ hadto terminate at the points or if they could be extended beyond the points.


Figure 5. Model included by the instructor in the discussion.

51 Sheila: So what is a line?52 Katy: So that one triangle (Figure 1), that whole thing, is a line.

Again, the class erupts spontaneously in (untranscribable) argument overKaty’s claim, invoking the axioms recorded on the board (see Table II)to support their positions. Finally, Beth’s objection (53) to students’arguments that a line cannot contain points A, B, and C is distinguishableabove the rest.

53 Beth: But you can say that ABC is one line.54 Katy: If you did say that, if you saw it as one line, then axiom 3

would not be satisfied. That’s what I’m saying. That model(referring to Figure 1). But with the one below it (referring toFigure 4), it is satisfied.

From statements 50, 52, and 54, it seems that Katy is arguing that one couldinterpret a line as containing three points when there is no visual notionof distinct lines, such as in Figure 5. However, she notes that while the


representation may be characterized as a line (perhaps in some geometry),it cannot be a model for three-point geometry because the line containsthree points. Students agreed with Katy’s reasoning.

Univocal and dialogic functioning in students’ discourseWhat can be inferred from this excerpt about these students’ mathematicaldiscourse and how the function of that discourse supported their concep-tualization of mathematical ideas? Perhaps most significantly, an analysisof the function of discourse in this excerpt suggests that these studentswere able to use the collective knowledge of the class to generate newunderstanding, the essence of dialogic discourse. Consider the interactionbetween Laura and Brad (13–16). The speaker (Laura) articulated her pointof confusion (13, 15) and Brad responded by comparing it to his ownconception. This is inferred from his response (14) in which he validatedLaura’s position by identifying it as his own. He subsequently treated thisinteraction as a springboard for assessing Jia’s proposed representation(16). Later in the discussion, Beth questioned if a third point could alsobe on a line (26). Again, the respondent (Jia) took a questioning stancetowards Beth’s utterance and ultimately rejected it based on the axiomsof the geometry (27). As Beth’s belief that a line was characterized byits constitutive segments, not the number of points it contained, persisted(49, 53), Katy actively addressed this belief by assuming it to be true, thenarguing that a line could contain three points, but not in this particulargeometry. In this exchange, Katy seemed to treat Beth’s utterance as astarting point for arguing her own position. In particular, she began byvalidating Beth’s perspective that the symbol designated as line ABC couldbe a line (“If you did say that, if you saw it as one line . . .”), then pursuingthe implications for this assumption (“then axiom 3 would not be satis-fied. That’s what I’m saying . . . . But with the one below it [referring toFigure 5], it is satisfied”).

I would argue that these dialogic episodes were significant in howstudents came to conceptualize the axiomatic consistency of a model, andthat the significance of the episodes derives from the fact that they repre-sented socially constituted events, uniquely determined by the interactionsthemselves through acts of shared sense making. From the perspectivethat higher forms of thinking are rooted “in the individual’s participationin social behaviors that are mediated by speech” (Minick, 1996, p. 33),it seems that the absence of such dialogue would have altered funda-mentally how students came to think about these mathematical ideas.Instead, the data suggest that the dialogic episodes, because they wereabout actively interpreting utterances, generated an interaction between


public and private speech that could facilitate the internalization of aconcept. For example, in statement (9), Brad’s objection (“but on thatline”) to Laura’s claim (8) and his utterance “well” followed by a pause,suggests that Brad was actively thinking about Laura’s utterance, albeit atan internal or private level, while engaging with Laura publicly. Again,when Laura made her confusion explicit (13), Brad seemed to processit privately, then described how his thinking compared to Laura’s (14).Finally, as a result of his interaction with Laura, he offered a public assess-ment (16) based on his private (unspoken) thinking about the model. Thus,it seems that the active, or dialogic, interpretation of Laura’s utterancespropelled Brad to a different point of understanding about the consistencyof the model because it engaged him in alternating sequences of publicand private speech. While at this point Brad and Laura were not at aplace where they could fully interpret the axiomatic consistency of themodel, they did continue to work through their misconceptions (17–31).In general, it seems that without occasion for public, dialogic discoursewith more knowing others, students’ conceptualizations will depend on acapacity to engage in inner speech that is structurally equivalent to dialogicdiscourse while in the absence of its public counterpart. My sense was thatthese students were only beginning to develop the tools for this type ofautonomous thinking and that they were doing so by the social, public actof peer argumentation.

There also were points in the dialogue that functioned univocally inthat information was given or requested for clarification. For instance,in (19) I restated to Jia what I thought represented her prior comment(17) in order to confirm that our codes were aligned. At another point,Laura asked for clarification about what I meant by ‘two distinct points’(29). Interpreting her utterance univocally, I responded by clarifying theintended meaning (30). Finally, in 17–18 Jia and Andrea provided inform-ation about their representation (Figure 1). I characterize this as univocalbecause they seemed to be responding to my request for information, notactively addressing Brad’s conclusion about their representation (16). Inother words, it seems the intent of their comments was not to build onBrad’s notion, but instead to serve as a form of information we could useto assess how our ideas about the representation coordinated with theirs,thus facilitating the clear communication of our ideas. Although here Ihave described instances of univocal and dialogic functioning as disjointevents, it seemed in actual dialogue that they were tightly connected. Thatis, throughout this episode, there was a natural, ongoing transition betweenthese two functions that seemed to speak to Lotman’s claim that theyshould be dualistically balanced in practice. For example, I characterized


19–20 as univocal in its function, yet this interaction quickly merged intodialogic discourse when Laura and Beth joined the exchange. In partic-ular, rather than interpreting Laura’s incorrect response (22) univocallyby assuming a fault in Jia’s or my communication about the meaning oftwo distinct points on a line (17, 19), I questioned Laura (23) to try toengage her active participation. This required Laura to engage in her ownsense – making (25) rather than respond to an explanation I might havegiven. I would argue that this induced Laura to assume an active ratherthan passive stance in her participation in the discourse, thus creating adialogic context that seemed to engage her as a learner. Moreover, Bethdrew on this exchange to conjecture about the possibility of a third pointon a line (26), and, as noted earlier, Jia seemed to interpret Beth’s utterancedialogically in that she questioned and ultimately rejected the conjecture(27). Finally, the conversation merged back into discourse that functionedunivocally (28–30), as I questioned students to confirm that our codeswere aligned (28) and was consequently asked to explain the meaningof ‘two distinct points’ (29). My clarifying comment (30) suggests thatI interpreted Laura’s question univocally.

This type of transitioning is reflected in Figure 6, which providesan approximate representation of how univocal and dialogic functioningoccurred in a portion of the discourse excerpted here. It is approximate inthat the nature of a conversation dynamic makes it difficult in practice totease apart these functions as discrete events and in that identification ofthe function of discourse is interpretive and thus not an exact process. Thisgraphic is not intended to imply that all discourse should have approx-imately equal instances of univocal and dialogic functioning. The balancedepends on the purpose of the dialogue (e.g., giving someone directions toa location would be predominantly univocal). I propose that this represen-tation does reflect a balance in our conversational purpose that, if continuedover an extended period of time, could lead to a habit of mind wherebystudents actively interpret classroom utterances, that is, where dialogicfunctioning is a natural and viable part of classroom discourse.

Finally, it is significant that students’ perceptions about the three-pointgeometry discussion seemed consistent with this analysis. Their reflec-tions, written about the discussion after this class occurred and prior tothe next meeting, included the following observations:

(a) “What happened in class was a perfect example of dialogic func-tion. The students kept building on each other’s thoughts, making ita collaborative process and a real learning process.”;

(b) “Students posed their own interpretation of the rules which the otherstudents evaluated and either supported or rejected. . . . Had the teacher


Figure 6. Representation of the occurrence of univocal and dialogic functioning in aselection of classroom discourse.

introduced three-point geometry by discussing its rules, giving herown examples, and the model for the geometry, students would haveinevitably been bored.”;

(c) “Finite geometry was completely new to me. . . . Thus, [the teacher]could have just given us the facts . . . or let us explore on our own.Luckily, [the teacher] chose the latter approach. Thoughts were flyingall over the room about the basics of this geometry as we slowlyfigured it out on our own.”

Reflections on how these students might have benefited from the discourseThe analysis suggests that 1–54 could be characterized as an activityof eliminating perceived incongruities between one’s own and others’mathematical activity (Cobb, Yackel & Wood, 1992) through a processwhich included both univocal and dialogic discourse. At a minimum, thisexcerpt offers an existence proof that students could interrupt the inertiaof passive listening and create and sustain meaningful dialogue. In partic-ular, students were able to identify and articulate their points of confusion(e.g., 13, 15), form conjectures (e.g., 26), successfully argue conjecturesposed by teacher or student (e.g., 17, 27, 45–54), and develop precisionwith the language of mathematics (e.g., notions of ‘distinct’). As onestudent observed about the discussion (1–54), “Each student helped thenext to change or refine his ideas”, such as in 32–44 and 45–54, wherestudents negotiated a more rigorous notion about what constituted a line inthis geometry. The PSSM (2000, p. 348) argues that through the process


of students refining each others’ ideas, they are able to “sharpen theirskills in critiquing and following others’ logic” and consequently, “asstudents develop clearer and more coherent communication (using verbalexplanations and appropriate mathematical notation and representations),they will become better mathematical thinkers.” I would argue that thesestudents did develop a more complex understanding of the mathematics.For example, by proposing and justifying their representation of three-point geometry, students were able to confront openly their strongly heldnotions from Euclidean geometry (e.g., the representation of a line), pickapart the subtleties of meaning packed in a statement such as “two distinctpoints are on exactly one line”, and refine their notions of when a mathe-matical claim has been established and what that takes (a recurring issuewhich students reported in informal classroom conversations as unresolvedfrom their previous course work). In short, these students seemed to belearning how to think and argue mathematically by the sheer act of doing it.They also seemed to be learning that their own students need those types ofdiscursive experiences in order to become critical thinkers. As one studentobserved on her DAP, “I am beginning to realize now that it is going to bedifficult to expect my own students to think critically when I have neverbeen expected to do so.”

Emerging Models of Teaching

The discussion excerpted above (1–54) tells us about these studentsas participants in mathematical discourse. It remains to be establishedhow the organizing triad outlined earlier – (a) student, as a participantin discourse; (b) pre-service teacher, as a student in the pedagogy ofdiscourse; and (c) teacher, as an architect of discourse – challengedstudents’ internalized models of teaching with respect to the function androle of discourse in instructional practice. In this section, I draw froma video-recorded, in-class pedagogical discussion and students’ DAPs toexamine shifts in this aspect of students’ thinking.

Analyzing mathematical discourse in situBy design, mathematical discourse became an artifact for our in-classanalysis, the purpose being to use an authentic mathematical eventto expand students’ thinking about the discourse that constituted thatevent. As such, our conversations about discourse were based not onhypothetical referents (which could arguably be the case in a traditionalmethods course), but on mathematical discourse that was intenselystudents’ own, immediate experience. I conjecture that the occurrence ofthis type of pedagogical discussion (see 55–82) as a natural progression


of an experience with that pedagogy in an authentic mathematical setting(e.g., 1–54) fundamentally altered the ways these students thought aboutdiscourse. The frequency and length of these pedagogical discussionsvaried throughout the semester, occupying about 15 percent of in-classtime, or, on average, about 10 minutes of a 75-minute class. While thisseems like a brief amount of time, I still found that this, in conjunctionwith out-of-class activities, increased students’ sensitivity to classroomdiscourse. The selection below, which chronicles part of our collectivereflection on the mathematical discussion about representations for three-point geometry (1–54), is included to illustrate this type of discussion andto document students’ emergent thinking about the pedagogy of discourse.

55 Teacher: Sometimes teachers will think that in this type of interaction(i.e., 1–54) there’s a lot of confusion, which there seems tobe, which is good, I think. It’s very good, and [you mightthink] that you don’t get stuff done, it’s not efficient, you don’t“cover the curriculum”. But if you get from Chapter 1 toChapter 50 and your students have no idea what’s going on,I don’t know that you’ve been successful.

56 Brad: Yeah, but it does get frustrating after a certain period of time.Can you imagine middle school? I mean they would enjoy it,but there’s going to be some kid that’s going, “What in theheck?”

57 Sheila: That’s why you give them notes.58 Worth: You do this [type of discussion] to a point, but then you’ve

got to tell them what the right answer is.59 Brad: Yeah.60 Laura: You need a nice combination of both.61 Brad: I was going to say that the reason we think like that,

the reason we get frustrated is we’re used to getting rightanswers.

62 Teacher: Absolutely.63 Brad: If you start kids doing this at the beginning, I guess they

would be used to it and they wouldn’t care about right orwrong answers. But we’re so concerned about right or wronganswers, that’s why we get frustrated. (Students expressedtheir agreement.)

64 Teacher: It’s not that we’re not concerned with what the right answeris. I want to make that clear . . . . It is that there’s more thanthe right answer. Students need to experience what it meansto argue mathematics and think about ideas because when


they leave class, whether or not they can do it (by “do it”, Imean do mathematics in a deep, conceptual way, not followrote procedures) is going to depend on whether or not theycan argue with themselves and think through theorems, andso forth.

65 Laura: But can you do this with finite math? Do you think you canhave this much dialogic conversation with finite math?

66 Brad: I don’t know.67 Teacher: Do you mean, like, Euclidean geometry?

I was not sure what Laura meant here. I interpreted her use of the term‘finite’ as an overgeneralization from the ‘finite’ geometries we hadbeen studying. My response was intended to suggest an area (Euclideangeometry) in which Laura had experienced dialogic functioning as a wayto get her to clarify her question.

68 Laura: Yeah.69 Teacher: I certainly do.70 Laura: Like x + 6 = 10.71 Jan: I think you can because, like in those articles you gave us, . . .

the teacher would ask the kids how they came to a conclusionon an answer and I had to really think about some of thoseanswers because I was like, “How did they get that?”. ThenI [realized what they meant] and I thought, “It does makesense!”

72 Laura: It’s just like [the article (Blanton, 1998)] where [the teacher]was doing algebraic functions. She broke them up into groupsand she was like, “How did your group get that?” (Here,Laura is referring to an instance in which a teacher promptedstudents to justify their results). It’s not as much dialogic[conversation] as we give, but at least it’s a little bit.

73 Katy: The thing that I get from [the readings] is that it is good tohear different ways to arrive at the right solution, but I don’tunderstand why it’s helpful to explore how a student got thewrong solution. I mean, for the teacher, yeah, because you canlearn where they’re messing up, but for the rest of the class, itseems like . . . . I think in one of the articles a teacher exploredwhy a kid got [an incorrect answer] and I didn’t understandwhy it was necessary . . . .

74 Teacher: Why a teacher would want to do that?75 Katy: Yeah.


76 Brad: Well, there are probably others who are doing that (incorrectprocedure), first of all.

77 Andrea: And if you can recognize an incorrect procedure, that helpsyou. It’s like, even though your model (referring to a repre-sentation proposed by Katy during the three-point geometrydiscussion) wasn’t correct, it helped us all understand acorrect model.

78 Teacher: It’s a counterexample.79 Brad: Yeah and we see where they messed up.80 Andrea: So in a way, you did help us [by sharing an incorrect model

for the 3-point geometry] because we had to think about, “Isthis right?”

81 Brad: We thought it was and then we [realized it wasn’t] and thatreinforced these things (pointing to the axioms on the board).

82 Jia: And we can figure out why [the model] was wrong.

What can we glean from this conversation about the development of thesestudents’ notions about discourse? First, it seems that this pedagogicaldiscourse was structurally similar to our mathematical discourse in thatit reflects a balance of univocal and dialogic functioning, with studentsusing their peers’ utterances to generate new understanding (e.g., 73–82).For instance, when Katy questioned why it might be useful to explore astudent’s incorrect response (73), Andrea took a previous mathematicalepisode in which Katy had proposed an incorrect representation for three-point geometry and, building on Brad’s comment (76), argued that anincorrect response allows students to better understand a correct one (77).Brad and Jia supported Andrea’s argument by noting that exploring Katy’sincorrect model had allowed them to determine how and why the modelwas incorrect (79, 82). Furthermore, Katy’s question had stemmed fromher effort to make sense of the assigned reading, thus illustrating her activestance toward that material. I would describe this interaction (73–82) asessentially dialogic in its function. Moreover, I interpret students’ applica-tion of such discursive practices to a context other than mathematics asfurther evidence for their internalization of the ways they were beginningto think about and participate in discourse. Additionally, I interpret thefollowing as an indication that these students were beginning to reflectcritically on the pedagogy of discourse:

(a) Brad’s reflective activity, through which he identified characteristicsof classroom discourse as an attribute of school culture (56, 61, 63);

(b) reflections and observations by Jan (71), Laura (72), and Katy (73) thatbuilt on their knowledge of current research about discourse;


(c) Laura’s effort (65) to extend her knowledge about dialogic discourseto areas of mathematics she would soon teach (e.g., algebra);

(d) Andrea’s ability to connect her mathematical experience in the classwith a peer’s question about a particular discursive practice (i.e.,exploring a student’s incorrect response) (73, 77, 80); and

(e) students’ increasingly sophisticated language about discourse, inparticular, their use of the constructs ‘univocal’ and ‘dialogic’ as aframework by which they talked about discourse (65, 72). (This wasconfirmed by informal data as well: Students reported using univocaland dialogic functioning as a framework to analyze discourse in otherclasses.)

Students as architects of discourse: Results of the DAPThe DAPs provided a snapshot of students’ teaching practices, and moresignificantly, of students’ reflections on and analyses of the kinds ofdiscourse they created as well as their notions about what discourse shouldresemble. In addition, the experience itself of doing a DAP seemed to be apowerful mechanism for these pre-service teachers’ professional develop-ment. An analysis of the DAPs showed that individual results, presented inthis section, were highly comparable.

Students reported in their DAPs and in clinical interviews that it wasmore difficult to create dialogic discourse than they had anticipated. Theyrecognized constraints imposed by time as well as the patience it takes toallow students to lead the discourse. In his DAP, one student observed that

there were many times . . . I could see places that univocal functioning took place whendialogic functioning should have, . . . times when I jumped too quickly to answer questionsand give the right answer when I should have allowed the students to discuss the mattersand resolve them among themselves.

Students noted that, if at all, dialogic conversation occurred during groupor paired activities, or when the instructor was confused (because, as onestudent noted, he then “could not take the position of authority”). In thelatter case, that confusion essentially placed the pre-service teacher inthe role of learner; hence, the fact that dialogic discourse grew out ofthis dynamic is consonant with findings by Peressini and Knuth (1998).It suggests that teachers need to be especially sensitive that they donot cultivate predominantly univocal discourse when their mathematicalunderstanding seems most clear in their own thinking.

Without exception, students found that the function of discourse consti-tuting their respective classroom teaching experiences was essentiallyunivocal, although their perception prior to the DAP about how they would


teach their particular lesson was that it would be characterized by dialogicdiscourse. In her DAP, Andrea noted that

univocal text dominated the classroom dialogue. . . . Students are not allowed to developtheir own methods for approaching this proof and are instead led to the teacher’s (referringto herself) proof solution step by step. It is clear that no dialogic conversation takes placebecause the students do not use the teacher’s or each other’s utterances as a basis for re-interpretation.

Katy found that she was challenged by her beliefs about what constitutesgood teaching and her actual practice of teaching:

Students heard mainly what I planned for them to hear. If the discourse started movingaway from my script, I fought to get it back where I wanted it. I always believed in ateacher as a facilitator, [someone who] encourages thinking and does not “spoon feed”ideas to students. Yet, I did not think about how to facilitate thinking.”

As a result of their findings, students stated their intent (and wish) tocultivate more dialogic discourse in their practice.

In reflecting on the discourse in their teaching, some students suggestedthat it was essentially univocal because of a need for structure and control,or because of a perception about the nature of mathematics. Jia wrote, “Ibelieve univocal functioning [dominated] my lesson because of my needfor a direct, concise and organized presentation”, while Katy observedthat “with univocal discourse I do not have to risk confusion in theclassroom”. Sheila wrote “my view of mathematics . . . as a series ofsteps . . . contributed to how I [conducted] the lesson in a univocal style”.Through their activity in this course it seems as if students had come tobelieve in the necessity of dialogic discourse; however, they had difficultiesin interpreting these beliefs in their practice of teaching.

As a result of the DAP, students seemed to develop not only a moreaccurate perception of their practice of teaching, but also a more completeunderstanding of the pedagogy of discourse, specifically, of the mean-ings of dialogic and univocal functioning. Katy’s observation documents aclarification in her own thinking:

Before [the DAP], I thought two-way communication existed if the teacher and studentsalternated dialogue. Well, I was not the only person speaking during my lesson. Alternatingdialogue took place. Still, the discourse was mostly univocal.

In addition, students reported that understanding the function of discourse(Lotman, 1988) would better equip them to create active learning situationsfor their students because they had a tangible sense of how classroomdiscourse could invite active participation.

The awareness students gained from the DAP, coupled with theevolution in their understanding about discourse and its various forms,


converged to influence how they thought about their practice. For example,recognizing his habit of “giving answers”, Brad acknowledged the need toallow students to be the problem solvers. He wrote:

The fact that I believe this time (referring to the time students struggle with the mathe-matics) is lost is a misconception that many teachers have. This time is actually a timewhen students are attempting to generate new meaning and understand the material. Thisis an area in which I must become more comfortable . . . and must learn not to . . . give outanswers.

Evan’s thinking reflected a similar shift:

Students will learn more if they are allowed to share and explore their own thoughts. Also,students need to be able to interact with one another as well as with the teacher. It is veryeasy for the teacher to fall into the trap of spoon feeding material to the students withoutrealizing that it is happening. I now realize that it is okay if students struggle with problemssome or come up with different ways to solve problems.

Katy recognized that she needed to be listening rather than telling: “I musthear and understand my students’ thinking in order to help them learn.They must raise questions and share ideas different from my own to learnfrom each other”.

Worth, a middle school teacher seeking teacher certification, articulateda perspective about the role of dialogic discourse that was more conser-vative than that of his peers, yet consistent with his ongoing (and notuncommon) concern that dialogic discourse required more class time thanhe felt he could afford. In his DAP, he wrote

I was open to dialogic discourse when it did occur . . . . If the discourse shifted away frommy questions and game plan, I was patient and encouraged the discussion until I could getback to my planned lesson . . . . This is not always easy to do when the clock is slowlycreeping towards the end of class and we are only halfway through the lesson.

In Worth’s thinking, it seemed that the purpose of dialogic discourse wasat best a type of mental field trip for students, at worst an interruption. Itwas not a critical path by which students could construct an understandingof mathematics; it was an activity in which he allowed them to participate,though it was not the real purpose of his agenda. In spite of this uncertainty,he acknowledged that the “biggest weakness of the discourse [during histeaching experience] was the lack of extensive dialogic function and classdiscussion”. Moreover, he described his intent to cultivate a different typeof teaching practice, one that included dialogic discourse:

I (until now) looked at teaching very much in terms of teaching a lesson and asking if thereare any questions . . . . Based on [the DAP] and our discussions in class, I am going toattempt to change my way of thinking and be more open to discussion, even if it meansfalling behind in the curriculum.


With a perspective more cautious than that of his peers, it still seemsthat even Worth’s models of teaching, particularly his notions of discourse,were challenged. In fact, it seems implicit that the DAP was a usefulcatalyst for promoting change in these students’ thinking about discourse.Students argued this explicitly as well:

I think doing this [DAP] opened my eyes in a lot of ways. Applying what I had learnedto my own experience (i.e., teaching the class) allowed me to understand where I am andwhere I would like to be by the time I begin teaching.

[This DAP] has pointed out to me the importance of student input.

I can’t say enough how beneficial this [DAP] has been . . . . It showed me howimportant dialogic discourse is in a classroom.

A critical look at my own experienceAs I considered my own practice of teaching, issues arose that were similarto those described by students in their DAPs. For instance, I also found itdifficult to build dialogic discourse. Perhaps this was because, as studentsreported, they were more familiar with classrooms in which discoursewas predominantly univocal. Thus I often needed to prompt them to leadthe conversation, at times even physically withdrawing from the conver-sational space. Perhaps more significantly, this difficulty arose from anongoing conflict, in my own thinking, related to knowing when to step intothe discourse in order to influence its direction or when to allow students tomake their own sense of the mathematics. I often questioned whether I hadallowed a conversation to extend too long, or when it might be unnecessaryto follow a particular student-generated idea, or how I could get studentsto engage with each other. The result was a temptation to ‘tell’ studentswhat they needed to know. I found that a deliberate effort was required tocultivate dialogic discourse and I would anticipate that these students andothers like them would need professional support to create and maintain adiscourse that actively engages students’ thinking.

Did the Course Effectively Teach Geometry?

In response to an open-ended questionnaire and in their reflective writ-ings, students described the course as emphasizing (a) teacher-student andstudent-student discourse; (b) understanding how to teach geometry insecondary grades; (c) collaborative, hands-on explorations during class;(d) understanding, rather than memorizing, mathematics; (e) collectivethinking rather than copious note taking; and (f) alternative forms ofassessment. Students also reported that the level of mathematical diffi-culty was comparable to that of their other undergraduate mathematics


TABLE III

Student Performance on Course Assessments

Homework Discourse Final exam Final grade

assignments analysis project

94 96 89 93

97 96 78 92

95 92 88 92

90 95 70 87

91 95 75 88

90 96 91 92

97 97 87 95

100 97 98 98

92 96 90 93

94 98 86 93

98 97 97 98

courses. But how effectively did the course teach geometry by including adialogic mode of discourse? Are there comparable results, or even gains,in student understanding when discourse includes dialogic functioning?While more research is needed to answer these questions, this section doesprovide some quantitative results on the course assessments, describedearlier, of students’ mathematical learning. In particular, Table III givesthe percentage points that each student earned in the course. Homeworkassignments constituted 40 percent of the final grade, the DAP contributed35 percent, and the in-class final exam was 25 percent. The mathema-tical content for these assessments was taken or adapted from the textused by school faculty for this particular course (see sample problemsin Table I). Results of the homework assignments and final exam, onwhich geometry comprised respectively 75 and 70 percent of the concepts,indicate that students did master the mathematical content (a final gradebelow 60 percent was not a passing score). Mathematical proof consti-tuted 14 percent of the final exam, with student scores on these itemsranging from 100 percent (1 student), 93 percent (3 students), 79 percent (3students), 72 percent (2 students), 65 percent (1 student), and 22 percent (1student). Results on the 12 homework assignments also show that studentsperformed above mastery on problems requiring mathematical proof (seeTable III). Although more research is needed for more specific claims to bemade about differences in students’ mathematical performance in a course


such as this versus one that excludes an attention to and experience indialogic discourse, the results of this study do suggest that students learnedthe content effectively and were challenged in their perceptions aboutteaching mathematics. Given this, I would argue that these students wereserved better by a more in-depth study of fewer mathematical concepts,where that depth was derived in part from an analysis of the discourse thatconstituted their mathematical experiences. As such, this raises the issueof if, and how, mathematics classrooms can include careful deliberationswith students about the mathematics they are learning and the ways inwhich they are learning it.

CONCLUSIONS

This study explored an approach for challenging pre-service secondarymathematics teachers’ fundamental notions about discourse by using theundergraduate mathematics classroom to engage their thinking as parti-cipants in discourse, as students in the pedagogy of discourse, and asarchitects of discourse. Results suggest that the three components of thisorganizing triad converged to (a) shift pre-service teachers’ thinking toinclude an image of discourse as an active process in which students usethe collective knowledge of a group to build understanding (i.e., dialogicdiscourse); (b) strengthen pre-service teachers’ ability to participate, asstudents of mathematics, in discourse that reflects a balance of univocaland dialogic functioning; and (c) through an analysis of their own prac-tice, reveal their habits of discourse as classroom teachers and subsequentimplications for their own professional development.

But how robust can such shifts be and what does this imply for teachereducation? The data suggest that these students developed a perceptionabout what they wanted their practice to involve based on what theyobserved to be lacking in it. However, there is a wide chasm betweenrecognizing an area for growth and being able to act on that in thesecondary classroom. The imperatives students made about their practice(e.g., “I must learn not to give out answers”) reveal issues that, poten-tially, could impede their capacity to include dialogic discourse in theirinstruction. In particular, as teachers they reported experiencing discom-fort in sharing authority with students, in allowing peer collaborations, infocusing on students’ thinking by acknowledging and incorporating theirideas, in balancing the need for dialogic discourse with the time constraintsof ‘covering the curriculum’, and in allowing time for discussion to emergewithout the teacher’s immediate intervention. Students’ recognition ofthese areas is a first step, but can they effect a transfer of their own


learning into other less ideal settings such as the secondary classroom?Creating dialogic discourse is difficult; thus, it seems that the growth ofthese students will ultimately depend on how they are supported by teachermentors and university faculty in progressing through these issues. In otherwords, their perceptions and experiences need to be nurtured by a sustainedfocus that examines univocal and dialogic functioning in other contextssuch as other university coursework, the student teaching practicum, andlong-term teacher professional development. In fact, given the challengesnew teachers face in the classroom, it would be difficult for one coursealone to prevent previously established patterns of teaching from emergingin the face of opposition or reluctance on the part of students. Thus, furtherresearch is needed to determine how robust students’ learning in a coursesuch as this is, how that learning is shaped by their experiences subsequentto it, and how students’ continued development can be supported outsideof the university classroom.

This study suggests that, perhaps more than the traditional methodsclassroom, the mathematics classroom is a unique forum in which pre-service teachers can practice, articulate, and reflect collectively on reform-minded ways of teaching. In this sense, the incorporation of a mathematicseducator into the teaching structure can benefit pre-service teachers. Byvirtue of our discipline, mathematics educators are perhaps more preparedto extract the pedagogy of the mathematics classroom as an object ofreflection for pre-service teachers. It is our business to scrutinize thenuances of the classroom for pedagogical soundness and to be aware of andmodel reform-minded ways of teaching. That is, our research as educatorsrequires us to grapple with issues that can become a rich part of the mathe-matics classroom comprised, in part or whole, of pre-service teachers.Thus, teaching arrangements based on partnerships between mathematicseducators and content faculty (see e.g., Fallon & Murray, 1991) can enrichundergraduate mathematics for pre-service teachers.

Since this was a terminal course for my students, I did not feel thepressure that these students might as classroom teachers to ‘cover thecurriculum’. The result was a sense of freedom to explore topics withoutissue of reprisal from the external assessments that many classroomteachers face. An overabundance of topics can drive the discourse bygenerating dilemmas about content coverage. It would seem, therefore,that careful choices must be made about what classroom discourse shouldresemble and, subsequently, how topics can be chosen to support that.Finally, while results indicate that students did learn geometry effectivelyby incorporating dialogic discourse, more research is needed to understandthe nature of students’ learning in contexts where discourse is essentially


univocal versus those in which dialogic functioning plays a more signi-ficant role. That is, we need to understand the kinds of cognitive schemastudents build in these various contexts and how those schema differ acrosscontexts, as well as how higher psychological functioning emerges on asocial plane as students engage in and reflect on discourse.

The litmus test for this study will be how these pre-service teacherscreate discourse as instructors in their own classrooms. Given the shiftsobserved in these students’ thinking in such a limited time frame (onesemester), it is feasible that a focus on classroom discourse which extendsover a longer period of time and which includes secondary classrooms asa teaching site for pre-service teachers would intensify their development.At this point, I can attest only to their intent to have a teaching practice thatbalances classroom discourse in the manner argued by Lotman (1988) andcan report that they have an early understanding of and experience withwhat that means. It is in this sense that this study contributes an approachfor challenging pre-service teachers’ notions of classroom discourse asparticipants in and architects of that discourse.

ACKNOWLEDGEMENTS

The research reported herein was supported in part by a grant fromthe U.S. Department of Education, Office of Educational Research andImprovement, to the National Center for Improving Student Learningand Achievement in Mathematics and Science (R305A60007-98). Theopinions expressed here do not necessarily reflect the position, policy, orendorsement of the supporting agencies.

NOTES

1 Author Note: Partial results of this study were presented at the annual meeting of theInternational Group for the Psychology of Mathematics Education, Haifa, Israel, July 1999.2 By junior or senior year, I mean those students who were in their final two years of theacademic program.3 I use the term “secondary” here to include middle and upper grades (i.e., grades 6–12).4 The notion of discourse is specified here to denote talk, or utterances, made about mathe-matics or teaching mathematics by teacher and students in the classroom.5 Numbers indicate lines in the protocol.


REFERENCES

Blanton, M. (1998). Prospective teachers’ emerging pedagogical content knowledgeduring the professional semester: A Vygotskian perspective on teacher development.Unpublished doctoral dissertation, North Carolina State University.

Blanton, M., Berenson, S. & Norwood, K. (2001). Using classroom discourse to under-stand a prospective mathematics teacher’s developing practice. Teaching and TeacherEducation: An International Journal of Research Studies, 17(2), 227–242.

Cobb, P., Yackel, E. & Wood, T. (1992). Interaction and learning in mathematics classroomsituations. Educational Studies in Mathematics, 23(1), 99–122.

Fallon, D. & Murray, F. (1991). Project 30: Year two report. Newark: School of Education,University of Delaware.

Grossman, P. (1990). The making of a teacher: Teacher knowledge and teacher education.New York: Teachers College.

Hanna, G. & Yackel, E. (to appear). Reasoning and proof. In J. Kilpatrick, W.G. Martin &D. Shifter (Eds.), A research companion to NCTM’s standards. Reston, VA.

Kaput, J. & Dubinsky, E. (Eds.) (1994). Research issues in undergraduate mathe-matics learning: Preliminary analyses and results. Washington, D.C.: MathematicalAssociation of America.

Lortie, D. (1975). Schoolteacher: A sociological study. Chicago: University of Chicago.Lotman, Y. (1988). Text within a text. Soviet Psychology, 26(3), 32–51.Minick, N. (1996). The development of Vygotsky’s thought. In H. Daniels (Ed.), An

introduction to Vygotsky (pp. 28–52). London: Routledge.National Council of Teachers of Mathematics. (2000). Principles and standards for school

mathematics. Reston, VA: Author.National Council of Teachers of Mathematics. (1991). Professional standards for teaching

mathematics. Reston, VA: Author.Peressini, D. & Knuth, E. (1998). Why are you talking when you could be listening?

The role of discourse and reflection in the professional development of a secondarymathematics teacher. Teaching and Teacher Education, 14(1), 107–125.

Schoenfeld, A. (Ed.) (1990). A source book for college mathematics teaching: A reportfrom the MAA Committee on the Teaching of Undergraduate Mathematics. Washington,D.C.: Mathematical Association of America.

Smart, J.R. (1994). Modern geometries (4th ed.). Belmont, CA: Brooks/Cole.Steen, L.A. (Ed.) (1992). Heeding the call for change: Suggestions for curricular change.

Washington, D.C.: Mathematical Association of America.Vygotsky, L. (1978). Mind in society (M. Cole, S. Scribner, V. John-Steiner & E.

Souberman, Trans.). Cambridge, MA: Harvard University (Original work published in1934).

Vygotsky, L. (1986). Thought and language (A. Kozulin, Trans.). Cambridge, MA:Massachusetts Institute of Technology (Original work published in 1934).

Wertsch, J. (1988). L.S. Vygotsky’s “new” theory of mind. The American Scholar, 57,81–89.

Wertsch, J. & Toma, C. (1995). Discourse and learning in the classroom: A sociocul-tural approach. In L. Steffe & J. Gale (Eds.), Constructivism in education (159–174).Hillsdale, NJ: Lawrence Erlbaum.


Wood, T. (1995). An emerging practice of teaching. In P. Cobb & H. Bauersfeld (Eds.),The emergence of mathematical meaning: Interaction in classroom cultures (203–227).Hillsdale, NJ: Lawrence Erlbaum.

Zeichner, K. (1996). Designing educative practicum experiences for prospective teachers.In K. Zeichner, S. Melnick & M.L. Gomez (Eds.), Currents of reform in preserviceteacher education (215–233) New York: Teachers College.

Department of MathematicsUniversity of Massachusetts Dartmouth285 Old Westport RoadNorth Dartmouth, MA 02747-2300E-mail: [email protected]

BARBARA M. KINACH

UNDERSTANDING AND LEARNING-TO-EXPLAIN BYREPRESENTING MATHEMATICS: EPISTEMOLOGICALDILEMMAS FACING TEACHER EDUCATORS IN THESECONDARY MATHEMATICS “METHODS” COURSE

ABSTRACT. Building on the work of Ball and McDiarmid, this study provides an equiv-alent at the secondary level to the work of Liping Ma at the elementary level in thatit provides a better understanding of the conceptual knowledge of school mathematicsheld by prospective secondary teachers, along with examples of the sorts of knowledgeneeded to teach for understanding within the domain of integer subtraction. Part of aneight-year longitudinal study of secondary teacher candidates’ conceptions of instructionalexplanations, this analysis of interaction in the author’s methods course and its discus-sion of epistemological obstacles and changes combines subject-matter and interactionistperspectives. The author concludes that secondary teacher candidates can deepen theirrelational knowledge of secondary mathematics within a methods course by focusing oninstructional explanations.

RESEARCH ON INSTRUCTIONAL EXPLANATIONS

Ball (1988b) characterized prospective teachers’ explanations of divisionwith fractions, division by zero, and division with algebraic equations asprocedural in nature, lacking regard for meaning and based on memoriza-tion rather than understanding. McDiarmid (1990) documented one way toshift prospective teachers’ explanations from a procedural to a more mean-ingful conceptual orientation within a pre-student teaching field experiencecourse by having prospective teachers watch young children explain theirunderstanding of why (−8) – (−2) = (−6) in a constructivist-oriented third-grade classroom. Using a conceptual framework to distinguish differentdepths of mathematical understanding, Kinach (2001, 2002) developed acognitive strategy to transform prospective teachers’ procedural explana-tions across domains of representation for any mathematics topic. Ma’s(1999) widely acknowledged comparative study of Chinese and USteachers provides important examples of the sorts of mathematical under-standing elementary teachers need in order to teach for understandingwithin the domains of subtraction with regrouping, multidigit number


154 BARBARA M. KINACH

multiplication, and division by fractions. Following Ball and McDiarmid,the study I report here provides an equivalent at the secondary level to thework of Liping Ma at the elementary level in that it offers an understandingof the conceptual knowledge of school mathematics held by prospectivesecondary teachers within the domain of integer addition and subtraction.Part of an eight-year longitudinal study of secondary teacher candidates’conceptions of instructional explanations, this analysis of interaction in theauthor’s methods course with its discussion of epistemological obstaclesand changes combines subject-matter and interactionist perspectives. Theauthor concludes that secondary teacher candidates can deepen their rela-tional understanding of secondary mathematics within a methods courseby focusing on instructional explanations.

EPISTEMOLOGICAL DILEMMAS FACING TEACHEREDUCATORS IN MOVING TOWARD TEACHING FOR

UNDERSTANDING

How would you explain addition and subtraction of integers to someonelearning it for the first time? I have asked prospective teachers this ques-tion in my secondary mathematics methods course for the past eight years.Of 96 total students, one student (1%) responded with a story metaphor,ten (10%) responded with ways to use the number line to get the answer,while the majority of students (89%) responded with “sign” rules. Thefollowing are samples of rules students wrote on the board during our classdiscussion.

Rule A“I would teach the student to make all the problems addition or subtraction by removingthe negative sign from the second integer in the equation. Then remember that positive oraddition is moving right and negative or subtraction is moving left.”

Rule B“The two mark system: change subtraction to addition and change the sign of the followingnumber.”

Rule C“If X is to the right of Y then the sign of XY is positive; if X is to the left of Y then thesign of XY is negative.”

Rule D“Minus a minus is a plus.”

These research results point to some of the dilemmas teacher educatorsface in the mathematics methods course as prospective teachers learn toteach for understanding. I begin here by providing some articulation of

REPRESENTING MATHEMATICS 155

the attitude toward learning-to-teach that influences my writing and myinterpretation of the dilemma posed by prospective teachers’ explanationsin my methods course. Perhaps my main concern is the comfort and confid-ence prospective secondary mathematics teachers feel in putting forwardthese “sign” rules as explanations to students learning the topic for thefirst time. What strikes me is the absence of a reason for the rule, and thetendency not to engage the child’s reasoning ability to derive the rules.My main concern is not whether the reason for the rule comes in the formof a representative story, diagram, or activity with some manipulative orconcrete model as students begin to explore the characteristics of instruc-tional explanations. Rather, to me what is important is to move towardteaching for understanding by shifting prospective teachers’ preference for,and comfort with, promoting learning without meaning.

Mathematically this means sorting out why the sign rules are prob-lematic. It means going deeper to distinguish the mathematical conceptsand processes underlying integer addition and subtraction. In not discrimi-nating the different meanings of the “–” symbol (subtract, negative,opposite) and the “+” symbol (add, positive), these sign rules limit whatstudents are likely to learn mathematically. I acknowledge that prospectiveteachers’ future students may gain procedurally and learn how to find theanswer to integer addition and subtraction problems using these rules. Butthese students are not likely to learn the importance of using problemsyntax to distinguish the meaning of mathematical symbols (Pimm, 1995)and the power of representational thinking for developing understanding ofmathematical problem situations (Pape & Tchoshanov, 2001; Steinbring,1998).

Pedagogically, the dilemma for the teacher educator given this situationis twofold. On the one hand, there is the question of how to conduct mosteffectively the pedagogic process of identifying, and if necessary challen-ging and re-negotiating, teacher candidates’ knowledge and beliefs aboutmathematics and the aims of mathematics teaching. On the other hand,there is the teacher education curriculum, and a question about the kindsof epistemological knowledge new teachers need to teach mathematics forunderstanding. In this study, I explore epistemological questions relatedto instructional explanations. What is the preferred explanation for addingand subtracting positive and negative numbers, if any? What counts as a“good” explanation? What are the characteristics of good representationsfor mathematical concepts, processes, and relations? As Steinbring (1998)points out, epistemological knowledge is not a codified body of knowledgeand must be learned through experience. However, as a community, mathe-matics teacher educator-researchers have yet to explore fully and delineate


the parameters of epistemological knowledge for teachers making thedevelopment of this knowledge base a potential focus for research (Ball,1988a, 1988b, 1990, 1992; Ball & McDiarmid, 1990; Steinbring, 1998;Ma, 1999).

In summary then, instructional explanations pose (at least) threeepistemological dilemmas for teacher educator-researchers as they enableprospective teachers to move toward teaching for understanding in themethods course. First there is the question of what counts as a goodexplanation in mathematics teaching; secondly there is a question about therelationship between the mathematical understanding prospective teachershold and their conceptions of instructional explanations; and lastly thereis the question of the process whereby prospective teachers and theirinstructor interactively negotiate criteria for good instructional explana-tions. Drawing upon my prior analysis of the third question (Kinach,2001, 2002), the purpose of this paper is to explore the first two questionsby examining the epistemological obstacles and mathematical under-standings that study participants interactively discover and create whiledebating the characteristics of instructional explanations for reform mathe-matics teaching. Ultimately, I recommend instructional explanations andtheir characteristics as essential epistemological knowledge for teachers.Other topics researchers include in this emerging knowledge category forteachers are mathematical definitions and their characteristics (Winicki-Landman & Leikin, 2000); representations and their relationship to mathe-matical understanding (Pape & Tchoshanov, 2001); abstraction in realisticcontexts (Hershkowitz, Baruch & Dreyfus, 2001); and the structuriza-tion of teaching episodes to identify the development of mathematicalunderstanding (Steinbring, 1998).

CONCEPTUAL FRAMEWORK: LEVELS OF DISCIPLINARYUNDERSTANDING

Multiple frameworks exist for thinking about mathematics understanding.Skemp (1976, 1978) was among the first to point to the difficulty whenhe distinguished instrumental and relational views of mathematics under-standing. His work has been a foundation for other classifications ofmathematical understanding including the procedural/conceptual dicho-tomy of Hiebert (1986) and the distinction between knowledge of andabout mathematics by Ball (1988b). Multiple frameworks identifyingcommon types of knowledge across disciplines and school subjects alsoexist. Perhaps the most influential currently are Schwab’s (1978) cate-gorization of disciplinary knowledge into substantive, syntactic, and taxo-


nomic components (Shulman, 1987) and Perkins and Simmons’ (1988)levels of understanding framework as developed in the Teaching forUnderstanding Project at the Harvard Graduate School of Education(Wiske, 1998).

For the purposes of this study, and my work with teacher candidatesin the methods course, I have reconstructed Skemp’s distinction using amodification of Perkins and Simmons’ levels of disciplinary understandingframework. Central to this modified framework are five levels of discip-linary (or school subject matter) understanding: content, concept, problemsolving, epistemic, and inquiry. Content-level understanding of mathe-matics or any discipline refers to the basic vocabulary, skills, and factsof the discipline. This sort of “knowledge” is received knowledge. Notactively acquired by students, but given to them in the form of informationor isolated skills, this sort of “knowledge” is the most superficial under-standing one can have of mathematics or any discipline. Understandingdivision at the content level would mean, for example, that having beenshown how the long division algorithm works, students would be proficientat performing it but unable to illustrate what the algorithm means through adiagram or story. Were this possible, students would be seen as performingat the problem-solving or epistemic levels of understanding. Characteristicperformances at the content level include recall of facts, correct descrip-tions of instances using the vocabulary of the domain, acquisition of askill, or replication of a thinking strategy in the exact situation or contextin which it was learned.

The next four levels of disciplinary knowledge, which are deeper thanthe content level just described, emerge when students are actively engagedin their own inquiries identifying, analyzing, and synthesizing patternsand relationships to produce knowledge. The concept level of disciplinaryunderstanding is the first of these levels and refers to the abstract ideasand idea clusters that define, bound, and guide inquiry in mathematics orany discipline (Schwab, 1978). These are the ideas without which therewould be no discipline (e.g., no sociology without the concepts of iden-tity, role, and society; no biology without the concept of cell; no algebrawithout the concept of function). Characteristic understanding perform-ances, when students are operating at the concept level of understanding,include identifying patterns and relationships and categorizing, into a class,the phenomena possessing them.

What distinguishes concepts from the above-mentioned vocabulary ofthe content level is, in part, context. One way to illustrate the differencebetween vocabulary and concepts as used in the levels of understandingframework is to look more closely at the teaching and learning of concepts


versus vocabulary. To teach vocabulary is to give students a list of wordsand their definitions and ask students to demonstrate their understandingof the definition – perhaps by identifying examples and non-examplesof the vocabulary word. To teach concepts, on the other hand, entailscreating a problem or inquiry situation where students can learn some-thing about the pattern finding and categorization that leads to the creationof a concept in the first place. If one simply memorizes, for example,the vocabulary word mitosis and its definition as found at the backof a science textbook, one is acquiring content-level understanding ofmitosis. Comparatively, if one learns about mitosis as the result of ascience experiment in which the teacher names the process of cell divisionthat the class has been looking at in the microscope as mitosis, one isdeveloping concept-level understanding of mitosis as a category of cellbehavior. This point about context was first brought out by Joseph Schwabduring the 1960s, and later reinforced by Skemp during the 1970s. AsSchwab observed, and as Skemp implies with his notion of relationalunderstanding, context is what makes pattern finding, generalization, andcomparative and critical thinking possible. Moreover without context, thecategorization of phenomena would be impossible. And since it is from thecategorization of phenomena that concepts emerge, concepts would not bepossible without the context of inquiry.

Problem-solving level understanding refers to the analytic tools andmethods scholars and students use to pose and resolve the puzzles, ques-tions, and dilemmas of mathematics or any discipline. Generic problem-solving and analytic strategies such as inference, deductive thinking,and partitioning the decision space belong to this level as do morediscipline-specific problem-solving techniques like electron spectroscopyor mathematical modeling. Also included at this level are metacognitivestrategies for monitoring one’s own thinking (Perkins, 1992). Character-istic understanding performances at this level within mathematics includethinking abilities such as finding a pattern, working backwards, solving asimilar problem, applying X in situations different (to greater and lesserdegrees) from the one in which it was learned, or creating mathematicalrepresentations to model physical or social phenomena.

Epistemic-level understanding refers to the warrants for evidence in adiscipline. Students exhibiting epistemic-level understanding are able tojustify their mathematical thinking. Also included at this level is know-ledge about knowledge itself: the sources of knowledge; how it is testedand changes over time; what counts as evidence; and the nature of goodexplanations. Familiarity with competing schools of thought within adiscipline, the logical structure of the discipline itself, and its relation


to other fields of inquiry also belong to this level. This level providesthe reasons for thinking done at the concept and problem-solving levels(Perkins & Simmons, 1988).

Finally, inquiry-level understanding refers to the generation of newknowledge or theories in a discipline (Perkins, 1992; Perkins & Simmons,1988; Donald, 1991). Problem posing and theory building belong to thislevel, as do the value judgments about what is worthy of study and howit should be studied (Donald, 1991). Used in this sense of generatingnew knowledge, inquiry goes beyond the problem-solving level of discip-linary understanding described above. Inquiry is not likely to be found inclassrooms where a conception of teaching as telling prevails.

Within the levels of understanding framework, the deepest under-standing one can have of a topic is to exhibit understanding at the concept,problem-solving, epistemic, and/or inquiry levels while the shallowestunderstanding one can have is at the content level. By comparison, withinSkemp’s framework, relational understanding is the deepest understandingone can have of any mathematics topic while instrumental understandingis the most superficial. Within this study, I equate Skemp’s instrumentalunderstanding to content-level knowledge of a topic while relational under-standing will refer to comprehension at the concept, problem-solving,and epistemic levels. Relational understanding may, but does not neces-sarily, include inquiry-level understanding. One of my aims in using thelevels of understanding to reconstruct Skemp is to refine my assessment ofhow far teacher candidates’ subject-matter understanding and instructionalexplanations go toward promoting the profound understanding of schoolmathematics that researchers like Ma (1999) and the National Council ofTeachers of Mathematics (2000) advocate beyond memorized rules androutines. My refinement of Skemp makes it possible to identify four areaswhere relational understanding or knowledge can be enhanced: concepts,problem solving, epistemology, or inquiry.

Two points should be emphasized about these levels. The first pointI wish to make is that the levels are intended not to be hierarchical butonly to distinguish different kinds of knowledge or understanding. Onetype of knowledge is not inherently “better” than another; however, ofall the levels, content-level understanding or knowledge of a topic isthe most superficial. The second point I wish to emphasize is that thelevels are not intended to be mutually exclusive. Any level may possessaspects of another. For example, working backwards could belong to eitherthe concept or problem-solving levels depending on one’s focus. As aconcept, working backwards is a noun, only one of many problem-solvingstrategies. As a problem-solving strategy, working backwards is a process,


a procedure used to solve some puzzle or problem. The overall point I wishto make here is that the levels derive their prime characterization from thekind of knowledge under study.

Pedagogically, to refer to instrumental and relational understanding inmy methods course, I used more “user friendly” terms derived from twoclass readings. From Stodolsky’s (1985) Telling Math: Source of MathAnxiety and Aversion, I borrowed the term telling math (tm) to refer tocontent-level/instrumental understanding of mathematics. From Perkins’(1992) book Smart Schools: Better Thinking and Learning for Every Child,I use the term teaching for understanding (tfu) to refer to my reconstructionof relational understanding as concept, problem-solving, and epistemicunderstanding. As used in class, tm and tfu were views of teaching withassociated conceptions of understanding, learning, and educational aimsboth generally and as they relate specifically to mathematics.

DATA AND METHODOLOGY

Setting and Study Focus

Explaining Integer Addition and Subtraction is a teaching experimentinvolving a number of trials that I have conducted in my secondary mathe-matics methods course for the past eight years. Between 1993 and 2000,I conducted two versions of this teaching experiment. Version A invitesprospective teachers to devise and thoroughly debate different explana-tions for integer addition, using a number line or algebra tile unit squares,before they attempt to explain integer subtraction. It is anticipated that therepresentations for integer addition will transfer to explanations for integersubtraction. Version B (Kinach, 2002) invites prospective teachers toexplain integer addition and subtraction in anyway they choose. Followingthis, teachers are invited to explain both integer addition and subtrac-tion using representations on the number line. Any confusions arisingfrom attempting to explain integer addition and subtraction on the numberline are subsequently resolved when learning to represent and explaininteger addition and subtraction using the algebra tiles. In this paper,I report the results of the trial of the Version A teaching experimentI conducted in my secondary mathematics methods course during the1993–94 academic year. Although the discourse reported and analysed inthis paper is drawn from one teaching trial, teacher education students’responses are representative of all trials conducted during this longitudinalstudy.1


Data

Data for the current study consist of 5 hours of video recordings from 2of my secondary mathematics methods course classes for academic year1993–94, in addition to students’ written instructional explanations forinteger addition and subtraction; students’ written reflections assessingtheir learning at the end of the course; and course instructor field notesconsisting of the plan of learning experiences for each class and post-classreflections on the actual learning path taken during class discussion.

Participants

Participants in this teaching trial include 15 undergraduates, 6 graduatestudents, and the instructor. All but one of the students were pre-serviceteachers expecting to complete their student teaching during the semesterfollowing their completion of the methods course. The majority ofstudents’ disciplinary backgrounds were in mathematics, with 19 mathe-matics majors, 1 history major with a mathematics minor, and 1 engin-eering major. The latter two students were seeking certification to teachsecondary mathematics.

Data Analysis

Video-recordings of the class session on integer subtraction were tran-scribed and analyzed using the constant comparative method of Glaserand Strauss (1967) and a modification of Sierpinska’s (1990) “acts ofunderstanding” methodology. The aim of the Sierpinska methodology isto document the process of learning a concept by making explicit theepistemological obstacles encountered, and the resulting new understand-ings gained. Central to the Sierpinska methodology are four analyticguides: IDENTIFY previously unnoticed concepts; DISCRIMINATE amongpreviously conflated concepts, ideas, or relationships; GENERALIZE fromexamples to theoretical perspectives; and SYNTHESIZE relationshipsamong seemingly divergent ideas. What I have added to Sierpinska’s “actsof understanding” methodology are the four levels of relational under-standing: concept, problem solving, epistemic and inquiry. So in additionto looking for stumbling blocks in learning at the concept level, I alsoexamined difficulties encountered in learning at the other three levels.

Following Steinbring (1998), I also structured the teaching experimentinto phases and subphases to understand the interactive development ofconceptions of “good” explanations. Table I summarizes the emergentresearch methodology I developed for analyzing the renegotiation process.


TABLE I

Emergent Research Methodology for Analyzing Epistemological Changes during theRe-negotiation Process

1. ACTIVELY ENGAGE PRE-SERVICE TEACHERS IN SOME INVESTIGA-TION TO MAKE THEIR KNOWLEDGE AND BELIEFS (E.G., IN THISCASE, ABOUT INSTRUCTIONAL EXPLANATIONS FOR [z, –]) EXPLICIT.

2. IDENTIFY THE EPISTEMOLOGICAL OBSTACLES THAT TEACHERCANDIDATES ENCOUNTER IN CRITICAL INCIDENTS OF LEARNINGTO EXPLAIN INTEGER SUBTRACTION.

That is, identify moments when prospective teachers become aware that some-thing is wrong with their explanations of integer addition and subtraction. Ask:What is the epistemological stumbling block that teacher candidates must resolveto achieve understanding? Why are you, the instructor, dissatisfied with thisexplanation? Use the levels of disciplinary understanding to guide your analysis.

3. DESCRIBE THE NEW UNDERSTANDINGS THAT TEACHER CANDID-ATES REPORT

First, report new insights into integer subtraction at the concept, problem-solving,epistemic, or inquiry levels. For each of these levels, account for Sierpinska’sfour categories of acts of understanding: (a) IDENTIFY objects that belong to theconcept in question, or of a term as having scientific status, (b) DISCRIMINATEbetween two objects, properties, ideas that were confused before, (c) GENER-ALIZE or become aware that some assumption is not essential, or that the rangeof applications of a concept can be extended, and (d) SYNTHESIZE or grasprelations between two or more properties, facts, objects, by organizing them intoa consistent whole.

4. TO DESCRIBE THE RE-NEGOTIATION PROCESS, STRUCTURIZE THETEACHING EPISODE TO DEVELOP A CHRONOLOGY OF TEACHERCANDIDATES’ EVOLVING EPISTEMOLOGICAL OBSTACLES ANDRELATED NEW INSIGHTS.

RESULTSTHE TEACHING TRIAL: LEARNING TO EXPLAIN INTEGER

SUBTRACTION

Structurization of the Teaching Trial

Structurization (Steinbring, 1998) of classroom discourse for the teachingtrial reported in this paper suggests re-negotiation took place in threeparts: Phase I, setting the standard for instructional explanations; Phase II,debating the standard; and Phase III, instantiating the standard fortfu/relational over tm/instrumental explanations. During Phase I, teachereducation students explored ways to represent and explain integer addi-tion and subtraction using the algebra tiles. After setting the standard for


tfu/relational explanations in the algebra-tile context during Phase I to bewhat to do and why, the instructor shifted the instructional context forexplaining integer addition and subtraction to the number line. DuringPhase II, teacher candidates reverted back to their tm/instrumental concep-tion of explanation as what to do. Debates ensued about the purpose(s) ofexplanation, examples of it, and arguments for and against different typesof explanation. By the end of class (Phase III), the debate was resolved infavor of the standard and tfu/relational explanations.

Next, I provide background knowledge of methods course activitiesleading up to this teaching trial before describing each phase of theteaching trial in detail.

Prior Knowledge, Course Activities, and Homework

Prior to the teaching trial reported here, teacher candidates had participatedin several activities designed to challenge their notions of mathematicsteaching and learning. These activities laid the foundation for our in-class discussion of integer subtraction in that teacher candidates hadalready (1) talked about telling-math and teaching-for-understanding astwo competing views of mathematics teaching and understanding; (2)developed an understanding of the different levels of disciplinary know-ledge through video analyses; and (3) considered representations forinteger addition on the number line and in the algebra-tile context. Withrespect to the latter, teacher candidates had been asked to prepare aresponse to the following question for class discussion:

How would you explain integer addition and subtraction to eighth graders learning it forthe first time?

Use these eight addition and subtraction problems to develop your explanations:5 + 3 = 8 5 – 3 = 2

5 + (−3) = 2 5 – (−3) = 8−5 + 3 = −2 −5 – 3 = −8−5 + (−3) = −8 −5 – (−3) = −2

During our class discussion, we established a representation for integeraddition with algebra tiles, where O represents positive one (+1) and• represents negative one (−1). We also established representations forinteger addition on the number line, where a right vector of magnitude fiverepresents positive five (+5) and a left vector of magnitude three representsnegative three (−3). The operation of addition is represented by vectoraddition.

To prepare for the class on explaining integer subtraction, I askedteacher candidates as a written homework assignment to apply the under-


standing they had developed of “good” explanations for integer addition tointeger subtraction.

The Re-negotiation Process

In this section, I describe the renegotiation process with all its phasesand sub-phases highlighting both my own and my students’ beliefs about‘good’ explanations for mathematics teaching. My structurization is asfollows:

Phase I (1–9): Setting the standard for “good” instructional explanations in thealgebra-tile context

Phase II (10–71): Debating the standard in a new instructional context, the numberline

Phase IIa (10–46): Comprehensiveness and consistency, characteristics of explana-tions

Phase IIb (47–54): Completeness, a characteristic of explanations

Phase IIc (55–71): Isomorphism, a characteristic of explanations:

Disagreements about which concepts need explication anddistinct representations

Phase III (72–80): Instantiating the standard for ‘good” explanations:

Interactively distinguishing explaining how from explaining why

In the Discussion section following, I offer a more detailed analysis ofthe different conceptions of instructional explanations that emerged in thisstudy. I conclude the paper with the instructor’s post-class reflection on there-negotiation process and why it was successful.

Phase I: Setting the Standard for “Good” Instructional Explanations inthe Algebra-tile Context

During Phase I, teacher candidates began to apply what they had learnedpreviously about explaining integer addition to integer subtraction. Aftersharing ideas in small groups, the whole class reconvened to considerBetty’s explanation for integer subtraction in 5 – (−3) = 8 which sheillustrated on the overhead projector using algebra tiles. Ultimately, thisdiscussion served the instructor’s pedagogical goal of setting a standard fortfu explanations. Retrospective analysis shows that instructor and studentsinteractively constituted four criteria for mathematical explanations (ascan be seen in Table II below) during this phase of the re-negotiationprocess.


TABLE II

Setting the Standard for “Good” Instructional Explanations

Criteria for tfu/relational explanations

1. A mathematical explanation is an argument that answers the question:

Why is this the way it is?

2. Mathematical reasons are the basis for explanations.

3. Explanations employ problem syntax and the logic of the problem context todiscriminate the different meanings of mathematical symbols.

4. Explanations involving manipulatives or other representations discriminatedifferent mathematical meanings by symbolizing each mathematical idea differ-ently. That is, objects/activity in the manipulative environment are isomorphic totheir corresponding mathematical objects/processes.

1 Betty We have positive five and you want to subtract, or take away, a negativethree.

You get a little more difficulty [than we did with 5 – 3].

You start with positive five ( O O O O O ), [Five yellow tiles]

and you want to take away a negative three which is red.

And if you look up here, you don’t see any negative,

you don’t see any red ones ( O O O O O ).

So what you can do is remember a yellow and a red make zero.

You can add zero to this (O O O O O), and add as many chips as youwant.

So since we want to take away three,

we’re going to add three reds (O O O O O • • •). [5 yellow and 3 redtiles]

But to make this still 5 positive, we want to have three yellows,

(O O O O O O• O• O•). [5 yellow, 3 red, and 3 yellow tiles]

So if we look at this, we have five here (OOOOO),

and then what does this make here? (O•O•O•)

2 Class Zero.

3 Betty And so we still only have 5.

Now if we do the equation (sic),

5 (OOOOO O•O•O•) and we want to take away –3,

you can take away the negative three, which are the red ones

(OOOOO O -•- O -•- O -•-).

And then if you look up there, how many total do we have?

4 Class Eight (OOOOOOOO).

5 Instructor So in this model, the number 5 is represented how?

6 Class 5 yellow blocks.


7 Instructor And subtraction is represented by?

8 Class Taking away.

9 Instructor That’s right, subtraction is the action of removing, as opposed to addi-tion, which was represented by the action of adding more squares(whether red or yellow). So what’s nice about this model, I think, is thataddition and subtraction as operations are distinguished from positiveand negative integers by the actions that you perform on the tiles. Thetiles symbolize the numbers/mathematical objects, while activity on thetiles symbolizes mathematical operations.

Participating in this exchange, teacher candidates learned that as theinstructor I legitimized explanations that discriminate between thedifferent meanings of mathematical symbols: specifically between opera-tions and signed numbers, subtraction and negative numbers, and additionand subtraction. Generally, I learned that my teacher-education studentseither ignored, or claimed never to have learned, the different meaningsof the “–” symbol (subtract, negative, opposite) and of the “+” symbol(addition, positive). Indeed most considered both symbols to have onlyone meaning at this point in the re-negotiation process.

In this exchange, teacher candidates also learned to discriminate thedifferent meanings of mathematical symbols through representations withmanipulatives. Consider, for example, how Betty symbolized the subtrac-tion problem 5 – (−3) with algebra tiles. Each component of the subtractionproblem (i.e., numerals, subtraction symbol, positive/negative signs) hadits own distinct representation (or symbol), that is, 5 was symbolized byfive yellow tiles, subtraction by the action of removing tiles, and −3 bythree red tiles. Activity on the tiles was then determined by interpretingthe meaning of subtract to be “take away.” How do we remove three redsfrom five yellows? Obviously, we can’t do that, so the question becomeshow to represent the number five differently. Once students realize that fivecan be represented as

1 + 1 + 1 + 1 + 1 or 1 + 1 + 1 + 1 + 1 + (1 + −1),

it is a small logical leap to realize that five represented as

1 + 1 + 1 + 1 + 1 + (1 + −1) + (1 + −1) + (1 + −1)

is useful for removing three red (−1) tiles. The norm being estab-lished here thus requires explanations involving manipulative represent-ations to discriminate among mathematical meanings by symbolizing(or representing) each mathematical idea differently, so that mathema-tical concepts/processes exist in one-to-one correspondence to manipu-


lative objects/activity. In other words, there is an isomorphic relation-ship between mathematical objects/processes and objects/activity of therepresentation, respectively.

Setting the standard: Explanation as giving mathematical reasons

Our class discussion about explanations for integer subtraction in thealgebra-tile context established the beginnings of an intuitive under-standing (Tall, 1978) of “good” explanations as arguments that answer thequestion: Why is this the way it is? Several other criteria for explanationwere also being established during this first phase of the re-negotiationprocess. One criterion was the notion that mathematical reasons, andnot tricks, are the basis of mathematical explanations. Another was thatmathematical activity derives its meaning from the meaning of mathe-matical symbols and the logic of the problem context. A final criterionfor explanations using manipulative environments as explanatory toolsrequired that this type of explanation discriminate among mathema-tical meanings by symbolizing each mathematical idea differently. Thatis, manipulative objects/activity are isomorphic to their correspondingmathematical objects/processes. After providing teacher candidates withopportunities to develop an intuitive sense for “good” explanations in thealgebra-tile context, the instructor would develop these intuitions furtherin the number-line context during the next phase of the debate. Table IIsummarizes the criteria for explanation operating implicitly at this point inthe re-negotiation process.

Phase II: Debating the Standard in a New Instructional Context, theNumber Line

Clash of presuppositions

It was our number-line debates that first alerted me to the profound philo-sophical disagreements that existed between teacher candidates and me.Up to this point, class discussions had been lively when I insisted on distin-guishing the concept of subtraction from negative and opposite, but ourdifferences about explanation were obscured in the algebra-tile context.Symbolizing integer subtraction in that discrete environment posed littleconceptual difficulty. On the number line, this was not the case. In thisenvironment, disagreements about what counts as an explanation continu-ously arose between us as teacher candidates strongly disagreed withmy assessment that their “tricks” were not explanations. We struggledrepeatedly to differentiate teaching how from explaining why it works.


Figure 1. Subtraction of integers on the number line.

Tricks or explanations?

To check prospective teachers’ understanding of mathematical explana-tions for teaching, I asked them to explain integer subtraction in a differentinstructional context. I chose the number line because it is commonly usedin textbooks. I expected explanations for integer subtraction on the numberline to parallel the criteria for explanations that we had developed in thealgebra-tile context. For example, I anticipated that teacher candidatesmight argue as follows.

Represent positive integers with right-vectors, negative integers with left-vectors, addi-tion as vector addition, and subtraction as removing vectors. To symbolize 5 – (−3), firstrepresent the minuend five, the quantity from which you want to take away the subtrahendnegative three. Draw a right-vector starting at the origin whose magnitude is five. From thisquantity five, we want to remove a left-vector whose magnitude is three. Since we can’tdo this, represent the quantity five as the sum of three vectors: 5, 3, and −3. Now to takeaway −3 from 5, remove the left-vector of magnitude three. The sum (right-vector withmagnitude eight) of the remaining two right-vectors, whose magnitudes are five and three,is the answer to the subtraction problem.

According to this line of thinking, the number-line representation of5 – (−3) would appear as in Figure 1.

Teacher candidates made no such argument. Instead they reverted backto their telling-math explanations. What teacher candidates offered werewhat I called “tricks,” not explanations. Most common was the analogy:symbol “–” means change direction and symbol “+” means same direc-tion. Another was: subtraction means go left, addition means go right,and negative sign means go in the opposite direction (i.e., positive has norepresentation here). Yet another was: assume you will always go forwardand change direction when you see the “–” symbol. Compared to mathe-matical explanations in the algebra-tile context, these logistical “tricks”did not distinguish concepts like negative and positive from subtraction andaddition on the number line. Neither did they justify activity on the numberline with mathematical reasons. Despite our algebra-tile work, my studentsreverted to a procedural explanation of integer subtraction. Further debate,clarification, and reinforcement of the criteria for instructional explana-


Figure 2. A problematic explanation for −5 + 3.

tions that we had established in the algebra-tile context was necessary ifthey were to learn to teach-for-understanding.

Phase IIa: Comprehensiveness and consistency, characteristics ofexplanations. The most common “explanation” for integer subtraction onthe number line was based on the rule: symbol “–” means change directionand symbol “+” means same direction. Feelings of (logical) paradox firstarose when Beth discovered what seemed to be a flaw in the rule as sheattempted to apply it to the eight addition and subtraction problems theclass had discussed at the beginning of this unit on explaining integeraddition and subtraction, i.e., 5 + 3 = 8, 5 + (−3) = 2, −5 + 3 = −2,−5 + (−3) = −8, 5 – 3 = 2, 5 – (−3) = 2, −5 – 3 = −8, and −5 – (−3) = −2.In the exchange below, teacher candidates who had been sensitized tothe characteristics of “good” explanations in the algebra-tile context,begin to question the adequacy of explanations on both mathematical andpedagogical grounds. In particular, Beth shares her intuitive sense withthe class that the number line seems to be a less desirable representationthan the algebra tiles for integer subtraction. In particular, she maintainsthat the change direction/same direction rule above seems to work for allof the eight addition and subtraction problems except −5 + 3.

10 Instructor What about representing and explaining integer subtraction on thenumber line? What was your reaction to trying to do that?

11 Beth The distinction between negative and subtraction is kind of difficult.The problem I ran into was if you did −5 + 3 and you are trying tokeep them [i.e. + and –] different directions, I kept seeing that a studentwould get –8 by doing that because they wouldn’t realize that addinga positive . . . I don’t know . . . [see Figure 2].

12 Instructor Why don’t you do those four subtraction problems on the board, andshow us what the difficulty seems to be.

13 Beth [After drawing a number line on the board with units numbered from−5 to +5]

Ok. . . . the first question is where do we start when we work on anumber line?

14 Class Zero.

Ok. The first problem is 5 – 3, so which way do I go?

15 Class Right.


16 Beth You go right, and how many times do I go?

17 Class Five.

18 Beth Ok. Now what do we have to do? We’re subtracting 3. So how do Isubtract three from where I am right now?

19 Class Switch directions.

20 Beth Switch directions, and go back 3 spots?

21 Class Uh-hum.

22 Beth Where does that put us?

23 Class At 2.

24 Beth [Pointing to 2 on the number line] So this is where we want to be.

So what happens with the second problem, 5 – (−3)?

25 Class Start at zero.

26 Beth So do we go to the same place we did before? Ok. So we’re at 5. Sohow do I subtract a negative three (−3)?

27 Class Switch directions twice.

28 Beth Switch directions twice. So the subtraction tells me to go back thisway, but the negative tells me to switch it back and go three. So thatputs us at positive eight. So what does that tell us about subtracting anegative number? What do you actually do?

29 Class You’re adding.

30 Beth Adding. Ok. So what if we start in the other direction. What if we startwith a negative five? How do we do that [i.e., −5 – (+3)] ?

31 Class Go to the left.

32 Beth Go left? And how many times?

33 Class Five.

34 Beth Ok. So this is our new starting point [i.e., negative 5]. And we’re gonnasubtract three, so which direction do we go? [Pause]. Do we stay thatdirection, or do we flip back?

35 Class Stay that direction.

36 Beth Stay that direction. Why?

37 Class Because you’re subtracting.

38 Beth Because you’re subtracting. But what kind of a number is it?

39 Class Positive.

40 Beth Positive number. So you stay going that way [i.e., to the left].

Ok. The next one is negative five minus negative three [i.e., −5 – (−3)].So we start at zero, and where do we end up after the first one?

41 Class Negative five.

42 Beth [Beth moves from zero one unit left to negative one on the numberline, from negative one another unit left to the point negative two onthe number line, etc., until she reaches negative five on the numberline.] Ok. Now how do we do ‘subtract a negative three?’ What doesthat mean?

43 Class Go right.


44 Beth Go right. But why do you go right?

Because it’s a subtract.

Because it’s a subtract, but it’s a subtract a negative number. Right. But. . . I don’t . . .

45 Instructor Yes . . . what do you feel about that?

46 Beth Yeah. Well. That one. It just doesn’t seem to work as well [i.e., asthe algebra tiles]. The only . . . I don’t know. . . . Especially what Iwas talking about before – the negative five plus three . . . If they gethere [i.e. at negative five], I can see where they would keep going leftbecause nothing told them to reverse direction.

Mathematically, this rule fails because it is not comprehensive. The ruledoes not produce the correct answer to −5 + 3 even though it worksfor other integer arithmetic problems in the assignment. Pedagogically,the rule is also problematic because prospective teachers anticipate howit could be conceptually confusing for students. As Beth observed: “Itjust doesn’t seem to work as well (as the algebra tiles). . . . I can seewhere they (i.e., her future students) will continue going right becausenothing tells them to change direction.” Analytically, in the explanatorynotion E is an explanation of X for A (Martin, 1970), Beth is ques-tioning both the deductive relation between E and X, as well as theadequacy of the explanatory notion (E) for her future students (A). Basedon the logical contradiction, teacher candidates ultimately decide that thechange-direction/same-direction trick does not work. These epistemolo-gical obstacles led us to IDENTIFY and GENERALIZE a new insightabout mathematical explanations, namely that like mathematical proofs,mathematical explanations must be comprehensive and work for all cases.

In the heat of class discussion, other opportunities to invalidate Beth’srule actually went unnoticed. Consistency, for example, was a problem.On several occasions, prospective teachers interpreted the subtraction “–”symbol in different ways to make sure they got the right answer. In oneinstance, when explaining −5 – 3, teacher candidates’ prior knowledge ofinteger subtraction seems to obscure the fact that to get their same direc-tion/change direction trick to work, they had to interpret the subtractionsymbol “–” to mean “go left,” when in all other problems they had assignedthe meaning “change direction” to the operation of subtraction. Clearly,this would be a problem for their future students who, not knowing theanswer to the integer subtraction problem, would not know when to usethe different interpretations of the symbol. Had we noticed, these logicaldifficulties could have led us to IDENTIFY and GENERALIZE anotherepistemic-level insight about requiring explanations to be consistent so that


symbols are interpreted and applied in the same way within any problemor related set of problems.

Phase IIb. Completeness, a characteristic of explanations. In this sub-phase of the teaching trial on explaining integer subtraction, Jasonresponds to Beth’s concern about the failure of the change direction/samedirection trick to explain −5 + 3 by proposing an alternative metaphorlinking subtraction to the number line representation: subtraction meansgo left, addition means go right, and negative sign means go in theopposite direction (i.e., positive has no representation here). The classdebates the adequacy of his “explanation” in the excerpt below. Noticethat, like Beth’s rule, Jason’s proposal does not fit the previouslyestablished isomorphic criterion for explanations.

47 Jason I made a distinction between the big negative and the superscript fornegative 3. Because usually most texts write the negative three witha superscript or parentheses around it. And, uh, I made a distinctionbetween . . . the big one, like the big negative, that’s like the minussign, and the little negative.

48 Instructor It is a minus sign.

49 Jason Right. Ok.

Because I would call those two just directional, telling them whatdirection to go.

Like if you add, the plus sign means “go to the right,”

and then if you get to the superscript, like plus negative 3, that wouldbe “the opposite of.”

So the opposite of right would be left. So when you get to −5 + 3,

you’re at negative 5, and the plus says you have to go to the right. Sogo to negative 2.

50 Instructor [To the class], what do you think about that? . . .

51 Ed Well, you may have difficulty with the first number. Before you get tothe plus sign [5 + −3 or −5 + 3], it’s rather ambiguous what you’regoing to do when get to five or superscript negative five.

52 Jason Alright. Just starting out. There’s no. This is how I’ve always done itwith the number line. The idea that the first number, unless there’s asign in front of it, that just always means start out from zero and go tothe right. Unless there’s something that tells you otherwise.

53 Instructor What would tell you otherwise?

54 Jason Like the superscript negative sign.

By participating in exchanges like this, teacher candidates learn toIDENTIFY completeness of a representation (or model) as an importantcharacteristic of explanations. By completeness, I mean that all parts ofthe mathematical entity being represented must be symbolized.


Figure 3. Debates about explanations for 5 – 3 and 5 + (–3).

Continuing the exchange below, teacher candidates next deal with theproblem of using a single symbol to represent several mathematical ideas.Specifically, Jason’s instructional explanation conflates three concepts byrepresenting positive three (+3), negative three (−3), and subtract three(x – 3) with the same symbol (see Figure 3). Again, this violates theisomorphic criterion. In the exchange below, teacher candidates object toJason’s representation of 5 – 3 on grounds that it does not discriminatebetween 5 – 3 and 5 + −3.

Phase IIc: Isomorphism, a characteristic of explanations. Disagreementsabout which concepts need explication and distinct representations

55 Jason Ok, 5 – 3. Since there’s nothing in front [of the numeral 5], I’m goingto start from zero. And the plus sign tells me I’m going to go to theright five units, So I’m over here [i.e., at positive five on the numberline]. Then the minus tells me go to the left starting from where I justended up I’m going to go back three units [Figure 3a].

56 Jodie But that’s not a negative three.

57 Jason That’s just a . . . that’s not a little . . . that’s just a little this (drawing asuperscript dash) not a negative, it’s just the direction.

58 Instructor What’s the symbol before the 3 [for the example 5 – 3]?

59 Jason This? What is that minus? That’s minus 3, I could write minus 3 likethat

[On the left vector, he writes −3. See Figure 3b.], or I could just leaveit as three. . . .

60 Jodie You couldn’t just leave it as 3 like you just said [Figure 3a].

You’d have to have the negative symbol [i.e., because the arrow pointsleft].

61 Instructor See the confusion here [Figure 3]? . . . If you look at that vector goingto the left, it looks like negative 3. But in this setting, we’re trying tofind a physical action, or some sort of representation for subtraction.So the question is: If, in this setting, we want to say this [left vector] isminus 3, that this means subtract 3, then how can you tell the differencebetween that vector and another vector of magnitude 3 going to the leftwhich means negative 3?

62 Jason What’s the difference?


63 Instructor This model is not distinguishing “subtracting 3” from “adding negative3” . . .

64 Jason I’m not sure what you mean. . . .

65 Instructor The idea is that with models, for different ideas, you have differentphysical representations.

66 Jason So do you mean, how would I represent 5 plus negative 3?

67 Instructor Yes, how would the representation for 5 + (−3) differ from 5 – 3?

68 Jason . . . because it’s the same distance and the same magnitude, I don’tsee a difference between plus negative three [and subtracting 3]. Plusnegative three is starting from some point and going (in this case it’sgoing the opposite of right which is the left 3 units) . . . that’s the samething as going 3 to the left.

69 Instructor But the point is . . . that for a child learning this for the first time whatyou’re suggesting is that that arrow which has a distance of three unitsand going to the left . . . is both positive 3 and negative 3 depending onthe setting. That’s going to be confusing conceptually. This model isnot distinguishing subtracting 3 from adding negative 3. . . . If the childis going to be able to think about the mathematics with the model, thenthere needs to be a one-to-one correspondence between each part ofthe physical model (and activity on it) and the mathematical idea.

70 Jason I don’t see why it’s a different idea. Why is it a different idea? I don’tsee that it’s a different idea?

71 Instructor Subtracting positive 3 versus adding negative 3 are two different typesof operations involved with two different types of numbers. It justso happens that when you operate on the same number with thosetwo things [i.e., adding −3 and subtracting +3], you get the sameanswer. But conceptually they are two totally different ball games.When teachers teach the rule “subtracting positive three is like addingnegative three,” what they are really doing is teaching the kids to getthe answer to the subtraction problem by adding, but they are neverreally subtracting . . . .

By participating in this exchange, teacher candidates grappled with somefundamental questions about the nature of representations and mathe-matical models. Jodie and the instructor argued for distinct represent-ations for different mathematical ideas and processes. Jason, however,did not find this argument convincing because he did not see the needto DISCRIMINATE the different mathematical ideas undergirding thesubtraction problem. This conversation was an opportunity for him toextend his mathematical understanding to the epistemic and problem-solving levels. But at this point in the debate, these arguments did notappeal to his sense of what counts as a “good reason” so the instructortried another tack, this time appealing to Jason’s imagination of what willmake sense to a child’s reason. What will make learning integer subtrac-


tion conceptually clear and mathematically accurate to someone learningit for the first time? This was the question the instructor posed for Jasonto consider. We see, however, that Jason is unable to relate to the child’sreason argument. He is still fixed on the idea that the two problems 5 – 3and 5 + −3 represent the same idea because they produce the same result.

Following this extended discussion of explaining integer subtractionon the number line, teacher candidates IDENTIFIED problem syntax as atool for DIFFERENTIATING symbolic meaning. Subsequently the classdiscussed the fact that students learning integer subtraction for the firsttime need to know how to tell when the symbol “–” means subtract ornegative. In other words, students must be able to determine when thesymbol “–” functions like an adjective from when it functions like a verb.Thus, rather than apply rules like “minus a minus is a plus,” as they wereinclined to do, prospective teachers learned to analyze the syntax of theexpression 5 – (−3) to mean the difference of two integers a – b, where “–”represents the verb subtract and the signs of a and b represent the adjectivesnegative or positive.

Phase III: Instantiating the Standard for Explanations: Toward anExpanded View of Mathematical Explanation and Teaching

Of the number-line “explanations” prospective teachers created, only oneactually “worked” for all integer subtraction problems: Assume You WillAlways Go Forward and Change Direction When You See the “–” Symbol.Yet this “explanation” did not distinguish the different mathematicalmeanings of the “–” symbol, giving rise once again to the question:What counts as a mathematical explanation? In response, the instructorattempts below to make the case that teaching-for-understanding requiresexplanations to account for why a procedure works, not just how.

72 Instructor [To recap], your point was . . . don’t distinguish [the different meaningsof the “–” symbol], and just tell them what the negative sign means,i.e., every time you see a negative sign do the opposite. . . . [Supposeyour students ask you], “Why, Mr. Smith, do we do that?”

73 Mike Well, why not? [everyone laughs]

74 Instructor [Try the Go Forward rule again for] 3 – (−3), what does that mean onthe number line?

75 Mike We’re considering all of these to be addition problems. So, alright Ican’t explain what it means, but I can figure out how to solve this usingthe number line.

76 Instructor This is a great example of teaching at the content level only. If youcan’t provide a rationale for the procedure, then until the kids have arationale, it’s still teaching a trick.


77 Mike Are we using the number line to try to explain what it means, or to tryand help them solve the problem?

78 Instructor Both . . .

79 Mike Really? Oh . . . alright . . . I thought that this is just a way to show themhow to do it.

80 Instructor Knowing how to do it, and why that works are the two things you wantto hook together.

Teaching for understanding requires that you explain why, whereastelling math requires you to explain only how. Actually, you want toget your students to explain why something is the case mathematically,and not just how they executed some procedure to get the answer to amathematics problem.

This episode won the debate for tfu explanations. In this exchange, teachercandidates learned to discriminate between instrumental and relationalexplanations, founding their explanations on mathematical reasons, nottricks. Educated themselves in the telling-math tradition, teacher candid-ates had interpreted my request for explanations of integer subtractionto be a request for information about how to find the answer. It was acritical moment of profound insight when Mike realized that explana-tions clarify both how and why. With this established, we proceeded infuture classes to implement these insights about instructional explana-tions into teacher candidates’ lesson plans as they continued their goal ofteaching-for-understanding.

Prospective teachers’ reflect on their learning

At the end of the course, prospective teachers were asked to reflect and tocomment upon the mathematical insights gained during the course. Theirreflections suggest the sorts of relational knowledge prospective secondaryteachers identified as missing from their own mathematical education. Themost commonly cited shortfalls in understanding related to the multiplemeanings of mathematical symbols (Pimm, 1995), the conceptual under-pinnings of integer subtraction, and different conceptions of mathematicsand mathematics teaching.

By the end of the re-negotiation process, students’ ideas about symbolicmeanings changed in that all twenty-one students identified changes intheir understanding of mathematics in the area of symbols and theirmultiple meanings (Pimm, 1995). One student’s final journal entry ischaracteristic of others’ end-of-course reflections:

Over the past semester, there have been a few times where I have stopped and realized thatthere are some math concepts that I thought I knew, but actually didn’t. The first schismoccurred with the colored tiles. When I was taught addition and subtraction of integers, I


was simply shown the trick of canceling negatives (subtracting a negative is like adding apositive). Sure, in doing so I was able to solve problems correctly. But, if pressed, I couldnot explain why this was true. The tiles work showed that I had learned a trick and hadforgotten that subtracting meant “take away.” I believed that a minus sign and a negativesign were the same thing.

Teacher candidates also learned to use problem syntax and contextto DIFFERENTIATE symbolic meanings. Students’ end-of-course reflec-tions provide further evidence that syntax and problem context playedlittle, if any, role in their thinking about how to explain integer subtractionprior to our class discussions.

Before taking the course, I considered the ‘–’ symbol to have only one meaning. This beliefwas probably the result of having teachers explain 6 – (−4) as “six minus minus four.”Now I realize that it can represent both a negative sign and the operation of subtraction.This realization was made possible by the algebra tile model, which assigned an adjectiveto the negative meaning (red), and an action to the operation (removing tiles). This modelhas shown me that mathematics, like English, is expressed using a complex grammaticalstructure. By teaching the true meaning of symbols like ‘–’, I can give my students a morecomplete understanding of the concepts they describe.

All in all, the use of manipulatives has provided these new insights. These insights werealso “discovered” rather than shown to me. . . . Personally, these insights have helpedcement the beneficial aspects of using manipulatives in the classroom, of being able toexplain, justify and apply concepts rather than just reciting them.

Clearly, according to teacher candidates’ own accounts, the pedagogy offostering their instructional explanations and ability to represent schoolmathematics topics with manipulatives and diagrams results in their morerelational understanding of mathematics. This student’s comment that “byteaching the true meaning of symbols like “–” I can give my studentsa more complete understanding of concepts they describe” raises thepedagogic issue of how to do this. Some might ask, given this student’scomment, is the idea then that new teachers go and offer their ownexplanations to students in classrooms, so that these students learn throughtheir teachers’ explanations? There would be some pedagogical incon-sistency were this the case. This student’s comment suggests that havingbeen convinced of the need to teach for understanding, teacher candidatesneed next to consider the pedagogic process of facilitating the develop-ment of their students’ understanding. Indeed, my objective as a teachereducator is for teacher candidates to foster the deeper relational under-standing of integer subtraction (and other topics) at the concept, problem-solving, and epistemic levels through interactive discussions with theirown students. Having participated in such interactive discussions them-selves, they are better prepared to anticipate and use the cognitive puzzlestheir future students may encounter to develop relational understanding. In


fact, several students had opportunities in the field experience linked to mymethods course to teach a lesson on integer subtraction to eighth graderswhere they attempted to facilitate such a discussion using the algebra tilesand the eight addition and subtraction problems we used in methods classto spark this discussion of integer addition and subtraction.

Finally, all but one of the teacher candidates commented on theirgrowing awareness of different views of mathematical understandingand related conceptions of mathematics learning and teaching. Students’comments below suggest several epistemological distinctions that seem tobe effective ways to conceptualise these different views for prospectiveteachers. Growing awareness of the need to foster children’s thinking alsoseems to be an important insight for teacher candidates making the shiftfrom an instrumental to a relational view of mathematical understanding.

In all of these instances, my previous understanding was challenged. I learned math in the“telling math” method, and most of the math I learned, I just accepted as true information.I learned the “tricks,” rather than a true understanding of methods and procedures.

Before this class, I could “do” all of these types of problems. I know how to factor, doVenn diagrams, subtract, etc. However, I know how to do these things because a longtime ago I was told to, and in subsequent years, I have had a lot of practice with theseproblems. Exposure to these “new” methods, or alternative methods, has really changedmy belief on what it means to teach and learn mathematics. I believe it is absolutelyessential that students are given the opportunity to understand and generate knowledgeabout mathematics, rather than be told math. These methods learned this semester validate(for me, at least) the difference between knowledge and information.

In order to instil in students a true understanding of mathematics, secondary math teachersneed to create learning environments in which learning can occur. This can be accom-plished by using discovery and inquiry type lessons where students are challenged todevelop their own thinking skills. These thinking procedures (of the student, not teacher)can then be used to build up and generate generalizations about mathematics.

DISCUSSION AND CONCLUSION

This study indicates that creating instructional explanations is not a simpletask for teacher candidates. Neither is re-negotiating prospective teachers’epistemologies a prescriptive task for teacher educators. Below I discussthe two main results of this study relating to prospective teachers’ concep-tions of mathematical explanations for teaching and the re-negotiationprocess itself.


Conceptions of Explanations

Jane Roland Martin (1970), one of the few philosophers of education tocomment upon the relationship between understanding and explanationas they relate to teaching, provides an analytic perspective on the char-acteristics of explanations that is useful to orient this discussion. Martinidentifies five distinct notions of explanation for teaching: (1) A explainsE, (2) A explains E to B, (3) X is an explanation of E given by A, (4) Xis an explanation of E for A, and (5) X is an explanation of E. Accordingto Martin, explanations for teaching take form (4): X is an explanationof E for A. In such explanations, X must account for audience under-standing, as well as maintain a deductive relation between X and E. Inthe case of teaching, this suggests to me that the teacher’s explanationmust be adjusted and made understandable for students at different agesand grade levels, while still being deducible from the problem context. Inother words, explanations for teaching must account for learner epistemo-logy (i.e., how the student makes sense of the subject matter) and subjectepistemology (i.e., types of understanding; sources of knowledge and howit changes over time; how knowledge is evaluated and tested and logicallytied to the context of inquiry). One way to interpret the difficulty in thisstudy therefore is to observe that teacher candidates’ explanations, i.e.,their sign rules and their explanations with representations on the numberline, did not account for either subject or learner epistemology within theteaching-for-understanding worldview.

What this created for the instructor was an epistemological challengemuch like the one Brown (1977) describes:

One of the most striking characteristics of debates between thinkers who are working fromdifferent presuppositional bases is that along with disagreements on what problems needto be solved and what constitute adequate solutions of these problems, they also disagreeon which concepts do and which do not require explication and find it unnecessary (if notimpossible) to offer arguments for their choice (p. 57).

Working from different presuppositional bases, my students and I foundourselves to be in disagreement about what counted as an explanation, andeven more fundamentally, about what counted as knowing, and thereforelearning, mathematics. We disagreed about what it meant to understandand learn integer subtraction. We specifically disagreed about the conceptsunderlying integer subtraction and the sorts of thinking practices chil-dren engage in while learning to perform subtraction of positive andnegative numbers. As a result of these disagreements, our ideas aboutdesirable learning outcomes for students diverged considerably. I wantedmy students’ instructional explanations to develop understanding at theconcept, problem-solving, and epistemic levels of mathematical under-


standing whereas my students’ inclination was to stop at content-levelunderstanding. According to my epistemology, mathematical explanationsfor teaching ought to answer the question Why is this the way it is? whileaccording to my students’ epistemology, explanations for teaching oughtto provide directions to get the answer.

Changes in Epistemological Obstacles

Table III summarizes the epistemological obstacles and changes in under-standing prospective teachers and the instructor negotiated in shifting fromtm/instrumental explanations to tfu/relational explanations. In making theshift, teacher candidates’ relational understanding was enhanced at theconcept, problem-solving, epistemic, and inquiry levels. Specifically, atthe concept level, teacher candidates learned to distinguish the differentmathematical ideas underlying integer subtraction (subtraction, negative,opposite, addition, positive) by symbolizing these ideas differently acrossrepresentations (i.e., number line and algebra tiles). At the problem-solving level, teacher candidates learned to distinguish skill activity fromproblem solving. Skill activity refers to applying memorized (or given)routines while problem-solving activity entails deducing mathematicalactivity from the logic and syntax of the problem context and the meaningof the mathematical symbols. At the epistemic level, prospective teachersshifted from offering no justification for their sign rules to creating justi-ficatory arguments using representations on the number line and withalgebra tiles to explain integer subtraction. At the inquiry level, prospectiveteachers’ instrumental explanations offered little opportunity at the outsetfor problem posing, generative questions, and new inquiries outside thebox of the “sign” rules which, basically, were occasions for asking ques-tions of clarification to apply the rule correctly. As the shift in teachers’epistemologies took root, however, teacher candidates themselves engagedin a guided form of inquiry-level understanding as they debated theadequacy of their logistical tricks.

Pedagogically, these logistical tricks were extremely valuable to meas the instructor providing new insights into prospective teachers’ gapsin mathematical understanding. I learned particularly what aspects ofepistemological knowledge to emphasize for these pre-service secondarymathematics teachers (Steinbring, 1998). Consistency, completenessand comprehensiveness were characteristics of instructional explana-tions that teacher-education students had not previously considered(Table III). These were not qualities I anticipated having to emphasizewith my students given their mathematical background. The isomorphicrelationship between mathematical objects/activity and representation


TABLE III

Instrumental vs. Relational Explanations for Teaching: Results of Integrated Sierp-inska-Perkins ‘Acts of Understanding’ Methodology

Instrumental Explanations Relational UnderstandingEpistemological Obstacles Acts of UnderstandingTelling Math Teaching-for-Understanding

concept CONFLATE CONCEPTS DISCRIMINATE CONCEPTSIDENTIFY subtract addition subtractDISCRIMINATE negative positive negativeGENERALIZE opposite oppositeSYNTHESIZE addition

positive

problem solving SKILL ACTIVITY PROBLEM-SOLVING ACTIVITYIDENTIFY based on deduced fromDISCRIMINATE applying logic & suntaxGENERALIZE memorized rules of problem contextSYNTHESIZE

epistemic NO EVIDENCE NEEDED MATHEMATICAL REASONSIDENTIFY NEEDEDDISCRIMINATE just apply rules support claims & proceduresGENERALIZE with mathematical principlesSYNTHESIZE

Instrumental Explanations Relational Explanationsdescribe how offer reasons why

When instrumental explanations When relational explanationsemploy representations as employ representations asexplanatory tools the purpose of explanatory tools, objects objectsthe logistical tool is to get the and activity of the representationanswer. are isomorphic to the mathematical

object-activity being symbolized.

To be useful, the logistical tricks Relational explanations areof instrumental explanations comprehensive, consistent &must always produce the correct complete.

answer.

inquiry CLARIFICATION QUESTIONS GENERATIVE QUESTIONSIDENTIFY REPLICATE GIVEN CREATE NEW KNOWLEDGEDISCRIMINATE KNOWLEDGEGENERALIZESYNTHESIZE


objects/activity was another criterion that emerged during this teachingepisode. Additionally our class debates deepened teacher candidates’epistemic- and inquiry-level understanding by providing firsthand experi-ence of what it means to participate in a community of inquiry. Ulti-mately, prospective teachers’ logistical tricks were valuable for deepeningtheir subject-matter understanding because they led to deep discussionsabout the limits of logistical tricks as explanations. The power of alogistic, especially in the case of arithmetic computation, lies in itsability always to produce correct results. Teacher candidates learned thatlogistics are not a panacea as they are easily misapplied and not soeasily created. Out of these discussions also emerged insights for teachereducator-researchers into the kinds of epistemological knowledge pre-service secondary teachers need about instructional explanations if theyare to teach for (relational) understanding.

The Re-negotiation Process: Argumentation and Shared EducationalAims

Retrospective analysis of classroom video-recordings shows the overallflow of argumentation in the re-negotiation process to have proceeded asfollows.

Teacher candidates (TCs) argue X1 is explanation for (Z, –);

X2 is explanation for (Z, –);

X3 is explanation for (Z, –).

Instructor argues Y better than Xi because

1. Y promotes understanding and critical thinking more than Xi;

2. Xi only promotes memorization, and memorizing often createsmath anxiety because the learner who is dependent on memoryis not in control of his or her own thinking.

In reflecting on the argument, I asked myself: What did you have to appealto? How did you come to convince TCs to change their telling-math views?It seems there are at least four educational aims that we shared, andwhen we did not share these, conceptual change was more difficult, if notimpossible, to effect.

1. TCs sense that understanding is better than memorizing. This sharedvalue was created through our common reading of the article TellingMath: Origins of Math Aversion and Anxiety by Stodolosky (1985).

2. Understanding is better because students can reconstruct their ownthinking. By comparison, memorizing does not allow students to get


inside the object of study to make sense of it with their own built-ineveryday reasoning abilities.

3. Understanding is better because it’s more meaningful. Most TCsagreed that memorizing was a singularly boring and meaninglessactivity, and would not be appropriate as the sole learning outcomeof mathematics education. It took a while however to undo TCs’longstanding habit of accepting rules without reasons in mathematicsclasses.

4. Smart Schools Philosophy. It was important to have set up Perkins’Smart Schools philosophy at the beginning of the course. This gaveTCs and me ample opportunity to clarify the meaning of the levelsof understanding framework and the differences between telling mathand teaching for understanding. Having the leverage of a frameworkwas useful for going beyond personal opinions and maintaining a toneof reasoned debate.

Finally, one student, Jason, never seemed to be convinced of the valueand importance of tfu for students and clung to “getting the right answer”as the prime criterion for explanation. The fact that this student was amath minor and history major with a perfect 4.0 cumulative grade pointaverage raises the question of the degree to which students’ records ofsuccess make them resistant to change. Given his success in the telling-math system, I was unable to convince him that students’ thinking wasalso an important aspect of learning. Therefore, I realize in retrospect thatI was counting on the tension between these two worldviews (tm/tfu) togenerate TCs’ commitment to tfu values.

IMPLICATIONS

A decade ago, Ball and McDiarmid (1990) urged teacher educator-researchers to turn their attention to improving the subject-matter prepara-tion of teachers:

[c]ontinued documentation of the inadequacy of subject-matter preparation will not helpto improve the problems we face in teacher education and teaching, for the contributingviews of knowledge, teaching, and learning are deeply rooted in educational institutionsand in the wider culture. Altering these patterns will not be easy. We should turn our futureefforts and attention to the difficult task of improving teachers’ subject-matter preparation(p. 446).

And so, I close with the question of what has been accomplished here?How does this study contribute to the improvement of teachers’ subject-matter preparation in mathematics? The findings of this study call into


question a common assumption that “graduating teacher candidates wholack adequate subject-matter preparation . . . will develop deeper know-ledge as a result of having to explain it to others” (emphasis added, Ball &McDiarmid, 1990). On the contrary, this study shows that if prospectiveteachers’ telling-math/instrumental epistemologies go unchecked, theirinstructional explanations are not likely to lead to the sort of rela-tional understanding at the concept, problem-solving, and epistemic levelsadvocated by ‘reform’ mathematics.

The findings of this study also call into question the common assump-tion that secondary teachers “know” their mathematics. Building on thework of Ma (1999) at the elementary level, this study suggests that, whilesecondary teachers may know their mathematics well enough to “do”it and to debate the adequacy of different representations and instruc-tional explanations given a framework for thinking about deeper levels ofmathematical understanding, secondary teachers nevertheless do requireenhancement in their relational understanding of secondary school mathe-matics (Ma, 1999; Watanabe & Kinach, 1997; Ball, 1988a, 1988b, 1990;McDiarmid, 1990).

In addition to providing specific examples of the kinds of knowledgeteachers need to teach secondary mathematics for understanding, thisresearch recommends several ways to enhance teachers’ relational know-ledge of mathematics within the methods course. These include specificlearning experiences related to integer arithmetic, criteria for instructionalexplanations, and a levels of subject-matter understanding framework.Ultimately, this research suggests that re-conceptualizing the content,purposes, and organization of the methods course may be necessary toenable prospective teachers to teach for understanding (Thompson, 1984,1992; Kinach, 2002). Rather than its traditional emphasis on pedagogy, thenew tfu “methods” course might be conceived better as an integration ofcontent and pedagogy.

NOTE

1 Building on Kinach (1996), the analysis reported in the present study as I havesaid relates to Instructional Sequence A. Elsewhere I report the results of InstructionalSequence B (Kinach 2001, 2002). Although my analysis in this paper builds on this priorwork, there are two major differences that make this paper significant. First, whereasKinach (2001, 2002) proposes a process for re-negotiating prospective teachers’ instruc-tional explanations from an instrumental to a relational worldview, the present study reportson the mathematical criteria for instructional explanations that were interactively nego-tiated during that process. Second, whereas Kinach (2001, 2002) proposes a pedagogicalmethodology, based on subject matter and interactionist perspectives, for transforming


prospective teachers’ instrumental orientations to teaching, the present study proposes anemergent research methodology, based on subject matter and interactionist perspectives,for analyzing epistemological obstacles and changes during learning-to-teach episodeswithin teacher education classrooms.

REFERENCES

Ball, D.L. (1988a). The subject-matter preparation of prospective mathematics teachers:Challenging the myths. Issue paper 88-3. East Lansing, MI: National Center forResearch on Teacher Education.

Ball, D.L. (1988b). Unlearning to teach mathematics. Issue paper 88-1. East Lansing, MI:National Center for Research on Teacher Education.

Ball, D.L. (1990). Breaking with experience in learning to teach mathematics: The role ofa pre-service methods course. For the Learning of Mathematics, 10(2), 10–16.

Ball, D.L. (1992). The mathematical understandings that prospective teachers bring toteacher education. In J. Brophy (Ed.), Advances in research on teaching (Volume 2,1–48). Greenwich, CT: JAI.

Ball, D.L. & McDiarmid, G.W. (1990). The subject-matter preparation of teachers. In W.Robert Houston (Ed.), Handbook of research on teacher education: A project of theAssociation of Teacher Educators. New York: Macmillan.

Brown, H.I. (1977). Perception, theory and commitment: The new philosophy of science.Chicago: University of Chicago Press.

Dewey, J. (1916/1944). Democracy and education. New York: The Free Press.Donald, J.G. (1991). Knowledge and the university curriculum. In C.F. Conrad & J.G.

Haworth (Eds.), Curriculum in transition: Perspectives on the undergraduate experience(295–307). Needham Heights, MA: Ginn Publishing.

Glaser, B. & Strauss, A. (1967). The discovery of grounded theory. New York: Aldine.Hershkowitz, R., Baruch, B.S. & Dreyfus, T. (2001). Abstraction in context: Epistemic

actions. Journal for Research in Mathematics Education, 32(2), 195–222.Hiebert, J. (1986). Conceptual and procedural knowledge: The case of mathematics.

Hillsdale, NJ: Lawrence Erlbaum.Kinach, B.M. (1996). Logical trick or mathematical explanation? Re-negotiating the

epistemological stumbling blocks of pre-service teachers in the secondary mathematicsmethods course. Proceedings of the eighteenth Annual Meeting of the North AmericanChapter of the International Group for the Psychology of Mathematics Education,Volume 2 (414–420). Columbus, OH: Ohio State University.

Kinach, B.M. (2001). Assessing, challenging, and developing prospective teachers’pedagogical content knowledge and beliefs: A role for instructional explanations. In T.Ariav, A. Keinan & R. Zuzovsky (Eds.), The ongoing development of teacher education:Exchange of ideas. Tel Aviv, Israel: The Mofet Institute.

Kinach, B.M. (2002). A cognitive strategy for developing prospective teachers’ pedago-gical content knowledge in the secondary mathematics methods course: Toward a modelof effective practice. Teaching and Teacher Education, 18(1), 51–71.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understandingof fundamental mathematics in China and the United States. Mahwah, NJ: LawrenceErlbaum Associates.

Martin, J.R. (1970). Explaining, understanding and teaching. New York: McGraw Hill.McDiarmid, G.W. (1990). Challenging prospective teachers’ beliefs during an early field

experience: A quixotic undertaking? Journal of Teacher Education, 41(3), 12–20.


National Council of Teachers of Mathematics. (1989). Curriculum and evaluation stand-ards for mathematics. Reston, VA: author.

National Council of Teachers of Mathematics. (2000). Principles and standards for schoolmathematics. Reston, VA: author.

Pape, S.J. & Tchoshanov, M.A. (2001). The role of representation(s) in developingmathematical understanding. Theory into Practice, 40(2), 118–127.

Perkins, D.N. (1992). Smart schools: Better thinking and learning for every child. NewYork: Free Press.

Perkins, D.N. & Simmons, R. (1988). Patterns of misunderstanding: An integrative modelfor science, math, and programming. Review of Educational Research, 58(3), 303–326.

Pimm, D. (1995). Symbols and meanings in school mathematics. New York: University ofOxford Press.

Schwab, J.J. (1978). Education and the structure of the disciplines. In I. Westbury & N.J.Wilkof (Eds.), Science, curriculum and liberal education: Selected essays (229–272).Chicago: University of Chicago Press.

Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. HarvardEducational Review, 57(1), 1–22.

Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learningof Mathematics, 10(3), 24–41.

Skemp, R.R. (1976). Relational understanding and instrumental understanding. Mathe-matics Teaching, 77, 20–26.

Skemp, R.R. (1978). Relational understanding and instrumental understanding. ArithmeticTeacher, 26(3), 9–15.

Stodolsky, S.S. (1985). Telling math: Origins of math aversion and anxiety. EducationalPsychologist, 20(3), 125–133.

Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.Journal of Mathematics Teacher Education, 1, 157–189.

Tall, D. (1978). The dynamics of understanding mathematics. Mathematics Teaching, 84,50–52.

Thompson, A.G. (1984). The relationship of teachers’ conceptions of mathematicsteaching to instructional practice. Educational Studies in Mathematics, 15, 105–127.

Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. InDouglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning(127–146). New York: Macmillan.

Watanabe, T. & Kinach, B. (1997). Creating a community of inquiry in an undergraduatemathematics course for prospective middle grade teachers: Voices from MCTP. Paperpresented at the American Educational Research Association, Chicago.

Winicki-Landman, G. & Leikin, R. (2000). On equivalent and non-equivalent definitions:Part 1. For the Learning of Mathematics, 20(1), 17–21.

Wiske, M.S. (Ed.) (1998). Teaching for understanding: Linking research with practice. SanFrancisco, CA: Jossey-Bass.

Department of EducationUniversity of MarylandBaltimore County1000 Hilltop CircleBaltimore, MD 21250USA

BOOK REVIEW

Lampert, Magdalene (2001). Teaching problems and the problems ofteaching. New Haven: Yale University Press. ISBN: 0-300-08973-2

Magdalene Lampert’s Teaching problems and the problems of teaching,like teaching, its subject matter, is complex and deeply rooted in aparticular context. The context is comparatively easy to describe. On adaily basis during the 1989–1990 school year, just after lunch and recess,Lampert, then a faculty member at Michigan State University, taught thestandard, district-mandated, mathematics curriculum to a fifth grade class(children ten and eleven years old) at Spartan Village Elementary Schoolin East Lansing, Michigan.

But, the content of the book is harder to describe; it operates on manylevels. On the most basic level, Lampert helps readers come to know thisyear of teaching. Readers learn about Richard, Sandra, Awad, Tyrone,Ellie, Yasu, and the other children in the class, as the children learn to addfractions, to solve distance-rate-time problems, and to be people who studymathematics in school. Readers also learn about Lampert as a teacher – hergoals and her actions, as well as her reasons for them.

But, there is more to the book than a description of one year’s teaching.The book is clearly the fruit of many years’ work. It is a book that examinesmathematics and teaching, without limiting itself narrowly to mathematicseducation. It provides both an experienced practitioner’s insights into aparticular kind of elementary school mathematics teaching,1 the “teachingproblems” part of the title, and a well-respected teacher educator andresearcher’s attempts to represent teaching as a practice, the “problemsof teaching” aspect of the title. The book will be read by many outside ofmathematics education circles interested in understanding teaching moregenerally.

Chapters 1, 2, 3, and 14 of the book outline the research programfrom which this volume springs, the data collection procedures used tocollect information about a year of fifth grade mathematics teaching,key vision-based metaphors used to capture to complexity of teaching,and an instructional triangle (student-teacher-content) based description


188 BOOK REVIEW

of teaching. Chapters 4 through 13 provide opportunities to explore anddeepen one’s understandings of aspects the work of teaching. Thesechapters are grounded in particular acts of Lampert’s teaching of fifthgrade mathematics through problems during one school year. For example,Chapters 5 through 7 examine respectively, the work of teaching whilepreparing lessons, while students work independently, and while theclass is convened as a whole group. These domains of teaching areexamined through examples taken from Lampert’s work with her studentsin the conceptual field of multiplicative structures (she uses this termfollowing Vergnaud, 1994), as her students work on problems posed inthe context of ascertaining equivalent groupings. When analyzing theseexamples from her teaching, Lampert moves seamlessly from discussionsof teaching these particular students, teaching mathematics with problems,and teaching more generally.

This book is an ambitious undertaking that reminds readers at everyturn of the limitations of educators’ capacities to portray and theorize thework of teachers. It is a book that insists on scholarship that takes teachingpractice seriously. By doing so, it implicitly offers criticism of muchresearch on teaching and suggests alternative directions. Rarely does onemeet portrayals of teaching, like this one, that insist on taking the year, inmost cases the full duration of the teacher/class bond,2 as the fundamentalunit of time for the analysis of teaching. Rarely does one find descrip-tions of teaching that insist on the portrayal of students as individuals, asconstituents of constantly shifting groups, and as constituents of the class,that fictive entity that teachers often use to portray their teaching. Rarely isthe connectedness of skills and concepts in a curriculum assumed. Rarelydoes one read detailed descriptions of a teacher’s thinking that indicatehow that teacher has different curricular intentions for different studentsand groups of students at one and the same time. Rarely are teachers givenreason (in the sense of Duckworth, 1987), rarely is their point of viewvalued so highly. And, rarely does a teacher or researcher have records ofpractice that would support all of the above. This is truly an impressivepiece of scholarship.

The book offers many potential readings. As a mathematics teachereducator, one might be interested in it as a text for preservice teachercandidates. One might also be interested in it as a teacher educatoror researcher of teaching, to broaden and deepen one’s own ways ofportraying teaching and conceptualizing the work of teaching. As a result,the book asks much of its readers. Readers must continually ask them-selves: When is Lampert discussing particular practices of a certain kindof teaching? When is she discussing her teaching of this particular class?

BOOK REVIEW 189

And, when is she discussing teaching in general? The book also challengesreaders to consider more elaborate ways of portraying teaching than thosethat are often taken for granted. This review will briefly take up each ofthese challenges.

GENERALITIES THROUGH A PARTICULAR

Right from the start, Lampert emphasizes that she aims at generalities byexamining a particular:

By taking a close look at the actions of a single teacher, teaching a single subject to awhole class over an entire academic year, I attempt to identify the problems that must beaddressed in the work of teaching (p. 1).

The situation here is more complicated than Euclidean geometry wherea particular triangle is used as a generic example with which to reasonabout all triangles, without devoting attention to that triangle’s individualcharacteristics. In Lampert’s use, there is more than one level of generality.

The single teacher I study here is myself. Like all teachers, I take a particular approachto teaching [what she calls teaching with problems], and this book is also a study of thatapproach (p. 1).3

Lampert’s focus on these two levels of generality is especially fascinatingin chapters 4 through 13; chapters whose text includes substantial reportsof classroom activity. The chapter titles themselves work on these levels ofgenerality. There are titles that suggest aspects of the activity of teaching,what Lampert calls problem domains of teaching. Presumably, these activ-ities are common to most teaching and are not limited to particular kinds ofteaching: teaching while preparing a lesson, teaching while students workindependently, teaching while leading a whole-class discussion, teachingthe whole class, and teaching closure (at the end of an academic year, forexample). And, there are other titles that speak of goals. They can be readeither as the province of all teaching – though not all teachers might beequally articulate about their efforts – or as related to the particular kindof teaching a teacher does. These titles include: teaching to establish aclassroom culture, teaching to deliberately connect content across lessons,4

teaching to cover the curriculum, and teaching students to be people whostudy in school. They are intended to be general: “work must be done ineach of these domains by teachers of different subjects in different grades,in city schools, in rural schools, and in the suburbs” (p. 45). But, at thesame time, there is much in these chapters that is particular to Lampert’sapproach to teaching.

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For example, when exploring teaching to establish a classroom culture,Lampert asserts that “The establishment of a classroom culture thatcan support studying is a fundamental element of teaching practice. Allteachers do it, to a more or less deliberate extent” (p. 53). She then goes onto describe what she did with her students on the first day of class. Twelvepages later, as she prepares to zoom out to encompass the first week ofschool, she notes that “The problem I faced in doing the work I set formyself was something like establishing and maintaining a counterculturein the midst of a conventional school environment” (p. 65). She outlines indetail how the norms for what constitute studying mathematics in her classare different from what her students have learned so far in school. Thegeneral problem of establishing classroom culture has taken on a particularcomplexion, as a result of the goals and commitments of a teacher teachingin a particular way.

Similarly, the subtitles in each of these chapters and the text in thesesections also move back and forth between a particular practice and theproblems of teaching. For example, in the chapter on teaching to estab-lish a classroom culture, the majority of the text is in first person; inan “I” voice, Lampert talks about her own teaching. At the same time,the subtitles present more impersonal descriptions of her work with herstudents early in the school year: arranging the physical environment tosupport a classroom culture; choosing mathematical content to comple-ment culture-building practices; zooming out to the first week of school:introducing conditions, conjectures, and revision; and establishing normsfor written communication. These subtitles provide scaffolding for readersinterested in reading about Lampert’s work with her students as a case (asa prototype, precedent, or parable as outlined in Shulman, 1986). For thosereaders unfamiliar with teaching with problems, such titles can exemplifyhow instruction can be designed to meet the goal of establishing a partic-ular kind of classroom culture. For other readers intent on learning to createsuch a classroom culture, these titles – along with the text of the section“Problems in teaching in the domain of establishing classroom culture”– may provide heuristics for thinking about this problem of teaching, aswell as particular images of one teacher’s work in this domain. For thoseinterested in teaching more broadly, the particulars of Lampert’s classroomare one instantiation of the general problem of establishing a classroomculture that is consistent with a teacher’s desired way of working withstudents.

In contrast, in the section “What kind of work is this?” Lampert speaksto readers in a different voice. The teacher is now “she,” she who bendssocial and physical resources to her purposes. Yet, even in this section,

BOOK REVIEW 191

the particular approach that Lampert uses is explicitly named. In closingthe chapter, Lampert suggests that this chapter on establishing a classroomculture has examined:

. . . how teaching can work to establish these three conditions [that something that isappropriate to study is available to everyone, that teacher and students agree about whatconstitutes “studying,” and that the tools for studying are provided] in a way that iscongruent with students doing mathematics problems in school (p. 93).

This sort of movement is present throughout the volume. It challengesreaders as they interpret this rich and complex text. There are opportunitiesfor many readings, but at the same time an imperative to read rigorouslyand carefully.

PORTRAYING PRACTICE AND MODELING TEACHING

In addition to Lampert’s use of her own practice as a way to study teachingmore generally (e.g., 1989, 1990), and her use of her practice for thepurpose of teacher education (e.g., Lampert & Ball, 1998), Lampert isknown for her efforts to develop educators’ capacity to portray the work ofteaching. For example, her emphasis on representing problems in teachingas dilemmas that must be managed, rather than problems to be solved(Lampert, 1985), has influenced U.S. researchers (e.g., Ball, 1993), as wellas scholars in other countries (e.g., Adler, 2001). Yet, this text is her firstopportunity to portray her favored unit for the analysis of teaching, thewhole year.

With the chance to be more expansive in analyzing her practice comes achallenge, that of exploring and communicating her insights. Lampert usesthree mechanisms that for this task:

• Vision and terrain-based metaphors,• an explicit teacher-student-content model of the practice of teaching

that is introduced in chapter 3 and elaborated in chapter 14, and• an implicit organization of the practice of teaching into the problem

domains that organize chapters 4 through 13.

Examination of each of these mechanisms in turn serves to highlight theambition of Lampert’s goal of developing an analytic language for talkabout teaching.

In chapter two, Lampert’s first take on the ambitious task of repre-senting a full year’s teaching starts with her teaching of a lesson on rateon Monday, November 20th. In her words, Lampert is after “a representa-tion of the multiple levels of teaching action as they occur in different

192 BOOK REVIEW

social relationships over time to accomplish multiple goals simultan-eously” (p. 28). As she begins with this instance of teaching, in order tohelp readers integrate her text, Lampert begins to utilize a number of meta-phors involving vision and terrain. Explicitly modeling herself on the filmand book “Powers of Ten,” (Morrison, Morrison, & the Office of Charlesand Ray Eames, 1982), she proposes

. . . using the idea of adjusting a lens to multiple focal lengths while moving it around tolook at the terrain of a practice . . . [to] retain the idea that larger and smaller pieces of workare happening simultaneously and are intricately related, even though we can only “see” atone resolution at a time” (p. 44).

She begins with the “big picture” and introduces readers to the contextof the teaching she will discuss in the book. She then “zoom[s] into” alesson on rate. After introducing the task, a problem about cars moving at aconstant speed, and a kind of diagram that the students make to coordinateelapsed time and distance traveled, she “zoom[s] further in” to discussthree students, Richard, Catherine, and Awad. Staying at the same resolu-tion, she then “change[s] focus to Anthony and Tyrone, and does so againfor Ellie, Sam, and Yasu. This movement, the zooming in and the changeof focus, is done against the background of a multi-dimensional terrain.This terrain is made of:

• conceptual fields, like multiplicative structures, that make up thecontent to be taught;

• units of study, like the unit on time-speed-distance problems, anotherway to describe content that is organized by the problem contexts thathold units together;

• the calendar of the school year;• the individuals and groups that make up the class; and• teaching with its problem domains.

As a result of Lampert’s choice of metaphors, through her descriptions,aspects of teaching that are invisible to many observers become metaphor-ically visible. For example, for preservice teacher candidates used to seeingthe classroom from the perspective of a student, Lampert’s descriptionsmake available a teacher’s point of view on classroom interaction. Theynow might see aspects of practice that they may not have seen before.They might see relationships between Lampert’s actions at the beginningof the year to create a classroom culture of a particular kind and students’capacity at the end of the year to conjecture and revise their thinking.Similarly, they might be able to see relationships between teaching Sandrato be a person who studies in school and reasons to continue to work withthe whole class, rather than splitting the class into groups based on theirperformance on a quiz in April involving the addition of fractions.

BOOK REVIEW 193

How does choice of vision-based metaphors shape what can be learnedfrom Lampert’s descriptions? What might have been the influence of othersensory metaphors (e.g., hearing as in Davis, 1996)? Taking Lampert’sgoal seriously, these questions are important to consider.

Similarly, how might the choice of terrain as a way to describe teachingpractice, and time as an organizing feature of trips through this terrain,compare with, for example, the notion of a narrative unfolding with all ofits tools to support change of “focus” and time? What sort of terrain is itthat is made of conceptual fields, units of study, time, groupings of people,and problem domains in teaching? Is this terrain a surface or could it be aspace of a different kind as Lampert suggests (pp. 446–447)? What does itmean to travel on or in such a terrain? And, perhaps most importantly, whatdoes the choice of these metaphors say about the creatures, teachers, whotravel in such a terrain?5 What capacities and senses do teachers have ormight they want to have? And, how might those be developed? All of theseare questions stimulated by Lampert’s choices and her goal of creating alanguage for discussing teaching.

In her search for a language of analysis for discussing teaching, inaddition to these organizing metaphors, Lampert also offers readers whatshe calls a model of teaching,6 one in which teaching is represented as“working in relationships” (p. 30). Like others before her (e.g., Brousseau,1984; Hawkins, 1974; McDonald, 1992; Sizer, 1984), Lampert begins withthree nodes: teacher, student, and content, an instructional triangle. But,right from the beginning, she makes this model more complex. In additionto the edges of this triangle – relationships between teacher and student,teacher and content, and student and content – her instructional trianglehas an interior; there is a relationship between the teacher and the rela-tionship between student and content. The resulting three pronged modelof a teacher’s practice emphasizes the teacher’s “simultaneous relation-ships with students, with content, and with the student-content connection”(p. 423).7

From Lampert’s perspective, from inside the role of the teacher,however, this model must be further elaborated. With each elaborationcomes greater appreciation of the challenges of teaching, and also greaterunderstanding of resources that teachers may employ to reach their goals.Lampert is concerned that this three-pronged model of a teacher’s practicedoes not portray time. For example, time makes the content node morecomplex. Over time, a teacher teaches different material. The nature ofteachers and students relationships to content may change with differentcontent.8 And, the content node can be made more complex in otherways. Different kinds of teaching may interact with content differently;

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for example, teachers may have content goals that are syntactic, as well assubstantive (to use the terms of Schwab, 1978). Or, as Lampert argues:

The fundamental difference between teaching with problems and other kinds of teachingrevolves around the nature of content and what it means to study it in school. As it isenacted in classroom relationships while students work on problems, the content is morethan a series of topics. When students engage with mathematics in a problem, the contentis located in a mathematical territory where ideas are used and understood based on theirrelationships to one another within a field of study (p. 431, italics in original).

Similarly, Lampert argues that the student node is not representative ofteaching practice. The model Lampert begins with speaks of the studentin the singular and does not attend to social complexities of teaching.Teachers teach individual students, ever-shifting groups of students (forexample, the group of students who have their hands in the air at anygiven moment or students in a particular social group or assigned to aparticular set of desks), as well as the class as a whole. And, of course,there are interactions between all of these complexities. For example, aspeers, students have an influence on each other’s individual connectionsto the content, and their willingness to invest energy to be someone whostudies this content in school. Similarly, students can teach each othercontent.

But, if this were not enough, Lampert goes farther. The teacher andstudent nodes are not sufficiently complex in another way. These nodesrepresent the roles that individuals play, they do not capture the complex-ities of individual identity. Yet, attentiveness to individual traits, of one’sown or of one’s students, can help teachers identify resources for teaching.But, of course, such traits also change over time. Similarly, content is notfixed in time. In the U.S., curriculum is a matter of district policy, butit is influenced by the availability of instructional resources, as well asreform initiatives and other cultural currents. This ever-expanding web ofcomplexity leads Lampert to close the book by saying

Even though I know the model must be further revised and elaborated, I recognize thatthis endeavor is dangerous because it forces us back on the question of whether teaching –especially teaching with problems – is an impossible task (p. 448).

But, is it teaching that is the impossible task, or is it the task of describingteaching? Perhaps, the impossibility Lampert has identified lies with thenotion of a comprehensive and coherent model of teaching.

Lampert offers readers another, admittedly partial and less explicitdescription of teaching practice. This description builds on the moreexplicit model she has provided and on her terrain-based metaphors.Chapters 4 through 13 are organized around problem domains of teachingpractice. While these problem domains do not all easily map onto a terrain

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suggested by Lampert’s elaborated teacher-student-content model, thesechapters are quite helpful in expanding typical notions of teaching andfocusing attention on particular aspects of teaching. For example, thechapter “Teaching while preparing a lesson” uses the word “teaching” ina non-standard way; readers might tend to label this sort of work plan-ning for teaching, rather than teaching. But, Lampert demonstrates that thework that is done while planning a lesson is situated in the relationshipsbetween teacher, student, and content. She helps readers see ways in whichstudents are present in her planning, though they are not physically present,and how she uses her knowledge of students and of fifth graders as aresource to anticipate what will happen in the classroom. Similarly, thechapter “Teaching while students work independently” expands teachingbeyond instruction. Lampert helps readers see problems teachers face, asthey support students’ independent work. She illustrates, for example, howduring this time she works to help her students study collaboratively andbecome resources for each others’ learning. This chapter seems especiallyuseful for preservice teacher candidates.

While these problem domains are useful for talking about teaching,this description of teaching does not seek the completeness or coherenceof the model elaborated in chapter fourteen. For example, Lampert doesnot provide reasons for examining these particular domains, as opposedto others. And, there certainly are other domains that would be quiteuseful to articulate. For example, many people seem to think that teachingwith problems means that the teacher does not introduce new mathema-tical ideas into the teacher, student, content interaction (one exception isChazan & Ball, 1999). For this reason, it might have been useful to havea problem domain devoted to “Teaching while introducing new mathe-matical ideas.” Lampert clearly makes such moves. Her students did notinvent the coordinated distance-timeline in chapter two. In chapter four,she describes how, early in the year, she explicitly introduced the notions ofconjecture and revision, as a part of what doing mathematics meant in herroom. In chapter twelve, she debates when to begin a unit of study. And,in chapter nine when discussing her responsibilities to cover curriculum,she wonders about when to enter a discussion “to infuse something new”(p. 262), as opposed to “using students to take the class into new mathe-matical territory” (p. 164). But, the value in these problem domains inteaching is not that they completely map the domain of teaching. Rather,their value is in the attempt to develop a language within which it ispossible to articulate problems of teaching that are recognizable to a widerange of practitioners and researchers.

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CONCLUSION

In chapter two, after first describing an instance of her teaching practice,Lampert asserts that:

Practitioners and scholars are beginning to accumulate a corpus of first-person narrativedescriptions like the one above, portraying the complexities of teaching from the teachers’point of view . . . . But, there is as yet no coherent tradition of scholarship whose purposeis to look across stories and identify the complexities of practice in a way that is as multi-focal as the work itself; nor is there a professional language that goes very far beyond theanecdote or “case” for talking about practice in a way that captures the multiple ways inwhich any teaching action may be working to link students and content (pp. 27–28).

Lampert’s book is an ambitious attempt to rectify this situation. Byoffering readers “cases” from her teaching, as well as a number ofmetaphors, an elaborate teacher-student-content model of teaching, andproblem domains in the practice of teaching, Lampert offers readersdifferent ways of representing teaching. She also offers readers one partic-ular attempt to talk in deeply connected ways, about her teaching of agroup of students, about a particular way of teaching mathematics, andabout teaching more generally.

The question now is what others will do with these offerings. Will otherresearchers and teachers be able to see themselves in the general termsthat she offers for describing teaching? For example, will people be able toidentify a problem domain of teaching on which they are working? And,will they be able to use Lampert’s description of teaching as working inrelationships? Will people be able to identify how a teacher uses one set ofrelationships in the instructional triangle as a resource to work on another?

Reflecting on my attempts to write about my teaching of high schoolalgebra in the lower track of a tracked system makes me believe thatpeople might be able to use what Lampert has offered. In writing aboutthat teaching (e.g., Chazan, 2000), I describe my attempts to learn aboutwho my students were and what algebra might be, as a way of seekingresources to connect my students to the study of algebra in a compulsorysetting. It seems to me that this task is appropriately described as squarelyin the domain of “teaching students to be people who study in school.”

More particularly in that text (see especially pp. 92–107), I contrastthe notion of teacher-generated relevance with “psychologizing thecurriculum” (in the sense of Dewey, 1902/1990). The notion of teachersmaking content relevant suggests that the teacher’s knowledge of thecontent is a resource that is sufficient to the task of creating a relation-ship between student and content. In my case, and that of the high schoolteacher who shared this teaching assignment with me, Sandy Bethell, Ifound this not to be true. We needed to know more about our students than

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one typically does as a high school mathematics teacher. And, we felt weneeded to know algebra in a different way. As we investigated a shift inthe algebra curriculum, we tried to use our evolving understandings of thecontent, coupled with our understandings of our students, as resources toinfluence the relationship between our students and the content we taught.For example, we designed a task that asked students to join us in exploringthe relevance of algebra. We identified for our students the mathematicalobjects to be studied in our course. We then asked them to search for theseobjects in their hobbies and the workplaces of adults they knew. In sodoing, we sought to help our students understand what the course was allabout and to view the content of the course as something they might devoteenergy to studying.

While it is useful to see that I am able to cast one exploration ofa teaching problem in Lampert’s terms, the more important question iswhether the field as a whole will take up the language that she has offered,and continue to refine and sharpen it as well. It will be fascinating to see towhat degree future accounts of teaching, whether from inside or outside therole, will situate themselves in a problem domain of teaching and will tryto articulate how the teacher uses relationships in the instructional triangleas resources in addressing problems of practice. If teachers and researchersbegin to do so, then beyond what can be learned from this case, Lampertwill have contributed to improving the caliber of discussion of teachingand in helping connect research on teaching and teaching practice.

ACKNOWLEDGEMENT

I would like to thank David Pimm for his thoughtful comments on thisreview.

NOTES

1 Though many readers may well associate the sort of teaching Lampert does with theNCTM Standards movement, particularly the Professional Teaching Standards (1991), thebook does not highlight such a connection.2 Of course, there are situations where students and a teacher persist in the teacher/classrelationship over a number of years, as in a Steiner school, or a Chinese junior secondaryschool. And, there are many circumstances under which this relationship is of a shorterduration, for example, a university quarter.3 In addition to the two levels of generality that Lampert identifies, there might have beenothers. Her teaching could be thought of as elementary school teaching, as opposed to highschool or university teaching. Her teaching could be discussed as public school teaching,

198 BOOK REVIEW

as opposed to sectarian education. But, these perspectives do not play a central role in thetext. In the text, teaching with problems is implicitly contrasted with more standard waysof teaching mathematics. An open question is whether teaching with problems is a kind ofteaching that is best thought of as a way to teach mathematics, or whether it can be thoughtof more broadly.4 This chapter, for example, seems structured around a problem domain that some modesof instruction might reject. Lampert argues that teachers “can work to deliberately structurethe making of connections to enable the study of substantial and productive relationshipsin the content” (p. 179, italics in original). But, one question is whether all teachers do suchwork. For example, what of teachers using the US-based Saxon Math texts who attemptto carry out the image of teaching suggested in that series? Textbooks in that series havetwo page lessons, but no structure of units to connect between lessons. Is this componentof teaching part of this sort of instruction?5 I mean this question in senses suggested by Jonathan Crary’s (1990) introduction tohis study of vision in the nineteenth century. He suggests that vision-based metaphors aresymptomatic of a larger trend in a cybernetic world, a world in which people navigateweb sites. In his view, as in the nineteenth century, we are living through a change incultural notions of the observer. He suggests that the shift to cybernetic worlds involves“. . . relocating vision to a plane severed from a human observer” (p. 1).6 Lampert’s use of the term model does not seem to be similar to that of a cognitivepsychologist, a mathematician, or an economist. She does not seem to want to use thismodel to predict. Rather, the model is meant to understand the work of teaching and to beable to describe it. Yet, her desire to call this a model, as opposed to a description, suggeststhat it is meant to offer more than description.7 Lampert focuses on students’ “complementary work of making a relationship with thecontent to learn it” (p. 423), or studying. Students could similarly be seen as working in athree pronged way with the teacher, the content, and the teacher-content connection.8 On page 436, Lampert explores whether a map of the mathematical terrain could beplaced at the content node of her model. But, this choice would mean placing the teacher,who is in relationship with different students who are in different locations in this terrain,in several places at once. This seems to be one place where the vision and terrain-basedmetaphors reach their limit.

REFERENCES

Adler, J. (2001). Teaching mathematics in multilingual classrooms. Dordrecht: Kluwer.Ball, D.L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching

elementary school mathematics. The Elementary School Journal, 93(4), 373–397.Brousseau, G. (1984). The crucial role of the didactical contract in the analysis and

construction of situations in teaching and learning mathematics. In H.-G. Steiner(Ed.), Theory of mathematics education (pp. 110–119). Occasional paper 54. Bielefeld,Germany: IDM.

Chazan, D. (2000). Beyond formulas in mathematics teaching: Dynamics of the high schoolalgebra classroom. New York: Teachers College.

Chazan, D. & Ball, D. (1999). Beyond exhortations not to tell. For the Learning ofMathematics, 2–10.

Chazan, D. & Bethell, S. (1998). Working with algebra. In Mathematical SciencesEducation Board (Ed.), High school mathematics at work: Essays and examples from

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workplace contexts to strengthen the mathematical education of all students (35–41).Washington, DC: National Research Council.

Crary, J. (1990). Techniques of the observer: On vision and modernity in the nineteenthcentury. Cambridge, MA: MIT.

Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland.Dewey, J. (1902/1990). The school and society; The child and the curriculum. Chicago:

University of Chicago.Duckworth, E. (1987). “The having of wonderful ideas” and other essays on teaching and

learning. New York: Teachers College.Hawkins, D. (1974). The informed vision: Essays on learning and human nature. New

York: Agathon.Lampert, M. (1985). How do teachers manage to teach? Perspectives on problems in

practice. Harvard Educational Review, 55, 178–194.Lampert, M. (1989). Choosing and using mathematical tools in classroom discourse. In J.

Brophy (Ed.), Advances in Research on Teaching (Volume 1, 223–264). Greenwich, CT:JAI Press.

Lampert, M. (1990). When the problem is not the question and the solution is not theanswer: Mathematical knowing and teaching. American Educational Research Journal,27(1), 29–63.

Lampert, M. & Ball, D.L. (1998). Teaching multimedia, and mathematics: Investigationsof real practice. New York: Teachers College.

McDonald, J. (1992). Teaching: Making sense of an uncertain craft. New York: TeachersCollege.

Morrison, P., Morrison, P. and the Office of Charles and Ray Eames (1982). Powers of ten:About the relative size of things in the universe and the effect of adding a zero. NewYork: Scientific American.

National Council of Teachers of Mathematics (1991). Professional standards for teachingmathematics. Reston, VA: Author.

Schwab, J. (1978). Education and the structure of the disciplines. In I. Westbury and N.Wilkof (Eds.), Science, curriculum, and liberal education: Selected essays (229–274).Chicago: University of Chicago.


Sizer, T. (1984). Horace’s compromise: The dilemma of the American high school. Boston:Houghton Mifflin.

Vergnaud, G. (1994). Multiplicative conceptual field: What and why? In G. Harel andJ. Confrey (Eds.), The development of multiplicative reasoning in the learning ofmathematics (41–60). Albany: State University of New York.

Michigan State University Daniel Chazan

TERRY WOOD

EDITORIALDEMAND FOR COMPLEXITY AND SOPHISTICATION:

GENERATING AND SHARING KNOWLEDGE ABOUT TEACHING

In the past 15 years, there has been considerable concern about the qualityof mathematics education and significant efforts to change the way inwhich mathematics is learned. This change is not simply a local or nationalproblem but one that is internationally collective. The changes that areoccurring in the view of what constitutes the mathematics to be learned inschool and the view of how learning takes place necessarily mean changesin teaching. This is obvious, but for the most part, the vision of teaching isdrawn largely from theoretical and epistemological tenets of learning andsources of mathematical knowledge generalized to a hypothetical view ofteaching, rather than a perspective grounded in the practice of teachers. Forthose among us interested in teacher education, this combination of factorshas created a “devil of a problem” to put it politely, for the past decade.

As envisioned, these pedagogical changes demand a more complexand sophisticated form of mathematics teaching (e.g., National Council ofTeachers of Mathematics, 1991) while providing few examples or modelsin existence among teachers of mathematics at all levels of schooling.Moreover, the vision of teaching presents a form of pedagogy that isprincipally counterintuitive to time-honored practices of teaching. Forteachers, this means learning to teach in ways that are oppositional totheir more “natural” tendencies. Quite often, mathematical pedagogy isconsidered from the macro level of how the instructive environment isorganized, the tasks are selected and used, and the organization of thelesson. But the actual work of teaching is what the teacher does withstudents – discernible in the nature of interaction and discourse in bothpeer collaborative and whole class situations. Only recently have alternateforms of teaching, developed by a cadre of professional teachers inresponse to the demand for change, become available. This allows us tounderstand better the complex interplay between the norms teachers estab-lish with their students, the routine patterns of interaction, and students’mathematical thinking and reasoning in order to provide insights into whatis meant by the phrase beyond classical pedagogy.


202 TERRY WOOD

In this issue, Miriam Sherin makes visible the mathematics teaching ofDavid Louis, a secondary teacher. Although teachers’ work with studentsis agreed to be primarily interpretative, the meaning differs betweenconventional and alternative pedagogy. One view holds that interpreting iswhether students have or have not learned about the intended goal, whilethe other perspective believes interpreting is what students seeminglyunderstand at the moment about the intended goal. Sherin’s paper ablydescribes the dilemma that alternating views of interpretation create for ateacher who consistently tries to uphold his newly developed alternativepedagogy.

From a point of practicality, knowledge of the dimensions involved inmathematics teaching is crucial to knowing how and why different formsof teaching are effective in supporting student learning. As mathematicsteacher educators we are all too well aware, without a clear understandingof teaching, it is difficult to create approaches to help teachers developtheir practice and to decide what knowledge teachers need to have and forwhat purpose. Over the past decade, we have been confronted continuallywith the onerous task of creating approaches that successfully insure thatteachers can accomplish the substantial changes to meet the demands ofteaching and yet allow these amendments to be self-initiated.

In this issue two papers use knowledge of teaching drawn from empir-ical sources in teacher education approaches with prospective teachers.At issue for both is the lack of practicum opportunities for prospectiveteachers in classes of alternative teaching and the realization that preser-vice teachers need to do more than simply observe alternative teaching.They contend that instead of merely seeing what teachers do with studentmathematical thinking, they need to understand the reasoning that theteacher uses in eliciting student responses and asking probing questions.In one paper, Johanna Massingila and Helen Doerr provide insight into theuse of multi-media case studies of reform-oriented mathematics teachingwith middle school (11–13 years) prospective teachers. Their article illus-trates the ways in which these teachers interpret the case teacher’s workthat provides support in adapting and revising strategies in their practicum.In the other paper, Laura Van Zoest and Jeffrey Bohl focus on the “talk”that occurs between a prospective teacher and mentor teacher. They reasonthis talk needs to be more focused if it is to prove to be useful to prospectiveteachers’ learning. In the case study, Van Zoest and Bohl claim it is theaccess to a reform-oriented curriculum that enables the mentor teacherto focus the prospective teacher beyond issues of general teaching to thefundamental elements critical in mathematics teaching.

DEMAND FOR COMPLEXITY AND SOPHISTICATION 203

This is an exciting issue and perhaps readers of JMTE will find thisissue responsive to the concern about the existing state of conventionalmathematics teaching raised by Stigler and Hiebert (1999), in which theyassert, “To really improve teaching we must invest far more than we donow in generating and sharing knowledge about teaching” (p. 12).

REFERENCES


Stigler, J. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers forimproving education in the classroom. New York: The Free Press.

MIRIAM GAMORAN SHERIN

A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITYIN A MATHEMATICS CLASSROOM

ABSTRACT. This article examines the pedagogical tensions involved in trying to usestudents’ ideas as the basis for class discussion while also ensuring that discussion isproductive mathematically. The data for this study of the teaching of one middle-schoolteacher come from observations and videotapes of instruction across a school year as wellas interviews with the participating teacher. Specifically, the article describes the teacher’sattempts to support a student-centered process of mathematical discourse and, at the sametime, facilitate discussions of significant mathematical content. This tension in teachingwas not easily resolved; throughout the school year the teacher shifted his emphasisbetween maintaining the process and the content of the classroom discourse. Neverthe-less, at times, the teacher balanced these competing goals by using a “filtering approach”to classroom discourse. First multiple ideas are solicited from students to facilitate theprocess of student-centered mathematical discourse. Students are encouraged to elaboratetheir thinking, and to compare and evaluate their ideas with those that have already beensuggested. Then, to bring the content to the fore, the teacher filters the ideas, focusingstudents’ attention on a subset of the mathematical ideas that have been raised. Finally,the teacher encourages student-centered discourse about these ideas, thus maintaining abalance between process and content.

KEY WORDS: class discussion, discourse community, student-centered discourse, teachercognition, teacher’s role in discussion

A central goal of mathematics reform is for teachers to develop classroomlearning environments that support doing and talking about mathematics(National Council of Teachers of Mathematics [NCTM], 1991, 2000).However, creating and maintaining these environments is a complexendeavor for teachers. In particular, two key tensions are apparent. Onthe one hand, teachers are expected to encourage students to share theirideas and to use these ideas as the basis for discussion. At the same time,teachers are supposed to ensure that these discussions are mathematicallyproductive. The tension comes in trying to find a balance between havinga classroom environment that is open to student ideas and one whosepurpose is to learn specific mathematical content.

These tensions are explored through an investigation of one middle-school teacher’s attempts to implement mathematics education reform.The teacher, David Louis, worked hard to establish and then maintaina discourse community in his mathematics classroom. In doing so, he


206 MIRIAM GAMORAN SHERIN

struggled to facilitate class discussions in which student ideas were at thecenter and in which mathematics was discussed in a deep and meaningfulway. David explained this dilemma in a journal in which he reflected onhis teaching:

Today I was forced to consider an interesting issue. The issue is, ‘Do I sacrifice some . . .

content in order to foster discussions during class?’ . . . There were several different placestoday where discussion arose . . . I should have expected that considering I’m trying to seta culture of expressing one’s ideas, but it caught me by surprise a little. At first I tried topress on [to the content he had planned to cover], but students still had [new] ideas. In factat one point, their ideas [about the content] were quite different than mine . . . [and] whenI wanted to move on, they didn’t. (Louis, 1997a, p. 10)

Unable to resolve this tension, David moved back and forth in his emphasison student ideas and on mathematics learning – sometimes striking anexcellent balance, and sometimes finding his efforts less successful. Thepurpose of this article is to characterize how the tension played out inDavid’s classroom by contrasting the teacher’s focus on the process ofmathematical discourse with his focus on the content of mathematicaldiscourse. In brief, the process of mathematical discourse refers to the waythat the teacher and students participate in class discussions. This involveshow questions and comments are elicited and offered, and through whatmeans the class comes to consensus. In contrast, the content of mathe-matical discourse refers to the mathematical substance of the comments,questions, and responses that arise.

This research advances both our theoretical and practical understand-ings of the nature of the teacher’s role in a discourse community. Priorresearch on teacher cognition has explored the process through whichteachers learn to elicit and to monitor student ideas. This article extendssuch work by examining the tensions involved in this process, andthe manner in which teachers manage competing goals. The researchdescribed here can also provide teachers and teacher educators with onevision of a discourse community, and with a model for interpreting classdiscussions and the teacher’s role in such discourse.

BUILDING A DISCOURSE COMMUNITY

When researchers speak of classroom discourse, or discourse more gener-ally, they are referring to the processes through which groups of individualscommunicate (Cazden, 1986; Pimm, 1996). Analyses of discourse decom-pose these processes and underlying structures in different ways. Someresearchers attempt to enumerate norms that define aspects of classroomdiscourse. For instance, norms can govern who can speak and when

A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 207

(Mehan, 1979; Sinclair & Coulthard, 1975). Other researchers look toidentify discursive strategies used to support instruction. One example isthe work of O’Connor and Michaels (1996), who describe “revoicing” asa technique used by teachers to restate a student’s idea for the class. In stillanother approach, researchers examine the meaning of particular wordsand phrases in the context of instruction (Lampert, 1986; Lemke, 1990;Pimm, 1987).

Recently, mathematics educators and researchers have placed increas-ing emphasis on fostering classroom discourse that has certain properties(Elliott & Kenney, 1996; NCTM, 1989, 2000). Specifically, students areexpected to state and explain their ideas and to respond to the ideas oftheir classmates. Teachers are asked to facilitate these conversations and toelicit students’ ideas. In this article, classroom environments where suchdiscourse flourishes, are referred to as a discourse community. Further-more, the use of the term mathematical discourse community emphasizesthat this communication concerns mathematics in particular.

To examine the development of a mathematical discourse community,two related perspectives are examined. The first perspective examines whata mathematical discourse community might look like and evidence thatsuch a community can exist. A second viewpoint considers the teacher’srole in developing a discourse community and the teacher learning that isoften required as part of this process. Together, these lenses serve to framethe current study.

Visions of a Mathematical Discourse Community

Recent research demonstrates that a discourse community can exist in themathematics classroom. For example, Ball (1993) and Lampert (1990)share vignettes from their own classrooms in which students defend andargue for mathematical ideas. In these examples, students build on thethinking of their peers and the class works to come to consensus on themeaning of important mathematical ideas. Models such as these are crit-ical if we want to help teachers and researchers have a vision of what adiscourse community might look like in practice.

Additional research seeks to characterize the key components of adiscourse community. For example, in looking at how such a communitydevelops, Yackel and Cobb (1996) describe the importance of classroomnorms. In particular, they argue for the existence of sociomathematicalnorms, norms that are specific to participating in discussions of mathe-matics. Thus, while norms for justification and explanation might apply todiscourse in any subject matter, they argue that “what counts as an accept-able mathematical explanation and justification is a sociomathematical


norm” (p. 461). In their view, then, becoming a member of a mathematicaldiscourse community involves learning to talk about mathematics in waysthat are mathematically productive.

The Teacher’s Role in a Discourse Community

Considerable evidence shows that moving from teacher-directedclassrooms to more student-centered classrooms places complex demandson teachers (Fennema & Nelson, 1997). First, teachers have a verydifferent role to play in student-centered classrooms than they do in tradi-tional classrooms. In the past, teachers often relied on presenting facts andprocedures for students. Today, however, teachers are encouraged to moveaway from this format of instruction and “telling” is seen as only one ofseveral ways in which teachers can communicate and interact with studentsabout mathematics (Chazan & Ball, 1999). As a result of this shift, teachersneed to develop a new sense of what it means to teach mathematics, andof what it means to be an effective and successful mathematics teacher(Smith, 1996).

Second, leading a discourse community requires that teachers developnew understandings of content and pedagogy. For example, in studyingchanges in her own mathematics teaching, Heaton (2000) found that it wasrelatively easy for her to get students talking and sharing their ideas aboutmathematics. However, it was quite another matter to understand, from theteacher’s point of view, what to do with those ideas – where to go next,when to pursue an unexpected digression, and when to head off a poten-tial misconception. Heaton claims that she needed new understandings ofthe mathematics that she was teaching in order to facilitate the discourseeffectively.

Despite these obstacles, developing a discourse community in one’sclassroom can be a powerful form of professional development. Specific-ally, in a discourse community, it is not just the students who learn,but the teacher who learns as well (e.g., Fennema et al., 1996; Hufferd-Ackles, 1999; Schifter, 1998). And the fact that students are sharingand explaining their ideas seems to be a key factor in this learning.For example, previous research demonstrates that novel student ideasprompted teachers to rethink their understandings of mathematics and thepedagogical strategies that they use in teaching such ideas (Sherin, 1996).Because of the critical role that teachers play in the implementation ofmathematics education reform, exploring ways to support teacher learningis of great importance to the mathematics education community.


Supporting the Process and Content of Classroom Discourse

Before proceeding to a discussion of the research design, it is necessaryto elaborate on the tension that is the focus of this article – the diffi-culty that teachers face in trying to use students’ ideas as the basis fordiscussion while also ensuring that discussion is productive mathematic-ally. This challenge can be characterized as a tension between supportingthe process of mathematical discourse on the one hand, and the contentof mathematical discourse on the other hand. The term process refers tohow the teacher and the students interact in discussions – who talks towhom, when, and in what ways. An important component of the processof discourse involves the expectations for participation. For example, arestudents expected to share their ideas with their classmates? Is the normthat all comments are to be directed to the teacher or to one’s classmates?These questions concern the process of the classroom discourse.

The content of the discourse, in contrast, refers to the mathematicalsubstance of the ideas raised, to the depth and the complexity of theseideas in terms of the mathematical concepts under consideration. Further-more, the content of the discourse concerns how closely the ideas that areraised in discussion are aligned with the teacher’s curricular goals and withmathematics as it is understood by the mathematical community that existsbeyond the boundaries of the classroom.

A number of researchers discuss this tension (Ball, 1996; Jaworski,1994; Nathan, Knuth Elliott, 1998; Schifter, 1998; Silver & M. S. Smith,1996; Wood, Cobb & Yackel, 1991), and some make similar distinctions interminology. For example, Wood (1997) discusses the form and the contentof classroom discourse, where form refers to “knowing how to talk,” andcontent refers to “knowing what to say” (p. 170). Similarly, Williams andBaxter (1996) describe two types of scaffolding that teachers provide forclassroom discourse. First, teachers offer social scaffolding that helps toestablish and support classroom norms for how students should talk aboutmathematics. Second, teachers provide analytic scaffolding for structuringhow and what mathematical ideas are discussed in class. Both Wood’sform and Williams and Baxter’s social scaffolding are similar to what isdefined above as the process of discourse. Furthermore, Wood’s contentand William and Baxter’s analytic scaffolding are related to what thisresearch considers the content of discourse.

In discussing the tension between supporting the process and thecontent of classroom discourse, some researchers suggest that teachersmanage this tension by first turning their attention to the discourse process,and later, once classroom norms have been established, turning to issuesof the content of the discourse (Rittenhouse, 1998; Silver & Smith, 1996;


Wood, 1999; Wood, Cobb & Yackel, 1991). This research illustrates asomewhat different situation that occurred with David Louis. Though hedid lay a foundation of process in the first few weeks and then movedonto to content issues, maintaining the integrity of both the process andthe content of the mathematical discourse was a continuing struggle.Throughout the year, David moved back and forth in his emphases, alwaysstruggling to balance what proved to be competing goals. The purpose ofthis article is to characterize this struggle and to explain how and whyDavid ended up shifting his focus between the process and content ofdiscourse in his classroom.

CONTEXT AND RESULTS OF LARGER STUDY

This research took place in the context of the Fostering a community ofteachers as learners project (FCTL) (L. Shulman & J. Shulman, 1994).The central goal of the FCTL project was to examine how middle andhigh-school teachers from different subject areas might implement thepedagogical reform outlined by Brown and Campione (1992, 1996) intheir Community of Learners (COL) research. In addition, the researchersexplored the design of professional development and teacher educationactivities intended to support teachers’ efforts to implement the COLpedagogy.

The Teacher

The teacher, David Louis, taught middle-school mathematics in an upper-middle class suburb of the San Franciso Bay Area. During the 1995–96 school year, he explored how specific COL participant structures andprinciples might apply to a mathematics classroom. To do so, the teacherdesigned and tested curriculum units that incorporated many of the COLparticipant structures. For example, groups of students worked together tobecome experts in a specific area and were then organized into “jigsawgroups” comprised of experts in each of the different areas.

Context for the Study

In the summer of 1996, two researchers from the FCTL project (EdithPrentice Mendez and the author) met with David to discuss his experiencethus far with the COL pedagogy. David explained that despite imple-menting COL units, he did not believe that the COL principles had comealive in his classroom and he had yet to feel that he had successfullydeveloped a community. Furthermore, David had come to believe thatencouraging students to talk about their ideas was the critical element


in developing community in the mathematics classroom. Thus, for thecoming year, he planned to focus on developing a “mathematical discoursecommunity” rather than adhering strictly to what he thought was the struc-ture of typical COL units. He imagined a classroom in which studentswere “enthusiastic about sharing their ideas with their classmates” andin which “students would comment on and critique each other’s ideas”(Louis, 1997b, p. 4).

Data Collection

During the following school year, 1996–97, we observed and videotapedin the teacher’s classroom, choosing one eighth-grade class as the focus ofthe data collection. This class met four days a week. From September toDecember, an average of three of the four classes were observed. And fromJanuary through June, two classes a week were generally observed. In all,78 classes were observed and videotaped throughout the school year. Inorder to capture much of the discourse that took place in the classroom,we used multiple microphones and an audio mixer. The teacher wore awireless lapel microphone, and two additional microphones were placedaround the room on students’ tables. The sound was then fed throughan audio mixer to the video camera. In addition, we made copies of allassignments given in class and of all the overhead transparencies that wereused.

Field notes also were collected for the days observed. For all lessons,a lesson-structure summary was created during the observation. Thissummary listed the various activities that comprised the lesson on that day,gave brief descriptions of each and the times at which each activity began.In addition, for over 60% of the observations, more detailed notes weretaken during class. One focus of these notes was to track the mathema-tical ideas that were discussed in class and to record how these ideas wererepresented and by whom. The notes often contained snippets of transcriptsfrom class discussions. Similar notes were made for the rest of the lessonsusing the videotaped data.

In addition to the classroom data, the teacher kept a written journal inwhich he reflected on his teaching approximately three times a week fromSeptember to December, and twice a month after that. David, Edie, and Ialso met once a week to discuss what was happening in David’s class andto watch video excerpts from the class. Furthermore, the teacher was inter-viewed four times across the year. In these interviews, David discussed hisgoals for the coming year, his impressions of the discourse that existedin his classroom, and his perspective on what he and his students werelearning. The meetings and interviews were audiotaped and transcribed.


Data Analysis

The research described in this article seeks to understand teaching bylooking closely at classroom interactions across one school year. Ingeneral, the research reported is qualitative in nature, based on analysisof videotape data and interviews with the teacher. Furthermore, the teacherwhose classroom is the focus of this study, David Louis, was a collaboratorthroughout the project.

Analysis for this study focuses on class discussions. First, using video-tapes and observation notes, those lessons in which a class discussion wasone of the primary activities of the day were identified. This includeda total of 68 lessons across the school year1. Preliminary analysis theninvolved coding these lessons on a coarse scale (high vs. low) for theextent to which David focused on the process and the content of theclassroom discourse. Discussions in which his focus was rated high onprocess were those in which David consistently elicited students’ ideas andasked students to comment on each other’s ideas. In contrast, discussionsrated low on process were mainly teacher-centered with little room forstudents to contribute their ideas. Discussions in which David’s focus wasrated high on content were those in which his comments were intendedto move the discussion along mathematically. For example, David mightcompare and evaluate the mathematical substance of ideas that arose orask the students to do so, or he might direct their attention to a relevantmathematical issue. Discussions rated low on content were those in whichDavid allowed extended discussion of non-mathematical ideas or ideaswhich were only superficially mathematical.

Initially, one researcher coded the entire data set in this manner. Asecond researcher then reviewed the coding of each lesson. Agreementbetween the two researchers for each lesson was 91% and above. Casesof disagreement were reviewed together until the researchers reachedconsensus. Table 1 displays the results of this coding.

TABLE I

Distribution Across Lessons of David’s Focus on Process and Content

Low Process High Process

Low Content 28% (19 lessons)

High Content 15% (10 lessons) 57% (39 lessons)


A number of questions arose based on this preliminary analysis. First,what happened in those lessons in which David apparently focused on bothprocess and content? Was he able to use the students’ ideas to discuss thekey mathematical concepts in the lesson? And what affected whether andhow this was achieved? In addition, why was it that at some times Davidchose to focus on either process or content, but not on both? And how didthose lessons play out in class? Investigating these questions formed theresearch study that is reported in this article.

STUDYING THE TENSION BETWEEN PROCESS ANDCONTENT

In order to investigate these issues, a subset of 20 lessons from across theschool year were selected for more detailed analysis. In general, one lessonwas selected every other week from September through May. Because ofvarious school holidays that occurred throughout the year, the 20 lessonswere comprised of two lessons per month from September through May,with a third lesson included from the months of September and February.No lessons were selected from the month of June because school endedduring the first week of that month.

Class discussions from these 20 lessons were transcribed and a fine-grained analysis of video (Schoenfeld, Smith & Arcavi, 1993) was thenused to analyze the teacher’s role in these discussions. In particular,based on prior research on the role of discourse in the mathematics class,specific areas of discussion were identified to be the focus of the analysis(Ball, 1991; Brown & Campione, 1994; Mendez, 1998; NCTM, 1991;Silver, 1996). These areas included the questions raised by the teacher, theteacher’s responses to students’ questions, the mathematical content intro-duced by the students, and the mathematical content introduced duringdiscussion by the teacher. In addition, analysis examined the differentmathematical representations used during discussion. The results of thisanalysis are described in the next section.

RESULTS

The tension between process and content in David’s classroom played outat two time scales: (a) at a macro-level across the year, and (b) at a micro-level, within class discussion in individual lessons. First at the macro-level,David’s efforts to balance process and content across the school year arediscussed. These results draw from the coarse ratings of all 68 lessons as


being high or low on process and content as well as from analysis of the 20lessons selected for more detailed study. Following this, David’s efforts tobalance process and content within particular class discussions, the micro-level, are examined. The focus here is exclusively on the analysis of the 20selected lessons.

Process and Content at the Macro Level

Across the school year, David shifted his efforts between supporting theprocess and the content of the discourse community that he desired (Figure1). Initially, David’s interest was in process; the first seven lessons of theyear were coded as high on process and low on content. Two of thoselessons were the focus of detailed analysis. From the observer’s perspectivethey reveal that during this time, David’s goal was to establish the struc-ture for class discussion. In general, he did this by brainstorming withthe class about appropriate roles for the students during class discussionsand by experimenting with these roles in the context of non-mathematicalactivities. Thus, students were explaining and comparing ideas, howeverthese ideas were not mathematical in nature. For example, on the first dayof class, groups of students worked together to make shapes with a loopof yarn. The class then came together to discuss the activity and Davidencouraged the students to comment on working as a group rather than onthe different shapes that students had been able to make and why2:

D. Louis: How did it feel to [work in groups] today?Jason: Fine.D. Louis: Expand.Jason: It was easier because when we had a problem, it was easier to

work through if you had someone to talk to about it.D. Louis: What do other people think about what Jason said? Do you

agree or disagree?Ben: I agree.Julie: I agree too. Without group members you couldn’t hold the

corners [of the yarn].

In writing about this lesson in his journal, David was explicit that his goalsfor the day were to “debrief with attention to questioning techniques,” andto “comment on discussion skills.” Furthermore, he was not concerned that“we didn’t discuss too much mathematics [today]” (Louis, 1997a, p.2).Instead, David had chosen specifically for the start of the school year, tofocus on establishing the process of discourse in his classroom.


Figure 1. A sketch of David’s emphasis on process and content across the year.

After a few weeks, David was satisfied that the norms for discourse hehad envisioned were established and he was ready to add content to thisprocess. In an interview he explained:

[T]he students learned the protocol for talking to each other . . . and listening to ideas, andthey learned expectations for giving ideas . . . The basic skeleton of norms are there. Nowthey [need to] move past that and talk much more about [mathematics], to use the protocolthat I’ve tried to establish to learn math.

David began to prompt the students to talk about mathematics andthe students responded accordingly. For example, in the following lesson,students worked in groups to determine a method for estimating thenumber of dots placed randomly in a 9 × 14 cm rectangle (Lappan, Fey,Fitzgerald, Friel & Phillips, 1997). Several students then explained theirgroup’s method to the class:

Julie: We divided [it] up by one centimeter by one centimeter . . .

and then we’d have 126 little squares. So we counted [thedots in one of] the little squares and there’d be about 17 littledots in there. So then we multiplied 17 by 126.

D. Louis: Okay. What do people think about this group’s method?Robert: I think it’s a good idea but bigger squares would have been

more accurate.D. Louis: Why do you say that?Robert: Because . . . there may be a bunch of dots packed into a small

area. In just that particular area. Or, there might be not a lotof dots.

Amy: I agree . . . because there are not the same amount of dots inthe same place.

D. Louis: And why would that make a difference?


As seen in the above excerpt, in discussing the activity as a class, Davidfocused on the mathematics of the problem. He asked students to commenton and to compare the different groups’ methods. Furthermore, he encour-aged Amy to explain why the two different methods would producedifferent results.

Achieving balance. A balance had now been achieved with Davidfocusing on both the process and the content of the classroom discourse.Students were asked to share their ideas and to comment on the ideas ofothers, and they were expected to do so in the context of the mathematicsof the given activity. This balance lasted at the macro level for severalmonths.3 In fact, during the months of October, November, and December,over 85% of the lessons were rated as high on both process and content.David was pleased with the level of discourse that the class had achieved.He wrote:

There are several interesting things happening here. First, the [discourse] norms are hardat work. Students are building on each other’s knowledge and work. . . The second . . . isthe mathematics. I never would have expected to discuss [the mathematics] in such detailand depth. (Louis, 1997b, p. 21)

Shifts from balance to process or content. Mid-year however, thebalance shifted. In January and February, over 50% of the 68 total lessonswere rated high on process but low on content, similar to the beginning ofthe school year. This shift appears due to the fact that David had becomeconcerned with the level of justification that students offered in supportof their ideas and methods. In writing about his goals for the second halfof the school year, David stated that he wanted to improve the classroomdiscourse by “focusing with students on what counts as justification fora mathematical idea” (Louis, 1997b, p. 5). In discussing this goal in aninterview, David explained that he wondered if he had made it clear to thestudents that “you shouldn’t let things by without a justification . . . and[that] it’s the class’ responsibility to judge this.” He was also concerned attimes that “students would agree with each other, but without expressing adifferent viewpoint than the one first given.” For example, a student wouldrespond by saying, “I agree because of the same reasons that Amy gave.”

With this in mind, David once again turned his focus to processand encouraged students to take on new roles in the structure of classdiscussions. In particular, students were expected to contribute to classdiscussions not only by sharing their ideas, but also by providing thereasoning behind those ideas and by judging whether their classmateshad given sufficient justification for an idea. For example, David explic-itly discussed with students whether the statement, “I agree becausethat’s what I got,” is a “good” mathematical argument. Other prompts


that David used included “How could you verify that Jin’s conclusion iscorrect?” and “Does that make you more convinced?” In a sense, Davidwas attempting to renegotiate the classroom norms for participating indiscussion to include a sociomathematical norm for justification (Yackel &Cobb, 1996). However, David’s emphasis on justification occurred partlyat the expense of the mathematical content of the lessons. He worked hardto help students justify their ideas and was less concerned with the direc-tion that the discourse took in terms of the mathematical concepts underdiscussion. Thus, during this time, his main emphasis was on process, withless attention given to content.

A final shift occurred late in the year when David began to teacha unit on algebra. In contrast to lessons earlier in the year, during thealgebra unit over 85% of the coded lessons were rated low on processand high on content. At the end of the year, David chose to emphasize thecontent that he wished students to learn. Furthermore, David set aside thepattern of discourse that had developed in his class, and relied on moreteacher-directed instructional techniques to introduce the class to algeb-raic methods. For example, David wanted the students to create a tableshowing how the price of a pizza depended on the base price plus thecost per topping. Rather than asking students how they might representthis information, David gave the class explicit instructions for making aT-table and filling in the columns. David himself recognized this shift inpedagogical style and wrote about it in his journal:

[Today’s] lesson was quite different than what I was used to . . . What happened today wasmuch more directed instruction than usually exists in my classroom . . . I was telling thestudents what I wanted them to know about [algebra] . . . I just showed them what to doand why to do it. I didn’t provide a forum for discussion about student ideas or check forunderstanding via discourse. (Louis, 1997a, p. 39)

David’s reasons for focusing on content at this point in the year were two-fold. First, David was influenced by his beliefs about the nature of algebra.David explained to his class at the beginning of the algebra unit that hebelieved algebra was a highly structured domain and that learning algebrarequired a structured approach. Thus David believed that he needed tosacrifice the open-endedness of the discourse in order to help students learna set of predetermined algebraic procedures. In his journal he claimed thatthe shift in his pedagogical style was due to “the [math] that I wanted todiscuss today” (Louis, 1997a, p. 39). Second, David taught in a communitythat was highly political and was in the midst of a controversy concerningmathematics instruction. While there were many proponents of mathe-matics reform in this community, support came mainly for reform at theelementary and middle school levels. At the same time, a very vocal group


of parents and teachers at the high school level argued for more emphasison skill and computation in order to help prepare students for high schooland college mathematics. As a result of this controversy, David was partic-ularly sensitive about his teaching of algebra – a topic that is typicallytaught at the high school level, or only in an honors class at the eighthgrade. He wrote, “I could not help but think, what if a parent were to viewthis videotape?” (Louis, 1997c, p. 3).

Summary. This analysis shows that the tension between process andcontent was not something that David easily resolved as the school yearprogressed. It was not the case, as one might have imagined, that Davidbegan the year struggling to find a balance between process and content,but once a comfortable balance was reached, it was maintained for the restof the school year. On the contrary, Figure 1 illustrates that this dilemmawas ongoing throughout the year as David continued to shift his emphasisbetween the process and the content of classroom discourse.

Process and Content at the Micro-level

Fine-grained analysis of 20 lessons from across the year show that thetension between process and content existed not only at the macro levelas described above, but also arose within individual class discussions.Specifically, in the context of a single discussion, David shifted hisemphasis between the process and the content of the discourse. Further-more, it appears that moving back and forth in his emphasis at timeshelped David to facilitate meaningful discussions about mathematics inwhich students’ ideas were a key component of the discourse. How didthis occur?

Prior to the beginning of the school year, David identified three ques-tions that he planned to use to guide his comments during class discussions.Based in part on his viewing of a videotape of Deborah Ball teachingmathematics to third grade students (Ball, 1989), David planned to ask thefollowing questions: 1) “What do people think about this idea?” 2) “Why?”and 3) “What do other folks think about that?” David hoped that usingthese questions repeatedly would encourage students to share their ideasand to build on each other’s ideas. These questions represented a pattern ofdiscourse that was quite different from traditional classroom discourse inwhich discussion followed a pattern of IRE – 1) Initiation by the teacher,2) Reply from the student, followed by 3) an Evaluative comment fromthe teacher (Mehan, 1979). Instead, David planned to respond to students’comments with additional questions, either asking the student to elaborateor for other students to comment on the idea.


As a result of using these three questions, a structure for class discus-sions emerged. Specifically, many of the class discussions followed asimilar format involving three main components: (a) idea generation, (b)comparison and evaluation, and (c) filtering (Figure 2). In the first part ofdiscussion, idea generation, David elicited ideas from students concerningwhatever topic was being discussed. He used the three questions extens-ively to facilitate this initial brainstorming of ideas. David would elicit anidea from a student by asking, “What do you think?” After the studentresponded, David would ask for elaboration: “Why?” or “Can you explainthat?” David would then turn to the rest of the class and ask, “What doother people think?” Following this trio of questions, David would cycleback to the first question, “Okay. Other ideas on this?” and the cyclecontinued. As can be seen from this description, ideas were not only gener-ated, but were also preliminarily elaborated and evaluated by members ofthe class. At this point in the discussion, David was not particularly worriedabout taking control over the content that was being raised. Instead, as heexplained in an interview, he used the three questions at the beginning ofa discussion to “draw out kids’ ideas,” and to give the students a sense ofownership over the discourse.

Figure 2. Components of class discussion.

Once several ideas had been raised, the class generally shifted intoa second phase of discussion: comparison and evaluation. The shift wassomewhat subtle. Rather than asking for one new idea and then anothernew idea, and then another, David’s questions focused more on askingstudents to consider one idea in light of another, “So, is what you’re sayingthe same as Tina? What do you think?” Students’ comments also reflectedthis shift. Students were less likely to introduce new ideas at this stage, andwere more likely to state whether they agreed or disagreed with a particularidea that had been suggested.

The final structure was filtering. Here the class narrowed the space ofconsideration and developed a plan to investigate a few ideas in detail.Some ideas that had been raised were highlighted and pursued further,while others were set aside for the moment. This occurred as Davidfocused the class overtly on two or three specific ideas. In addition, David


introduced new mathematical content intended to help the class sort out theissues under consideration. David’s emphasis here was on content issues.He explained that he would “look for strategic, timely entries into theconversation to push the mathematics to a higher level . . . to tie together,or help make conclusions” (Louis, 1998, p. 5). The term filtering is usedto emphasize that any new content raised by the teacher is based on anarrowing of ideas raised already by the students. Other researchers alsoidentify this seeding of ideas as an important component of mathematicsinstruction. For example, Chazan & Ball (1999), argue that substantivemathematical comments on the part of the teacher can be a valuable cata-lyst for class discussions. Similarly, Wood (1994, 1995, 1997) talks ofteachers using a series of “focusing” questions that serve to direct students’attention to the key elements of a particular solution strategy. Yet afterasking these focusing questions, the teacher did not take an active rolein discussing the ideas with the class. In contrast, during filtering, Davidworked with the students to examine the narrow set of ideas that were nowunder consideration.

The three components shown in Figure 2 appeared quite regularlyin class discussions. Of the 20 lessons selected for detailed analysis 19involved idea generation, 16 involved comparison and evaluation, and18 of the lessons included filtering. Furthermore, 15 of the 20 lessonscontained all three structures. Despite the frequency with which thesestructures appeared, David’s class did not adhere rigidly to a prescribedformat for class discussions. On the contrary, class discussions proceededin a rather fluid manner. Although in general, the class progressed throughthe three structures in the order presented here, it was not always thecase. In particular, a single discussion might involve cycling throughthese components more than once, or repeating the first two componentsa number of times before moving to filtering. Furthermore, further ideageneration or additional comparison and evaluation of ideas often followedfiltering.4

Taken together, the three components can be thought of as a frameworkthat highlights the ways in which different processes were used by theteacher to make progress on content issues. As such, the framework isparticularly useful in exploring the tension that David faced in supportingboth the process and the content of classroom discourse. In particular, it ispossible to consider how control over each of the three processes shiftedamong the teacher and the students and the affect that this shift had on themathematics that was discussed.


AN EXAMPLE FROM THE CLASSROOM

To examine this framework more closely, consider the following examplefrom the classroom. This example comes from a lesson that was codedas high on both process and content. Thus, at the macro level, Davidwas trying to support the process and the content of the mathematicaldiscourse. However, at the micro level, David continued to shift betweenthese two goals throughout the discussion. In doing so, David was able todraw out student ideas and to use these ideas to pursue what he believedto be the mathematical content of the lesson. Far from being an anomaly,this example is representative of many class discussions that took placethroughout the school year. In discussions such as these, the claim isthat David achieved an effective balance between his goals of supportingstudent discourse and facilitating the learning of mathematical content.Furthermore, examining David’s use of the discourse structures outlinedin the previous section helps to explain why this is the case.

Background on the slingshot lesson. The slingshot lesson took placeduring a unit on functions in the second month of school. The lessonlasted for two and a half class periods. An important goal of the unit wasfor students to explore the relationship between changing quantities. Thislesson followed a format that was similar to several other lessons in theunit. Students would first collect some data, they would then graph thedata, and finally they would write an equation to represent the relationshipinvolved. For example, the previous week, the students had measured thechanging height of the water level as one, two, and then three cubes wereadded to a cup of water. To be clear, these students were not in a pre-algebra or algebra class, and the goal of the unit was not the standard y =mx + b material. Instead, the unit was intended to give students experienceexploring data, interpreting graphs, and writing simple linear equations.

During the slingshot lesson, small groups of students were given anapparatus that resembled a slingshot. The apparatus, which consisted of arubber band strung between two nails, rested on the floor. Using the rubberband, students were to measure the distance that a small ball made outof tinfoil traveled along the floor after being released from the slingshot(Figure 3). The groups were to begin by pulling the rubber band back onecentimeter and letting the ball go. They would then repeat the experimentfor two and three centimeters. Students were encouraged to take morethan one measurement for each of the three distances, and to average theirresults.

Unlike the cubes in a cup lesson, here David did not expect the class toproduce uniform data. While he believed that in an ideal physical world,increasing the stretch of the rubber band by a constant amount would result


Figure 3. The slingshot apparatus.

in a constant increase in the distance the ball traveled, he recognized thatthe classroom was not an ideal physical world.5 Thus David did not expectthe students’ data to exhibit a linear relationship perfectly. He explainedthat the class “was entering the wide world of a data collection” where“you never know what you’re going to get.” In particular, David believedthat it would not be a simple matter for the students to find an equation thatcorresponded to their data.

Fire away: The slingshot lesson in action. On the first day of theslingshot lesson, the students worked in groups to complete their datacollection. Following this, David held a brief discussion before the endof the period. During this time, the students raised a number of ques-tions regarding the procedures they had used for collecting their data: “Wecouldn’t make the ball go straight,” and “Our rubber band broke so westapled it. Does that matter?” Before handing out the homework, Davidencouraged the students to begin looking for patterns regarding how farthe ball traveled. He asked, “For every centimeter you pull it back, the ballgoes how far?” David explained that the class would pick up the discussionon the following day.

David began class on the second day of the lesson by reviewing thestudents’ homework. For homework, the students were asked to completea worksheet with six questions concerning the slingshot activity. The firsttwo questions, which are listed below, formed the basis for much of theclass discussion (Figure 4). This discussion is the focus of analysis.

Figure 4. The slingshot homework assignment.

The class quickly agreed that, in Patrice’s equation, y corresponded tothe distance that the ball traveled and x corresponded to the amount thatthe rubber band was stretched. Furthermore, it was clear that in Patrice’s


case that for every centimeter that you pulled the rubber band back, the balltraveled another 120 centimeters. David then asked, “Was it pretty accurateto say that it’s about 120 centimeters?” In response, students introduceda number of factors that they believed would affect whether or not theball traveled 120 centimeters. The following interaction is typical of theconversation that took place:

D. Louis: What do you think?Jeff: Depends on what floor it is.D. Louis: Okay, depends upon what floor it was. Why do you say that?Jeff: The more, the less, the less friction, the further it goes.D. Louis: Okay, what do other people think?

The students recognized that their data did not demonstrate the constantincrease suggested by Patrice’s equation. Thus, they suggested otherfactors as possible reasons for some variation within each group’s data.Additional variables mentioned included human error and the fact that theballs did not always travel in a straight line.

After a few minutes, Ben joined the conversation, raising an issue thatwas somewhat different from the types of comments made up to this point.Ben explained that while the factors that students had named already wouldaccount for some of the variation the groups encountered in collecting theirdata, there might also be another issue in play. Specifically, Ben wonderedif the increase in distance might not actually be constant. Another student,Robert, then explained that if this were the case, graphing the data wouldproduce a curve rather than a line.

Ben: I also think it depends like on how far you pull it back.D. Louis: What do you mean?Ben: Like if you pull it back to the one centimeter, and you do

that like three times, like it might be 120 centimeters. Butthen the first time that you pull it back it, say the secondone, it might be farther than 120 centimeters. It might justkeep going at a steady rate, but . . . it might be larger than120 centimeters apart.

D. Louis: Does anyone understand what Ben is saying because I don’tquite exactly understand . . .

Robert: I think he means that the graph might not be linear. If youmake a graph out of it, it might not go at a constant rate.

D. Louis: Is that what you’re saying?Ben: Yeah.D. Louis: What do other people think about that?


As the conversation continued, Jeff responded in agreement with Ben,“The change between zero cm and one cm will be less than the changebetween one cm and two cm.” In contrast, Sam argued that the variationwas due to human error and was not because the difference in the distancetraveled was increasing. At this point, David highlighted these two issuesfor the class:

D. Louis: So I hear people saying two things. One group of people [is]saying that you pull back a certain amount, and then it will gothat much farther each cm you pull it back. So each time itgoes 120 centimeters farther . . . the same amount farther eachtime. I hear another group of people saying that possibly, thefurther you pull it back each time, it goes a little farther. So ifyou pull it back the first time it goes 120, and you pull it backthe second time, or 2 cm back, it might go 140. You pull it 3centimeters back, it might, well the first was 120, then 140, andthen maybe 160. So it goes a little farther each time you pull itback. So what do you guys think about that idea?

To respond to David, students began to look at their data to see whichpattern fit most accurately. Soon David suggested that the class pursuethis issue using the graphing calculator. David introduced the notion of a“scatter plot” as a graph whose values do not make a perfectly straight line.With the students’ help, David entered one group’s data into a graphingcalculator that worked with the overhead projector. David selected thescatter plot function so that the data was now displayed in view of theentire class. The students discussed how to visually estimate which oneline would most accurately represent the data. In addition, they used thegraphing calculator to determine a “line of best fit.” In this way, the classwas beginning to deal with different ways to interpret the complex set ofdata that had been collected. In fact, students began to offer a number ofdifferent ideas about why the notion of scatter plot was useful for them,and how they could determine whether one of their own estimates was aline of best fit.

Analysis of the slingshot lesson: A filtering process. In this example,David achieves multiple goals. In particular, he achieves a balance betweenprocess and content by first allowing a great deal of open-ended discourse,and then by focusing the discussion himself, and thus taking more controlof the content. The beginning phase of the discussion is a typical exampleof idea generation. David uses the three questions to draw out students’ideas and to keep the conversation moving. The content of the discussionis clearly in the students’ hands at this point, as they are the ones suggestingwhich factors affect the data.


After students raised several ideas concerning why the distance mightnot consistently be 120 centimeters, there is evidence that the classshifts into the comparison and evaluation structure. In particular, Ben’scomments indicate that he has classified the ideas raised thus far as beingabout physical factors that affect data collection. In contrast, Ben had adifferent kind of argument to make. Following Ben’s comment, Davidencouraged further comparison among the students’ ideas including eval-uation of Ben’s proposal. Thus, in this phase of discussion, the studentsand David appear to share responsibility for the content of the lesson – yetopen-ended discourse is still a prominent feature of the discourse.

The beginning of the third phase, filtering, is much more obvious. Here,David shifts his position in the conversation somewhat and brings thestudents’ attention to two particular ideas. In addition, he seeds the ensuingdiscussion with the notion of a scatter plot and of finding a line of best fit.For a time then, David has taken control of the content of the conversation,and has narrowed the space of ideas being raised and discussed. Open-ended discourse is not closed off completely, in fact David often asksfor student input to explain the ideas he is presenting. However, this partof discussion resembles teacher-directed discourse more than that whichoccurred earlier.

The class then uses this filtering by David to redirect their attention, andreturn again to idea generation. Specifically, they began to discuss what a“line of best fit” would look like (e.g., “There must be the same numberof data points above and below the line.” “Should some data points passthrough the line?”) It is important to note that considering how to interpreta scatter plot and how to determine the features of a line of best fit, consti-tute significant mathematical content for these students. In the past, theyhad explored data intended to represent linear functions more precisely –the difference between data values was often consistently the same. Herethe students were dealing with a very different set of data and they neededa new set of mathematical tools to do so. The combination of the graphingcalculator with the notion, not of a line that fit perfectly, but rather of a lineof best fit, had the potential to help them explore these issues productively.

Examining the flow of ideas in the class, a pattern is evident. First, interms of the process of mathematical discourse, many ideas are encouragedearly on, a few are chosen for more focused attention, and then the classreturns to soliciting many ideas. This is a view of the process of mathe-matical discourse because it describes how and when ideas are solicited.Furthermore, this particular process involves a great deal of open-endeddiscourse in which students are encouraged to have control of the ideasbeing raised.


Figure 5. A representation of the process of the classroom discourse.

Second, this approach serves a very different purpose for the content ofmathematical discourse. Each time that the teacher narrows the scope ofideas that are considered during filtering, he takes control of some of themathematics that is discussed. And even though many ideas are then gener-ated about this filtered topic, the mathematical content has neverthelessbeen redirected and narrowed. As this cycle is repeated, the mathematicalcontent of the lesson moves from a broad arena to one that is more focused.The initial question or topic that the teacher raises is certainly an importantfactor in determining the direction of the content of the discussion. Yetin addition, the filtering process allows the teacher continually to refocusthe content of discussion in areas that he or she feels are mathematicallysignificant and that will be productive for the class to pursue (Figure 6).

Figure 6. A representation of the space of mathematical content.


Taken together, these two perspectives demonstrate how David was ableto balance the process and content of mathematical discourse in conversa-tions such as the one discussed here. Furthermore, this example illustratesthat this balance was achieved in part because David shifted his emphasisbetween process and content in the context of the discussion. Thus, ratherthan hindering his goals, at times, the ongoing tension between process andcontent was an important factor in enabling David to facilitate classroomdiscourse successfully.

DISCUSSION AND IMPLICATIONS

Teaching with open-ended discourse poses a problem for the learning ofcontent. On the one hand, students are expected to learn specific content,but on the other hand, students’ ideas are supposed to direct the discus-sion. How do teachers respond to the need to support both the processand the content of classroom discourse? Under what circumstances arethey able to manage both of these goals simultaneously? Based on theanalysis presented in this article, two issues are proposed as being at thecore of teachers’ efforts to meet these competing demands. For each issue,both theoretical implications and considerations for teacher education arediscussed.

New Structures for Classroom Discourse

First, the teacher does find ways to structure class discussion in order tosupport both the process and the content of classroom discourse. Specific-ally, a filtering approach involving a combination of three discourseprocesses is used to make progress on content issues. In this approach,multiple ideas are solicited from the students in the initial phase. Studentsare encouraged to elaborate their thinking, and then to compare and eval-uate their ideas with those that have already been suggested. The filteringpart of the discussion comes next, as the teacher focuses the students’attention on a subset of the ideas that have been raised. In addition, theteacher may introduce a new mathematical idea or approach that the classcan use to consider the focused content. This focusing on the part ofthe teacher is then followed by additional idea generation on the part ofthe students. A single class discussion may involve several cycles of thispattern.

This filtering approach can serve both process and content goals. Interms of process, the students have a great deal of opportunity to sharetheir thinking and the teacher’s “filtering of ideas” is based on the ideas


that students raised. Yet at the same time, this approach also enables theteacher to exert some control over the mathematical direction of the lesson.By selecting from among those ideas that have been raised, and seeding thediscussion with new ideas, the teacher is able to orchestrate discussionsthat are based on student ideas and that are also productive and worthwhilemathematically.

Identifying this filtering model contributes to our understanding ofteaching in several ways. To start, the model explores how teachers respondto the ideas that students raise in class – how teachers help other membersof the class interpret and respond to these ideas, how teachers coordinatediscussion of multiple ideas, and how teachers begin to pursue particularideas in more detail. This focus extends prior research by looking beyondwhether or not teachers pay attention to students’ ideas or how they learn todo so, towards an investigation of what teachers do with these ideas in thecontext of classroom discussion. If we want to understand how teachers’knowledge of student thinking influences classroom instruction, lookingat the ways in which teachers structure class discussions that are based onstudents’ ideas is a valuable first step.

In addition, the model provides insights concerning how teacherscoordinate some of the many overlapping demands with which they arefaced. Current views of teaching argue that teaching is a complex cognitiveactivity and that teachers hold multiple goals simultaneously (Schoenfeld,1998). The research reported here provides evidence that a teacher can usea series of discourse processes in order to achieve multiple goals. Specific-ally, the model offers an account of how process and content goals arecoordinated through a cycle of filtering of students’ ideas. Furthermore,the model suggests that the shifting of goals on the part of the teacher isa key feature in developing effective classroom discourse in which studentideas are at the center.

Finally, the model also has practical implications. Introducing teachersto the three discourse structures and encouraging them to experiment withthese structures has the potential to improve mathematics teaching. Whilethe filtering technique is clearly only one possible approach to balan-cing the process and content of classroom discussion, it could providean important first step for some teachers by providing a vision of prac-tice. Close investigations of classroom discourse on the part of teachershave also been shown to positively impact classroom instruction (Sherin,1998, 2001), and the model presented here could be used to structure suchinvestigation.


Influencing Teachers’ Focus on Process and Content

If as the above discussion suggests, teachers can support both the processand the content of classroom discourse, then why is this not consist-ently the case? As this research shows, at times a teacher may choose toemphasize process over content, or vice versa. For example, David Louischose to focus on process during the first few weeks of school because thisis how he believed that a discourse community should be established. Incontrast, later in the year, David decided to emphasize content over processin his teaching of algebra. This decision was due to his belief that the natureof algebra required a specific teacher-directed pedagogical approach.

Clearly, a variety of issues influence teachers’ decisions about theirapproach to classroom discourse. Based on this research, two specificissues come to mind that extend prior research on the developmentof discourse communities in the mathematics classroom. First, whileprevious research has argued that teachers’ beliefs about discourse influ-ence teachers’ practices (e.g., Nathan, Elliott, Knuth & French, 1997),here the more specific claim is made that beliefs about the relationshipbetween the process and the content of discourse are critical in determ-ining how teachers facilitate discussion. Similarly, there has been extensiveresearch on the ways in which teachers’ content knowledge constrainsand enables instruction (Fennema & Franke, 1992). Yet the relationshipbetween teachers’ knowledge of mathematics and the ways in whichteachers support classroom discourse has not been widely examined. Theresearch reported here contributes to understanding this relationship.

As teacher educators support teachers’ efforts to develop and maintainclassroom discourse communities, it is important not only to help teachersdevelop effective instructional strategies but also to address their beliefsabout discourse. Furthermore, teachers should have opportunities to usediscourse as a tool for their own learning of mathematics, and within avariety of mathematical domains. Doing so can provide a foundation fromwhich teachers view discourse as a viable tool for students’ learning ofdiverse mathematical concepts.

ACKNOWLEDGEMENTS

This research was supported in part by a post-doctoral fellowship from theJames S. McDonnell Foundation. Additional support was received fromthe Andrew W. Mellon Foundation for a grant to Lee S. Shulman andJudith Shulman for the Fostering a Community of Teachers as LearnersProject. The opinions expressed are those of the author and do not neces-


sarily reflect the views of the supporting agencies. The author wishes tothank Bruce Sherin, Margaret Schwan Smith, James Spillane, and threeanonymous reviewers for their thoughtful feedback on this article as wellas Sharon Liszanckie, Josh Devlin, and Jacob Rossmer for their researchassistance. The project on which this research is based would not havebeen possible without the assistance of my collaborators Edith PrenticeMendez and David Louis. An earlier version of this article was presentedat the Annual Meeting of the American Educational Research Association,April, 1999.

NOTES

1 Extended class discussions were not a feature of David’s class every day. For example,there were a number of lessons in which students worked in groups for the majority of a daycollecting and analyzing data. There were also a few days in which class discussion wasplanned but not executed because of administrative issues such as fire drills or school-wideassemblies. In all, of the 78 lessons that were videotaped, 68 included class discussions of10 minutes or more.2 Students’ names are pseudonyms.3 In her research on the structure of robust mathematical discussion, Mendez (1998)analyzed a set of discussions that occurred in David Louis’ class during the first fourmonths of school. Mendez analyzed the discourse along two dimensions, mathematicsand discussion, with attention to several components within each dimension. ThoughMendez focused mainly on student participation in these discussions, her data also showan increasing focus on mathematics during the first month of school.4 In the first month of school, the end of a discussion was often signaled by a reflectivesummary on the part of David. In these summaries, David would offer the class his assess-ment of the discussion that had occurred, with particular attention to the ways in which thestudents had participated in the discourse.5 In fact, it is not obvious whether the growth in this instance should be linear or expo-nential. Treated as a spring, the distance would increase as the square of the stretch of thespring.

REFERENCES

Ball, D.L. (1989). Beginnings: Transcript from Deborah Ball’s Classroom, September 11,12, 18, 1989. East Lansing, MI: M.A.T.H. Project, College of Education, Michigan StateUniversity. Used with permission.

Ball, D.L. (1991). What’s all this talk about “discourse”? Arithmetic Teacher, 39, 44–48.Ball, D.L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching

elementary school mathematics. Elementary School Journal, 93, 373–397.Ball, D.L. (1996). Teacher learning and the mathematics reforms: What we think we know

and what we need to learn. Phi Delta Kappan, 77, 500—508.


Brown, A.L. & Campione, J.C. (1992). Fostering a community of learners: A proposalsubmitted to the Andrew W. Mellon Foundation. Unpublished manuscript, University ofCalifornia, Berkeley.

Brown, A.L. & Campione, J.C. (1994). Guided discovery in a community of learners. In K.McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice(pp. 229–270). Cambridge, MA: MIT Press/Bradford Books.

Brown, A.L. & Campione, J.C. (1996). Psychological learning theory and the design ofinnovative environments: On procedures, principles, and systems. In L. Schauble & R.Glaser (Eds.), Innovations in learning: New environments for education (pp. 289–325).Hillsdale, NJ: Erlbaum.

Cazden, C. (1986). Classroom discourse. In M.C. Wittrock (Ed.), Handbook of researchon teaching (pp. 432–462). New York: Macmillan.

Chazan, D. & Ball, D.L. (1999). Beyond being told not to tell. For the Learning ofMathematics, 19(2), 2–10.

Elliott, P.C. & Kenny, M.J. (1996). 1996 Yearbook: Communication in mathematics, K–12and beyond. Reston, VA: NCTM.

Fennema, E., Carpenter, T.P., Franke, M.L., Levi, L., Jacobs, V.R. & Empson, S.B. (1996).A longitudinal study of learning to use children’s thinking in mathematics instruction.Journal for Research in Mathematics Education, 27, 458–477.

Fennema, E. & Franke, M.L. (1992). Teachers’ knowledge and its impact. In D.A. Grouws(Ed.), Handbook of research on mathematics teaching and learning (pp. 147–164). NewYork: Macmillan.

Fennema, E. & Nelson, B.S. (1997). Mathematics teachers in transition. Mahwah, NJ:Erlbaum.

Heaton, R.M. (2000). Teaching mathematics to the new standards: Relearning the dance.New York: Teachers College Press.

Hufferd-Ackles, K. (1999). Learning by all in a math-talk learning community. Unpub-lished doctoral dissertation. Northwestern University, Evanston, IL.


Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruc-tion, 3, 305–342.

Lampert, M. (1990). When the problem is not the question and the solution is not theanswer: Mathematical knowing and teaching. American Educational Research Journal,27, 29–63.

Lappan, G., Fey, T., Fitzgerald, W.M., Friel, S.N. & Phillips, E.D. (1997). Comparingand scaling: Ratio, Proportion, and Percent – The Connected Mathematics Project. PaloAlto, CA: Dale Seymour.

Lemke, J.L. (1990). Talking science: Language, learning, and values. Norwood, NJ:Ablex.

Louis, D. (1997a). Reflections on my teaching: My mathematics teaching journal. Unpub-lished manuscript.

Louis, D. (1997b). Talk about math: Understanding communities of learners in a math-ematics classroom, A proposal submitted to the Spencer Foundation. Unpublishedmanuscript.

Louis, D. (1997c). 400%: A case of talking about mathematics. Unpublished manuscript.Louis, D. (1998). Summary report to the Spencer Foundation. Unpublished manuscript.Mehan, H. (1979). Learning lessons: Social organization in the classroom. Cambridge:

Harvard University Press.


Mendez, E.P. (1998). Robust mathematical discussion. Unpublished doctoral dissertation,Stanford University, Stanford, CA.

Nathan, M.J., Knuth, E. & Elliott, R. (April, 1998). Analytic and social scaffolding in themathematics classroom: One teacher’s changing practices. Paper presented at the annualmeeting of the American Educational Research Association, San Diego, CA.

Nathan, M.J., Elliott, R., Knuth, E. & French, A. (March 1997). Self-reflection on teachergoals and actions in the mathematics classroom. Paper presented at the annual meetingof the American Educational Research Association annual meeting, Chicago.

National Council of Teachers of Mathematics (1991). Professional standards for teachingmathematics, Reston, VA.

National Council of Teachers of Mathematics. (2000). Principles and standards for schoolmathematics. Reston, VA.

O’Connor, M. & Michaels, S. (1996). Shifting participant framework: Orchestratingthinking practices in group discussions. In D. Hicks (Ed.), Discourse, learning, andschooling (pp. 63–103). New York: Cambridge University Press.

Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms.London: Routledge & Kegan Paul.

Pimm, D. (1996). Diverse communications. In P.C. Elliott, (Ed.), 1996 Yearbook:Communication in mathematics, K–12 and beyond (pp. 11–19). Reston, VA: NCTM.

Rittenhouse, P.S. (1998). The teacher’s role in mathematical conversation: Stepping inand stepping out. In M. Lampert & M.L. Blunk (Eds.), Talking mathematics in school:Studies of teaching and learning (pp. 163–189). New York: Cambridge University Press.

Schifter, D. (1998). Learning mathematics for teaching: From a teachers’ seminar to theclassroom. Journal of Mathematics Teacher Education, 1, 55–87.

Schoenfeld, A.H. (1998). Toward a theory of Teaching-In-Context. Issues in Education,4(1), 1–94.

Schoenfeld, A.H., Smith, J.P. & Arcavi, A. (1993). Learning: The microgenetic analysis ofone student’s evolving understanding of a complex subject matter domain. In R. Glaser(Ed.), Advances in instructural psychology (pp. 55–175). Hillsdale, NJ: Erlbaum.

Sherin, M.G. (1996, April). Novel student behavior as a trigger for change in teachers’content knowledge. Paper presented at the annual meeting of the American EducationalResearch Association, New York.

Sherin, M.G. (1998). Developing teachers’ ability to identify student conceptions duringinstruction. In S.B. Berenson, K.R. Dawkins, M. Blanton, W.N. Coulombe, J. Kolb, K.Norwood & L. Stiff (Eds.), Proceedings of the Twentieth Annual Meeting of the NorthAmerican Chapter of the International Group for the Psychology of Mathematics Educa-tion. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and EnvironmentalEducation.

Sherin, M.G. (2001). Professional vision of classroom events. In T. Wood, B. Nelson & J.Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics(pp. 75–93). Mahwah, NJ: Lawrence Erlbaum Associates.

Shulman, L.S. & Shulman, J.H. (1994). Fostering a Community of Teachers: A proposalsubmitted to the Andrew W. Mellon Foundation. Unpublished manuscript, StanfordUniversity.

Silver, E.A. (1996). Moving beyond learning alone and in silence: Observations from theQuasar project concerning communication in mathematics classrooms. In L. Schauble& R. Glaser (Eds.), Innovations in learning: New environments for education (pp. 127–159). Mahwah, NJ: Erlbaum.


Silver, E.A. & Smith, M.S. (1996). Builiding discourse communities in mathematicsclassrooms: A worthwhile but challenging journey. In P. C. Elliott (Ed.), 1996 Yearbook:Communication in mathematics, K–12 and beyond (pp. 20–28). Reston, VA: NCTM.

Sinclair, J. McH. & Coulthard, R.M. (1975). Towards an analysis of discourse: The Englishused by teachers and pupils. London: Oxford University Press.

Smith, J.P. (1996). Efficacy and teaching mathematics by telling: A challenge for reform.Journal for Research in Mathematics Education, 27, 458–477.

Williams, S. & Baxter, J. (1996). Dilemmas of discourse-oriented teaching in one middleschool mathematics classroom. Elementary School Journal, 97, 21–38.

Wood, T. (1994). Patterns of interaction and the culture of mathematics classrooms. InS. Lerman (Ed.), The culture of the mathematics classroom (pp. 149–168). Dordrecht,Netherlands: Kluwer.

Wood, T. (1995). An emerging practice of teaching. In P. Cobb & H. Bauersfeld (Eds.),The emergence of mathematical meaning (pp. 203–227). Hillsdale, NJ: Erlbaum.

Wood, T. (1997). Alternative patterns of communication in mathematics classes: Funnelingor focusing: In H. Steinbring, M. Bartolini-Bussi & A. Sierpinska (Eds.), Language andcommunication in the mathematics classroom (pp. 167–178). Reston, VA: NCTM.

Wood, T. (1999). Creating a context for argument in mathematics class. Journal forResearch in Mathematics Education, 30, 171–191.

Wood, T., Cobb, P. & Yackel, E. (1991). Change in teaching mathematics: A case study.American Educational Research Journal, 28, 587–616.

Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy inmathematics. Journal for Research in Mathematics Education, 27, 458–477.

School of Education and Social Policy, Northwestern University,2115 N. Campus Drive, Evanston, IL 60208-2610E-mail: [email protected]

JOANNA O. MASINGILA and HELEN M. DOERR

UNDERSTANDING PRE-SERVICE TEACHERS’ EMERGINGPRACTICES THROUGH THEIR ANALYSES OF A MULTIMEDIA

CASE STUDY OF PRACTICE1

ABSTRACT. The limits inherent in field experiences and the difficulties in learning howto use student thinking in instructional practice are significant challenges in pre-serviceteachers’ preparation. In this research, we have investigated how multimedia case studiesof practice can support pre-service teachers in making meaning of complex classroomexperiences and in developing strategies and rationales for using student thinking to guideinstruction. In this paper, we present a brief review of the research on case studies to situateour particular approach that builds on the notion that a multimedia case study can be a sitefor investigation, analysis and reflection by pre-service teachers. We then report the resultsof examining the issues that one cohort of pre-service mathematics teachers (grades 7–12)identified as meaningful for them in terms of their own emerging practice and the waysin which they connected the case study teacher’s practice to their own practice. We foundthat the pre-service teachers were able to use their perspectives on a common practiceto highlight some of the dilemmas and tensions found in teaching. In particular, these pre-service teachers focused on the difficulties encountered when trying to use student thinkingand to follow their own mathematical goals in a lesson. They were able to frame many ofthe issues that they encountered in their own practice (such as checking for student under-standing and the use of questioning) in terms of their analysis of the case study teachers’practice.

Mathematics teacher educators face several practical problems in theirwork with pre-service teachers. One set of difficult issues involves thecomplexities of field placement experiences. First, there are insufficientnumbers of high quality, reform-based classrooms available for pre-service teacher placements. Many teacher educators struggle with thedifficulty that their pre-service teachers face when confronted with thedisparity between what is taught in a methods course as “best-practices” or“research-based practice” and the reality that pre-service teachers find inthe classrooms in which they begin their observations and teaching exper-iences. Second, even when placed in exemplary classrooms to work andobserve, student teachers lack the experience necessary to observe mean-ingfully the complex and rapid interactions that can occur. The subtleties ofmultiple human interactions, the unspoken rules and agendas in an exper-ienced teacher’s classroom, and apparent ease with which an exemplaryteacher can interact with students and their ideas make it exceedingly diffi-


236 JOANNA O. MASINGILA AND HELEN M. DOERR

cult for an inexperienced pre-service teacher even to know what to focuson or what questions to ask of the experienced teacher. Third, during theirfield placement experiences, pre-service teachers lack a common experi-ence upon which they can reflect with their peers as they strive to makesense of classrooms. This situation limits their ability to reflect on theirown practice and also limits their opportunities to analyze the processesof teaching and learning more generally. Without a shared experience,pre-service teachers are less able to appreciate the multiplicity of interpret-ations and perspectives on classroom interactions. These problems haveslowed the pace of change envisioned in mathematics reform documents(National Council of Teachers of Mathematics [NCTM], 1991; NationalResearch Council, 1990).

A second set of issues facing those who seek to improve the systemof school practices involves the gap between educational research onstudent learning on the one hand and changes in teachers’ practice on theother. For example, research on teacher development indicates that pre-service teachers need opportunities to observe and reflect on students’mathematical thinking and on how effective teachers build on students’thinking (Ball, 1993; Simon, 1995). However, research has only begunto document effective ways for pre-service teachers to investigate studentlearning and incorporate this into their developing ideas of teaching (cf.,Fennema, Carpenter, Franke, Levi, Jacobs & Empson, 1996). Even if pre-service teachers were to begin to appreciate the significant role that studentthinking can play in a classroom, the gap between an appreciation ofthat role and the implementation in practice is non-trivial. Moreover, theknowledge that pre-service teachers need to develop includes much morethan seeing what it is that effective teachers do with student thinking.Pre-service teachers need to understand the reasoning that an experiencedteacher uses when eliciting student responses, asking probing questions,deciding on instructional materials, and assessing the quality of studentwork.

These two sets of issues, namely the difficulties inherent in pre-serviceteachers’ field placement experiences and the challenges in understandinghow to use student thinking in instructional practice, have led us toinvestigate the use of multimedia case studies to support the professionaldevelopment of pre-service teachers. Carefully designed, multimedia casestudies that capture the complexities and richness of exemplary, reformed-based classrooms may be of value to teacher educators in their work withpre-service teachers to the extent that such case studies can create newimages of practice. However, the critical importance of such images is thatthey can become sites for investigation, reflection and study by pre-service

UNDERSTANDING PRE-SERVICE TEACHERS’ EMERGING PRACTICES 237

teachers in ways that are not easily accomplished with actual classroomexperience. In other words, these multimedia “records of practice,” toborrow a phrase from Lampert and Ball (1998), provide artifacts that canbe examined by pre-service teachers and support their reflection on boththe case study teacher’s practice and their own emerging practice. To thisend, we have developed several multimedia case studies for use with grade7–12 pre-service mathematics teachers. In this paper, we focus on how onecohort of intern teachers used one of these case studies to reflect on theirown emerging teaching practices through their analyses of a case studyteacher’s practice. In particular, we examine the kinds and the nature ofthe connections that the pre-service teachers made between their practiceand the case study teacher’s practice. Such connections reveal pre-serviceteachers’ ways of thinking about the practice of teaching and hold thepotential to support the development of that thinking in ways that leadto more nuanced and sophisticated understandings of practice.

PERSPECTIVES AND GUIDING FRAMEWORKS

Two areas of research have informed this study: (a) research on teacherdevelopment, and (b) research on the use of case studies in supportingthis development. Research on teacher development has identified severalkey issues with respect to the preparation of pre-service teachers. Someresearchers have focused explicitly on pre-service teachers’ content know-ledge (e.g., Even & Lappan, 1994); many researchers have focused onthe relationship between pre-service teachers’ pedagogical beliefs and theimpact of those beliefs on their developing knowledge of teaching (e.g.,Eisenhart, Borko, Underhill, Brown, Jones & Agard, 1993; Meredith,1993; Thompson, 1992). Following Shulman’s early work (1986), otherresearchers have investigated pre-service teachers’ pedagogical contentknowledge (e.g., Ball & Wilson, 1990; Barnett, 1991). This set of issues,however, does not fully capture the complexity of what it is that teachersneed to know and the situations in which they know these ideas. Teachers’knowledge needs to include at least the knowledge of psychological devel-opment of children’s thinking, the logical (or mathematical) developmentof concepts, the curricular instantiation of ideas in instructional materials,the representational (or media) development of mathematical ideas, thepractical development of discourse, and strategies for management in theclassroom.

Research on the use of cases in pre- and in-service teacher develop-ment courses has suggested that case-based instruction has potential foraddressing these issues. Yet, the reported data on the use of text, video,


and multimedia case studies is rather limited. In part, this is due to thecomplexities of teaching and the complexities of teaching about teaching.It is also due to the fact that the use of case studies (in whatever media)will only be one component of a pre-service teacher’s preparation program.Any use of case studies will be necessarily deeply embedded in a largercontext of field observations, methods courses, supervised practice, andcontent courses. In addition, pre-service teachers entering their preparationprograms have already had 12 or more years of first hand experience inclassrooms. Within the context of preparation programs, it is our goal tounderstand how case studies can support the development of case-basedpedagogical understandings by pre-service teachers by providing a sharedfocus for their reflection on their own emerging practices.

Research on Teacher Development

Prospective teachers’ visions of effective teaching are largely informedby their prior experiences as students and, to a lesser extent, by theirunderstandings of how children learn. As Thompson (1992) notes, oneconsequence of these prior experiences as students in that pre-serviceteachers’ beliefs about effective teaching are often difficult to change. Forexample, a common belief among pre-service teachers is that mathema-tics should be taught in a straightforward and procedural way by clearlyexplaining how traditional algorithms are applied (Eisenhart et al., 1993).Based on this finding, one might assume that videos of children from tradi-tional classrooms who are struggling with non-traditional tasks could beused to illustrate the dangers of cultivating procedural rather than concep-tual orientations (cf., Thompson, Philipp, Thompson & Boyd, 1994).However, showing videos of students who may be struggling does notaddress pre-service teachers’ queries regarding alternative ways to teach.Such videos leave unanswered the questions about what an effectiveclassroom would look like, and even more importantly, how a successfulteacher can create such an effective classroom. As Lampert and Ball (1998)and others have noted, pre-service teachers come to their teacher educa-tion programs with years of experience in observing what teachers do andthis experience is, in fact, an obstacle for novice teachers to overcome indeveloping their own effective practice. The crux of the difficulty is notwhether or not these pre-service teachers have observed poor teaching. Thedifficulty is that the years of observing what it is that teachers do seldomprovide pre-service teachers with insight into how teachers think or evenwhat aspects of the teaching and learning process teachers are thinkingabout (Doerr & Lesh, in press).


Recent research efforts to address the difficulty inherent in teacherchange have described a shift towards strategies in which teachersare encouraged to become reflective about their practice (Cooney &Krainer, 1996; Simon, 1995). In examining the frameworks that pre-service teachers use to understand the practice of teaching, Cooney (1999)has highlighted the importance of examining the contexts through whichteachers develop and use their knowledge. He states that “whatever lenswe use to describe teachers’ knowledge, that lens must account for theway in which knowledge is held and the ability of the teacher to use thatknowledge in a reflective, adaptive way” (p. 171). This view of teachers’knowledge is consistent with current developments in cognitive scienceand situated learning that suggest that knowledge is situated and groundedin the contexts and constraints of practice (e.g., Borko, Mayfield, Marion,Flexner & Cumbo, 1997; Lave & Wenger, 1991; Leinhardt, 1990). Animportant characteristic of case studies is that they are embedded in thecontexts of practice and, at the same time, provide an opportunity for pre-service teachers to engage in the analysis of elements of the case. Theissue we will examine is whether (and/or how) this analysis can becomea basis for supporting pre-service teachers in reflecting on their emergingpractices. To understand better the reflective thinking of teachers, we drawon the theoretical work of Schön (1983, 1987, 1991) and his analysis ofreflective practitioners.

According to Schön (1995), the knowledge of competent professionalsis tacit and implicit in their actions as they make judgments and performtasks in everyday settings that are characterized by uncertainty, complexity,uniqueness, and conflict. Schön also observes that effective practitionersare often at a loss to produce adequate descriptions of what it is thatthey know. This tacit and implicit knowledge of the effective practi-tioner is especially problematic for the pre-service teacher, who lacks theexperience of the professional in action. The practice of the competentteacher may well appear smooth, easy and unproblematic to the pre-serviceteacher. There may be no obvious queries to make. Even if the pre-serviceteacher did ask the experienced teacher about her understandings of aproblem situation or reasons for strategies of action, it is unlikely that thedescriptions would be adequate for the pre-service teacher.

Rich descriptions of reasons for actions or of strategies for decidingwhat elements of a situation to attend to are not necessarily part ofthe reflection-in-action. According to Schön (1995), the reflection onreflection-in-action makes explicit the action strategies, the assumptions,and the aspects of the problem-settings that were implicit in reflection-in-action. By reflecting on reflecting-in-action, the practitioner restructures


her or his understanding of the problem situation and of the strategies(“Why didn’t that work?” or “What should I try next time?”), the examina-tion of assumptions (“How was I thinking about that student’s ideas?”), andthe understanding of variations in problem-settings (“What does this meanfor my teaching of some other content to a different group of students?”).Teaching takes place in contexts that are filled with complexity anduncertainty; expertise in such ill-structured domains requires the flexibleuse of cognitive structures to accommodate partial information, chan-ging or unclear goals, multiple perspectives, and uncertain consequences(Feltovich, Spiro & Coulson, 1997). Within this complex context, teachersmust select instructional materials, design lesson plans, evaluate studentlearning, and create opportunities for children to extend their thinkingabout mathematical ideas. In traversing such complex domains, the skilledpractitioner not only reflects-in-action but also reflects back on thosereflections-in-action.

Research on the Use of Case Studies

Case studies have had a long and rich history of use in preparing profes-sionals for the practices of engineering, medicine, law and business. Morerecently, there has been interest in the use of case studies for bringingnew practitioners into the profession of teaching. Most of the researchbase that examines the use of cases in teacher education has focused onthe use of text-based cases (e.g., Barnett, 1991; Merseth & Lacey, 1993;Shulman, 1992; Stein, Smith, Henningsen & Silver, 2000; Wassermann,1993a), with more recent work using video cases and multimedia casematerials (e.g., Copeland & Decker, 1996; Lampert & Ball, 1998). Indeveloping approaches for using case studies in the professional prepar-ation and development of teachers, broad and varying appeals are madeto the potential of case studies to: (a) promote the epistemological devel-opment of teachers, (b) support the development of teachers as reflectivepractitioners, (c) provide a means of understanding theoretical principles,while bridging the gap between theory and practice, (d) enable teachers toanalyze and reason effectively about the complex particulars of practice,(e) support the development of teachers as decision-makers, (f) provideparadigmatic exemplars of practice, and (g) overcome the limitationsof field experiences. With this wide range of rationales for using casestudies, it is not surprising to find that research studies and professionaldevelopment efforts vary considerably, depending on the ways that casestudies have been adapted for the particular instructional goals of a givenprogram.


After reviewing the historical role of case studies in law, medicine, andbusiness. McAninch (1993) argues that the role of the case study in teachereducation should be to move teachers beyond the “clinical consciousness,”which is characterized as being tied to the immediacy of practical actionand a distrust of generalization. McAninch suggests that the instructionalgoals for teacher education should include “the promotion of theoreticalunderstanding, and the strengthening of the disposition to bring theor-etical principles to practice” (1993, p. 59). That is, the goal of the useof the case study in teacher education is to promote the development oftheoretically grounded reasoning. As Shulman (1986) has also argued, acase is not simply the description of an event, but to be understood itmust be understood as an instance of a larger class, or a “case of some-thing.” Shulman sees cases as events in practice that are theoreticallyinterpreted and he states, “There is no real case knowledge without theoret-ical understanding” (p. 12). The generalized understanding of what a givenphenomena is “a case of” is shaped as the case is explained, interpreted,argued and reconstructed by those studying the case. The development ofcase knowledge goes hand in hand with the development of theoreticalunderstanding.

Others have suggested that the primary purpose of case studies is tosupport the development of critical analysis by teachers and informeddecision-making. A key characteristic of such a case study is that it isembedded in the context of teaching (and schooling) with all its concom-itant complexity, ambiguity, and incomplete information. As Spiro et al.(1988) have argued, the knowledge base of teaching is an ill-structureddomain and as such is best learned by a criss-crossing of the landscapethrough the study of cases of practice. It is precisely within the complex,ambiguous, and partially understood context of practice that teachershave to make reasoned judgments and decisions for action. Learningthrough case studies, it is argued, promotes teachers’ understanding ofthe complexities of practice and of the need to become more analyticalabout the data of classroom practice (Wassermann, 1993b). In their work,Lampert and Ball (1998) have used video and digital technology to capture,catalog, and access the records of practice of their own fifth- and third-grade classrooms over a year’s time. These records of practice, in turn,became a site for the investigation of practice by various groups of pre-service teachers. While finding that the pre-service teachers’ investigationsdid push their thinking beyond where it started, Lampert and Ball acknow-ledge that they do not now to what extent the pre-service teachers tookwhat they learned into their own practices or if the changes in their thinkingwere in large part influenced by other factors in their preparation program.


Drawing heavily from the tradition in business education, Merseth(1992) argues that cases in teacher education can function as catalystsfor the development of decision-making skills. As in business, cases thatemphasize decision-making often pose a dilemma that requires a course ofaction to be taken and consequences of actions to be analyzed within thecontext of the variables of the case. Within teacher preparation programs,unlike business, such cases rarely include the case writer’s “solution” orcommentary on what actually happened. The role and function of expertcommentary on cases is an area of case study research that is little exploredand seldom even commented on. The intent of such dilemma-based casesis to evoke multiple perspectives, an appreciation of multiple variables, andmultiple levels of analysis, that in turn lead to the serious consideration ofseveral alternative courses of actions.

While the literature on case studies is replete with examples of casesand, to a lesser extent, how they can be used (e.g., Barnett, 1998; Colbert,Trimble & Desberg, 1996; Lynn, 1999; Wassermann, 1993a, 1993b), theliterature on the effectiveness of case-based methodologies on the profes-sional development of teachers is considerably more sparse. We wouldlike to discuss briefly the results of two such studies that are particularlygermane to our work. In a study of the use of dilemma-based cases to gaininsight into the development of teachers’ reasoning, Harrington (1995)examined students’ written case analyses over the course of a semester. Shefound that students initially had difficulty dealing with the ill-structurednature of the cases, but moved toward more grounded analyses that allowedthem to see key facts and issues in the case. By the end of the semester halfof the students were “drawing clear connections between the issues in thecase, the actions that should be taken and the reasons why” (p. 209). Yet,at the same time, over a third of the students provided little substantiationfor their recommendations. Over the course of the semester, most of thestudents still did not consider how perspectives other than those of keyparticipants might be relevant. Harrington recognized the limits of writtenwork in providing insight into students’ reasoning, and in particular thatthe reasoning seen in written work may not be reflected in the work that thestudents do in the schools as teachers. The shift toward analyses groundedin the key issues and facts of the case and toward the expression of thereasons why a decision would be made suggest that case studies are usefulas sites for the development of analytic reasoning by pre-service teachers.In our study, we intend to build on this, while at the same time explicitlyextending the connections from the case study teacher’s practice into thepractice of the novice teacher. In multimedia case studies, observation offacts and issues is much more complex than with text-based cases; the data


of the case is not already laid out in the pre-structured form of text. Ratherfacts and issues are in the edited form of video, which presents the illusionof faithfully capturing all (or at least most) of what actually happened in theclassroom. In addition, many of the artifacts of the classroom (such as theteacher’s lesson plan, student work, and the background of the school) areavailable for teacher educators and/or their pre-service teachers to selectand structure in ways that make sense to them.

In a study of Copeland and Decker (1996), researchers explicitlyattempted to see how a case discussion would impact the meaning whichpre-service teachers made of a video vignette. In this study, studentswere interviewed before and after they participated in a group discus-sion on the vignette. The participants’ group discussion was not mediatedby a case facilitator; the participants’ task was to examine and discussthe lesson segment and then write a group statement that describes theimportant aspects of the lesson. These researchers found that the pre-service teachers did identity and discuss issues that might be consideredof import. However, the discussion of many of the topics and the extentto which the pre-service teachers explored implications appeared to bequite limited. It is important to note that these discussions were without thebenefit of facilitation or interaction with other groups. These researcherscontinue to argue that over one third of the topics from the case discus-sion session were adopted, transformed or created by the participants intheir description of the meaning of the case in the second interview. Thissuggests the potential of the video-based case for supporting the reflectivemeaning making of pre-service teachers. Both of these studies, as well asthe work of Lampert and Ball (1998) point to the difficulties in examiningthe development of teachers’ reasoning and the attribution of any discernedchanges to a particular event (or set of events) such as the use of a casestudy.

METHODS AND DATA SOURCES

We used a multimedia case study with a cohort group of pre-servicemathematics teachers (grades 7–12), who were concurrently studentteaching. In this section, we describe the case study materials and how theywere used, the participants in this study, the data sources and the methodsof analysis.


The Case Study Materials

We and our colleagues have developed two multimedia case studies foruse with pre-service mathematics teachers (Bowers, Doerr, Masingila &McClain, 1999, 2000). The materials of these cases were not posed asspecific dilemma for novice teachers to analyze and propose resolutions,but rather these materials were designed to be sites for investigation,analysis and reflection on the part of those teachers. We designed thesecases to capture the records of practice in a classroom in a way thatwould reflect the complexities of classroom interactions, teacher decisions,and students’ mathematical thinking. The case lessons were designedto engage the students in actively learning mathematics, generating andcritiquing mathematical arguments, and expressing their mathematicalideas about problematic situations. We chose to focus on a teacher whowas particularly reflective on her own practice and who readily expressedher understanding of student ideas and showed how she expected to usestudent thinking to further her mathematical agenda. We intended that thiswould provide impetus for the pre-service teachers to reflect on their ownpractice in light of the reflections of an experienced teacher.

The multimedia case study that was used in this study involved a four-day lesson sequence in an eighth-grade mathematics class in an urbanpublic middle school. The mathematical focus of the lesson sequencewas on making decisions by ranking and weighting data. The case studymaterials include a video overview of the school setting, the teacher’slesson plans, video of class lesson, student written work, and a videojournal of the teacher’s reflections and anticipations on each lesson, andthe transcripts of all video. In addition, to help use these materials, thecase includes an issues matrix with links to video clips and text organ-ized around four sets of issues: planning, facilitating, student thinking, andmathematical content and context. A search tool allows the user to searchthe transcripts. A book marking feature supports the tagging of video clipsand a notebook allows the pre-service teacher to make annotations as s/heis using the materials. Study guide questions, mathematical activities forthe pre-service teachers, and bibliographic resources are also included. Theclassroom video contains about 40 minutes of edited video for each day ofthe four-day lesson sequence.

The first lesson begins with the teacher introducing the “Sneakers”problem in which student groups are asked to rank criteria for selectingsneakers. The video contains edited segments showing how three groupsof students ordered the criteria. The teacher then poses the crucial questionfor the second part of the problem – how to combine the ranked lists of datathat each of the groups generated. This is followed by video of the same


three groups of students working on this problem and then the whole-classdiscussion of different groups’ solutions. The second lesson begins with anintroduction of the “Crime” problem, where the students are asked to makea recommendation on the city budget based on whether or not their city,Nashville, is sufficiently safe. The students apply their ideas about rankingand combining data to develop a system for determining if Nashville issafe. In the third and fourth lessons, the teacher builds on the students’notion that not all crimes are equally important to introduce the concept ofweighting. She introduces a second data set and a way of symbolizing thestudents’ systems as they compare and contrast various approaches.

The video journal contains the teacher’s anticipations of each lessonand her reflections on the lesson that just occurred. The teacher discussesher concerns in planning the lessons, how she will build on studentunderstanding, her approaches for engaging all students in the lesson,her thoughts as she is monitoring small group activities, ideas forassessing how well students are understanding the mathematical issues,and her strategies for orchestrating whole class discussion and developingclassroom norms for participation, explanation, and justification.

An “issue matrix” was designed to provide some structured accessto the episodes in the case; the matrix was organized about the fourmain issues (a) planning the lesson, (b) facilitating group interactions andwhole class discussion, (c) understanding and using student thinking in thelesson, and (d) the mathematical content and context of the lessons. Thematrix consisted of links to selected portions of the classroom video andthe teacher’s reflections for each of these issues (and related sub-issues)across the four days of the lesson. However, these tools for accessing therecords of practice in this classroom were not intended as an exhaustivemeans of investigating any particular issue, but rather were intended as apossible starting point for a pre-service teacher’s investigation of practice.

THE SETTING

The participants in this research study were nine pre-service mathema-tics teachers (grades 7–12) who were completing an eleven-week, full-daystudent teaching experience. They were enrolled concurrently in a seminarclass that met once a week for two hours in the afternoon after studentteaching. These student teachers had completed a six-week, half-daystudent teaching experience during the previous semester while concur-rently enrolled in a mathematics methods course. Prior to the first studentteaching experience, they had completed sixty hours of field experiencedivided between at least two different classrooms. Three of these pre-


service teachers were graduate students who had already earned an under-graduate degree in mathematics and were completing a Masters degreein mathematics education that included a teacher certification program.The other six pre-service teachers were undergraduate students who wereearning an undergraduate degree in mathematics while completing ateacher certification program. This group of pre-service teachers formeda cohort in the program and had been together as a group in both studentteaching semesters and in education classes.

The seminar (taught by the first author) engaged the student teachers inthinking about and discussing issues of assessment, teaching and learning,and developing one’s teaching practice. Early in the course, the pre-serviceteachers identified goals for themselves that they were addressing in theirown student teaching practice. For five weeks near the end of the seminarclass, the multimedia case study was used. The pre-service teachers wereasked to identify a specific issue that they had been working on in their ownpractice and that they saw addressed in the case study teacher’s practice.They traced this issue throughout the case study and through their ownteaching practice. For each of the first four weeks of the five-week periodthat the multimedia case study was used, these pre-service teachers viewedthe video on the multimedia case study from one day of the four-day lessonsequence and the teacher’s journal. The pre-service teachers watched thevideo separately from the seminar class, with an assigned journal question.This viewing of the classroom data and the teacher’s reflections became thebasis for their discussion in the seminar class. This culminated in a paperdiscussing this issue and giving evidence from the case study and fromtheir own teaching practice. During the fifth week, the pre-service teacherspresented their papers in class and used video from the multimedia casestudy (using a book marking feature) and video and/or dialogue examplesfrom their student teaching experience as evidence.

Data Sources and Analysis

The data for this study consisted of transcripts of all class discussions, tran-scripts of the video of the presentations and the students’ bookmarks in themultimedia case study, the students’ written journal assignments related toeach day of the case study, the students’ final papers discussing the issuethat they linked to their own practice, a questionnaire concerning the casestudy completed by the students at the end of the semester, a journal keptby the instructor of the seminar and field observations made by one of theresearchers.

The analysis of the data was conducted in three phrases. The first phaseconsisted of developing a coding scheme by each researcher independently


coding, using inductive methods, the transcripts from the first two classdiscussions. The detailed codes were compared and notes generated thatdescribed the meanings of each code. The detailed codes were classifiedinto larger categories that reflected the emergent issues for the students inthe class. The researchers then continued with the independent coding ofthe remaining class transcripts. Codes were compared and discrepancieswere resolved by reference to the earlier coding schema. We carefullyidentified those issues that were raised by the teacher educator and distin-guished them from the issues raised by the pre-service teacher. In thisway, we were able to make the distinction between pre-service teachers’responses to the case and the responses directly related to the goals of theteacher educator. In this phrase, we also identified pre-service teachers’types of reasoning about the case study.

In the second phase of the analysis, we took the issues that were iden-tified in the first phrase and completed a detailed analysis of the issuesof concern to the pre-service teachers from the perspective of each pre-service teacher. The three primary sources of data for this analysis werethe coded transcripts of the class discussions, the transcript of the students’presentation and the written paper by the student. Each researcher againindependently coded the transcripts using revised codes. The codings werecompared and a small number of new categories were added. Discrepan-cies were resolved by a cross-comparison with previous codes. The finalwritten papers of the students were first coded according to the previous,detailed coding scheme. This was followed by the creation of writtensyntheses of their issue in practice and their reasoning about that issuefor each pre-service teacher.

In the final phase of the analysis, we compared the synthesis of eachpre-service teacher’s issue and reasoning with the issues and reasoningthat had been identified in the class transcripts. We examined the students’journals and the instructors’ journals for disconfirming evidence or alter-native perspectives on the data. The syntheses of the students’ issues andreasoning were then categorized into clusters of common issues and differ-ences. We went back to the coded transcripts to seek other supportingor disconfirming evidence where the pre-service teacher made explicitconnections to his/her own practice. Note that in coding the data, wecounted only explicit references to their own teaching practice as consti-tuting links to their practice. This analysis then became the basis for theresults that follow in the next section.


TABLE I

Categorization of Pre-service Teachers’ Links from Case Teacher Practice to EmergingPractice

Pedagogical Issues Not Pedagogical Issues from a Mathematical Issues from aLimited to Mathematics Mathematical Perspective Pedagogical Perspective

1. Keeping students 1. Checking for student 1. Introductions andparticipating in class by mathematical transitions to mathematicalkeeping comfort levels high understanding ideas

2. Facilitating classroom 2. Role of questioning ininteractions promoting student

mathematical thinking

3. Use of student responses infurthering the teacher’smathematical agenda

RESULTS

Throughout the four one-hour class discussions and the writing andpresentations of their issue papers, the nine pre-service teachers made avariety of links between the case study teacher’s practice and their ownemerging practices. When analyzing the data, we found that the data fitinto three categories: (a) pedagogical issues, (b) pedagogical issues froma mathematical perspective, and (c) mathematical issues from a pedago-gical perspective. We note that no issue arose that we found to be purelymathematical. We categorized discussion about the case study teacher andother issues raised that are reported in Doerr and Masingila (2001) andThompson and Doerr (2001). Table 1 summarizes our categorization ofthe links made by the pre-service teachers.

Table 1 reports the six issues that were identified as primary findings.There were several other secondary findings that arose in class discussions;however, we discuss here only the primary findings that came from theissues chosen by the pre-service teachers. In the section that follows, weillustrate each of these six issues from the perspective of the pre-serviceteachers.

Pedagogical Issues not Limited to Mathematics

We categorized the links made by the pre-service teachers to their ownpractice as pedagogical issues when they dealt exclusively with issuesrelated to teaching, independent of teaching a specific content such as


mathematics. The issues that we found were (1) student participation, and(2) facilitation of classroom interactions.

The issue of student participation was chosen by one of the pre-serviceteachers, Mark, as the issue he traced through his student teaching exper-ience and the case study. We describe this issue as keeping studentsparticipating in class by keeping their comfort levels high. Mark wantedstudents in his class to be active participants. He recognized that studentsoften get off track in solving a problem or become confused, and he argued,“[I]t is entirely in the teacher’s hands to make sure that the misunderstoodstudent does not shut down and end his or her participation”.

Mark laid out several approaches for keeping students participatingthat were based on his reflections on his own practice and the case studyteacher’s practice. We described his approaches in terms of two inde-pendent courses of action: (a) supporting student affect and ideas, and (b)following the teacher’s agenda. Mark’s overriding concern was the first ofthese two. He noted in class that the case study teacher very frequentlygave positive feedback to students:

Did you hear what she said right at the end? She said, “But that is a really good point.” Shesaid basically you don’t know what you are talking about but that is a really good point.So she uses positive reinforcement and she, I don’t know if many of you noticed this – I’venever heard someone say, “That is a really good point” more in my life than in watchingthis video. No matter what they can say, she says “That is a very good point but . . .” andthat is the way she goes on to talk about that. But it seems to work, even when the studentsare off, she would say, “That’s a really good point.”

Mark’s discussion indicates that he valued having students feelcomfortable in the classroom and that student comfort is a prerequisite forthe teacher being able to address student mistakes in class. His primaryconcern is that students do not “shut down” but keep participating inclass. The idea implicit in this is that keeping students participating willresult in increased or improved learning; however, Mark did not discussthis idea explicitly and he did not draw any relationships between studentparticipation in a lesson and its effect on student learning.

A secondary concern that Mark discussed was the need to follow theteacher’s agenda, but still keep students’ comfort level high. Mark notedthat the case study teacher had an agenda for the lesson sequence but alsoinvolved students by using their ideas, and one way in which she did thiswas by using positive feedback and then moving on. This became a keypoint for Mark’s further analysis of ways to keep students participating.In discussing his approaches, Mark made several connections between hisown practice and that of the case study teacher. He described how he usedthe method of telling a student that he/she was wrong by using humor.He carefully qualified the use of humor by noting that the prerequisite


comfort level must be present. Mark also described how he used “positivereinforcement” to tell a student that he or she had a good idea but it was notcorrect; this drew him closer to the students. Another method he describedwas to go along with a student’s idea and then to work with the rest ofthe class to see if they agreed. Mark noted that students found it easierto hear that they were wrong from a peer rather than from the teacher.We see this as evidence that Mark learned from adapting a strategy heobserved in the case study teacher’s practice, and found a way to resolvethis pedagogical issue by shifting the authority from the teacher to thestudents for evaluating their thinking.

Building specifically on the actions of the case study teacher that firstcaught his attention, Mark noted that he used the idea of “moving on”(keeping to the teacher’s agenda) in his own teaching and the way hehandled validating a student’s contribution with the need to move on. Healso noted that he did not often use a method that the case study teacherused of taking something from a student who is confused and clearing itup by basically stating the idea correctly. Mark stated that he would ratherguide students through by providing hints but found a need to use thismethod when time was short. For Mark, the value of student participationand comfort was a given. To Mark, the case study teacher’s practice seemedto reinforce that value as well as help him devise a new strategy – shiftingthe responsibility for resolution to the student – about “moving on” andproviding positive feedback and support to students.

The second pedagogical issue link made by the pre-service teacher totheir own practice involved the issue of facilitating classroom interac-tions. We illustrate this link with an example of Molly, another pre-serviceteacher. The issue that Molly selected to examine through her studentteaching experience and the case study was the role of the teacher in gettinggroups to work effectively. She commented:

[T]hroughout my teaching experiences, I have realized that getting groups to work effec-tively is not as easy as it sounds. By examining the case study and learning from myown teaching, I have realized there are some techniques to use to make group work in theclassroom more efficient.

Molly described some general guidelines and principles for practicerather than specific “how to’s” to address problem situations. Mollyexplained that effective group work could be accomplished by attendingto three general aspects of practice: “the role that the teacher plays inthe classroom, the size of the groups, and the way that the groups areformed.” Molly then made several direct connections from the case studyteacher’s practice to her own practice. Two aspects of practice that thecase study teacher emphasized in her reflections included establishing


classroom norms for participation and using student ideas to build thelesson. In her practice, Molly described how she tried to take on the dualrole of monitoring the group work for their ideas as well as establishingnorms for group participation. She described how she did this when herstudents were using motion detectors to complete an investigation intomotion. Molly also described how she laid out three specific roles for groupmembers to assume and rotate. She explained how she realized the import-ance of allowing students to come up with their own solutions instead ofgiving them the answers.

In analyzing the case study, Molly chose to focus on one particulargroup for the four-day lesson sequence and analyzed in great depth theirinteractions with each other and with the case study teacher. Molly notedthat the case study teacher had discussed that she had chosen to havestudents work in groups of three so that she had a total of six groups.This, in turn, created a more manageable number of group products. Mollyobserved that the interaction patterns of the group of students she wasclosely monitoring changed when on one day a student was added to thegroup because the student’s other group members were absent. She notedthat one student appeared to stop participating when a fourth person wasadded to the group. Molly connected this observation to her own practiceby noting that for the motion detector lab, she had students work in groupsof three, as larger groups would not have worked as well. She noted thatthe decision of how many students to have in a group was based upon theactivity, the social interaction of the students, and the goals the teacher hasfor the activity.

Molly’s discussion in her issue paper indicates that through her analysisof the case study teacher’s interaction with one group and her analysis ofher own teaching practice, she came to several realizations about facil-itating effective group work concerning the role of the teacher, the sizeof the groups, and how groups are formed. The case study supportedher analysis of her issue by allowing her to analyze and reflect on thecase study teacher’s actions and interactions with one group, and to makeinferences about the teacher’s reasons for making the decisions she did.

Pedagogical Issues from a Mathematical Perspective

We categorized links made by the pre-service teachers as pedagogicalissues from a mathematical perspective when they dealt with issues relatedto teaching mathematics and the teaching aspect was primarily the focusof the issue. We found three related sets of links in this category: (a)checking for student mathematical understanding, (b) the role of ques-tioning in promoting student mathematical thinking, and (c) using student


responses in furthering the teacher’s mathematical agenda. An exampleof a link in the first category was the issue Susan chose to trace throughher student teaching experience and the case study – checking for studentmathematical understanding. Susan stated, “Throughout the case study,the teacher continually checks for student understanding in a variety ofways, some similar to those I have used.” She discussed strategies thecase study teacher used to check for understanding and made a numberof connections to her own practice. Susan described methods she used inher student teaching to check for student understanding, such as asking forexplanation, and asking students to give the main ideas in the lesson at theend of class. She explained how she worked on using questioning to assessstudents’ work in groups in order to facilitate the whole class discussionand assess the individual student’s progress as well as the group’s progress.Susan noted that she also “realized the need to check on my students’understanding also by listening to their questions” and said that she “sawthe need to be aware of their understandings so I was able to achieve thegoals of the lesson, regardless of how the lesson actually played out.”

Susan described that she found she needed to adjust homework assign-ments based on students’ understandings. This included an examinationof how she used questioning to get input from students, which incorpor-ated “[trying] to ask questions that required higher level thinking and. . . [extending] a problem situation to assess and establish their under-standing”. She discussed how it did not seem to be problematic for the casestudy teacher to proceed with a task after giving directions, whereas Susansaid that she found that she needed to check her students’ understanding ofthe requirement of the task before beginning the task in order for the taskto go well. She noted that she “concentrated on checking my students’understanding by seeking their explanations of agreement or disagreementin class discussions” and that this was a strategy that the case study teacherhad used with much success.

As she discussed in her issue paper, Susan came to understand thatsimply asking students if they had questions or if they understood was notenough to assess their understanding. She came to realize that students,and particularly she noted that middle school students, will not offer thatinformation to a teacher, and that the “do you understand” type of ques-tions does not elicit the information she was interested in gaining as ateacher:

This issue I chose to work on was understanding, checking for understanding. I thought itwas an important issue, something that I looked at in my student teaching, and I thought itwas important because basically students are just not going to offer that information to you.They’re not going to tell you readily that they specifically understand or don’t understandsomething. Even questions like, “Do you understand?”or, “Do you have any questions?”


are not going to get that information from them. If I didn’t realize that before the studentteaching experience, I do now. Middle schoolers are not going to raise their hands; they arenot going to tell you, “I don’t understand. Please go over that.” They will not risk that inthe classroom setting.

As with the other pre-service teachers in this cohort, this was an issuethat Susan was working on prior to viewing the case study. She beganto see in her student teaching that asking if students had questions or ifthey understood did not help her in assessing students’ understanding. Thecase study allowed Susan to scrutinize an experienced teacher’s methodsof checking for understanding in great detail. Because of the teacher’sreflections and the classroom video showing the teacher’s interactions withlarge and small groups of students, Susan was able to see many instancesand ways in which the case study teacher checked if her student wereunderstanding the material. The case study provided a context throughwhich she could reason about her issue (assessing student understanding),and draw comparisons with her own teaching practice. Susan saw the casestudy teacher as quite successful in continually assessing her students’understanding of the activities, and was able to identify strategies the casestudy teacher used to analyze her own practice and ways she did and couldassess students’ understanding.

A second pedagogical issue from a mathematical perspective link madeby several student teachers to their own practice involved the role of ques-tioning in promoting student mathematical thinking. For example, Jeffcategorized three levels of questioning – low, middle and high – and usedexamples from his own practice and from the case study teacher’s practiceto describe the kinds and purposes of such questions. Jeff made severalconnections to his own practice as he expressed his concern about his useof too many low-level questions. Jeff noted in contrast that the case studyteacher used many levels of questioning but that he did not use as many.he referred to his own practice of trying to use middle level questions tohave students give more extended explanations and of borrowing the casestudy teacher’s question, “what is your job?” when he put a problem onthe board in order to engage students in thinking about what is involved insolving the problem.

Jeff made an explicit connection between the case study teacher’s useof high level questions and his own, but in doing so, he revealed that hehad a serious misunderstanding of the nature of conjectures and the roleof data vis-à-vis mathematical argumentation. He described an activity inwhich he had students form conjectures about the role of the discriminantin predicting the number of solutions for a quadratic function. He statedthat he told his class “a conjecture is something that will always be true for


the data they have” and that he had the students check their conjectures tosee if they were always true with the equations that were given. Thus, whileJeff engaged his students in a form of higher-level questioning (formingand testing conjectures), this activity revealed that his own understandingof what it means to prove that something is always true is flawed.

We found that the case supported Jeff in his need to address the levelof questioning he was doing in his own practice. In some instances, headopted the case study teacher’s questions directly. He also acknowledgedthat compared to the case study teacher, his level of questioning was lowerthan what he felt it should be. We found that Jeff developed a way ofthinking about questioning through three levels that was powerful for himin terms of thinking about his enacted practices.

Other pre-service teachers noted similarities between the ways thecase study teacher used questioning and the way they themselves usedquestioning. Rich stated:

I thought I did similar to her, like the way she . . . is always saying, “What do you thinkwe would do if we did this?” It seems like she hardly ever just says to them, “This is whatwe do. This is what you do.” She is always trying to use questions to kind of get themto discover what she wants them to find out, and I think that is, I try to do that a lot also. . . [N]ormally what I am trying to say is . . . if I want the students to get a point . . . I askquestions . . . I want them to be like they are discovering, and they are thinking about itmathematically themselves.

From these comments during class discussions and his issue paper,we found that the case study enabled Rich to think more deeply aboutthe purposes of questions. In his issue paper, he focused on the role ofquestioning and developed three categories of questions that he felt wererelated to the teacher’s goal for the lesson. He framed these categoriesin terms of the purposes of the questions: (a) for discovery learning, (b)leading to a discussion of ideas or methods, and (c) as ways of checkingfor student understanding. Rich’s ways of thinking about the role of ques-tioning were linked to his observations about how the case study teacheruses questioning and emerged from his thinking about his own use ofquestioning. This reflection led to a careful analysis of how questioningis closely connected to what the teacher is trying to achieve at differentpoints in the lesson.

A third link in this issue category is the use of student responses infurthering the teacher’s mathematical agenda. The case study teacher wasexplicit in her reflections and in her lesson plans about the importance toher of using student ideas as the basis for building the sequence of lessons.Two of the pre-service teachers selected this issue to examine. We presentone here. In her issue paper and presentation, Jennie asserted that it isimportant to pay close attention to individual student responses.


Jennie made several connections to her own practice. She described anexample using motion detectors for a ball toss activity in which she usedstudent responses to guide the lesson. She discussed another lesson shetaught in which a student saw a 3–4–5 right triangle where she herself hadnot seen it and that that gave her insight into the student’s thinking as wellas a mathematical insight for herself and the rest of the class. She describedan example in her own class where a student’s explanation of why a graphlooks the way it does helped other students’ understanding of the situation.She also discussed that she had experience using student responses, evenwhen they were incorrect, to help students sort mathematical ideas out forthemselves.

The issue of using student responses often took the form of thetension between balancing the teacher’s agenda with the desire to buildoff students’ ideas and responses. This tension was especially evident inthe case study, wherein the pre-service teachers attempted to make senseof how and why the case study teacher chose to follow her “mathematicalagenda” while attending to and using students’ thinking.

In analyzing Jennie’s issue paper, we found that she framed her ownteaching through the case study teacher’s practice and made sense of thecase study teacher’s practice by reflecting on her own. Jenny identified andanalyzed issues related to student responses in both arenas. She saw thetensions and complexities in the use of student responses; for example,she noted it is hard “to keep everything the students are saying straight.Sometimes it’s hard to decide when to address which things, if you want toaddress them all”. We found that Jennie valued what students say duringclass, but she also recognized the difficulties in using student responses.She observed “enhancing the lesson by using what students say is a skillmastered over time.” Jennie also observed these difficulties in the casestudy teacher’s practice. For example, she wrote in her paper:

It is clear to me that Jeff is having trouble understanding the concept of rate. In her explan-ation, the teacher never mentions the word rate to make this connection of Jeff. This is akey area of the lesson; she should be sure he and others fully understand the purpose ofthese rates before moving to the next question.

The tension between balancing the teacher’s agenda with the desire tobuild on students’ responses was a major point throughout the seminardiscussions related to the case study. The pre-service teachers discussedthis issue with some connections to their own teaching, but primarily theydiscussed the tension as they saw it evidenced in the case study. Recog-nizing this tension and seeing it in many situations within the case studyhelped the pre-service teachers to come to grips with the complexities ofteaching mathematics.


Mathematical Issues from a Pedagogical Perspective

We categorized links made by the pre-service teachers to their own practiceas mathematical issues from a pedagogical perspective when the mathe-matics aspect was the primary focus of the teaching issue. We foundone link that a pre-service teacher made to his own practice. The issuethat James chose to examine through his student teaching and the casestudy was introductions and transitions to mathematical ideas. Jamesclaimed, “In an effort to foster mathematical thinking and to encourageconceptualization of mathematics, there need to be strong introductionsand transitions to content material.” He supported his claim by appealingto the National Council of Teachers of Mathematics (NCTM) CurriculumStandards (NCTM, 1989), citing Standard 4: Mathematical Connections.He used this lens to examine the transitions and introduction of contentmaterial as evidenced in the case study and in his own teaching practice.

James made several links to his own practice. He asserted that he “foundthat it was extremely important to be creative in teaching content material”and gave an example of how he used “the concept of an assortment ofpersons whose consecutive tasks would help in the process of buildingand completing a move into a house” to introduce his students to theconcept of order of operations. He described how he “used the idea ofa situation comic in order to portray a scenario that would utilize algebrawithout the students initially recognizing this.” James identified that themain problem he “encountered when striving to attain these worthy object-ives [Standard 4: Mathematical Connections] [was] the situation where thestudents [became] too caught up in the context of the scenarios created,[leading to] confusion in the classroom.”

This was the same issue that the case study teacher addressed in thesame language when discussing the context of the Sneakers problem andthe Crime problem that were the focus of the lesson sequence in the multi-media materials: “I’ve learned that if I am not careful about the contextthat wraps around the problem, then I spend a great deal of my time eitherunwrapping or wrapping the problem the whole time.” Thus, while Jamesportrayed his discussion of introducing content largely in ways that helpedstudents make connections in terms of the NCTM standards, we found astrong similarity between his concerns and strategies and those used in thecase study. In this way, it appears that the case study supported his effortsto understand his practice in terms of meeting the NCTM standards.


DISCUSSION

In analyzing the class discussions, the pre-service teachers’ written work,and their oral presentations, we found that the nine pre-service teachersused the case study teacher’s practice to support their analysis and reflec-tion on their own emerging practices. The pre-service teacher’s reflec-tions were in three broad categories of practice: (a) pedagogical issuesnot limited to mathematics, (b) pedagogical issues from a mathematicalperspective, and (c) mathematical issues from a pedagogical perspective.In each of these categories, the pre-service teachers were able to analyzethe case study teacher’s practice over four days of lessons, in whole classand small group interactions, in introducing an activity, in facilitating classdiscussion, and in her reflections before and after each class. The useof a multimedia case enabled pre-service teachers to delve more deeplyinto issues revealing the complexities of teaching through guidance by theseminar instructor. The pre-service teachers commented that in analyzingthe case they could (a) stop and replay video clips again and again, (b)move easily among the classroom video, the teacher’s reflections on thelessons, and the teacher’s lesson plans, (c) search for uses of particularwords or phrases (such as “explain”) used by the case study teacher, (d)use the bookmarking features to mark places of “evidence” for their issue,and (e) use the issues matrix to focus their observations.

Unlike observational field experiences where inexperienced teachersare on their own to observe and interpret practice, the analysis of thecase study provided a common experience for these pre-service teachersto observe and interpret. This shared experience enabled the pre-serviceteachers to analyze pedagogical issues in ways that appeared to go beyondthe usual concerns with classroom management issues and allowed them tofocus instead on more complex classroom issues. In particular, we foundthat the pre-service teachers began to understand the issue of supportingstudent ideas and thereby keeping participation at a high level, while at thesame time needing strategies that would simultaneously allow the teacherto provide this positive, affective support for the students and to move thelesson forward to reach particular mathematical goals.

As we argued earlier in this paper, a critical difficulty for inexperi-enced teachers is to understand how to use student thinking in teachingmathematics. In the category of pedagogical issues from a mathematicalperspective, all of the issues that the pre-service teachers raised can beseen as related to the notion of how to use student thinking while teachinga lesson. The fact that all of these pre-service teachers’ issues relate to thisimportant aspect of learning how to teach points to the power of this multi-media case study and how the pre-service teachers experienced the case.


The case study teacher’s video journals allowed the pre-service teachers tohear her discuss how she was using students’ ideas, and the video of threegroups of students working in small groups over four days provided thepre-service teachers with multiple opportunities to try to understand thestudents’ thinking and to see how the case study teacher made use of thesestudents’ ideas in class discussions.

An important pedagogical insight that all of the pre-service teachersexpressed was understanding the tension between moving forward with theteacher’s agenda and using student ideas and responses as the basis for thelesson. The pre-service teachers identified this dilemma in the case studyteacher’s practice, where she guided the students to bring up ideas that shewanted and guided others away from methods that she did not want. At thesame time, they recognized how difficult it was to “keep everything that thestudents are saying straight.” The pre-service teachers recognized that theyneeded to make decisions about when to move forward with a lesson andwhen to stop at critical points before moving on. The case study providedan opportunity to see how an experienced teacher made these decisionsand to hear the reasons she gave for her decisions at the time and howshe reflected back on those decisions. As in Harrington’s (1995) study, wefound that the pre-service teachers’ analyses were grounded in the facts ofthe case, but we found in addition that the pre-service teachers were ableto express their reasoning in terms of their own practice.

We found that none of the issues that were linked by the pre-serviceteachers to their own practice were primarily mathematical issues. Aswe have just noted, the mathematical issues that were raised appearedto emphasize teaching aspects, rather than the mathematical ideas them-selves. This result with pre-service grades 7–12 teachers confirms Lampertand Ball’s (1998) findings with pre-service elementary teachers. Sincethese pre-service teachers were mathematics majors and beyond, they didnot see the mathematics content of grades 7–12 as problematic. However,we found several instances where their case study connections revealedtheir faulty mathematical thinking. In one case, in his analysis of levelsof questions, a pre-service teacher revealed his own misunderstandingsof the nature of conjecture and proof and the role of data as evidence.In other cases, we found that the pre-service teachers stopped short of adeeper analysis of the difficult conceptual issues around the developmentof a robust understanding of the concept of rate. In still other instances,while the pre-service teachers appreciated the power of certain mathemat-ical examples in the case study, they had not yet had sufficient experiencewith such examples in their own student teaching so that they could makespecific connections.


In our analysis of the pre-service teachers’ use of the multimediacase study we found that the case study enabled these novice teachersto understand better through reflection the tacit and implicit knowledgethat teachers have when reflecting on reflection-in-action (Schön, 1995).The case study teacher reflections, the classroom video, and the lessonplans, coupled together with the assignment of tracing an issue through thecase and their own teaching practice, promoted the pre-service teachers’reflection and helped make explicit the action strategies, the assumptions,and aspects of the problem-settings that were implicit in the case studyteachers’ reflection-in-action. We found that this provided impetus forthe pre-service teachers to reflect on their own practice in light of thereflections of an experienced teacher. In each of the illustrations above,there is evidence that the pre-service teacher analyzed the issue and madegeneralizations that go beyond the case.

We recognize that this reflection was promoted not only by the multi-media case study, but also by the task given to the pre-service teachers oftracing an issue through the case and their own teaching practice. Thus,the way a case study is used as directed by a teacher educator may greatlyinfluence what benefit the pre-service teachers derive from the case. Wecannot make claims about whether the reflectiveness on the part of the pre-service teachers prompted by the use of the multimedia case study changedtheir teaching; we did not investigate this. This may be an area for furtherresearch.

In conclusion, we found that having the multimedia case study as a siteto investigate, analyze, and reflect on another teacher’s practice supportedone cohort of pre-service teachers in (a) focusing on issues that weremeaningful to their own teaching practice, and (b) thinking critically aboutanother teacher’s practice which in turn promoted critical thinking abouttheir own practice. The task of selecting and tracing an issue in both theirstudent teaching practice and in the case study teachers’ practice resulted inthe pre-service teachers making claims about their issues that they substan-tiated with evidence from the case study and their own student teaching.Through this process of forming a connection between the two practices,the pre-service teachers identified issues that were of concern to them andwere able to use their reflections on the case study teacher’s practice toprovide a perspective on their own emerging practice.

We found that the pre-service teachers were able to share perspectiveson a common practice and use that practice to highlight some of thedilemmas and tensions found in teaching. In particular, this group of pre-service teachers focused on the difficulties encountered when trying to usestudent thinking and to follow their own mathematical goals in a lesson.


They were also able to frame many of the issues that they encounteredin their own practice (such as checking for student understanding and theuse of questioning) in terms of their analysis of the case study teachers’practice. In other instances, the case study teacher provided strategies thatthey were able to adapt and extend in their own teaching. The multimedianature of the case allowed students to gain insight in ways that may not bepossible with written or video-only cases. In this way, we found the casestudy enabled the pre-service teachers to begin to understand the complex-ities of practice, in particular the use of student ideas when teaching, andreflect upon their own teaching practice through the case study teacher’spractice, and enabled us to understand better the pre-service teachers’understanding of teaching, learning and mathematics.

This research study has contributed to the growing body of research onthe effectiveness of case-based methodologies on the professional devel-opment of teachers. We found evidence that the multimedia case study andthe assigned task promoted pre-service teacher reflection on reflection-in-action (Schön, 1995). Furthermore, as teacher educators, we have gainedmore insight into pre-service teacher development and how multimediacase studies may be a vehicle for the transformation of educational practice(Cohen, 1998; Shulman, 1992) to the extent that they can provide a visionof educational practice that is beyond the limits of field experience andlong years as learners in classrooms where a primary strategy for teachingis telling. This, in turn, allows novice teachers access to experiencedteachers’ reflection-in-action.

NOTE

1 The research was funded by a National Science Foundation grant (REC-9725512)awarded to Janet S. Bowers (San Diego State University), Helen M. Doerr (SyracuseUniversity), Joanna O. Masingila (Syracuse University), and Kay McClain (VanderbiltUniversity). Any opinions, findings, and conclusions or recommendations expressed inthis paper are those of the authors and do not necessarily reflect the views of the NSF.

REFERENCES

Ball, D.L. (1993). Halves, pieces, and twoths: Constructing representational contexts inteaching fractions. In T. Carpenter, E. Fennema & T. Romberg (Eds.), Rational numbers:An integration of research (pp. 157–196). Hillsdale, NJ: Erlbaum.

Ball, D. & Wilson, S. (1990). Knowing the subject and learning to teach it: Examiningassumptions about becoming a mathematics teacher, Technical Report. National Centerfor Research on Teacher Education: Michigan State University. Report 90–7.


Barnett, C. (1991). Building a case-based curriculum to enhance the pedagogical contentknowledge of mathematics teachers. Journal of Teacher Education, 42(4), 263–271.

Barnett, C. (1998). Mathematics teaching cases as a catalyst for informed strategic inquiry.Teaching and Teacher Education, 14(1), 81–93.

Borko, H., Mayfield, V., Marion, S., Flexner, R. & Cumbo, K. (1997). Teachers’ developingideas and practices about mathematics performance assessment: Successes, stumblingblocks, and implications for professional development. Teaching and Teacher Education,13, 259–278.

Bowers, J., Doerr, H.M., Masingila, J.O. & McClain, K. (1999). Multimedia case studiesfor teacher development: Case I: Sneakers [CD-ROM].

Bowers, J., Doerr, H.M., Masingila, J.O. & McClain, K. (2000). Multimedia case studiesfor teacher development: Case II: Making weighty decisions [CD-ROM].

Cohen, D. (1998). Afterword: Experience and education: Learning to teach. In M. Lampert& D.L. Ball (Eds.), Teaching, multimedia, and mathematics: Investigations of realpractice (pp. 167–187). New York: Teachers College Press.

Colbert, J., Trimble, K. & Desberg, P. (Eds.) (1996). The case for education: Contemporaryapproaches for using case methods. Needham Heights, MA: Allyn & Bacon.

Conney, T. & Krainer, K. (1996). Inservice mathematics teacher education: The import-ance of listening. In A.J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde(Eds.), International handbook of mathematics education (pp. 1155–1185). Dordrecht,Netherlands: Kluwer Academic.

Cooney, T.J. (1999). Conceptualizing teachers’ ways of knowing. Educational Studies inMathematics, 38, 163–187.

Copeland, W.D. & Decker, L.D. (1996). Videocases and the development of meaningmaking in pre-service teachers. Teaching and Teacher Education, 12(5), 467–481.

Doerr, H.M. & Lesh, R. (in press). A modeling perspective on teacher development.In H.M. Doerr & R. Lesh (Eds.), Beyond constructivism: A models and modelingperspective on mathematics teaching, learning and problem solving. Mahwah, NJ:Lawrence Erlbaum Associates.

Doerr, H.M. & Masingila, J.O. (2001). Unpacking a case study: Understanding teachereducators as they understand their pre-service secondary teachers. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Annual Meeting of the International Group forthe Psychology of Mathematics Education (Vol. 2, pp. 369–376). Utrecht, Netherlands:Freudenthal Institute, Utrecht University.

Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D. & Agard, P. (1993). Concep-tual knowledge falls through the cracks: Complexities of learning to teach mathematicsfor understanding. Journal for Research in Mathematics Education, 24(1), 8–40.

Even, R. & Lappan, G. (1994). Constructing meaningful understanding of mathema-tics content. In D.B. Aichele & A.F. Coxford (Eds.), Professional development forteachers of mathematics (pp. 102–115). Reston, VA: National Council of Teachers ofMathematics.

Feltovich, P.J., Spiro, R.J. & Coulson, R.L. (1997). Issues of expert flexibility in contextscharacterized by complexity and change. In P.J. Feltovich, K.M. Ford & R.R. Hoffman(Eds.), Expertise in context: Human and machine (pp. 125–146). Cambridge, MA:AAAI/MIT Press.

Fennema, L., Carpenter, T., Franke, M., Levi, M., Jacobs, V. & Empson, S. (1996). Alongitudinal study of learning to use children’s thinking in mathematics instruction.Journal for Research in Mathematics Education, 27(4), 403–434.


Harrington, H.L. (1995). Fostering reasoned decisions: Case-based pedagogy and theprofessional development of teachers. Teaching and Teacher Education, 11(3), 203–214.

Lampert, M. & Ball, D. (1998). Teaching, multimedia, and mathematics: Investigations ofreal practice. New York: Teachers College Press.

Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation.Cambridge, England: Cambridge University Press.

Leinhardt, G. (1990). Capturing craft knowledge in teaching. Educational Researcher,19(2), 18–25.

Lynn, L.E. (1999). Teaching and learning with cases: A guidebook. Chappaqua, NY: SevenBridges Press, LLC.

McAninch, A.R. (1993). Teacher thinking and the case method: Theory and futuredirections. New York: Teachers College Press.

Meredith, A. (1993). Knowledge for teaching mathematics: Some student teachers’ views.Journal of Education for Teaching, 19(3), 325–338.

Merseth, K. (1992). Cases for decision making in teacher education. In J. Shulman (Ed.),Case methods in teacher education (pp. 50–63). New York: Teachers College Press.

Merseth, K. & Lacey, C.A. (1993). Weaving stronger fabric: The pedagogical promise ofhypermedia and case method in teacher education. Teacher and Teacher Education, 9(3),289–299.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation stand-ards for school mathematics. Reston, VA: Author.


National Research Council (1990). Reshaping school mathematics: A philosophy andframework for curriculum. Washington, D.C.: National Academy Press.

Schön, D.A. (1983). The reflective practitioner: How professionals think in action. NewYork: Basic Books.

Schön, D.A. (1987). Educating the reflective practitioner: Toward a new design forteaching and learning in the professions. San Francisco: Jossey-Bass.

Schön, D.A. (Ed.) (1991). The reflective turn: Case studies in and on educational practice.New York: Teachers College Press.

Schön, D.A. (1995, Nov./Dec.). The new scholarship requires a new epistemology. Change,27, 26–34.


Shulman, L. (1992). Toward a pedagogy of cases. In J.H. Shulman (Ed.), Case methods inteacher education (pp. 1–30). New York: Teachers College Press.

Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivistperspective. Journal for Research in Mathematics Education, 26(2), 114–145.

Spiro, R.J., Coulson, R.L., Feltovich, P.J. & Anderson, D.K. (1988). Cognitive flexibilitytheory: Advanced knowledge acquisition in ill-structured domains. In Tenth AnnualConference of the Cognitive Science Society (pp. 375–383). Hillsdale, NJ: Erlbaum.

Stein, M.K., Smith, M.S., Henningsen, M.A. & Silver, E.A. (2000). Implementingstandards-based mathematics instruction: A casebook for professional development.New York: Teachers College Press.

Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. InD. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.127–146). New York: Macmillan.


Thompson, A.G., Philipp, R.A., Thompson, P.W. & Boyd, B. (1994). Calculational andconceptual orietnations in teaching mathematics. In D.B. Aichele & A.F. Coxford (Eds.),Professional development for teachers of mathematics (pp. 79–92). Reston, VA: NationalCouncil of Teachers of Mathematics.

Thompson, T. & Doerr, H.M. (2001, April). In what ways does multimedia case-basedinstruction influence pre-service teachers’ conceptions of effective teaching? Paperpresented at the Annual Meeting of the American Educational Research Association,Seattle, WA.

Wassermann, S. (1993a). Getting down to cases: Learning to teach with case studies. NewYork: Teachers College Press.

Wassermann, S. (1993b). Growing teachers: Learning about teaching from studying cases.New York: Teachers College Press.

Joanna O. MasingilaSyracuse University215 CarnegieSyracuse, NY 13244-1150E-mail: [email protected]

LAURA R. VAN ZOEST and JEFFREY V. BOHL

THE ROLE OF REFORM CURRICULAR MATERIALS IN ANINTERNSHIP: THE CASE OF ALICE AND GREGORY

ABSTRACT. This report describes a case study of a secondary school mathematics intern-ship where most of the elements of the internship ecology were aligned with calls by theNational Council of Teachers of Mathematics [NCTM] for reform. Much research has beendone on the lack of impact of teacher education programs on teacher classroom behavior.Often this results from dissonance between the philosophy and goals of university teachereducation programs and those of schools. The case documented here offers an instanceof alignment between the university program and the school internship site. The findingsshowed that within this supportive atmosphere the reform curricular materials being used inthe classroom had a strong and largely positive influence on the character of the internshipexperience, and on the intern’s learning. The form and content of the materials – Core-PlusMathematics Project’s Contemporary Mathematics in Context – supported reform-orienteddevelopment for the intern by providing particular foci to the planning discussions betweenthe intern and the mentor. These areas of focus were: (a) the mathematics content itself,(b) conceptual understanding as a main goal for instruction, and (c) the use of questioningas the best means of guiding students to that understanding. The findings highlight theimportance of developing reform-supportive environments for intern teaching placements.

KEY WORDS: curriculum, internship, reform, secondary teacher education, teacherlearning contexts

The way an individual becomes a classroom teacher can be viewed asa process of socialization (Gregg, 1995; Zeichner & Gore, 1989). Suchprocesses take place within ecologies – relationships among contextualelements that include all of the people, programs and settings within whichlearning to become a teacher takes place (Feiman-Nemser & Buchmann,1987; Zeichner, 1985). In the United States prospective teachers completea fifteen-week, or longer, practice teaching experience in an experiencedteacher’s classroom at the end of their university teacher preparationcoursework. This internship, which serves as both the culmination of theformal aspect of teacher preparation and the beginning of the experientialaspect of classroom learning, encompasses a particularly critical intersec-tion of many of these contextual elements. As such, it is not surprisingthat teachers often refer to their internships as the most valuable part oftheir teacher preparation (Feiman-Nemser & Buchmann, 1987; Guyton &McIntyre, 1990; Lampert, 1988). The contextual elements that comprise


266 LAURA R. VAN ZOEST AND JEFFREY V. BOHL

the ecology within which this pivotal event transpires are, however, oftennot in harmony with one another. This is likely to be true particularlywhen the intern is committed to teaching in a reform manner. Reform-minded education students often confront a sense of disharmony when theyintern in school contexts that do not support reform-oriented approaches(Parmalee, 1992).

Often this is the case for interns who accept the reform movementin the United States that is articulated in the Standards documents putforth by the National Council of Teachers of Mathematics [NCTM] (1989,1991, 1995, 2000). These documents have served as catalysts for a nation-wide effort to reformulate K-12 mathematics education, largely aroundone central concern: teaching mathematics for student understanding.Such teaching involves moving students beyond surface-level rote andalgorithm-based mathematical knowledge, and towards a deeper concep-tual and problem-solving-based knowledge (Wilson & Goldenberg, 1998).The end goal of such instruction is students who are strong problem solversand who can reason and communicate mathematically (NCTM, 1989).1

Historically mathematics has been taught in the United States as setsof algorithmic skills for solving well-formatted problems that are sepa-rated into the somewhat arbitrary courses of Algebra, Geometry, AdvancedAlgebra/Trigonometry, and Pre-calculus. Teaching methods and curriculaassociated with this skill-based, single-subject approach are prevalent inmany of the classrooms where education students do their interning, andfrequently run counter to reform approaches. Thus, mathematics internsare often subject to the potentially negative influences of interning in ecolo-gies that compel them to conform to a non-reform status quo (Cohen, 1991;Frykholm, 1995; Graham, 1997; Zeichner & Liston, 1987).

This situation presents a problem for mathematics teacher educatorswho support the reform movement, as they are the members of the mathe-matics community with the most responsibility for directing and coordina-ting internship experiences. Thus, it is important for teacher educators tounderstand how the internship ecology can best be configured to facilitatethe development of strong reform-oriented teachers. However, given thescarcity of school locations where reform instruction is being successfullyundertaken, it has been difficult for teacher educators to ascertain howparticular characteristics of internship ecologies effect reform-orienteddevelopment in interns. Further, it has been difficult for university-basededucators to develop reform-oriented school sites, precisely because of thedifficulty in developing reform-oriented teachers without reform-orientedsites in which to train them. Simon (2000) refers to this situation as thecatch-22 that currently faces those of us working to foster reform-oriented

REFORM CURRICULAR MATERIALS IN AN INTERNSHIP 267

growth in mathematics teachers. In the hope of offering a look beyond thisconundrum, we undertook a case study of an internship where the intern,Alice, and her mentor, Gregory, worked within an ecology where mostof the elements were at least generally aligned with the tenets of reformmathematics.

Many of the elements that comprise the ecology of internship and theirimpact on the intern’s development have been the topic of prior research.These elements include the university training of interns (Ensor, 2001;Goodman, 1997; Guyton & McIntyre, 1990), the beliefs of the mentorteacher and the intern (Feiman-Nemser & Buchmann, 1986; Franke &Dahlgren, 1996), and the effects of the overall social context of the school(Borko, Lalik & Tomchin, 1987; McNally, Cope, Inglis & Stronach, 1997).Although these elements were at play in the internship we studied, wefocus our attention on the element that emerged as the most striking – thecurricular materials. Our goal is to provide an initial conception of how thereform materials used during the internship impacted the character of Aliceand Gregory’s interactions and Alice’s development as a reform-orientedmathematics teacher.

THEORETICAL PERSPECTIVES

This research project is part of our ongoing efforts to determine how bestto prepare teachers who are willing and able to teach in a manner reflectiveof the NCTM Standards. From a socio-constructivist perspective (Ernest,1996), an intern develops his or her own understandings in response topersonal interactions with the elements of the social context within whichhe or she is embedded. Overwhelming evidence, both research-based andexperiential, suggests that interns’ field experiences are the most powerfulinfluences on their future teaching (Brown & Borko, 1992; Evertson,Hawley & Zlotnik, 1985; Parmalee, 1992; Tabachnick & Zeichner, 1984).In particular, fieldwork has a tendency to invalidate university courseworkadvocating reform in teaching (Ball, 1990; Guyton & McIntyre, 1990).Most of the fieldwork in prior studies took place in environments that eitherdid not reflect the reform ideas advocated in the interns’ coursework (e.g.,Parmalee, 1992) or that seemed somehow “special” and not replicable inthe interns’ eyes (e.g., Ball, 1990). However, when a pre-intern experiencefor prospective elementary school teachers was aligned with the universityprogram and NCTM reforms, Mewborn (2000) found that the effects werepositive. Thus, it seems that aligning the internship with the universitycoursework has the potential to overcome these obstacles and contributeto the development of reform teachers.


The learning that interns experience is in many ways framed by theirmentor teachers (Feiman-Nemser & Buchmann, 1987; Haggarty, 1995;Jaworski & Watson, 1994). One important physical part of the ecologyof internship is the mentor’s classroom, within which the intern’s planningwill culminate in the instruction of students. The classroom norms andatmosphere are usually set, however, by the mentor teacher before internteaching begins, and the mentor is the person with the most immediatecontrol over what the intern will and will not be allowed to do as a teacher.Ideally mentors are purposeful in their interactions with and support ofinterns. Indeed, the recent change in the descriptors used for the parti-cipants in this event from student teacher and cooperating teacher to internand mentor has been, in part, an attempt to capture the nature of the desiredrelationship between them. From our perspective that relationship shouldbe a cooperative one in which each participant views her- or himself asboth a learner and a teacher.

In addition to the experiential classroom knowledge that mentors bringwith them to the mentor-intern relationship, they also have many of thecommon tools of teaching at their disposal. For several reasons the text-books being used in the courses taught are among the most influential.In general teachers in the United States rely heavily on textbooks for thecontent they teach and the approaches they use (Ornstein, 1994). Studies inmathematics and science education have shown that textbooks can stronglyinfluence both what and how teachers teach, as well as what and howstudents learn (Alexander & Kulikowich, 1994; Begle, 1979; Usiskin,1985). Results from the Second International Mathematics Study indicatethat inclusion of a given topic in the curriculum or textbook (generallysynonymous in the United States) provided by a school is one of the mostcommon justifications given by secondary school mathematics teachersfor their decisions about whether or not to teach that topic (Glidden,1991). There are undoubtedly many reasons for such heavy reliance ontextbooks, including both efficiency in preparation and ease of use (e.g.,Schug, Western & Enochs, 1997). Regardless of the reasons, textbookshave had a powerful effect, historically, on both the content and structureof classroom instruction.

The recent publication of curricular materials designed to instantiate theNCTM Standards raises the question of whether this reliance on textbookswill persist when the textbooks change and, if it does, whether or not it willmove classroom instruction toward reform. Historically mathematics text-books have played the role of a mathematical authority and reference forstudents and teachers in mathematics classrooms. They have also tendedto direct teachers’ activities by providing order and structure to the content


and by providing exercises which enable the students to practise and, hope-fully, master the mathematics. Some Standards-based texts are, in part,an attempt to change the role the text plays for students – moving frombeing the source of knowledge and practice towards serving as the providerof mathematical investigations. Such texts are designed to shift the roleof mathematical authority from the teacher and text to whomever hasevidence to support their mathematical claims. Not all teachers, however,have the beliefs, knowledge and technical expertise necessary to translatereform-inspired curricula into reform practice in the classroom (Cohen,1990; Martens, 1992). Even so, these texts have the potential to influencegreatly the classroom ecology, and thus the learning of new teachers.

DESIGN OF THE STUDY

Participants and Research Site

The intern, Alice,2 was a very determined and self-reflective student ata large United States state university known for its teacher preparationprograms. She characterized herself as an “ideal student” in secondaryschool who took advanced classes. She had been particularly successfulin mathematics. However, she had a revelation during her undergraduatework when she realized that, although she had always earned good gradesin mathematics courses, including those at the university, she actuallyhad developed very little understanding of the content of those courses.She had, rather, learned processes for arriving at correct answers withoutunderstanding the mathematics behind the processes. At that point shebegan to commit herself to reform in mathematics education, and madeit a personal goal to teach her own students so as to enhance their under-standing as well as their performance. Her subscription to this idealresulted, in part, from her undergraduate teacher education program insecondary school mathematics. The program was designed with the goalof preparing reform-minded teachers who would serve as change agentsin their future schools. During three 16-week-long mathematics educationmethods courses she was given information about, and the opportunity toexplore, many of the concerns and methodologies of reform mathematicsteaching. In addition to focusing on reform-related content, the coursesthemselves were taught with a reform pedagogy. One activity relevant tothis study was the critical analysis of different textbook options, includingthose claiming to be reflective of the NCTM Standards.

Gregory had 31 years of experience as a mathematics teacher and hadworked with many interns. He described his teaching for most of his career


as fairly traditional. Over the years he developed a desire for an approachto teaching mathematics that was more relevant to contemporary students.While he personally enjoyed doing mathematics for its own sake (thatis, without a real-life contextual basis), he believed that such traditionalapproaches did not serve the majority of the students well. He felt that themathematics education community was moving in the right direction withthe NCTM Standards, and viewed his own learning of new approaches asa natural and integral part of his job as an educator.

During the internship Alice worked with Gregory on the first twocourses of his school’s four-course sequence developed by the Core-PlusMathematics Project (CPMP: Coxford et al., 1997; described below). Eventhough he had participated in professional development on the new mate-rials, Gregory was teaching the first course of the sequence for only thesecond time, and the second course for the first time. Thus, he was still inthe preliminary stages of learning to orchestrate the program’s approaches.He was committed to learning how to do so, and was comfortable actingin the role of mentor in the midst of ongoing reforms and as a learner inthe process himself.

The internship location was Ransfield High School, a school of approxi-mately 1000 students in grades 9–12 (ages 14 to 18 years) serving alargely white middle class and upper-middle class section of a small urbanarea in the United States. The school was in the process of reforming itsinstruction in at least two relevant ways. First, in the year prior to thisstudy the school and its mathematics department had begun to offer theCPMP integrated mathematics program as an alternative to their moretraditional single-subject mathematics sequence. Second, the year of thisstudy was the first year of a school-wide change to block scheduling. Theschool switched from a traditional schedule, comprised of seven year-longcourses taught in daily 55-minute class periods, to a block schedule offour 90-minute daily class periods for half-year-long courses. This movewas largely supported by the school’s staff, and particularly supported bythe mathematics teachers who felt that the extra time would facilitate theteaching of the new investigation-oriented CPMP program.

This pair of structural changes lent an overall atmosphere of reform-in-motion to the mathematics department and its instruction. The mathe-matics faculty was brought together over the course of two years ofprofessional development investigating and preparing to teach the newmaterials. They also worked together to learn how best to deal with thechallenges of the new materials and class period format. These experi-ences helped the faculty develop a sense of professional community wherethe atmosphere was generally one of communal support for professional


growth and curricular reform. This made it a prime site to locate the internunder study.

The Curriculum

The CPMP materials consist of a four-year unified curriculum3 developedto implement the recommendations for reform in the NCTM Stan-dards documents. The materials replace the Algebra–Geometry–AdvancedAlgebra/Trigonometry–Pre-calculus sequence that is typical in the UnitedStates. Each course features interwoven strands of algebra and functions,statistics and probability, geometry and trigonometry, and discrete mathe-matics. These strands are each developed within focused units connectedby fundamental ideas such as symmetry, functions, matrices, and dataanalysis. The curriculum emphasizes mathematical modeling and capita-lizes on graphing calculator technology (Hirsch, Coxford, Fey & Schoen,1995). These changes and additions mean that portions of the curricularmaterial are not immediately familiar to most United States mathematicsteachers and must be learned or relearned.

Along with these changes in content and sequence, implementingCPMP as the authors intended requires changes in teaching methods.Classroom tasks are designed as investigations where students workcollaboratively in small groups either to discover or construct the mathe-matical ideas needed to understand real-world situations. In the processstudents are expected to communicate about that mathematics both orallyand in writing. Thus, teachers need to learn to utilize cooperative groupsin the classroom, as well as to facilitate rich classroom discussions thatwill help to improve students’ understanding of the mathematics theyare doing. The demands of making all of these reforms at the sametime add another dimension of adjustment for teachers, particularly thoseaccustomed to teaching students in the common United States format ofdemonstration-elaboration followed by individual seat work.

In addition to being investigation-based, the CPMP materials aredifferent from single-subject mathematics textbooks in other significantways. First, questions in the textbook are written in sentence/paragraphform and can be challenging for students to read and comprehend. Second,the problems used for class work are sufficiently challenging that manycannot be done by individuals alone, but rather demand that students workcollaboratively to solve them. Thus, the materials require the teacher tomove fluidly between whole-class, small group, and individual instruc-tion. Third, most of the problems require conceptual understanding as wellas mastery of calculation algorithms. This requires instruction that dealswith the mathematics at a deeper level than is usually necessary to ensure


success in more traditional single-subject sequences. Finally, the mode ofteaching required is much more that of facilitator than expert dispenserof knowledge. Because students are expected to arrive at mathematicalconclusions on their own, the desired teaching method is more to guidethan to direct.


Data for this study were collected prior to, throughout and followingthe internship studied. Individual (II and MI) and joint (JI) interviewswere conducted with the intern, Alice, and her mentor, Gregory, priorto the study to determine their expectations of the experience and theirbeliefs about teaching and learning mathematics. During the internshipAlice and Gregory documented eight joint planning sessions during whichthey prepared for forthcoming lessons (P). Each of the four classroomobservations was followed by an interview (CI) with Alice and, at times,Gregory, which focused on both the events of that particular class and theevolving internship experience. At the conclusion of the internship indi-vidual and joint interviews were conducted again. Alice wrote 25 reflectivejournal entries (J) during the internship and maintained ongoing audio-taped reflections of her experiences during the first several months ofher first teaching position (R). Finally, Alice was interviewed during thefourth month of that job to investigate the impact the internship had on herclassroom teaching (PI).

All interviews and planning sessions were audiotaped; tapes andjournals were transcribed and the transcriptions were coded and analyzedwith the assistance of a qualitative data analysis program (NUD•IST).Through the initial stages of the process of analysis we identified andcoded patterns in the concerns and foci of the discussions between theintern and mentor, and within the intern’s journals. As further patternsbegan to emerge, we refined the coding scheme and re-coded when neces-sary to determine the topics that were of greatest concern to both Aliceand Gregory. We focused specifically on issues related to the teachingand understanding of mathematics, and how and when the participantsreferred to and discussed such issues. We also documented evidence ofAlice’s thoughts, actions and development, as related to reform teaching,and the ways that the contextual factors – particularly those of Gregoryand the CPMP curriculum – were helping to shape those thoughts, actionsand development.

Throughout the coding process, reflections, findings and concernsfor further exploration were formally written up as memos, and werereviewed and discussed by the authors. After coding was complete reports


consisting of all like-coded data were created for each category of coding.These reports included cross-references to all relevant codes. Each codingcategory was then reviewed to determine trends and patterns, as wellas to determine trends in interaction between categories. Findings foreach report were written up in detail, and again shared and discussed.The reports and the resulting discussions were used as the basis forthe case study that follows. We, along with the study participants, readand commented on drafts of this article to strengthen the validity of theinterpretations of particular events and of the overall conclusions. Thetriangulation provided by the multiple data sources, the two researchers’views on coding and interpretation processes, and the views of participantsguarded against drawing invalid conclusions from the data (see Eisenhart,1988, for a discussion of triangulation).

FINDINGS

Alice came to her internship with strong feelings about reform mathe-matics and a great desire to learn how to enact a reform practice in herown teaching. Our data provided evidence that the overall atmosphere ofthe school created a context within which she began learning to do just that.The supportive professional climate, the move to block scheduling, and herpro-reform mentor were key aspects of this climate. From an ecologicalpoint of view, most elements of the context appear to have been workinggenerally in support of one another, and in support of reform mathematics.

Within this context the CPMP materials emerged as having played acentral role in determining the character of the interactions between Aliceand Gregory. One reason for this is that both Alice and Gregory held theCPMP materials in very high regard. Alice’s knowledge about the programcame from her university coursework. Although the materials were notapproached uncritically in her courses, they were offered as one of only afew existing examples of 9–12 curricular materials supporting the NCTMStandards. Alice felt that they were far superior to the materials she hadused in her own secondary schooling, and that teaching with them wouldresult in students “learning to reason and question better than I ever did atthat age” (J2, 57–58).4 Gregory came to his support of the CPMP materialsthrough numerous professional development activities in the years prior tohis department’s adoption of them. He believed that what was happening inhis classroom since adoption was “excellent,” and that it resulted in deeperstudent understanding (CI4, 604).

This great respect for the CPMP materials was shared by other membersof the faculty, and meant that the teachers worked hard to achieve the


authors’ original intent. Thus, the CPMP materials directed nearly all ofthe decisions made concerning what, how, and when to teach given topics.In the following sections we look at how the textbook served as a script forAlice and Gregory’s joint planning sessions, the way in which it causedthem to focus on the mathematics and student understanding of mathe-matics concepts, and the manner in which it directed their attention towardsquestioning as an instructional technique. Finally, we step back and look atthe overall impact the experience had on Alice’s development as a reformmathematics teacher.

Textbook as Planning Session Script

The investigative format of the text and the difficulty of its questions almostrequire that teachers use cooperative groups during instruction. Further, theconversational nature of the approach and the mathematical richness of thequestions create an atmosphere where mathematically-rich tangents can betaken by students at every turn. Gregory described the problem this way:

It’s easy to let this material drag out. You can have wonderful discussions and all that,but you’re really not accomplishing what you need to accomplish. It’s almost like you’rehaving too much fun, if there is such a thing, and not really getting the job [done]. (JI1,206–211)

This meant that Gregory and Alice spent much of their planning timewalking through each of the next day’s investigations in order to determinewhich parts students could answer without guidance, which parts might beskipped or glossed over to increase the pace of student progress, and howto avoid being overly directive while at the same time maintaining studentfocus on the day’s main mathematical concerns. The following discussionof Alice’s initial plans for a lesson is illustrative:

A: And, then, just have them go through 1 and 2 on their own in their groups, because Ithink those should go fairly quickly. I don’t think they should take any more than 7 or8 minutes . . . .

G: Okay. And I would put a time limit on it.A: Yeah, that I will. Then, I said do 1 and 2, read this. Then I wanted to go over “What is

lower quartile”?G: What you might want to do is have them do 1. Then come back and maybe have them

start working on 2 [pause]. Yeah. Because I think you’re going to want to maybe talkabout the percentiles here before you get in and get them with percentile/quartile. So ifyou stopped at the end of 1 and just do a little summing up. You see what I’m saying?

A: Uh-huh.G: And then let them do 2. And maybe just give them just a short [pause]. Because that

should be real quick, in terms of them coming up with the answers to that.A: Yeah. I struggled with [pause]; I didn’t know whether [pause].


G: I think if you have them go right from here to here, they’re going to get in here and getreal confused. But if you talk about the percentiles, then come back and talk about thesignificance of those percentile ranks. (P2, 717–743)

Here the textbook questions and the mathematics they contain serve asthe framework for the entire interchange. Although such conversationsoften also addressed mathematical concerns (as will become apparentbelow), this type of “how can we best get through this” discussion was notuncommon. Thus, while it might be correct to say that Alice and Gregorywere attempting to develop a reform mathematics practice, a more accuratestatement is that they were attempting to teach CPMP as it was designed.They were both committed to reform teaching and felt they could bestachieve that by moving students through the investigations in the text-book. The extent to which this was true was made clear by Alice. Shebelieved that “to change [the CPMP textbook] would be to imperfect it,”even though she desired to be more creative in her planning (PI, 1463). Shecommented, “that is one of the things I miss a bit with [CPMP] . . . I seemto be just following the book and directing what should be done . . . I feeldisappointed that [planning] is dull” (J3, 18–19; J23, 80–81).

Aside from viewing the textbook-directed planning process as boring,Alice was also frustrated by her need to rethink her plans every day. Shecommented that,

It is the night before or sometimes the day before that I sit down and formally write outthe lesson plan. I do this because I can never seem to predict where we will end up onthe next day. I find it so awfully hard to navigate the timing of this stuff, because mostof the time I just don’t know how the kids are going to react and how things will go. Icannot stress the amount of frustrated energy I have spent on this. This material manytimes seems to depend so much on the students themselves and whether or not I can hitupon the appropriate strategy of questioning and explanation. (J11, 15–26)

Such quotes are balanced in the data with others that indicate Alice’spositive feelings for, and experiences with, CPMP. She believed that it was“wonderful” and “inspiring” (JI2, 1462; II2, 38). Even so, statements likethe one above indicate the level of frustration that she felt as a result ofneeding to follow the text so closely, and speak to the strength of theirshared belief that it should be followed.

A Focus on Mathematics and Student Understanding of MathematicsConcepts

The use of the CPMP textbooks demanded more than a simple conside-ration of how best to progress through the material. The newness of someof the content meant that Alice and Gregory were, at times, learning andclarifying the mathematical content for themselves. Here is an interchange


concerning a graph theory activity in which students were to explain amethod for finding a spanning tree for a graph:

G: And then I could go from C to B. And that would be six segments.A: Or C to D.G: Or go, yeah. No, no, because I haven’t gotten to C. I went from G to F, G to E, G to D,

and G to A.A: Or I meant from D to C then.G: Okay. So I could go from [pause] But I could also go from A to B to C. I mean there’s

just a ton of them.A: Yeah. That’s why I’m not sure how to write down a method. (P7, 938–966)

This need to learn new mathematics indicates one of the challenges thesematerials present even to experienced teachers. However, given Gregory’s31 years of teaching mathematics, it was more often Alice who was in needof more content knowledge. Such occasions allowed Gregory to play therole of mathematics teacher for Alice. For example:

A: Now when you say like 40 percent or 40th percentile, cause I get a little bit confusedand I just have to go through it. Is it correct to say that you are taller [pause]

G: than 40 percent of the students. And 60 percent of the students are taller than you. Nowthat can be a little bit misleading because it probably should be 40 percent are as tallor shorter than you, [pause] and [it could be] 60 percent are as tall or taller than you.Because there could be a whole bunch of people tied for that spot. (P1, 211–220)

Throughout the experience, when there was a need to understand somemathematical content better, Gregory seemed very willing to spend thenecessary time clarifying it either for Alice or for the both of them.Gregory’s practice of using the textbook as the road map for theirdiscussions, and his steering them toward the mathematics involved, wascommon. And it often went well beyond simply clarifying the contentto focusing on what Shulman (1987) referred to as pedagogical contentknowledge.

Much of Gregory’s focus was on trying to anticipate how best to helpstudents understand the material, and he regularly presented Alice withideas that he felt would help her support students’ understanding of mathe-matical concepts. He did this by drawing on his vast teaching experience,and particularly his previous year working with CPMP, to develop Alice’spedagogical content knowledge. Specifically he regularly addressed waysthat students tended either to understand or misunderstand particular ideas,and offered her advice to help her steer students around predictable pitfalls.

To illustrate how Gregory incorporated a focus on content and pedago-gical content knowledge, consider the middle section of an investigationfrom the textbook on measuring variability using the five-number summaryas shown in Figure 1. During their walk-through of this investigation Aliceand Gregory had the following interchange:


The quartiles together with the median give some indication of the center and spreadof a set of data. A more complete picture of the distribution of a set of data is given bythe five-number summary: the minimum value, the lower quartile (Q1), themedian (Q2), the upper quartile (Q3), and the maximum value.

4. From the charts,5 estimate the five-number summary for 13-year-old girls’heights and for 13-year-old boys’ heights. Some estimates will be more difficultthan others.

The distance between the first and third quartiles is called the interquartile range(IQR). The IQR is a measure of how spread out or variable the data are. The distancebetween the minimum value and the maximum value is called the range. The range isanother, typically less useful, measure of how variable the data are.

5. a. What is the interquartile range of the heights of 13-year-old girls? Of13-year-old boys?

b. What happens to the interquartile range of heights as children get older?c. In general, do boys’ heights or girls’ heights have the larger interquartile

range or are they about the same?d. What happens to the interquartile range of weights as children get older?

Figure 1. Except from Coxford et al. (1997, p. 47): Contemporary Mathematics inContext: A Unified Approach (Course 1).

G: What blows them away is “interquartile range.” It’s those words.A: I was hoping that we could split it up. What does “inter” mean – interstate. Then when

you hear the word “quar” quartile and relate it back to the quarters . . .

G: It’s a word that they will be very uncomfortable with. It’s just one that we’re going tohave to keep coming back and talking about the IQR and interquartile range over andover again. We really want to emphasize the word range, it’s the length of that box.Rather than it being some THING [pause]. We don’t want them to get into the ideathat it is the box. It is a number. It’s the length of that box. It’s a range.

A: Okay.G: I think that’s a good idea – to talk about breaking it up. That’s excellent.A: Okay. Then have them do #4. Start #5, yeah, because they can do #5 without [pause]

and then talk about 4 and 5. And then I want to make sure that with the interquartilerange, when we talk about 4 and 5, stress that the range is all of the data, the wholespread. IQR gives you where the majority [pause]

G: 50 percent of the data. 50 percent of the data is in that interquartile range. Becausethat’s the difference [referring to diagram], see your first and your fourth quartiles areyour 25th and 75th percentiles and so 50% of your data is in that box. (P1, 788–824)

This passage gives insight into the level at which both Gregory andAlice wanted students to understand the concept, and the ways in whichGregory elaborated on the mathematics in the textbook. They did notspend time discussing how to help students calculate the IQR, but focused


on making sure that students understand what it represents. With hiscomments Gregory seems both to be ensuring that Alice understands theconcept herself and offering pedagogical suggestions to strengthen studentunderstanding of it. Here again his focus on the mathematical concernsof the textbook content played a part in determining the character of theirrelationship.

The influence of Gregory’s use of the textbook to focus on issuesof mathematical content and pedagogical content knowledge becameapparent in a comparison of the topics of their planning sessions andthe topics of Alice’s journals. The focus of student-related comments inAlice’s journals was by far most often on student behavior. Here is anexample:

I have to find a way to make the students more responsible for their actions. It has beensuch a long road to haul in this area. Two steps forward, one step back . . . . We started anew name writing on the board policy of discipline. It is kind of third-gradish, but they areacting like third graders . . . . I never knew that the feeling of being in front of class wasso negative towards teachers. The students do not seem eager to learn, but rather they areeager to be defensive . . . . It is such a big power struggle all the time. (J13, 34–54)

On the other hand, her joint planning sessions with Gregory showed aninverse relationship, with understanding mathematics being by far the mainfocus of student-related discussion between them. Alice’s frustration withthis discrepancy is captured by her explanation of the difference betweenher and Gregory’s approaches:

. . . he was more concerned with, “well does this power model fit it [better] with 1/x2, or isthe 1/x3 [better]?” To me, I wasn’t concerned about which one was actually better, I justwanted [the students] to be doing it. So it’s hard because I get impatient because I reallywant to know how to teach it. (CI3, 496–501)

She referred to this difference as her being “more concerned withprocedure,” and his being “more concerned with math” (CI3, 475–495).In her journals Alice described how she wanted students to be thinkingabout mathematics, using such terms as “conceptualizing rules,” getting“at the heart of a concept,” “conceptualizing” what symbols represent,and “grasping the ideas of the problems” (J5, 3; J11, 108; J18, 71; J19,6). However, these were used in a very general sense and in reference toentire classrooms of students or about entire lessons. In contrast, duringthe planning sessions Alice’s comments about student understanding weremore often about specific instances of understanding related to particularproblems or mathematical concepts, such as in the quote above. This indi-cates that Gregory’s use of the textbook as a guide and concentration onmathematical concerns helped create a context that aided Alice in focusingon the particulars of understanding the mathematics.


Gregory believed that the conversations he was having with Alice(and one intern the previous year) while using the CPMP materials weredifferent from conversations with other interns while using more tradi-tional single-subject textbooks. He felt that with CPMP the discussionstended to be “more about what’s happening with the students in theirlearning. And I think with the old curriculum it was more talking aboutwhat’s happening with you and your teaching” (MI2, 741–744). Thefollowing section highlights how working with the new materials helpedsupport a new understanding of the role of the teacher.

A Focus on Questioning

Gregory and Alice both believed that the changes required in the classroomby the CPMP materials involved redefining the roles of student and teacher.CPMP describes the role of the teacher as that of facilitator, and bothGregory and Alice used that word to describe what they as teachers oughtto be doing. Both also agreed that since students were expected to arriveat mathematical conclusions through their own reasoning, the desiredteaching method was based on the asking of good questions. Gregoryillustrated their views of good instruction in the following critique of bothhis own and Alice’s development about two-thirds of the way through theinternship:

I think the main thing that we both struggle with is we still haven’t gotten to the pointwhere we are letting the students do enough. We’re still doing too much directed teachingand not enough of getting them totally engaged in the discovery process and coming upwith [pause], but I think, again, that’s just going to come with time. And practice. Andagain, a lot of it is just learning, with time and experience, those questions to ask that aregoing to solicit that response that we need to get from them. It’s so much easier to just tellthem. And the problem with that obviously is, generally speaking, they don’t learn it frombeing told. (P3, 456–467)

Here the active role desired from students is highlighted alongside thefacilitative role of the teacher. And learning how to ask good questionsis proposed as the method by which they could both improve their work.

The strength of their shared belief in good questioning as a key toteaching the materials was evidenced throughout the planning sessions bytheir own usage of questions in conversations about how to teach specificconcepts. In the following passage Alice and Gregory were discussing theintroduction to a lesson on variability within data sets:

A: . . . and then, ask them to think about it, questions. Just by “Who thinks that you couldcall him taller than average?”, “Why do you think that?”, “Who thinks you can callhim tall?”, “What can you say about that?”, “Why?” And I also wanted to somehowdraw out of them that graphs don’t always tell you what you think they tell you. Bysaying, “You know, this is the mean, this doesn’t tell you all of them.” By somehow


maybe relating to the fact that like, when I stand with my roommates you think I’m thetallest, but when I stand with my sister and cousins, I’m the shortest. And so, “Whatkind of criteria do you need to have?”, “Does it matter in what types of situationsyou’re looking at the time”?

G: Okay . . . . And what you might be able to do is pick out a couple of classmates andjust have them stand. “Do you consider so-and-so tall?”, “Do you consider so-and-soshort?” (P1, 610–631)

Here the discussion consists largely of what they believe are the keyquestions that will help develop students’ understanding of the conceptof variability in relationship to the idea of mean. They are not discussingthe questions themselves so much as holding a discussion with questions.Such interchanges were frequent and seemed to serve as something of apractice run for classroom questioning of students.

This focus on questions and the use of questioning techniques wasdefinitely supported by, and in some senses determined by, the CPMPtextbooks. In the texts each lesson is framed by questions that are meantto focus student attention on the lesson’s central ideas. Each lesson islaunched with a Think About This Situation that provides a set of ques-tions about a problem situation to be studied through investigations. Thequestions are to be discussed in a whole-class setting. To conclude eachlesson, rather than giving a summary of the content, the textbook offersanother set of questions called the Checkpoint. The answers to these ques-tions are to be determined in small group and whole-class discussions,and are meant to require understanding of the key mathematical ideas andskills developed during the lesson. Further, conceptual-level questions areasked of the students throughout the investigations. Alice and Gregory’sdiscussions about and uses of questions were nearly always framed by thequestions posed in the textbook investigations themselves.

There was some evidence that Alice developed in her ability touse questioning in her classroom to promote student understanding. Inher journals she made many references indicating a conscious effort toimprove her questioning. Toward the end of the experience she noted thatshe was feeling more at ease “being able to think while I’m up there,”specifically as related to formulating productive questions in response tostudents’ questions and their answers to questions (P7, 426–426). That is,she believed that she was becoming a better facilitator of discussions thatwould help students develop their own understanding. Gregory’s evalua-tions and our observations of her classroom corroborate her perceptions.

The Experience’s Impact on Alice

In Alice’s perception, use of the CPMP curriculum was of particularimportance in terms of her focus on student understanding. She stated


during her internship that she believed CPMP helped students developtheir mathematical understanding. This belief was bolstered after she hadworked for a few months in her first full-time teaching position at ConcordMiddle School. She started at Concord in the middle of the school yearteaching two sections of Algebra, two of a lower track called TransitionMathematics, and one of Geometry. When reflecting on how her newstudents approached mathematics, she noted that her students at Ransfieldhad become better at problem solving and thinking about mathematics thanher new students were. She believed that this was because:

[CPMP] is really big-thing oriented and works down to the smaller parts of math. [Theseries at my new school] really starts small and builds up. [Students] don’t see the bigpicture. They don’t see why they have to have math or what we’re using graphing for. It’snot until you get to that section before the chapter review that things start to come togetherfor them. For them, math is a bunch of memorizing and taking notes. (R, 138–145)

During the first three months of her new position she did much work toadjust the available traditional single-subject curriculum so that studentswould do more of the types of thinking that she believed CPMP demanded.This included creating activities and projects that required students to linkmathematics to realistic situations, and using cooperative group activitiesthat forced students to communicate their thinking with one another. Muchof her focus in this reform effort was on her lower track Transition Mathe-matics students, who were being targeted to take the first year of CPMPat the high school the following year. The course had previously beentaught as a hodge-podge of mathematical skills with a file cabinet fullof skill worksheets serving as the curriculum. One senior mathematicsfaculty, who was vehemently opposed to CPMP, suggested that Aliceshould not change anything about the way her assigned courses weretaught. Alice decided, in spite of this pressure and with the support ofone school administrator, to rework the curriculum so that the studentswould be better prepared for what they would encounter in CPMP. She didthis by analyzing the CPMP materials and making a list of the skills andconcepts she thought would most benefit her students. She then organizedher teaching to help students develop these prerequisites.

In her closing interview, Alice attributed her willingness to undertakethis type of reform to the knowledge and confidence she had gainedinterning at Ransfield. Specifically, she felt that because of her experi-ences her “opinions [about reform mathematics instruction] are worthsomething” (PI, 1525–1526). She held her experiences with the CPMPcurriculum as exemplars of good reform teaching and learning, and utilizedthem as a baseline for comparison with the curriculum she was developing.


In the years since, Alice has continued to play the role of mathematicseducation reformer at Concord Middle. Partly as a result of her consider-able effort to align the middle school’s curriculum with its high school’sCPMP sequence, Concord is now piloting the Connected MathematicsProject (CMP) materials. These materials are also aligned with the NCTMStandards, and Alice played a key role in convincing her administrationthat CMP would serve its students well.

DISCUSSION

Alice’s internship experience helped her to develop some understanding ofwhat reform mathematics education involves and a considerable level offocus on the reform’s key concerns. Through Gregory’s use of the CPMPtextbook as a planning session script and source of classroom activities,these materials played a central role in formatting the character and contentof their interactions as mentor and intern. In general, it provided themwith a vision of what reform teaching might entail. More specifically, itchallenged them to learn (or relearn) some mathematics content, offeredthem conceptually thick material that directed them to focus on developingdeeper levels of student understanding, and provided them with models ofgood questions which they emulated in their own planning and instruction.Overall, it focused their conversations on substantive issues related to theteaching and learning of mathematics – something that past research hasfound lacking in internships (Bush, 1986; Feiman-Nemser & Buchmann,1987; Guyton & McIntyre, 1990; Parmalee, 1992).

The ways in which the CPMP textbook affected Alice and Gregory’swork also seem to run counter to how textbooks are usually understood toaffect teachers and their instruction. Critiques of teachers for the overuseof textbooks are common in many curricular arenas, and recent studies(Boaler, 1998; McNeal, 1995) have shown that in mathematics classroomsdependence on textbooks for instruction can lead to an overemphasis onmemorization and skill-focused learning. However, in the case of Aliceand Gregory, a seemingly total dependence on the CPMP textbook pushedthem to be more concerned with conceptual issues, and to focus more ondeveloping student understanding. One explanation for this apparent ironymay lie in the differences between the CPMP materials and more tradi-tional textbooks. Specifically, teaching the CPMP materials as intendedrequires of teachers both a deeper knowledge of content and of pedagogicalcontent knowledge than has been expected in the past. Fortunately, at leastin the case of Alice and Gregory, using the materials also offers teachersopportunities to increase their knowledge.


The new approach and content of the CPMP materials framed the jointplanning sessions in ways that allowed Gregory to tap into his experientialand mathematical knowledge to help Alice deal with issues of contentand pedagogical content knowledge. Although Alice desired to help herstudents understand mathematics better by engaging them with it, shedid not seem to see the need to engage with it herself in the same waythat Gregory did. Given her inexperience, it is not surprising that Alice’sprimary concerns dealt with instructional and classroom managementprocedures. The fact that Alice did focus on content, despite her othermore dominant concerns, reflects on both the mathematical strengths ofher mentor and the potential of curricula to have a positive effect on theintern teaching experience.

Strong pedagogical content knowledge is another key factor in goodteaching, allowing teachers to extend students’ knowledge beyond what isprovided in a textbook (Alexander & Kulikowich, 1994). Feiman-Nemser& Buchmann (1986, p. 239) refer to this type of thinking as “puzzlingabout what is going on inside the heads of young people,” and suggestthat making such thinking a habit is a very difficult transition for noviceteachers. Alice’s persistent personal focus on student behavior indicatesthat she was not fully prepared to move in that direction. By focusing onissues of content and pedagogical content knowledge, and by maintaininga focus on questioning to help students understand, Gregory seemed topush Alice to begin that important transition earlier than she might haveotherwise. However, consistent with other research (e.g., Feiman-Nemser& Buchmann, 1987; Parmalee, 1992), it appears that Alice might have hadeven greater learning had Gregory been more explicit and transparent inhis interactions with her. Alice didn’t understand the benefits of Gregory’sfocus and, therefore, was frustrated by what she felt she wasn’t learningand didn’t always give her full attention to what she was learning.

It is also possible that Gregory’s position as a learner with respect to thenew curriculum may have impacted upon his interactions with Alice. Eventhough Gregory was comfortable in the role of learner, the nature of theirdiscussions might have been different had he not been in the position ofhaving had so much to learn. In particular, Gregory’s approach with Alicetended to be more directive than with the students. This may be becausehe was still learning the materials himself and didn’t have sufficient know-ledge to facilitate her learning. It may also be that Gregory had not fullytaken on the role of teacher educator. That is, it did not appear that he putthe same kind of thinking into facilitating Alice’s learning as he did withthat of his students.


CONCLUSION AND IMPLICATIONS

Fullan (1993) suggests that many reform efforts fail because they addressissues of content, but not the most critical area of concern: “the culturalcore of curriculum and instruction.” Materials written in support of theNCTM’s Standards pose special challenges because ideally they do exactlythat. CPMP offers an additional challenge in that it also demands differenttypes of content knowledge. Our study suggests that the form, focus, andcontent of the CPMP materials may have aided a focus on students’ under-standing of concepts. This points to two potentially rich areas for furtherresearch. First, a determination of the specific characteristics of textbooksthat might promote this type of teacher focus and development wouldbe invaluable to textbook authors who are attempting to support reformmathematics teaching through the creation of curricular materials. Second,determining what particular characteristics of internship contexts mightsupport the full-blown use of such materials would aid teacher educatorsin their work to develop reform-supportive internship sites.

We want to be careful not to suggest that the use of any particular setof curricular materials will necessarily have such a strong impact on ateacher’s development. Because this particular school setting was involvedin reform consistent with that of the teacher education program, there weremany other elements within the ecology that impacted upon the nature ofthe experience. In particular, the participants’ shared desire to reform theirteaching was clear, and they were supported by the social contexts of thedepartment and school in their work. Many studies have indicated that thecharacter of the overall social context of an internship site bears stronglyon its impact on an intern (Calderhead, 1988; Frykholm, 1998; McNallyet al., 1997). The fact that there was a social network of jointly-engagededucators working towards the same goals that Alice and Gregory werepursuing had great impact, as Alice commented on several occasions.

In the broader sense, this study validates calls (e.g., Brown & Borko,1992) for teacher educators to place interns in school contexts that supportand extend the work begun in teacher education programs. However,we are not unrealistic about the chances of teacher education programslocating enough reform programs in which to place all of their interns.The lack of locations where reform is being undertaken to the extent, andwith the level of commitment, that existed at Ransfield poses us with agreat challenge. As university-based teacher educators we are not able toguide interns and determine the day-to-day experiences of their intern-ships. However, teacher educators can take on roles as activists to developand sustain efforts in reforming mathematics teaching in public schools(Liston & Zeichner, 1988). This would, of course, require something of


a redefinition and expansion of the role of teacher educator, so that moretime and resources could be made available for the development of place-ment schools, including the preparation of mentors to be teacher educators(Feimen-Nemser & Buchmann, 1987; Maynard & Furlong, 1993). Thiswork, although very time consuming and politically entangling, couldhelp increase the number of reform-supportive environments. Our studysuggests that reform-supportive environments can have a positive impacton reform-minded interns. Thus, whatever the work requires, the payoffmay be well worth it.

ACKNOWLEDGEMENTS

This research is based on work supported in part by the Western MichiganUniversity Faculty Research and Creative Activities Support Fund undergrant No. 96-020 and by the National Science Foundation under grant No.ESI-9618896. Any opinions, findings, and conclusions are those of theauthors and do not necessarily reflect the views of the funding organi-zations. An earlier version of the paper was presented at the AnnualConference of the American Educational Research Association.

NOTES

1 From here onward, mathematics taught in alignment with the reforms proposed bythe National Council of Teachers of Mathematics Standards documents (1989, 1991,1995, 2000) will be referred to as “reform mathematics,” and teaching in support of theStandards will be referred to as “reform teaching”.2 The participants and schools names are pseudonyms to comply with university humansubjects research requirements.3 The complete CPMP series is now available from Glencoe/McGraw-Hill PublishingCompany.4 All quotes from the data are referenced by the document name followed by the linenumbers in which the quote appears.5 Charts of boys’ and girls’ physical growth percentiles for ages 2 to 18 years areprovided in the text.

REFERENCES

Alexander, P.A. & Kulikowich, J.M. (1994). Learning from physics text: A synthesis ofrecent research. Journal of Research in Science Teaching, 31, 895–911.

Ball, D.L. (1990). The mathematical understandings that prospective teachers bring toteacher education. The Elementary School Journal, 90, 449–466.


Begle, E.G. (1979). Critical variables in mathematics education: Findings from a surveyof the empirical literature. Washington, DC: Mathematical Association of America.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings.Journal for Research in Mathematics Education, 29, 41–46.

Borko, H., Lalik, R. & Tomchin, E. (1987). Student teachers’ understandings of successfuland unsuccessful teaching. Teaching and Teacher Education, 3, 77–90.

Brown, C.A. & Borko, H. (1992). Becoming a mathematics teacher. In D.A. Grouws (Ed.),Handbook of research on mathematics teaching and learning (pp. 209–239). New York,NY: Macmillan.

Bush, W.S. (1986). Preservice teacher’s sources of decisions in teaching secondarymathematics. Journal for Research in Mathematics Education, 17, 21–30.

Calderhead, J. (1988). The development of knowledge structures in learning to teach. In J.Calderhead (Ed.), Teachers’ professional learning (pp. 51–63). New York: Falmer.

Cohen, D.K. (1990). A revolution in one classroom: The case of Mrs. Oublier. EducationalEvaluation and Policy Analysis, 12, 327–345.

Cohen, D.K. (1991). Revolution in one classroom (or, then again, was it?). AmericanEducator, 15(2), 16–23, 44–48.

Coxford, A.F, Fey, J.T., Hirsch, C.R., Schoen, H.L., Burrill, G., Hart, E.W., Watkins, A.E.,with Messenger, M.J. & Ritsema, B. (1997). Contemporary mathematics in context: Aunified approach (Course 1). Chicago, IL: Janson Publications, Inc.

Eisenhart, M.A. (1988). The ethnographic research tradition and mathematics educationresearch. Journal for Research in Mathematics Education, 19, 99–114.

Ensor, P. (2001). From preservice mathematics teacher education to beginning teaching: Astudy in recontextualizing. Journal for Rearch in Mathematics Education, 32, 296–320.

Ernest, P. (1996). Varieties of constructivism: A framework for comparison. In L.P. Steffe,P. Nesher, P. Cobb, G.A. Goldin & B. Greer (Eds.), Theories of mathematical learning(pp. 335–350). Mahwah, NJ: Lawrence Erlbaum Associates.

Evertson, C., Hawley, W.D. & Zlotnik, M. (1985). Making a difference in educationalquality through teacher education. Journal of Teacher Education, 36(3), 2–12.

Feiman-Nemser, S. & Buchmann, M. (1986). The first year of teacher preparation:Transition to pedagogical thinking? Journal of Curriculum Studies, 239–256.

Feiman-Nemser, S. & Buchmann, M. (1987). When is student teaching teacher education?Teaching and Teacher Education, 3, 255–273.

Franke, A. & Dahlgren, L.O. (1996). Conceptions of mentoring: An empirical studyof conceptions of mentoring during the school-based teacher education. Teaching andTeacher Education, 12, 627–641.

Frykholm, J.A. (1998). Beyond supervision: Learning to teach mathematics in community.Teaching and Teacher Education, 14, 305–322.

Frykholm, J.A. (1995, April). The impact of the NCTM Standards on pre-service teachers’beliefs and practices. Paper presented at the meeting of the American EducationResearch Association, San Francisco, CA.

Fullan, M.G. (1993). Why teachers must become change agents. Educational Leadership,50(6), 12–17.

Glidden, P.L. (1991). Teachers’ reasons for instructional decisions. Mathematics Teacher,84, 610–614.

Goodman, J. (1997). Against-the-grain teacher education: A study of coursework, fieldexperience, and perspectives. Journal of Teacher Education, 48(2), 96–107.


Graham, P. (1997). Tensions in the mentor teacher-student teacher relationship: Creatingproductive sites for learning within a high school English teacher education program.Teaching and Teacher Education, 513–527.

Gregg, J. (1995). The tensions and contradictions of the school mathematics tradition.Journal for Research in Mathematics Education, 26, 442–466.

Guyton, E. & McIntyre, D.J. (1990). Student teaching and school experiences. In W.R.Houston (Ed.), Handbook of research on teacher education (pp. 514–534). New York,NY: Macmillan.

Haggarty, L. (1995). The use of content analysis to explore conversations between schoolteacher mentors and student teachers. British Educational Research Journal, 21, 183–197.

Hirsch, C.R., Coxford, A.F., Fey, J.T. & Schoen, H.L. (1995). Teaching sensible mathe-matics in sense-making ways with the CPMP. Mathematics Teacher, 88, 694–700.

Jaworski, B. & Watson, A. (Eds.) (1994). Mentoring in mathematics teaching. London:Falmer.

Lampert, M. (1988). What can research on teacher education tell us about improvingquality in mathematics education? Teaching and Teacher Education, 4, 157–170.

Liston, D.P. & Zeichner, K.M. (April, 1988). Critical pedagogy and teacher education.Paper presented at the meeting of the American Educational Research Association, NewOrleans, LA.

Martens, M.L. (1992). Inhibitors to implementing a problem-solving approach to teachingelementary science: Case study of a teacher in change. School Science and Mathematics,92(3), 150–156.

Maynard, T. & Furlong, J. (1993). Learning to teach and models of mentoring. In D.McIntyre, H. Hagger & M. Wilkin (Eds.), Mentoring: Perspectives on school-basedteacher education (pp. 69–85). London: Kogan Page.

McNally, J., Cope, P., Inglis, B. & Stronach, I. (1997). The student teacher in school:Conditions for development. Teaching and Teacher Education, 13, 485–498.

McNeal, B. (1995). Learning not to think in a textbook-based mathematics class. Journalof Mathematical Behavior, 14, 205–234.

Mewborn, D.S. (2000). Learning to teach elementary mathematics: Ecological elements ofa field experience. Journal of Mathematics Teacher Education, 3, 27–46.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation stan-dards for school mathematics. Reston, VA: Author.


National Council of Teachers of Mathematics (1995). Assessment standards for schoolmathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (2000). Principles and standards for schoolmathematics. Reston, VA: Author.

NUD•IST (1993). Qualitative data analysis solutions for research professionals[Computer software]. Melbourne, Australia: Replee Pty. Ltd.

Ornstein, A.C. (1994). The textbook-driven curriculum. Peabody Journal of Education,69(3), 70–85.

Parmalee, J. (1992). Instructional patterns of student teachers of middle school mathe-matics: An ethnographic study. Unpublished doctoral dissertation, Illinois State Univer-sity, Normal.

Schug, M.C., Western, R.D. & Enochs, L.G. (1997). Why do social studies teachers usetextbooks? The answer may lie in economic theory. Social Education, 61(2), 97–101.


Shulman, L.S. (1987). Knowledge and teaching: Foundations of the new reform. HarvardEducational Review, 57, 1–22.

Simon, M.A. (2000). Research on mathematics teacher development: The teacher devel-opment experiment. In R. Lesh & E. Kelly (Eds.), Innovative research designs inmathematics and science education (pp. 335–359). Dordrecht, Netherlands: Kluwer.

Tabachnick, B.R. & Zeichner, K.M. (1984). The impact of the student teaching experienceon the development of teacher perspectives. Journal of Teacher Education, 35(6), 28–36.

Usiskin, Z. (1985). We need another revolution in secondary school mathematics. In C.R.Hirsch (Ed.), The secondary school mathematics curriculum: 1985 Yearbook (pp. 1–21).Reston, VA: National Council of Teachers of Mathematics.

Wilson, M. & Goldenberg, M.P. (1998). Some conceptions are difficult to change:One middle school mathematics teacher’s struggle. Journal of Mathematics TeacherEducation, 1, 269–293.

Zeichner, K.M. (1985). The ecology of field experience: Toward an understanding of therole of field experiences in teacher development. Journal of Research and Developmentin Education, 18, 44–52.

Zeichner, K.M. & Gore, J.M. (1989). Teacher socialization (Issue Paper 89-7). Wash-ington, DC: Office of Educational Research and Improvement (ERIC DocumentReproduction Service No. ED 326 486).

Zeichner, K.M. & Liston, D.P. (1987). Teaching student teachers to reflect. HarvardEducational Review, 57, 23–48.

LAURA R. VAN ZOEST

Western Michigan UniversityDepartment of MathematicsKalamazoo, MI 49008U.S.A.E-mail: [email protected]

JEFFREY V. BOHL

Battle Creek Area Mathematics and Science Center765 Upton Ave.Battle Creek, MI 49015U.S.A.E-mail: [email protected]

PETER SULLIVAN

EDITORIAL: USING THE STUDY OF PRACTICEAS A LEARNING STRATEGY WITHIN

MATHEMATICS TEACHER EDUCATION PROGRAMS

The current growth in interest in the methods and practice of educatingmathematics teachers is both the result of, and a stimulus for, greaterdiscussion of the nature of experiences that prospective and practicingmathematics teachers can have that contribute to their growth as teachers,and the nature of learning by reflecting on those experiences. It seems thatexperiences that represent or simulate practice are important, particularlyif participants study or critique those practices. This issue of JMTE makesa contribution to these discussions by examining the way that the study ofpractice can contribute to mathematics teacher education.

Of course, using the study of practice as a prompt for learning toteach assumes the adoption of a particular perspective on teacher educa-tion. Merseth and Lacey (1993) summarised some possible approaches toteacher education, including:

• an academic orientation that sees the teacher as the transmitter ofknowledge and the focus of teacher education as being to increasethe knowledge of the teacher;

• a personal development orientation where the focus is to foster self-knowledge in the teacher, who in turn will emphasise that with thepupils;

• a technical orientation that identifies and assesses competence in arange of discrete skills;

• a practical orientation where the emphasis is on thinking, decisionmaking, and the study of practice; and

• a social or critical approach that sees the teacher as a political actorforming part of a larger stage.

Merseth and Lacey argued for the fourth of these, a practical orientationfor teacher education. It seems to me that some policy makers inter-pret this orientation to mean that pre-service teacher education coursesshould maximise the amount of time prospective teachers spend in schools.However, this practical orientation requires much more than the provi-sion of unreflective field based experiences. It refers also to the intensive


290 PETER SULLIVAN

and intelligent study of practice; the critique of practice both within itsown context and within the light of other factors; the encouragement ofcritical reflection; the development of orientations toward looking pastthe obvious and moving beyond merely describing practice to analysingactions, responses, and pedagogical practices. In these respects it drawsalso on the second and fifth points in Merseth and Lacey’s list. Experiencein schools is clearly necessary for the practical orientation to the studyof teaching but it is not, by itself, sufficient. Not only is an orientationto reflection and change necessary, but so also is awareness of a need tothink and to grow. Mousley and Sullivan (1997) suggested that creatingthis awareness involves engaging with practice or exemplars, modelling ofalternatives, discussion of alternatives within collaborative teams, a feelingof ownership of the process, and external stimuli. They argued:

. . . teacher education programs need to find ways to perturb (participants’) existing concep-tions of . . . teaching and learning, as well as the wider contexts of schooling and society, tocreate a milieu in which change is a desired state. This needs to be done, however, in waysthat retain (participants’) control over the content, direction and pace of change. (p. 32)

Very much related to the approach to teacher education is the perspectivetaken on the nature of teaching. If teaching is seen primarily as predeter-mining objectives and devising activities that move students towardthe achievement of those objectives, then the study of curriculum andclassroom management is probably enough. If teaching and learning aretaken to include an interactive component and to be dependent on the waysthat teachers interact with students then some study of practice becomesimportant. To elaborate this point, consider the issue of the teachers’intention. Often teachers (and lecturers) are encouraged to be proactiveand to lead their students (and prospective and practising teachers) to theintended goals. Yet Brophy (1983) suggested that teachers should insteadbe reactive. He meant that, rather than prejudging what students know, andpredetermining a pathway to lead students to new understandings, teachersshould listen to what students are able to say, work with students as theycome to understand, and adapt their teaching approaches in reaction to theresponses of the students. If such approaches are expected to be part ofteaching, then clearly prospective and practising teachers need practicalexperiences of them in mathematics learning situations.

A challenge for mathematics teacher educators is that the overall timefor interacting with prospective and practising teachers is very limited, andso the learning experiences need to be powerful, easy to arrange, and ableto be assessed. One of the intentions of the Journal of Mathematics TeacherEducation is to provide opportunities for mathematics educators to report

EDITORIAL 291

on research into such experiences. This issue includes important articlesthat offer insights into the practice of mathematics teacher education.

The article by Patricia Moyer and Elizabeth Milewicz describes anapproach to this practical orientation in which prospective teachers inter-view students one on one. The advantages in their approach include thateach prospective teacher has a small scale experience in which some of thecomplexity of the classroom is minimised, with opportunities to interactwith, and respond to, a student. The prospective teachers also had anopportunity to analyse their experience, to reflect on it critically with theirpeers in a supportive environment, and to consider the nature of the ques-tions they asked. Essentially this process is about critiquing practice. Theirapproach is powerful, manageable, and assessable.

Also addressing the issue of practice, Pi-Jen Lin reports on a projectwith practising teachers in Taiwan. She uses case stories, written by theteachers, that address issues and episodes from real classroom events,as prompts for study, critique and reflection on aspects of practice. Inthis project the complexity of classrooms is considered, as is appropriatewith experienced teachers, yet it is also powerful and readily able to beevaluated.

Extending the notion of the study of practice, Despina Potari andBarbara Jaworski report on a detailed study of two teachers, using a toolthey term a teaching triad to provide a framework for study, discussion andreflection. They present the triad predominantly as a research tool and theirarticle elaborates the potential of the framework. It can readily be used asa prompt for thinking about and discussing teaching. It is clearly powerful,and also provides a tool for study and evaluation.

The review, written by Paul White, of the book Teaching mathematics insecondary schools, edited by Linda Haggarty, offers an interesting comple-mentary perspective. While there is a chapter that elaborates experiencesfor prospective teachers, other chapters remind us that mathematics teachereducation also includes consideration of the discipline of mathematics,issues of teaching, learning and assessing mathematics, and the socialcontext in which such teaching and learning occurs.

In summary, the argument here is that neither university-based theoret-ical study of teaching nor learning to teach through unreflective apprentice-ship, either separately or together, are likely to produce a career-longorientation to professional learning. Studying teaching in simulated or realsituations offers considerable potential for stimulating thinking not onlyabout the application of theory to practice but also for creating personaltheories for the study of practice.

292 PETER SULLIVAN

Collectively the articles offer readers the opportunity to reflect on theirbeliefs about what is teaching and what it means to learn to teach, and alsoto consider the strategies offered for the study of practice.

REFERENCES

Brophy, J.E. (1983). Research on the self-fulfilling prophecy and teacher expectations.Journal of Educational Psychology, 75(5), 631–661.

Merseth, K.K. & Lacey, C.A. (1993). Weaving stronger fabric: The pedagogical promiseof hypermedia and case methods in teacher education. Teacher and Teacher Education,9(3), 283–299.

Mousley, J. & Sullivan, P. (1997). Dilemmas in the professional education of mathematicsteachers. In E. Pekhonnen (Ed.), Proceedings of the 21st Conference of the InternationalGroup for the Psychology of Mathematics Education (pp. 131–147). Lahti, Finland:University of Helsinki, Lahti Research and Training Centre.

La Trobe UniversityP.O. Box 199Bendigo Victoria 3552Australia

PATRICIA S. MOYER and ELIZABETH MILEWICZ

LEARNING TO QUESTION: CATEGORIES OF QUESTIONINGUSED BY PRESERVICE TEACHERS DURING DIAGNOSTIC

MATHEMATICS INTERVIEWS

(Accepted 6 August 2002)

ABSTRACT. Developing appropriate questioning techniques is an important part ofmathematics teaching and assessment. This study examined the questioning strategiesused by 48 preservice teachers during one-on-one diagnostic mathematics interviews withchildren. Each participant conducted an audiotaped interview with one child, followedby an analysis and reflection of the interview. Data were analyzed to develop generalcategories of questions used by the preservice teachers. These categories included: 1)checklisting, 2) instructing rather than assessing, and 3) probing and follow-up questions.The analyses and reflections completed by preservice teachers indicated that using thediagnostic interview format allowed them to recognize and reflect on effective questioningtechniques. Through an examination of these categories of questions, we offer suggestionsfor teaching the skill of mathematics questioning in preservice teacher education courses.

KEY WORDS: interviewing, mathematics questions, preservice teacher education, ques-tioning techniques

The kinds of knowledge children construct and communicate during amathematics lesson may be prompted by teachers’ questions. Teacherswho can question effectively at various levels within the cognitive domain,such as knowledge, comprehension, application, analysis, synthesis, andevaluation (Bloom, 1956), are better able to discern the range and depthof children’s thinking. A good question may mean the difference betweenconstraining thinking and encouraging new ideas, and between recallingtrivial facts and constructing meaning (Kamii & DeVries, 1978; Kamii &Warrington, 1999; Schwartz, 1996; Stone, 1993). Some researchers arguethat a teacher’s verbal behavior is a strong indicator of their total teachingbehavior (Adams, 1994). Recent focus on the use of questioning inteaching mathematics (e.g., Carpenter, Fennema, Franke, Levi & Empson,1999, 2000; Mewborn & Huberty, 1999) supports the idea that a teacher’squestioning strategies are pivotal to the instructional process because ques-tioning is the most frequently used instructional tool (Wassermann, 1991).Developing appropriate questioning techniques is an important part ofteaching and assessing mathematics. Much of the research on questioningtechniques provides evidence of the types of questions used by classroom


294 PATRICIA S. MOYER AND ELIZABETH MILEWICZ

teachers (Buschman, 2001; Carpenter et al., 1999, 2000; Mewborn &Huberty, 1999). Fewer research studies document ways to support thedevelopment of questioning skills at the preservice level (Ralph, 1999a).

Although research in recent years has seen a surge of interest in the rela-tionship between teacher questioning and children’s thinking (Baroody &Ginsburg, 1990; Buschman, 2001; Carpenter, Fennema, Peterson, Chiang& Loef, 1989; Fennema, Carpenter, Franke, & Carey, 1993; Fennema,Franke, Carpenter & Carey, 1993), more than just an understanding of thechild’s knowledge can be gained from using questioning as an assessment.The diagnostic interview of a child (combining questioning and obser-vation) can also provide a record of how interviewers select questionsthat probe children’s mathematical thinking. We propose that preserviceteachers can benefit from these types of interactions with children byscrutinizing their own performance and reflecting on the questions theyuse in these interviews. After the interview, preservice teachers can reflecton why they use particular questions and use this self-reflection process todevelop better questioning strategies. Few studies address ways to cultivatethis fundamental skill in preservice teachers, and no research has catego-rized the types of questions that might be expected of these beginningteachers.

The purpose of this project was to (1) examine the questioningstrategies used by preservice teachers during one-on-one diagnostic mathe-matics interviews with children and (2) engage the preservice teachers inan analysis of, and reflection on, their own questioning. We believed thepreservice teachers would use a variety of questions during the interviews,ranging from basic factual questions to those questions that more deeplyprobed students’ mathematical thinking. We also believed that guidingpreservice teachers through an analysis of the questions they asked andthe responses they received from children would enable them to recog-nize both effective and ineffective questioning throughout the interviews.In this paper we categorize and label the types of questions used by thepreservice teachers during their mathematics interviews with the children.We propose that the use of these categories for discussion is an effectivestrategy for teaching the skill of mathematics questioning in preserviceteacher education courses.

LEARNING TO QUESTION 295

THEORETICAL PERSPECTIVE

The Interview

The use of alternative forms of assessment in mathematics, such as inter-viewing (a combination of questioning and observing), has grown inpopularity as a result of the Standards movement and other calls forreform in mathematics education (Huinker, 1993; NCTM, 1991, 1995,2000; Stenmark, 1991). Classroom teachers have used student interviewsto guide and inform their own instruction in mathematics (Buschman,2001). Teachers are encouraged to ask children questions in mathematicsthat help them to work together and make sense of mathematics; to learnto reason mathematically; to learn to conjecture, invent, and solve prob-lems; and to connect mathematics, its ideas and its applications (NCTM,1991). Verbal interactions and performance-based assessments are seen asan important part of the mathematics teaching and learning process.

Historically, one-on-one diagnostic interviews are derived primarilyfrom the clinical method of interviewing developed and perfected bySwiss psychologist Jean Piaget (1926, 1929) who used the interviewtechnique to investigate the nature and extent of children’s knowledgeon a variety of topics, including mathematics. Interviews have also beenused as an important means of diagnosing children’s misconceptions anderror patterns in computation (Ashlock, 2002). Posner and Gertzog (1982)identify the interview’s chief goal as ascertaining “the nature and extentof an individual’s knowledge about a particular domain by identifying therelevant conceptions he or she holds and the perceived relationships amongthose conceptions” (p. 195). As face-to-face meetings or conversations,interviews, by definition, rely on verbal communication as the primarymeans for eliciting this information from the interviewee.

Developing Effective Questioning Strategies

Typical interactions between teachers and students in mathematicsclassrooms in America are characterized by Stigler and Hiebert (1999)in their book The Teaching Gap. Their analysis of the cross-cultural ThirdInternational Mathematics and Science Study (TIMSS) research revealsthat the “script” for teaching mathematics in the United States involvesacquiring isolated skills through repeated practice (Stigler & Hiebert,1999). The TIMSS videos also demonstrate teachers’ rapid-fire questionsthat require one-word responses and the way in which “the nature and toneof teachers’ questions often give away the answer . . .” (Stigler & Hiebert,1999, p. 45). Developing effective questioning skills in mathematicsclassrooms requires shifting the practices and beliefs of the individuals


engaged in those interactions. For this reason, some classroom teachersand university researchers are working collaboratively in action researchprojects to study how effective questioning techniques help teachers under-stand student thinking and guide classroom instruction (Buschman, 2001),specifically in the mathematics classroom (Mewborn & Huberty, 1999).

Although seemingly a basic activity that requires little expertise,effective questioning in mathematics actually requires well-developedoral-questioning skills – in many cases, the same skills teachers mustuse during classroom interactions. Based on a synthesis of questioningresearch, Ralph (1999a, 1999b) proposes that these basic oral-questioningskills should include (among others) preparing important questions aheadof time, delivering questions clearly and concisely, posing questions tochildren that stimulate thought, and giving children enough time to thinkabout and prepare an answer. As Mewborn and Huberty conclude fromtheir classroom research on questioning, “allowing children to explain theirthinking takes more time than simply asking for one-word answers andtelling children whether or not they are correct” (Mewborn & Huberty,1999, p. 245).

Research has shown that using the interactive structure of dialoguein teaching mathematics is difficult for preservice teachers (Nilssen,Gudmundsdottir & Wangsmo-Cappelen, 1995). When open-ended ques-tioning is used and there are many right answers, the learning environmentbecomes complex and less predictable as teachers attempt to interpretand understand children’s responses. To do this effectively requires prin-cipled knowledge of mathematical concepts and an understanding of howstudents think and reason mathematically (Ball, 1991; Lampert, 1986;Leinhardt & Greeno, 1986; Ma, 1999). Whereas experienced teachers havea repertoire of easily accessible strategies and pedagogical content knowl-edge, preservice teachers have difficulty interpreting and responding tounexpected answers from children (Nilssen et al., 1995).

Developing questioning skills can be an integral focus of preservicemathematics education coursework. The one-on-one diagnostic interviewof a child offers an alternative staging area for practicing these skills, onethat simultaneously duplicates the uncertainty of classroom interactions,focuses on the child’s thinking, and forces preservice teachers to use ques-tioning to get at that thinking. Just as providing a scripted lesson does notguarantee competent teaching, providing an interview protocol does notguarantee competence in questioning. While the interview protocol andthe goal of the interview may be clear, preservice teachers cannot planwhat the children will say. In the process of interpreting and responding


to unexpected answers, preservice teachers practice developing questionsthat take into account children’s thinking.

Shifting Beliefs

Teacher education and professional development programs alike generallyagree that underlying beliefs guide a teacher’s adoption and use of instruc-tional techniques. The CGI (Cognitively Guided Instruction) (Carpenter etal., 1999) Professional Development Program attempts to shift teachers’beliefs about children’s mathematical thinking by demonstrating howchildren think mathematically and by encouraging teachers to invite, listento, consider, and incorporate children’s divergent solutions (Carpenter etal., 2000). Teachers in CGI classrooms who have not previously attendedto their students’ invented strategies “are often surprised at what studentssay and do,” suggesting that these interactions impact teachers’ underlyingbeliefs (Bowman, Bright & Vacc, 1998).

A number of studies indicate that when mathematics teachers listen toand comprehend their students’ thinking, they expand their understandingof mathematics, shift their beliefs about how it should be taught, andmodify their teaching practices (Fennema & Carpenter, 1996). One studyin particular determined that “a focus on children’s thinking” and “consid-erable reflection on both one’s beliefs and behavior” were instrumental inshifting preservice teachers’ beliefs about mathematics instruction (Bright& Vacc, 1994, p. 10). Other research (Moyer & Moody, 1998) conductedon the use of preservice teacher-conducted interviews found the one-on-one interactions to be a useful strategy in understanding what childrenwere thinking and shifting preservice teachers’ beliefs about mathematicsassessment in general.

Action research projects in classrooms have demonstrated that indi-vidual interviews with children make teachers more aware of what childrenknow, help teachers understand how children learn mathematics, and influ-ence teachers’ instructional practices (Buschman, 2001). Although muchof the research on questioning focuses on the inservice teacher, fewerstudies document the questioning skills of the preservice teacher. Webelieve that the types of questions preservice teachers choose to use duringmathematics interviews are worthy of inquiry, and that these categoriesof questions can guide discussions on the development of questioningtechniques. Having preservice teachers focus on the skill of questioningin a one-on-one diagnostic interview may be an effective starting pointfor developing the mathematics questioning skills they will use as futureclassroom teachers.


METHODOLOGY

Participants

This study occurred during the fall of one academic year. Participantswere 48 elementary preservice teachers enrolled in a mathematics methodscourse in their senior year as undergraduates, prior to their final internshipplacement for teacher certification. The participants were both male (3)and female (45). Participants were told they would be audiotaped and thatthese audiotapes would be transcribed and used in a self-reflection on theirquestioning during the mathematics interviews.

Procedures

The preservice teachers conducted mathematics interviews with childrenfrom local elementary schools. Rational numbers were chosen as the topicfor the assessment interview assignment. The instructor selected a varietyof fraction tasks developmentally appropriate for elementary children,and an interview protocol was designed using these tasks. The interviewprotocol included the following tasks: (1) use a region and a set modelto show 1/2, 1/3, 1/4, and 3/4; (2) use a pictorial model (shapes and sets)to shade or draw 1/2, 1/3, 1/4, and 3/4; (3) express what you know aboutfractions in symbolic form; (4) examine different models of 1/2 and explainwhy all of the models are called “one-half”; and (5) use a drawing to showhow to share a pizza. The region model uses a region (such as a circle,square or rectangle) as the whole, and the parts of the region are congruent(same shape and size). The set model uses a set of objects as the whole, andthe parts of the set are the individual objects (such as counters, buttons, orcoins). The interview protocols included several suggested follow-up ques-tions; however, preservice teachers were told that the main focus during theself-reflections would be on the types of questions they developed and usedthemselves during the interviews.

During one 3-hour class session, preservice teachers watched a videoin which the instructor interviewed two second-grade children. Duringthe video, follow-up questions used by the interviewer with the childrenwere highlighted as specific examples of questions that would be appro-priate for the responses given by the children. Following the viewingof the video, preservice teachers and the instructor discussed examplesof questions that elicited the children’s thinking and conceptual under-standing about the tasks. The video was used to identify questions thatprobed children’s thinking in several categories: 1) questions that helpedchildren to make sense of mathematics (i.e., Can you explain to me whythat makes sense?), 2) questions that helped children rely more on them-


selves to determine whether something was mathematically correct (i.e.,How did you reach that conclusion?), 3) questions that helped childrenlearn to reason mathematically (i.e., How could you prove that to me?),4) questions that helped children to conjecture, invent and solve problems(i.e., What would happen if . . .?), and 5) questions that helped children toconnect mathematics, its ideas, and its applications (i.e., Have we solvedany problems like this one before?) (Reys, Suydam, Lindquist & Smith,1998, p. 45; Stenmark, 1991, pp. 31–32). The preservice teachers usedthese question categories when they conducted analyses of their own inter-views. There was also discussion about ways to address a child’s potentialanxieties regarding the interview process (i.e., nervousness about beingtaped, fear of being graded, etc.) so that preservice teachers could try toprevent these anxieties from interfering with the interview. Each preserviceteacher was assigned to a classroom placement during the time they wereenrolled in the course and specific questions about children at differentgrade levels were also addressed during the class discussion.

Preservice teachers were given the protocols, including specific tasksand sample questions, and were asked to select a child from their schoolplacement with whom to conduct the interview. Some materials for theinterview were provided (reproducible sheets of region models) but preser-vice teachers were required to obtain some additional materials on theirown (e.g. counters). The children selected by the preservice teachersto participate in the mathematics assessment interviews were generallyregarded as average to above average students by the classroom teachers.There were 48 children ranging in age from 5 to 12, distributed amonggrades K-6 as follows: Kindergarten (6), Grade 1 (8), Grade 2 (10), Grade3 (10), Grade 4 (5), Grade 5 (6), and Grade 6 (3).

The interviews were designed so that the number of tasks completedwith follow-up questions would take approximately 20–30 minutes,depending on the responses of each child. The actual interviews variedin length from 10–60 minutes, with the age level of the child having noconsistent effect on interview length. That is, one child in Kindergartenmay have been interviewed for 10 minutes, while another was interviewedfor 30 minutes, and yet another was interviewed for 50 minutes. The inter-viewers determined the length of each interview, regardless of the age ofthe child being interviewed.

Preservice teachers were encouraged to conduct several practice inter-views before submitting their final interview for reflection. For the finalinterview, they audiotaped their interviews, recorded behaviors during theinterviews, and collected student writings and drawings completed duringthe interviews. These materials were used to provide a holistic picture of


the interactions that occurred between each preservice teacher and childduring the interviews. Following the interviews, preservice teachers tran-scribed their audiotapes fully, conducted guided analyses of their owninterviews and completed written reflections based on their questioningand interactions with the children during the interviews.

The procedures used in this research were those of a descriptive casestudy, which is useful in describing the details of an innovative practice(Merriam, 1988). This method of qualitative research was used to collectdetailed information on a single phenomenon using a variety of sources forthe purpose of explaining and evaluating the experience. It was importantto the researchers to collect these data in a natural context and to includepreservice teachers’ verbatim comments and points of view.

The study used three sources of data: complete transcriptions of theaudiotaped interviews, preservice teachers’ written reflections of the inter-view process, and descriptive data on non-verbal behaviors that occurredduring the interview. The interview transcripts and the written reflec-tions were read and coded separately by two independent readers usingan interpretational analysis to examine the data for constructs, themes,and patterns that may be used to explain the questioning strategies ofthe preservice teachers during the interviews (Gall, Borg & Gall, 1996).The researchers coded the themes using a modified constant comparativemethod (Strauss, 1987) that included an iterative process of reading andre-reading to identify categories and tagging preservice teachers’ writtenreflections and interview transcripts. Analyses of the transcripts focusedon preservice teachers’ questions that were not a part of the interviewprotocol. Following the first phase of coding data segments were clusteredaround the most salient and recurring themes. These themes were orga-nized into predominant categories of questioning strategies commonlyused across the 48 interviews, and data were re-analyzed against thesecategories.

RESULTS: CATEGORIES OF QUESTIONS

Several patterns emerged in the interviewing verbalizations that seemedindicative of differences in questioning techniques. We categorized theseverbalization patterns as:

(1) Checklisting, where the interviewer proceeded from one question tothe next with little regard for the child’s response, which included(a) no follow-up questions, and (b) questions with verbal check-marks;


(2) Instructing rather than assessing, which included (a) leading questionsthat direct the child’s response, and (b) abandoning questioning andteaching the concept; and

(3) Probing and follow-up, where different types of questions were usedto invite or further investigate the child’s answer, which included (a)questioning only the incorrect response, (b) non-specific questioning,and (c) competent questioning.

We acknowledge that there is a great deal of overlap among thesecategories and that, frequently, interviewers use more than one techniquesimultaneously within the same interview. In the following sections, wediscuss examples and descriptions of these categories using verbatimquotes from transcripts of the audiotaped interviews (where “T” stands forthe preservice teacher interviewer and “C” stands for the child interviewee)and preservice teachers’ written reflection comments.

Checklisting

A common behavior of preservice teachers during interviews was“checklisting.” The “checklisting” interviewer reads the questions onthe interview protocol one after the other, relying on the script to directthe interview rather than acknowledging the responses of the child. Nomatter what answer the child gives, the interviewer simply moves onto the next question. In essence the interviewer appears to be listening,not to the child’s thinking, but for a response which then allows theinterview to continue. The resulting interview is often fast-paced,marked by a lack of follow-up questions, and frequently accompanied byverbal “checkmarks.” Interviewers who used this questioning techniquerepeatedly often completed their interviews rapidly (in 10–15 minutes).

No follow-up questions. “Checklisting” interviews are characterized by thedegree to which their strategy precludes the child from expanding on ananswer. When interviewers ask no follow-up questions they risk obtainingno information about the child’s mathematical thinking by not specifi-cally inviting it. Several interviewers asked no follow-up questions of thechildren they interviewed. They simply read through the list of questionson the protocol, obtained an answer from the child, and moved on withoutprobing the child’s thinking, as in the following example:

T: What part is shaded and what part is not?C: The left.T: How much of it is shaded?C: The whole.


T: How much is shaded in the circle?C: Uh, half. (Grade 2, Interview 2, p. 1)

During one portion of the assessment interview, the interviewerpresents a half square piece and a half circle piece of construction paperand asks the child to explain why both are called half when they don’t lookalike. The purpose of this question is to assess the child’s informal under-standing of the concept of “half” even in the face of empirical evidencethat seems to suggest that the two models are different. This idea requiresa generalization of the concept of half and requires children to explain theirthinking. The interviewer should force the child to examine and defend hisexplanation. Here is how one interviewer handles the question:

T: If you have a circle and you halved it, how many pieces would you have?C: Two.T: What about the square?C: Two.T: Does it matter about the shape?C: No.T: Good. (Grade 6, Interview 1, p. 5)

Verbal checkmarks. In addition to its pace, checklisting may be distin-guished by the interviewer’s specific and repetitive use of verbal “check-marks.” These one- or two-word verbalizations, such as “OK,” “Right,”and “Good,” indicate to the child that it is no longer necessary to continuethinking about the question because the question is completed, and there-fore “checked off” the list. In essence a word or phrase becomes averbal checkmark ending one task to begin another. The written reflectionsindicate that interviewers often recognized their use of these verbal check-marks; as one preservice teacher commented, “I used the word OK to maketransitions from question to question” (Grade 2, Interview 5, Reflection).

In the following exchange, “Good” becomes the verbal checkmark thatends a task and begins a new one. This example also shows how inter-viewers who are checklisting move rapidly from one question to the next,allowing little time for a complete response from the child. Notice how thechild’s responses are brief and how the interviewer gives the verbal signalbefore moving on to the next question.

T: You made how many pieces?C: Two.T: Good, good job. I have given you six counters. I want you to give me half and you

keep half. [Child moves counters] Good, how many counters are half . . .?C: Three.T: Good, because three plus three equals six, right? [child nods “yes”] Good job.

(Grade 6, Interview 1, p. 5)


In the previous example, the verbal checkmark “good” signals that theinterviewer is moving on to the next question. One preservice teacherrecognized her checklisting behavior in her reflection, stating, “Afterlistening to the interview, I noticed that I rewarded the student’s correctresponses by saying ‘Good’ or ‘Very good’ . . . I feel I did this out ofinstinct for praise” (Grade 2, Interview 7, Reflection). These signal wordsend discussion of the question by indicating to the child that the intervieweris not waiting for any additional information.

During the questioning process, this child appears to recognize therapid pace of the interview. At one point, when the interviewer proceedstoo quickly, the child attempts to slow down the interview by asking theinterviewer to “wait” for him.

T: OK, now with these squares can you show me 1/2 on this square right here?[Childbegins coloring the square] OK . . .

C: Wait, wait! [child is still working]T: OK, on that second square right there . . . (Grade 5, Interview 3, p. 2)

After a brief pause, the interviewer resumes the checklisting of tasks. Later, the childagain asks the interviewer to slow down.

T: OK, now I am going to take these twelve counters right here and I want you to showme half, what would be half. [child moves counters] OK . . .

C: Wait! [child is still working] (Grade 5, Interview 3, p. 2)

For the child in this interview, the word “OK” has become a verbalcheckmark – an indicator that the interviewer is moving to the next taskand does not expect the child to think about or respond to the previousquestion anymore. In this exchange, the word “OK” has specific meaningto the child: the child does not ask the interviewer to “Wait” because he hasheard the next question; he asks the interviewer to wait as soon as he hearsthe interviewer’s signal word, “OK.” The interviewer proceeded throughall the questions at this rapid pace, consistently saying “OK” after shefinished one question and moved on to the next, for a total of 25 “OK”s inthis brief ten-minute interview. Reflecting on this interview, the preserviceteacher wrote, “The student may have felt a little rushed. . . . When I askquestions to students or anybody, I need to give them time to think andthen respond. If I were to assess another student, I would slow down thepace of the interview a little bit” (Grade 5, Interview 3, pp. 4–5).

Instructing Rather than Assessing

Leading questions. Some preservice teachers attempted to instruct childrenduring interviews instead of assessing their mathematical knowledge. In avariety of instances they used leading questions that directed the child’sresponse or provided hints about the answer, or they simply abandoned the


strategy of questioning and attempted to teach the concept by explainingor telling the solution.

Several preservice teachers use questioning strategies that are leadingand in essence attempt to guide and prompt the child to the correct answer.This strategy results in a guessing game in which the child concentratesmore on puzzling out what the interviewer is thinking rather than onexplaining his/her own thinking. In one example, when the problem is todivide a square in half, the interviewer asks, “What were you trying to do?Give me what and you keep what?”; the child fills in the blanks: “I give youone and I take one” (Kindergarten, Interview 1, p. 11). Reflecting on herinterview experience, this preservice teacher observed that “with the rightquestions . . . [the student] showed some knowledge of fractions”; she alsonoted that “to be an effective interviewer . . . you have to be ready to askquestions to prompt students” (Kindergarten, Interview 1, Reflection).

After coaching one child to the correct answer, this interviewer asksthe child to explain his thinking: “So what did you have to think about toknow that? You had to think about it as having what?” (Grade 1, Interview4, p. 7). The interviewer then supplies an answer for the child, delivered inthe form of a question: “You had to think about that whole thing, the wholeset, you had to think about it kind of being in four different groups and thenyou could take one of them?” The child agrees, “Yes,” and the interviewergoes on to the next problem (Grade 1, Interview 4, p. 7). Some interviewersacknowledged their leading behaviours: “After hearing myself in the tape,I feel I may have spoon-fed a few answers to [the child] without realizingit” (Grade 2, Interview 3, Reflection).

Interviewers often presented answers as “yes or no” questions, cueingchildren by ending their questions with “right?” During an interview with afirst-grader, this preservice teacher offers an elaborate leading explanationwith which the child is encouraged to agree:

T: Okay, since we had these divided, I had them in three different rows and you knewthat you had to have an equal amount in all three groups, right?

C: Yes, ma’am.T: So is that what you were thinking?C: Yes, ma’am. (Grade 1, Interview 4, p. 6)

Rather than persisting with questioning to extricate the child’s answer,interviewers who lead assume they know what the child is thinking, andin essence, attempt to verbalize this thinking for the child.

Teaching and telling. About one-fourth of the interviewers moved fromquestioning children to teaching the concepts rather than assessing thechild’s level of knowledge on the topic, adopting a more directive and


explanatory method of interacting with the child. Often “leading ques-tions” and “teaching” worked together to provide answers for childrenrather than encouraging children to think or elaborate on a response.

The following is an example of an interviewer who leaves the roleof questioner and takes over the role of problem-solver. When the childis unable to divide a drawing of a pizza equally among five people, theinterviewer steps in. Confronted with the child’s confusion about how tosolve the problem, the interviewer presents the solution for the child:

T: What if we didn’t draw a line all the way down the middle. What if we just drewit to the half and made some equal pieces this way. Do you think that might work?See we could get more equal pieces if we didn’t have to draw all the lines. We didn’thave to draw straight down. . . . What if we did that? (Grade 4, Interview 4, p. 6)

Another interviewer asks the child to write, in numbers, what one-fourth looks like; when the child responds that the number would be“Five,” the interviewer replies, “No. I just said to write one-fourth . . .”(Grade 1, Interview 8, p. 5). When the child writes “three plus four” forthree-fourths, the interviewer stops questioning and begins teaching thechild how to write fractions as numerals:

T: Ok, let me show you. Let me show you what I can write. I could write this – writeright here – one over two and that would stand for one-half. I could one over three[sic] and that would stand for one-third, one over four and that would be one-fourth,and three over four, so it’s a little bit different. (Grade 1, Interview 8, p. 5)

When asked to divide up two pizzas for five people so that each personwill get the same amount, a sixth-grader divides one pizza into five piecesand the other pizza into six. After questioning the amount of pieces thechild made for each pizza, the interviewer says, “Well, there are five peopleso you forgot that the two pizzas are going to have to be divide [sic] thesame way. You have to distribute the pizza to all five people. Each personhas to get the same amount” (Grade 6, Interview 1, p. 6). Having offeredthis reasoning to the child, the interviewer then returns to questioning thechild’s thinking about the solution. The child basically reiterates what theinterviewer has told him was incorrect about his answer, supplying theexplanation that the interviewer already gave.

Probing and Follow-up Questions

In contrast to the checklisting and instructing strategies, the use ofprobing and follow-up questions during an interview demonstrates theinterviewer’s greater attention to the child’s thinking. While the checklisteroften asks no follow-up questions at all, the probing interviewer responds


to the child’s answer with another relevant question in an attempt to getthe child to expound on the answer or think about it further. Rather thansignalling the end of the task, this strategy communicates to the child thatthe answer is still open for discussion. Yet not all probing and follow-upquestions adequately or appropriately assess what the child is thinking.Follow-up and probing questioning in the mathematical interviewswith children in this study included (1) questioning of only incorrectresponses, (2) non-specific questioning, and (3) competent questioningthat specifically and consistently probed a child’s answer.

Questioning only the incorrect response. There were several instanceswhere preservice teachers only questioned children when they gave anincorrect response; many did not use follow-up when a child’s responsewas correct. This practice works with the assumption that because the childproduced the correct answer, the child must understand the concept. Inthe following interaction, the interviewer probes only the child’s incorrectresponse.

T: Where is the fraction circle that shows 1/3?C: There [child points to 1/3].T: Good, what about 1/4?C: [child points to 1/4].T: Good, what about 3/4? Can you show me 3/4?C: [child points to 1/8].T: Is this 3/4? Eight pieces out of one? If I give you this circle [fraction circle divided

in fourths], can you show me 3/4 of it? (Grade 3, Interview 4, p. 1)

Using a probing question only when an incorrect response has been givenlimits the child to explaining only wrong answers, thus bypassing anopportunity for the child to articulate and defend accurate solution routes.

In another example, the interviewer moves rapidly through the ques-tions until the child gives an incorrect response to the interviewer’squestion to show 1/3 of 12 counters:

T: Show me 3/4.C: [Child uses fraction circles to show 3/4]T: Good, good job. Now with these 12 counters show me 1/2 of the counters.C: [Child shows 6 counters]T: Show me 1/3.C: Three? [Child shows 3 counters]T: You sure? Think about it. If you want to rearrange them, you can. (Grade 3,

Interview 3, p. 1)

As in the previous exchange, the progression stops when the child gives anincorrect answer, and only then does the interviewer begin to question the


child’s thinking. As one interviewer reflected, “I learned that it sometimeshelps to keep questioning to get an answer that you are looking for”(Grade 5, Interview 1, Reflection). For this preservice teacher, questioningwas merely a means of getting the “right answer.”

Non-specific questioning. Even those interviewers who consistentlyfollowed up children’s answers often did so with questions that lackedspecificity. Often questioning did not acknowledge the child’s specificresponse, resorting instead to general follow-up questions such as “Whatwere you thinking?” As part of the preparation for interviewing children,preservice teachers were given some general open-ended sample questionsone might ask children in order to encourage them to explain their thinking.In many cases, the interviewers simply used one or more of the suggestedquestions repeatedly rather than tailoring the questions to fit individualchildren’s responses. Upon reflection several preservice teachers recog-nized the lack of specificity in their questioning: “I think that I couldhave used better questions to get at what [the child] was thinking” (Grade2, Interview 1, Reflection) and “If I were to give this interview again, Iwould make sure that my questions were clearer” (Grade 2, Interview 4,Reflection).

For instance, some neutral questions provided by the instructor were,“How did you figure out the answer?”, “Can you explain to me what youwere thinking when you were trying to figure out the answer?”, and “Canyou give me another example to explain what you mean?” It is evidentfrom the following excerpts of different transcripts that these four differentpreservice teachers were simply using one of the sample questions ratherthan creating questions tailored to individual children’s responses.

T: Can you explain to me what you were thinking when you tried to figure that out?(Kindergarten, Interview 5, p. 1)

T: Can you explain to me what you were thinking when you were trying to figure outthe answer? (Grade 1, Interview 1, p. 1)

T: What were you thinking when you tried to figure out the answer? (Grade 5, Interview4, p. 1)

T: Can you explain to me what you were thinking when you tried to figure out theseanswers? (Grade 3, Interview 5, p. 4)

Several children gave interesting or unusual responses to interviewquestions. For example, one child described the concept of fourths as“knowing a window” and as a “lion’s cage” (Grade 4, Interview 5, pp. 1,4) and two other students described fractions as parts of a moon (Grade3, Interview 6, p. 7; Kindergarten, Interview 6, p. 1). These responsesseem to warrant follow-up by the interviewers to determine explicitly


what the children were thinking when they used these analogies in theiranswers. Unfortunately some of the interviewers did not use a specificfollow-up question to explore the response and the children’s analogieswere lost. Several preservice teachers recognized this in their reflections:“I also discovered after listening to myself, that there were plenty ofopportunities to add something or ask more thought-provoking questionsabout something the student said . . .” (Grade 2, Interview 9, Reflection);“I realized interviewing students is much more difficult then [sic] I hadexpected, especially when the student gives answers I did not expect tohear” (Grade 2, Interview 3, Reflection).

Competent questioning. In contrast there were interviewers who listenedto children and used their responses to construct a specific probe for moreinformation about children’s answers. For example, this interviewer usesinformation from the child’s accurate drawing of one-third in a squareto ask a specific probing question: “How did you figure that out? Howdid you know you had to put two lines to make three parts?” (Grade 5,Interview 5, p. 2). In the following conversation with a second grader, theinterviewer probes a correct answer by using the child’s response as partof the question:

T: Which is larger, 1/3 or 1/4?C: 1/3.T: How did you figure out that answer?C: I looked at the pieces.T: How did the pieces show you that 1/3 was larger than 1/4? (Grade 2, Interview 6,

p. 1)

Whereas other interviewers might have stopped questioning the childafter the child’s correct response (“1/3”), this interviewer continueswith two additional questions to probe the child’s initial response. Thisinterviewer is not looking for a correct answer or simply listeningfor a response. The specifically tailored follow-up questioning reflectsthe child’s answer and stimulates relevant discussion about the child’sthinking.

In this last example, the interviewer uses several skills of competentquestioning. She specifically probes for more information and follows uppersistently on an intriguing response from the child with questioning thatdemonstrates she has listened to the child’s response.

T: Can you explain to me what you were thinking about the half-circle piece and thehalf-square piece? What were you thinking?

C: Um . . . well, I knew that half would be half the circle and, um, it looked like a halfmoon.


T: Oh, that’s interesting. Can you tell me more about that?C: Um . . . no.T: About the moon and its shape, it does look like the moon, doesn’t it?C: Uh-huh. The moon can sometimes look like a half, but it changes shape a lot. (Grade

3, Interview 6, p. 7)

In this interaction, the child attempts to terminate discussion with thewords “um . . . no.” Even though the child seems to be finished explaining,the interviewer persists with an additional question specific to the child’sresponse and consequently gains greater insight into what the child isthinking.

DISCUSSION

The interview as a medium relies heavily on verbal language, especiallyquestioning, to carry information between the interviewer and the inter-viewee. Although an interview (with a set of guiding questions and tasks)is quite different from the teaching and learning processes in a classroomsetting, the skill of interacting with children through questioning is animportant component of both processes. For example, the use of questionsin a “checklisting” style, as exhibited by the preservice teachers, is oftenobserved during whole-group mathematics instruction when teachers areasking students brief factual questions. Knowing when different types ofquestions are appropriate to use is an important skill for preservice teachersto develop.

We expected that the preservice teachers conducted these interviewswith little or no prior experience in interviewing children or using ques-tioning skills in a mathematics classroom and that their underlying beliefswould guide the adoption and use of questioning techniques. The rapidpace of the questioning during several of the interviews may have mirroredthe experiences of some preservice teachers in previous mathematicsclasses. Because their experiences in this area were limited, the behaviorsof the children often dictated their actions, prompting them to speed upan interview when a child seemed bored. The level of questioning skillof the preservice teachers in this study is reflective of their status asnovices. We would not expect that these beginning teachers would havewell-developed questioning skills; many classroom teachers are them-selves working on developing their skills in asking effective questions inmathematics (Buschman, 2001; Mewborn & Huberty, 1999).

Yet, it is important for preservice teachers to recognize that there arevarious types of questioning that can be used to assess and understand


children’s thinking in mathematics. Furthermore, different types of ques-tions are more appropriate for different mathematical situations. On thebasis of the data presented above, these preservice elementary teachersbrought a variety of questioning techniques to their interviews with thechildren. Although preservice teachers generally do not use competentquestioning techniques (Ralph, 1999a, 1999b), these data demonstratethat the preservice teachers who participated in this project did exhibitsome important beginning characteristics of competent questioning. Forexample, asking appropriate follow-up questions is an important skill todevelop for teaching mathematics, and many of the preservice teachers didattempt to use follow-up questions. Although some used follow-up ques-tions that were non-specific or questioned only incorrect responses, othersdid specifically target children’s thinking and used follow-up questionsto probe that thinking. These preservice teachers were able to recognizethe various patterns of questioning strategies they used when reviewingtheir own transcripts. The logical next steps are for teacher educators toexamine how to develop better questioning skills and when different typesof questioning in mathematics might be more appropriate.

Reflecting on the Interview to Develop Questioning Competence

Questioning interactions are a significant part of mathematics teaching andlearning, both in structured interview settings and in the classroom, andpreservice teachers need opportunities to practice their questioning tech-niques. As a staging area for developing questioning skills and using thosequestioning skills with a child, the interview may serve as an importantvehicle for developing preservice teachers’ questioning techniques. Byrecording the questions they select, preservice teachers may use the one-on-one interview to reflect on their own questioning. This realistic, yetcontrolled interaction with a child allows preservice teachers to examinetheir own patterns of verbal interaction with children prior to using thoseverbal patterns in the classroom.

Structured opportunities that engage preservice teachers in learningappropriate questioning strategies in mathematics and that provide direc-tion in analysis and reflection can be valuable experiences in preparingfor future classroom situations. Through discussion about vague orgeneral follow-up questions and examples of specific questioning tech-niques, preservice teachers have opportunities to devise better questioningstrategies. For example, if a child has just said that a half circle is likethe moon, asking her “How is a half circle like the moon?” acknowledgesher answer and directs her toward a specific elaboration, helping to keepher focused on the mathematical question at hand without compromising


her thinking. As one interviewer stated, “With practice, I feel like I will bebetter able to think on my feet and guide without telling the answer” (Grade2, Interview 9, Reflection). When preservice teachers have questioning“practice” with children and a chance to reflect on those interactions, theexperience may lead to the recognition of those questioning strategies thatare more effective in certain situations.

The effectiveness of the one-on-one interview lies in its ability to docu-ment preservice teachers’ reactions to children’s unpredictable responsesin a mathematics interaction. Many of the preservice teachers reflec-tions’ expressed the lack of opportunity for explaining mathematicalconcepts during their own school experiences. Therefore, when a childgave an unexpected response they were unsure what questions to ask.Using the one-on-one interview may provide opportunities to interpret andunderstand unusual or unexpected solutions presented by children. Forprospective teachers who will soon be required to “think on their feet,”the interview is a place to interpret and respond to various solution routesin a controlled setting with one child. Because it provides a controlled,yet realistic, interaction with a child, it gives preservice teachers a chanceto reflect on their own questioning during that interaction. Learning torespond to children’s unexpected answers in a one-on-one structuredinterview is a first step towards developing the questioning strategiesthat will be used in the multi-dimensional, simultaneous, unpredictableenvironment of the classroom.

Implications for Teacher Education

These question categories provide an important contribution to the liter-ature on questioning techniques used by the beginner. The findings mayserve as a framework for beginning levels of questioning skills. Identifyingand labelling typical questioning patterns allows educators to have a shareddiscussion about the kinds of questions to expect of the beginner andstrategies for developing higher level questioning skills for classroom use.The examples from the data can be used to discuss forms of questioningin mathematics and the types of responses these questions typically elicitfrom children.

Preservice teacher-conducted assessment interviews can be effectiveperformance-based assessment tools for teacher educators. Documentingthe types of questions a preservice teacher uses during an initial interviewwith a child at the beginning of a course and comparing these questionswith interview questions used later in the semester are one way to docu-ment growth in developing questioning techniques. The course instructorcan use this documentation as an assessment that shows evidence of growth


by the preservice teacher. In these analyses, the preservice teacher might beasked to identify and reflect on specific questioning strategies that wouldsupport student learning.

The categories identified in this project can be used during aself-analysis of one’s own interview. Interviewers could document thefrequency with which they use “checklisting” or “specific follow-up”questioning strategies. The categories and names for question typeswith examples make it easier for educators to discuss the variety ofquestion types that might be used during an interview. This providesthe opportunity for a shared discussion about the kinds of questioningpreservice teachers are using in mathematical situations with children.Subsequent class sessions might be used to identify the different kindsof responses elicited from children when different types of questions areasked.

Real experiences with a single child provide a valuable learning contextfor a preservice teacher. One of the ways educators and preservice teachersmight use these questioning categories is as a framework for coding mathe-matics interviews with children. Preservice teachers could code their owntranscripts of an interview with a child as a self-reflection on the typesof questioning they used during the interview. The preservice teachermight audio- or videotape himself/ herself and then engage in conversa-tion with an experienced educator for an examination of the questions theinterviewer used with the child.

Additional research on the types of questioning categories preserviceteachers and in-service teachers use during teaching and learning inter-actions in typical classrooms would also contribute to our knowledgebase on the types of questions used and their appropriateness in differentmathematical situations. These examinations would lead to a better under-standing of the types of questioning skills that could be developed inpreservice teacher education courses to support questioning skills for theclassroom. This research brings focus to the importance of preparingpreservice teachers in the skill of questioning during their mathematicseducation coursework.

REFERENCES

Adams, N.H. (1994, March). Ask, don’t tell: The value of asking young children questions.Paper presented at the annual conference of the Association for Childhood EducationInternational.

Ashlock, R.B. (2002). Error patterns in computation: Using error patterns to improveinstruction. Upper Saddle River, NJ: Merrill Prentice Hall.


Ball, D. (1991). Research on teaching mathematics: Making subject matter knowledge partof the equation. In J. Brophy (Ed.), Advances in research on teaching, Vol. 2 (pp. 1–41).Greenwich: JAI Press.

Baroody, A.J. & Ginsburg, H.P. (1990). Children’s mathematical learning: A cognitiveview. In R.B. Davis, C.A. Maher & N. Noddings (Eds.), Constructivist views on theteaching and learning of mathematics (pp. 51–64). Reston, VA: NCTM.

Bloom, B.S. (1956). Taxonomy of educational objectives: The classification of educa-tional goals (Handbook I: Cognitive domain). New York: David McKay Company,Inc.

Bowman, A.H., Bright, G.W. & Vacc, N.N. (1998, April). Teachers’ beliefs across thefirst two years of implementation of cognitively guided instruction. Paper presentedat the annual meeting of the American Educational Research Association, San Diego,CA.

Bright, G.W. & Vacc, N.N. (1994, April). Changes in undergraduate preservice teachers’beliefs during an elementary teacher-certification program. Paper presented at theannual meeting of the American Educational Research Association, New Orleans,LA.

Buschman, L. (2001). Using student interviews to guide classroom instruction: An actionresearch project. Teaching Children Mathematics, 8(4), 222–227.

Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L. & Empson, S.B. (1999). Children’smathematics: Cognitively guided instruction. Portsmouth, N.H.: Heinemann.

Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (2000, September).Cognitively guided instruction: A research-based teacher professional developmentprogram for elementary school mathematics. National Center for Improving StudentLearning and Achievement in Mathematics and Science, Report No. 003. Madison,WI: Wisconsin Centre for Education Research, The University of Wisconsin-Madison.Available: http://www.wcer.wisc.edu/ncisla/publications

Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang, C. & Loef, M. (1989). Usingchildren’s mathematics thinking in classroom teaching: An experimental study. Amer-ican Educational Research Journal, 26, 499–531.

Fennema, E. & Carpenter, T.P. (1996). A longitudinal study of learning to use children’sthinking in mathematics instruction. Journal for Research in Mathematics Education,27(4), 403–434.

Fennema, E., Carpenter, T.P., Franke, M.L. & Carey, D.A. (1993). Learning to usechildren’s mathematics thinking: A case study. In C. Maher & R. Davis (Eds.), schools,mathematics, and the world of reality (pp. 93–118). Needham Heights, MA: AllynBacon.

Fennema, E., Franke, M.L., Carpenter, T.P. & Carey, D.A. (1993). Using children’smathematical knowledge in instruction. American Educational Research Journal, 30,555–585.

Gall, M.D., Borg, W.R. & Gall, J.P. (1996). Educational research: An introduction. WhitePlains, NY: Longman.

Huinker, D.M. (1993). Interview: A window to students’ conceptual knowledge of theoperations. In N.L. Webb (Ed.), Assessment in the mathematics classroom (pp. 80–86).Reston, VA: NCTM.

Kamii, C. & DeVries, R. (1978). Physical knowledge in preschool education: Implicationsof Piaget’s theory. Englewood Cliffs, NJ: Prentice Hall.


Kamii, C. & Warrington, M.A. (1999). Teaching fractions: Fostering children’s ownreasoning. In L. V. Stiff & F.R. Curcio (Eds.), Developing mathematical reasoninggrades K-12: 1999 Yearbook (pp. 82–92). Reston, VA: NCTM.

Lampert, M. (1986). Knowing, doing and teaching multiplication. Cognition and Instruc-tion, 3(4), 305–342.

Leinhardt, G. & Greeno, J. (1986). The cognitive skill of teaching. Journal of EducationalPsychology, 78(2), 75–95.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understandingof fundamental mathematics in China and the United States. Hillsdale, NJ: LawrenceErlbaum Associates.

Merriam, S.B. (1988). Case study research in education: A qualitative approach. SanFrancisco: Jossey-Bass Publishers.

Mewborn, D.S. & Huberty, P.D. (1999). Questioning your way to the standards. TeachingChildren Mathematics, 6(4), 226–227, 243–246.

Moyer, P.S. & Moody, V.R. (1998). Shifting beliefs: Preservice teacher’s reflections onassessing students’ mathematical ideas. In S.B. Berenson & K.R. Dawkins (Eds.),Proceedings of the Twentieth Annual Meeting of the North American Chapter of theInternational Group of the Psychology of Mathematics Education, Vol. 2 (pp. 613–619).Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and EnvironmentalEducation.

National Council of Teacher of Mathematics (1991). Professional standards for teachingmathematics. Reston, VA: Author.

National Council of Teacher of Mathematics (1995). Assessment standards for schoolmathematics. Reston, VA: Author.

National Council of Teacher of Mathematics (2000). Principles and standards for schoolmathematics. Reston, VA: Author.

Nilssen, V., Gudmundsdottir, S. & Wangsmo-Cappelen, V. (1995, April). Unexpectedanswers: Case study of a student teacher derailing in a math lesson. Paper presentedat the annual meeting of the American Educational Research Association, San Fran-cisco.

Piaget, J. (1926). The language and thought of the child (Preface by E. Claparede;translated by Marjorie and Ruth Gabain). London: Routledge and Kegan Paul.

Piaget, J. (1929). The child’s conception of the world (translated by Joan and AndrewTomlinson). London: Kegan Paul, Trench, Taubner, & Company.

Posner, G.J. & Gertzog, W.A. (1982). The clinical interview and the measurement ofconceptual change. Science Education, 66(2), 195–209.

Ralph, E.G. (1999a). Developing novice teachers’ oral-questioning skills. McGill Journalof Education, 34(1), 29–47.

Ralph, E.G. (1999b). Oral-questioning skills of novice teachers: . . . any questions? Journalof Instructional Psychology, 26(4), 286–296.

Reys, R.E., Suydam, M.N., Lindquist, M.M., & Smith, N.L. (1998). Helping children learnmathematics. Needham Heights, MA: Allyn and Bacon.

Schwartz, S.L. (1996). Hidden messages in teacher talk: Praise and empowerment.Teaching Children Mathematics, 2(7), 396–401.

Stenmark, J.K. (1991). Mathematics assessment: Myths, models, good questions, andpractical suggestions. Reston, VA: NCTM.

Stigler, J.W. & Hiebert, J. (1999). The teaching gap. New York: The Free Press.Stone, J. (1993). Caregiver and teacher language: Responsive or restrictive? Young

Children, 48(4), 12–18.


Strauss, A. (1987). Qualitative analysis for social scientists. New York: CambridgeUniversity Press.

Wassermann, S. (1991). The art of the question. Childhood Education, 67(4), 257–259.

PATRICIA S. MOYER

George Mason University14337 Uniform DriveCentreville, VA 20121USAE-mail: [email protected]

ELIZABETH MILEWICZ

Jacksonville State UniversityE-mail: [email protected]

PI-JEN LIN

ON ENHANCING TEACHERS’ KNOWLEDGEBY CONSTRUCTING CASES IN CLASSROOMS1

ABSTRACT. This study was designed to enhance teachers’ knowledge by constructingcases as part of a school-based professional development project in Taiwan. Cases,involving episodes and issues from real classroom events, were constructed collaborativelyby a school-based team consisting of the researcher and four classroom teachers. Theprocess of constructing cases, characterization of teachers’ understanding of cases, andtheir skills for case writing were developed in the course of the study. In the process ofconstructing these cases, teachers improved their awareness of and their competence indealing with the difficulties students encountered in the learning of mathematics; theyenhanced their pedagogical content knowledge and their ability to reflect on classroompractices.

KEY WORDS: classroom cases, pedagogical content knowledge, reflective teaching,teacher development

PROBLEMS RELATED TO TEACHER EDUCATION

Reacting to societal pressures and drastic changes in the educationalconditions of Taiwan, the curriculum standards issued by the Ministry ofEducation of Taiwan (1993) articulate a national vision of what constitutesreformed mathematics teaching and learning. The philosophy underpin-ning the curriculum reform reflects a constructivist perspective of learning.Problem solving, communication, reasoning, and mathematical connec-tions, as the focuses of the Standards of the United States (NationalCouncil of Teachers of Mathematics, NCTM, 1989, 2000), are fourcomponents emphasized in the new curriculum. The standards-orientedapproach to teaching has shifted its focus to become learner-centred ratherthan teacher-centred. Teachers are encouraged to cultivate an atmosphereof student discussion in classrooms in which teachers’ roles shift fromproblem solver to problem poser. Students shift from being copiers ofteacher’s solutions to problem solvers in their own right. Thus, teachereducation programs need to provide opportunities for teachers to learn totransform these emphases of curriculum into classroom practices.

Traditionally, in international contexts, reacting to teachers’ limitedunderstanding of a reformed curriculum, innovators have sought to develop


318 PI-JEN LIN

teachers’ understanding by offering short workshops. However, althoughteachers are exposed to theories of learning and teaching in some teachereducation programs, they are often not able to apply this knowledge toclassroom practice (Cooney, 1999; Jaworski & Wood, 1999). One result, asrevealed in research in Taiwan, is that teachers are limited by a weak under-standing of mathematics, a limited understanding of students’ thinking andlearning processes, and a limited knowledge of pedagogical alternatives inmathematics classrooms (Huang, 2000; Lin, 1999).

Recently, some research on teacher education has focused on providingteachers with the opportunity to reflect on their own practice for increasingknowledge of teaching (Krainer, 1999; Lin, 2001; Wood et al., 2001). Balland Cohen (1999) view professional practice both as a site for teacherlearning and as a stimulus for promoting inquiry from which many teacherscould learn. To achieve these goals, the possible situations a teacher mightencounter and strategies and skills for analysing and acting in differentsettings should serve as the core of professional education.

DEVELOPING CASES FOR TEACHERS’ LEARNING

One of the ways we learn from the experiences of others and help otherslearn from our own experience is through narratives or cases that reflectaspects of normal classroom experience and raise issues. Although manyteacher educators promote the use of cases as an alternative method ofinstruction in teacher education development, not all are in agreement onthe definition of terms (see for example, Doyle, 1990; Merseth, 1996).One commonly used definition, offered by Merseth (1996), suggests casesare characterized in three essential ways: (a) cases are real, (b) theyrely on careful research, and (c) they foster the development of multipleperspectives by users.

Cases used in teacher education may teach more effectively than tradi-tional expository approaches to teaching since cases reflect real situationsand pose problems, issues, and challenges for teachers and are vehiclesfor establishing a dialogic model of connecting theory and practice (Ball& Cohen, 1999; Shulman, 1992). Barnett (1998) used cases extensivelyin work with elementary school teachers. More recently, Schifter and hercolleagues (1996a, 1996b) have used narratives written by elementaryschool teachers for addressing dilemmas in teaching mathematics. Silverand his colleagues (1999), as part of the COMET (Cases of Mathe-matics Instruction to Enhancing Teaching) project, developed cases foruse with middle grade mathematics teachers. The cases developed in thisresearch were used by teachers to (1) develop knowledge of a particular

ENHANCING TEACHERS’ KNOWLEDGE BY CONSTRUCTING CASES 319

theory or build new theories, (2) practice analysis and assimilate differentperspectives; and (3) stimulate personal reflection.

The use of cases in teacher education includes both case discussionand case writing. Case discussion can play a critical role in expandingand deepening pedagogical content knowledge (Barnett, 1991, 1998).Discussing cases fosters personal reflection through an external process(Shulman & Colbert, 1989). While some research has examined theinfluence of discussing cases on what and how teachers learn to teach(Richardson, 1993; Merseth, 1996), a limited amount of research has beenconducted on the process of case writing. In Shulman’s (1992) study, casewriting was developed by taking a teacher’s self-report and turning it intoa case. Shulman’s research shows that individual case writing based onpersonal experience is limited as an exemplar. Cases constructed by acollaborative team consisting of various backgrounds and experiences forsharing multiple perspectives and comments are more likely to provideenriching exemplars.

THE USE OF CASES IN THE REPORTED STUDY

The study reported here was initiated by the researcher, who is a universityprofessor and a curriculum developer, as a response to a need for math-ematics curriculum reform. The researcher initiated a three-year researchproject that was designed to develop cases with a school-based collabora-tive action research approach to assist teachers in implementing the spiritof curriculum standards into classroom practices and to reduce the gapbetween theory and practice. That is, the goals of the research were:

(1) to enhance the rethinking of mathematics teaching in classrooms in thespirit of the curriculum standards;

(2) to foster teachers’ awareness of children’s learning;(3) to support teachers as they began to put into practice their new vision

of a learner-centred approach to teaching mathematics; and(4) to promote teachers’ ability to reflect on their teaching experiences.

The study reported here was designed to examine the effects ofconstructing cases with a collaborative research team in order to developknowledge central to teaching. Cases, here, are accounts of episodes fromclassroom teaching that were found to raise issues or dilemmas for theteachers. The cases were constructed from observing participant teachers’teaching, discussing the pedagogical issues arising from the observations,and writing the cases into a narrative form.

320 PI-JEN LIN

The action-oriented approach was used in the study to construct casessince it has potential to bring about classroom reform and its results aremore context specific and meaningful to teachers. As Miller and Pine(1990) explain, participation in collaborative action research providesopportunities for teachers to examine their own teaching and students’learning by way of purposeful conversation, collegial sharing, and criticalreflection for the purpose of improving classroom practice.

THEORETICAL FRAMEWORK OFTEACHERS’ LEARNING TO TEACH

Reflection, cognitive conflict, and social interaction are the three elementsof the framework for teachers learning to teach in the study. This studyassumes that teachers will learn new ways of teaching mathematics only ifthey have opportunities to reflect on how they construct their own knowl-edge and integrate it into their existing knowledge structures. Therefore,teachers were invited to participate in the process of experiencing practicalteaching, reflecting on their experiences in weekly meetings, and devel-oping their own insights into teaching through the interaction betweenpersonal reflection and theoretical notions offered by other teachers of thecollaborative team. Teachers involved in the teacher education programtook the role not only of teachers but also that of learners. Teachers, aslearners, can change or increase their knowledge when their cognitivestructures are developed through social interaction and reflection.

Teachers’ learning to teach is seen to be situated in interactions withstudents (Cobb & McClain, 1999). The classroom is the primary site forteachers’ learning, as learning to teach develops in part by focusing onunderstanding the dilemmas of teaching (Harrington, 1995). As defined bySullivan and Mousley (1999), a dilemma refers to an argument with twoalternatives, each conclusive, against an opponent, or to a difficult choice.Discussion of a dilemma allows teachers to articulate their own thinkingabout teaching and to learn from others’ varied perspectives and interpreta-tions of a particular event. In this sense, it helps teachers to understand thecomplexity of teaching.

Understanding the dilemmas of teaching is similar to overcomingcognitive conflicts. Conflicts of cognition result from considering twoconcepts which both seem plausible and yet are contradictory, or consid-ering concepts that become insufficient given new evidence. In each case,contradiction causes an imbalance providing the internal motivation foran accommodation. Piaget (1971) asserts that the way one accommo-


dates cognitive structures which are in disequilibria is to modify and toreorganize one’s current schemas, thus achieving cognitive development.

Social constructivists claim that the social constructing of knowledgeoccurs through its negotiation and mediation with others. Vygotsky’s lawof cultural development indicates that children’s development appearson the social plane followed by the psychological plane. According toVygotsky’s theory, “the zone of proximal development (ZPD)” is

the distance between the actual developmental level as determined by independent problemsolving and the level of potential development as determined through problems solvingunder adult guidance, or in collaboration with more capable peers. (Vygotsky, 1978, p. 86)

Vygotsky’s words here can be interpreted as suggesting that cognitiveconflicts, caused by discussing, debating and negotiating in interactionsbetween learners and more capable peers, act as a catalyst for reaching ahigher development level.

I envision teachers involved in a professional program as constructivelearners in the same way as students in the classroom are learners. Withthis in mind, adopting a social constructivist view of knowledge, inwhich knowledge is the product of social interaction via communicationwithin a collaborative team, this study was designed to create opportuni-ties for teachers to develop more specific and deeper mathematical andpedagogical content understanding through observation and discussion.Activities were structured to ensure that knowledge was actively developedby teachers rather than imposed by the researcher. The cases involvedin the study were developed by focusing on the dilemmas of teachingengendered by teachers’ professional dialogues and by providing teacherswith opportunities to examine their teaching practice. To achieve this goal,there were two research questions to be answered: (1) What are teachers’conceptions of cases? (2) How does the construction of cases influenceteacher’s knowledge?

CONTEXT OF THE STUDY

Setting and Participants

To answer the research questions, the researcher initiated a three-yearresearch project funded by a grant contract from the National ScienceCouncil of Taiwan. An aim of the research was to encourage the partici-pants to implement what they learned from the research without losing theauthentic context of practice. A school, Din-Pu, with about 780 studentsand 36 staff, was selected to take part in this study. The school was

322 PI-JEN LIN

selected because the researcher was invited by the school as a consultantin learner-oriented mathematics teaching in which the school was under-going a school-based project for the purpose of solving problems related tothe reformed curriculum. Further, the teachers of the school were willingto learn and there was support for the research from the principal andadministrators.

Thus, the three-year research project was integrated into the school’sproject with an action-oriented approach to provide the teachers withopportunities for examining their classroom practice by way of collegialsharing and critical reflection. Each teacher of the team took responsibilityfor planning, practicing, and modifying the processes of the research, butthe researcher was the pilot of the study and worked with the team todiscuss teachers’ implementation needs before the research started. Theresearcher was expected to contribute more theory than practice, while thefour teachers were expected to share more classroom experiences. Theresearcher acted as a partner to the teachers in helping them put ideasgenerated in discussion into practice.

The four teachers represent distinct experiences of teaching andacademic backgrounds. The name, age and years of teaching for eachteacher are respectively; Huei (32, 8), Jong (45, 12), Ling (38, 14), andSue (34, 13). Huei and Jong graduated from the Teachers College, whilethe others graduated from universities with teacher education programs.Jong is the only one with a masters degree and she is a consultant forassisting school teachers in Hsin-Chu city with implementing a standards-oriented curriculum to classroom practices. The teachers were selectedfrom the staff of teachers who were teaching in the first grade because theywere willing to learn and because they were using the mandate reformedcurriculum emphasizing a learner-oriented approach (Ministry of Educa-tion of Taiwan, 1993).2 For confidentiality, the name of each teacher is apseudonym, as is the name of the school.

In accordance with theoretical perspectives outlined previously, thisstudy was based on providing teachers with the opportunities for dialogueon critical pedagogical issues related to the mandated curriculum. It wastherefore necessary to create an environment for teachers’ learning inwhich, through professional dialogues, teachers could communicate whatthey were learning in their own classrooms to their colleagues. A profes-sional collaborative team was set up to discuss the situations that occurredin a particular teacher’s classroom and to compare them to others.

The first-grade classrooms were the primary contexts for these teachersto frame problems, analyze situations, and argue the advantages anddisadvantages of various ways of teaching. In addition, the contexts of


teachers’ learning included participation in regular weekly meetings. Therewere two reasons for selecting teachers from the same grade to partici-pate in this study. The first reason was that the participants, teachingthe same mathematics topics, confronted similar pedagogical problems.Similar mathematical content lent itself readily as a focal point when theteachers met together after observing each other’s lessons to address issuesand solve pedagogical problems. Secondly, similar pedagogical issuesaddressed in the regular meetings drew attention and concern from eachparticipant, leading to in-depth discussions.


Classroom observation was used as a means of initiating a case that wasthen developed by discussing issues of teaching. To develop a case inwhich teachers’ instructional approaches could be compared, it was neces-sary for the teacher group to observe the teaching of the same lesson. It wasagreed that the lessons of each participant teacher would be observed andthat all participants would observe simultaneously in the same classroomin which the instructor was one of the participants. To develop teachers’ability in writing cases, each teacher wrote at least one case based on herown teaching after a regular meeting for discussing the lesson. The lessonsof all participant teachers’ were scheduled to be observed in turn.

The teachers observed two participants’ teaching routinely on Mondaymorning every three or four weeks and had a meeting lasting for 3 hoursimmediately after the teaching. Although lessons were not observed everyweek, meetings were held weekly, providing opportunities for partici-pants to learn from one another’s concerns. The participants shared anddiscussed issues of pedagogy and students’ learning: the instructor wasasked to reflect on her own teaching and the rest of the participants wereinvited to articulate what they had observed. Thirty meetings were held intotal throughout the school year. Each teacher was observed teaching threedistinct lessons.

In addition, each teacher was interviewed individually three times intotal by the researcher to collect teachers’ responses to the processes ofconducting cases. The first interview was in the second month of the studyand the other two were in the middle and at the end of the study, respec-tively. The questions included: (1) What value do you give to the casesfor your teaching and students learning? (2) How does the opportunity forconstructing and discussing cases have impact on your understanding ofmathematics and teaching?

324 PI-JEN LIN

The phases of constructing cases, time and instructor of lessonobserved, interview, and weekly meetings of the professional team aresummarized in Table I as follows.

TABLE I

A Summary of Phases of Constructing Cases, Time and Instructor of Lesson Observed,Interview, and Weekly Meetings

Phases of Teaching observed Weekly meetings Time

constructing cases (Monday morning) (Monday afternoon) (month, year)

Instructor Observers

The first phase Sue, Huei Huei, Ling, 4 meetings 10, 1998

Jong, Sue,

researcher

Jong, Ling Sue, Ling, 4 meetings 11, 1998

Jong, Huei,

researcher

Interview I (1) What value do you give to the cases for your 11, 1998

teaching and students learning?

(2) How does the opportunity for constructing and

discussing cases have impact on your under-

standing of mathematics and teaching?

The first phase Sue, Ling, Sue, Huei, 4 meetings 12, 1998

Jong, Ling,

researcher

Huei, Jong, Sue, Huei, 4 meetings 1, 1999

Jong, Ling,

researcher

Interview II Interview questions same as Interview I 2, 1999

The second phase (No 2 meetings to discuss: 3, 1999

observations) (1) How do you think the cases we

developed have helped you

understand your own teaching

and students learning?

(2) What are the distinctions among

the cases provided?

(3) What content might be included

in each case?

(4) How can one construct a case?


TABLE I

Continued

Phases of Teaching observed Weekly meetings Time

constructing cases (Monday morning) (Monday afternoon) (month, year)

Instructor Observers

The third phase Jong, Ling, Sue, Huei, 8 meetings for 3–4, 1999

Jong, Ling, revising 4

researcher versions of

two cases

Sue, Huei Sue, Huei, 8 meetings for 4–5, 1999

Jong, Ling, revising 4

researcher versions of

two cases

Interview III Interview questions same as Interview I and II 6, 1999

The weekly group meetings and interviews were audio-recorded andthe lessons were video-recorded. The audio- and video-tapes were tran-scribed literally in Chinese. The Chinese transcriptions were producedto be as faithful as possible to the teachers’ exact words. The parts oftranscriptions required for this paper for international readers were thentranslated into English with adjustment to produce readable English. Anexcerpt from translating the transcriptions from Chinese into English isillustrated as follows (see Figure 1).

Figure 1. An excerpt from transcriptions translating the Chinese version into English.

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Triangulation of the data used multiple sources, multiple methods,and multiple analysts. The data collected from weekly group meetingsthroughout the entire year and individual interviews at three different timeswere for validating the data of teachers’ valuing of cases from multiplesources. The multiple analysts consisted of the researcher, a graduatestudent and one of the first-grade teachers of the collaborative researchteam. Since the reality of a case is the major principle for developing it,we examined carefully, for each case, whether the contexts of the case orthe students’ solutions described in the case did happen in real teaching.Each case produced and revised relied on the transcriptions of the weeklymeetings. To validate the case, each teacher examined a draft of the caseby rereading repeatedly the transcripts of audio and video recordings.

Because the study had been designed to develop understanding aboutthe effects of cases on teachers’ development of professional knowledge,the data was analysed using a grounded-theory approach, as described byStrauss and Corbin (1994). In this approach, the researcher is the primaryinstrument of data collection and analysis, applying inductive methods andstriving to derive meaning from the data. In keeping with this approach,there were no predetermined criteria or coding system in the analysis. Todocument teachers’ growth of knowledge, the transcripts of interviews,group meetings, and observations were analysed using a procedure inwhich all documents were reviewed and comments were made in themargins. All comments included in the margin of the observation sheetor transcription sheet were a part of ongoing analysis. Each transcriptionwas coded by the researcher and two graduate students. The results werereciprocally examined to see if the codes from paragraph to paragraphencoded by each analyst were consistent among the analysts.

When patterns were detected and analysed, there were seen to bethree categories emerging that are relevant to teachers’ conceptions; thesewere the value teachers gave to cases, the types of cases, and criteriafor constructing a case. Three sub-themes emerged showing the influ-ence of cases on teachers’ professional knowledge. The first sub-themeis concerned with the cases influencing teachers’ careful considerationsof the sequence of posing problems to be solved. The second sub-themeis related to the cases improving teachers’ awareness and competence inhelping students’ learning. The third sub-theme relates to the way that thecases help teachers become more reflective practitioners.

Construction of Cases

To develop a greater knowledge of cases, a teacher must develop adeeper understanding of the effects of variations in use. The purposes of


constructing cases varied with the phases of the study, and were guidedby the research questions. There were three phases. The purpose of thefirst phase was to help teachers understand what a case, developed by thecollaborative team and written by the researcher into a case form, couldlook like. The researcher did the writing of the case in this phase to demon-strate a possibility for the teachers. The purpose of the second phase wasteachers’ construction of a conception of cases including perspectives ofthe significance and value of the cases. Developing teachers’ own casewriting ability was the purpose of the third phase.

The first phase. The first phase included the construction of cases that weredeveloped collaboratively by the researcher and classroom teachers andlater written in case form by the researcher. Four cases were constructedin this phase, one for each teacher, each written into a case formally bythe researcher. These cases were inspired by observations of the teachers;first by discussing the issues addressed in one participant’s lesson, andthen developing them into a complete case form. Sue’s lesson serves as anexample which illustrates the process of construction. In our first observa-tion, Sue’s students learned to solve subtraction word problems by usingmagnetic counters. She asked students to count ten counters and then shehid some of them. Sue had placed the counters on the blackboard as shownin Figure 2 below:

Figure 2. Sue’s first arrangement of counters.

Then, Sue asked the students to answer the question: “How manycounters are hidden?” The students were unable to answer the questioncorrectly, with the exception of one student who answered correctly butwas unable to offer a reasonable explanation. Afterwards, Sue posed thesecond problem to help students understand subtraction better. The secondproblem was: “There are ten counters on the blackboard. If some of themare hidden (as in Figure 3, below), how many counters are hidden?”

Figure 3. Sue’s second arrangement of counters.

Her students still had the same difficulty with the second problem. Inthe rest of the lesson, Sue asked students, orally, three more subtractionproblems in word problem format:

Problem 3: There are 6 baseballs in the playground. Wei-Dert took 2 ofthem away. How many baseballs are there now?

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Problem 4: Mei-Huei ate 2 from 9 apples. How many apples are left?Problem 5: There are 8 children swinging and 3 children skating. How

many more children are swinging than skating?

It was observed that the students who were unable to answer the firsttwo problems were able to solve Problems 3, 4 and 5; also there were fewerstudents raising their hands to answer Problem 5 than to answer Problems 3and 4.

There was a group meeting immediately after Sue’s teaching had beenobserved. One of the main issues addressed by the members of the teamwas the question of why students had more difficulties with the twocounter problems than with the other three subtraction word problems.The students’ difficulties with the counter problems became the focus of acase. The following excerpt, taken from the transcription of the interactionbetween the researcher and teachers in the meeting, shows some of thediscussion which led to the case.

Researcher: Do you agree with the sequence of the five problems forstudents’ learning as arranged in Sue’s lesson? What isthe distinction between the five problems posed in Sue’steaching?

Huei: The first two problems, Problems 1 and 2, belong to decom-position problems which are easier for the first graders thanthe other three problems, since students just learned decom-position in their latest lesson. For example, eight can be splitinto two groups, two and six.

Jong: [Shaking her head], I don’t think so. Problems 1 and 2 do notbelong to decomposition problems. It’s hard for first gradersto visualize the quantities of the two groups presented inProblems 1 and 2 at one time. For instance, 10 magnets can besplit into 4 magnets and 6 magnets. This is a decompositionproblem. After 4 magnets are removed out of 10 magnets, 6magnets are left. The decomposition problem is much easierthan the magnet problem. What we observed was that onlyone student raised his hand and answered Problem 1 with anunexpected answer because he saw the number of magnetsthe teacher hid.I think the first magnet problem which was given by theteacher has been changed into another type of subtractionword problem, There are 10 magnets, Wu-Mei lost some ofthem, and 6 magnets are left. How many magnets did Wu-Meilose? This is a type of problem with unknown subtrahend sothat it can be expressed in 10 – ( ) = 6. A decomposition


problem, like 10 can be split as 6 and what, can be repre-sented as 10 = 6 + ( ). The magnetic problem is harderthan the decomposition problem. I think that students need tounderstand the inverse relationship between subtraction andaddition for solving the two magnet problems successfully.

Sue: Before today, I am not conscious of the differences betweenthe magnet problems and the decomposition problems. Ilearned the distinction between them from what Jong saidearlier. Moreover, although these five word problems belongto subtraction problems, they are not at the same difficultylevel for first-graders.

Ling: I don’t think they are the same level of difficulty for first-graders. Problem 3, 4 and 5 are easier for students than theothers, since the context of each of them is relevant to reallife. We can tell the difference from our observations, therewere fewer students raising their hands to answer Problem5 than Problems 3 and 4. Based on my past experience ofteaching, more students will use a one-one correspondencestrategy to solve Problem 5 involving the comparison of twoquantities. Students solve Problems 3 and 4 by using a takeaway strategy. Problems 3 and 4 belong to a kind of take awaycategory which is mentioned in the teachers’ guides, whileProblem 5 is a kind of compare problem.

Based on these professional dialogues, the issues of students’ diffi-culties with some of the subtraction problems were developed as majormathematical concepts, included in a case and written by the researcherin case form (as shown in Figure 4). This case portrayed the interac-tions between students and the teacher. Also, it reported the participants’perspectives. Three other cases were produced in a similar manner, one foreach teacher. These cases were also intended to offer other school teacherswho were not involved in this study some of the key mathematical ideasembedded in the case.

The second phase. This phase – for probing teachers’ responses to casesdeveloped in the first phase – was aimed at deciding if the construction ofcases is valuable for mathematics teaching and at assisting participants indeveloping their conceptions of cases. In this phase, there was no teachingor observation of lessons.

To understand how teachers valued case-based learning and theirconceptions of cases, four questions were discussed in two weekly meet-ings: (1) How do you think the cases we developed have helped you

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Figure 4. Case (from Phase 1) showing students’ difficulties with some subtractionproblems.


understand your own teaching and students’ learning? (2) What are thedistinctions among the cases provided? (3) What content might be includedin each case? (4) How can one construct a case? Outcomes from discussionof these questions will be addressed further in the Results section below.

The third phase. The purpose of the third phase of constructing caseswas to develop the teachers’ own case writing ability. Each case to beconstructed, from first draft to finalized version, took four weekly meetingsto develop completely. After observation, the instructor was encouragedto decide what a case from her own teaching would look like in accord-ance with the issues addressed from her own teaching and discussed inthe meetings of research team. Afterwards, she produced a first versionof the case in written form. This was brought to the second meetingfollowing her observed teaching. The teachers offered suggestions to theinstructor on the first version, suggesting, for example, that the contextof the case should be clearer than an outline form. Then, the instructorproduced a second version of the case and brought it to the third meetingfollowing her teaching. The teachers commented on the second versionagain, suggesting, for example, that the text of the case should be moresystematically organized, including perhaps the goal of the lesson, the casebackground, a context of mathematics activity, and a concluding set ofdiscussion points. After this, the instructor produced the third version andbrought it into the fourth meeting to finalize it.

Example of Constructing a Case

The case “Using a Number Sentence for a Subtraction Word Problem”constructed by the collaborative team and written by Ling, one of theteachers in the team, is as an example illustrating the steps of constructinga case. The goal of the teaching to be constructed in the case was torepresent a subtraction word problem with a number sentence. All fourfirst-grade teachers had to teach the same mathematics content which waspresented in the fourth lesson of the first-grade mathematics textbook.Before Ling’s teaching, Huei and Sue had taught the lesson. Jong’s andLing’s teaching the same content were both observed by other teachers onthe same Monday morning. Thus, the focus of the case is the comparisonof two instructional approaches between Ling and Jong. The goal of thecase was to help teachers understand the correspondence of the numbersentences with the semantic structure of subtraction word problems. Thecase was constructed throughout four steps in four successive meetings.

Step 1. Observing the instructor’s lesson is the first step of writing a case.In this case, the instructor involved in the case is Ling. Each observer jotted

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down what s/he observed on a prepared observation sheet. The contents tobe observed were listed in three columns: instructor’s activities, learners’activities, and comments. The format and content of the observation sheetwere to remind observers to attend to agreed focuses, which were designedby the teachers in the collaborative team.

Step 2. The second step of writing a case is to form the first version of thecase. After the observation, the instructor (Ling in this example) reflectedon her own teaching and had a group discussion in the first meeting inthe afternoon of the same day. In reflecting on her own teaching, Lingmentioned the status of the lesson and students’ use of number sequencescorresponding to additive word problems which were presented in herlesson. Ling said

. . . the lesson was initiated to review the meaning of symbols, addition, subtraction,and equality (+, –, =) [brackets () here are added for clarity]. Then, students solved thefollowing problem, proposed by the teacher, by drawing or by using counters: Wei-Ming’sfreezer has 8 eggs. She fried 5 of them. How many eggs are left in Wei-Ming’s freezer?[Italics are added to delineate the problem.] Students were encouraged to solve the problemby using multiple strategies. However, I noticed that some of my students represented theproblem as a number sentence followed by using counters or by drawings and some ofthem represented the word problem by drawing circles followed by a number sentence.Close to the end of the lesson, I posed the second problem for students to represent it byusing a number sentence instead of by drawings (Ling, Group discussion, 03/08/1999).

After Ling’s reflection on her own teaching, Jong addressed an issueto the group meeting relating to encouraging students’ multiple ways ofthinking that is drawn from comparing her own teaching with Ling’s.3

This issue drew other teachers’ attention after Jong raised it in the groupmeeting. Jong was concerned with the encouragement of multiple ways ofthinking, while Ling’s emphasis was on mastering the use of a numbersentence corresponding to a word problem in which understanding themeaning of the number sentence is the objective of the lesson.4 Jong statedthat

My way of teaching was different from Ling’s, although the content I taught was the sameas that of Ling. I encouraged my students to use various representations. Number sentenceshave been taught in the latest lesson prior to the lesson I taught today. The objective ofthis lesson was to encourage students to use multiple strategies to solve a word problem.Thus, in my lesson, I did not restrict my students to use a number sentence to representthe problem that I offered students. My students in the entire lesson still used variousrepresentations to represent the problem. The varieties of ways of solving a problem thatstudents were allowed to use was the emphasis of my teaching, while mastering a simplestrategy was the focus in Ling’s teaching (Jong, Group discussion, 03/08/1999).

Ling followed Jong’s account and mentioned that some of her studentsused a number sentence and pictures or hands-on in differing order. She


observed that some students gave the number sentence 8 – 5 = 3 prior tothe circles they drew. After discussion about the two teachers’ teaching,Huei started to wonder whether her cognition about encouraging students’ways of thinking is correct. Huei said

I did not expect my students to use a number sentence to resolve the subtraction wordproblem unless they were asked, in the text of the given word problem, to use the numbersentence to represent it. What I taught in my lesson was the same as Jong’s lesson. Ialso focused on the encouragement of students’ multiple ways of thinking at the time ofthe lesson. Does the focus not match the aim of student-centered emphasis in curriculumreform? (Huei, Group discussion, 03/08/1999).

The researcher gave a hint for developing the discussion into a casebased on the content of their dialogue. She said:

From the comparison between Ling’s and Jong’s lessons you addressed earlier, the discus-sion revealed the difference between Ling’s and Jong’s instructional approaches. Thesedialogues among all of you could be developed into a good case. Moreover, this case wouldalso help the awareness of alternative approaches to a specific topic for those teachers whowere not involved in this study (The researcher, Group discussion, 03/08/1999).

Afterwards, the instructor (Ling) incorporated the issue teachersreported in the meeting into her first draft of the case (as shown in Figure 5below).

Step 3. The third step of writing a case was for producing the secondversion. To revise the first draft, Ling brought the first draft to the secondmeeting after she taught the lesson. Then, she revised the first version andproduced the second version relying on the dialogues of the teachers in thesecond meeting. The revising of the first draft which teachers addressed inthe second step of constructing a case included: (1) the text of a case witha narrative form is clearer than an outline form, (2) the demonstration ofdifference between two instructional approaches needs to be concise, (3)the questions for discussion should address one or two important issues ofpedagogy. The second version of the case was produced by Ling as shownin Figure 6 and was brought to the third meeting after Ling taught thelesson.

Step 4. The fourth step of constructing a case was for finalizing the case.The final version was revised from the second version in the third meetingafter Ling’s teaching. The main concern addressed in the discussion wasthat the text of the second version should be more concisely organized.Teachers also pointed out that the components of each case included:the title of the case, students’ pre-knowledge required for learning thelesson, an instructional objective, case background, a context of mathe-matics activity or students’ responses to the activity, and a concluding set

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Figure 5. First draft of constructing a case ‘Using a number sentence for a subtractionword problem’ (in Phase 2).

of discussion questions. The mathematics activity was intended to givesome of the key mathematical ideas that are embedded in the case forother school teachers who were not involved in this study. The issuesteachers should be concerned with were to be specially identified. Theset of discussion questions was to stimulate critical reflection on the casewith respect to key issues related to students’ mathematics thinking andways of representing and formulating mathematical concepts. The ques-tions regarding the case were to encourage the users’ to think critically.Teachers’ preference for the format of a case was summarized briefly asFigure 7.


Figure 6. Second version of the case ‘Using a number sentence for a subtraction wordproblem’ (in Phase II).

RESULTS

Teachers’ Conceptions of Cases

This section addresses the questions asked in the second phase ofconstructing a case. Teachers’ valuing of cases, types of cases, and criteriafor constructing cases are described as follows.

Significance and value of cases. In terms of teachers’ perspectives ofsignificance and value of cases, the cases developed in the first phase wereable to improve the quality of group discussion that immediately followed

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Figure 7. The format of a case.

a lesson observed. Discussion in the second phase of constructing casesfocused more on students’ thinking instead of the instructor’s behavior andinstructional environment. Sue expressed her commitment to the value ofcases in one weekly meeting. She said:

Generally speaking, the focus of group discussion for reflecting on the lesson observedtended towards the instructor’s teaching behaviors rather than towards students’ learning.The cases for presenting a teacher’s analysis of children’s strategies and solutions are ableto assist me in understanding the variety of ways of thinking. In addition, the use of casesimproves my abilities to examine if the sequence of teaching material matches my students’level of learning (Sue, Group discussion, 12/13/1998).

Sue’s statement seems very much related to what was found in theCognitively Guided Instruction (CGI) project, that teachers’ increasedawareness of student’s mathematical thinking had a positive influence ontheir instructional decision-making (Carpenter & Fennema, 1988).

Moreover, cases helped teachers explore children’s development ofmathematical ideas in an in-depth manner. The following is an excerptfrom Huei’s perspective as expressed in a group discussion.

When observing a lesson, we only observed on the surface without grasping children’sthinking processes. The strategies of solving problems listed in the cases make me think


deeper about children’s development of mathematics ideas. I am able to discern those whounderstood or misunderstood and to know the ways of children’s thinking (Huei, Groupdiscussion, 11/23/1998).

Teachers agreed that, through using cases, they learned ways to resolveproblems encountered in instruction with cognitively challenging tasksand they became more sensitive to students learning. For instance, Sue’sawareness of the ways students think enhanced her ability to examinewhether the teaching and the organization of the curriculum matchedchildren’s cognition. As Sue stated, a case “Identifying Instances of aShape” developed from Huei’s geometry class helped her to understandthe reason why students were not able to follow Huei’s directions foridentifying a shape by its relevant features (angles, straight lines, curves);instead, they referred to its irrelevant features (color, size, materials). Suewas not aware of students’ difficulty with identifying geometric configur-ations, due to an inappropriate organization of the curriculum, until sheread the cases, because she is accustomed to using a particular sequenceof curricular materials to teach. Sue abstracted the main idea from the caseand said:

To identify a geometric configuration, sorting is a better way to discard the irrelevantattributes of shape than by identifying a single configuration at a time. Since sorting acollection of configurations, students easily recognized the similarity of various shapesfrom their distinctions. However, identifying a single configuration at a time prior to sortinga collection of shapes is the approach used in the learning sequence in the textbook (Sue,Group discussion, 12/14/1998).

A set of discussion questions accompanying the case helped Jongreflect on her own teaching. Thus, she learned through cases to reflect onwhat she did in a session of instruction. For example, questions describedin the last section, “Using a Number Sentence for a Subtraction WordProblem” let Jong reflect on her own teaching comparing with Ling’steaching. One of the significant aspects of cases, as mentioned by Jong,is that a case enables her to clarify her thoughts. She also remarked thatcases are beneficial to new teachers in particular. She elaborated this pointin a weekly meeting.

Cases seem well-suited to new teachers, since one’s instructional blind spots embedded inthe case have been clearly proposed in advance. I can predict that new teachers could easilyperceive the main ideas from reading through cases prior to teaching. Narratives are theaccumulation of others’ experiences situated in real contexts (Jong, Interview, 11/25/1998).

Types of cases. When teachers were asked to distinguish the four casesthat were constructed in the first phase, they categorized them into threetypes. Each type varies with the purpose of the case. The first type ofcase presented a teacher’s analysis of children’s strategies and solutions,

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focusing teachers’ attention on students’ multiple representations. Thecase given in the first phase belongs to this type. The second type presentedstudents’ interpretation of other students’ thinking. The purpose of thistype of case is to expand teachers’ views of students’ critical thinkingability. The third type of case shows two different approaches to teacha specific topic, intending to enhance teachers’ awareness of alternativeapproaches. The case described in Figure 6 belongs to the third type.

Criteria for constructing cases. One significant feature of the casesreferred to in this study was that the cases served as an instructional methodfor developing teachers’ professional knowledge. Cases constructed in thestudy were centered on the topics of geometry, number, measurement, andso forth. From the teachers’ perspectives, good cases were drawn aroundproblems or big ideas in which would be found significant issues in a topicthat warrant serious, in-depth examination. The cases were constructedfrom classroom teachers’ practice encountered in mathematics teaching.A good case kept the group discussion grounded in some of the persistentproblems that must be faced in classroom practice.

From teachers’ perspectives, the characteristics of a good case referredto in this study include: (1) a good case is irritating to promote the users’reflective thinking, (2) a good case ended not with a satisfying solution, butrather with some critical thinking, (3) a good case does not have to reportgood teaching, (4) a good case portrays the users’ active involvement andreaches a climax that highlights the dilemma. The questions to be asked insome cases include: what happens now? what should be done next? how isthis going to be resolved? These questions are described as “Questions forDiscussion” and have the purpose of spurring active involvement throughdiscussion.

To construct a good case, the teachers participating in the studysuggested the following procedure: (1) reading teachers’ guides and objec-tives of the lesson, (2) performing teaching, (3) observing one’s ownteaching carefully, and (4) observing each other’s teaching. They alsoargued that discussing one’s own teaching with others is essential forconstructing the third type of case. The teachers had a preference forreading a shorter text in the case; so constructing a case with a short concisetext was the target.

Influences of Cases on Teachers’ Professional Knowledge

Teachers’ carefully inspecting the sequence of problems posed. Teachers’awareness of the need for careful inspection of the sequence of posingproblems was one effect of constructing cases on teachers’ developing


knowledge. For example, Sue learned from the case, entitled “Additionand Subtraction” (see Figure 4) that students’ difficulties with solvingproblems vary with different structures of problems. She could also predictthe kinds of solutions students would arrive at in specific problems and herunderstanding of semantic structures of subtraction word problems wasconsidered in her planning of later lessons. Sue’s improved understandingis shown in an episode of a lesson which was observed a half year later thanthe lesson described in Phase 1. In the later lesson, Sue posed the followingthree problems in which the instructional objective was to enable studentsto represent the compared problems with the use of hands-on materialssuch as counters.

Problem 1: There are 6 baseballs on the playground. Wei-Ming took 2 ofthem away. How many baseballs are there now?

Problem 2: There are 8 boys playing in the sand and 3 girls playing onswings. How many more boys are there than girls?

Problem 3: There are 3 red and 8 green chips. How many fewer red chipsare there than green chips?

After teaching, Sue explained her intention of the arrangement ofthe sequence of the three problems in group discussion. She intendedto examine how the semantic structure influenced students’ strategies ofsolving the three word problems, so she attended to the size of numbers,the location of numbers presented in the text of the problem, and the rela-tional terms, in her posing of the problems. The semantic structure of thesubtraction word problems in the lesson drew other participants’ attention;thus, it was developed into a case in the same manner as the case describedin Phase 3 (see Figures 5 and 6).

She distinguished the differences among the three problems from thesyntactic and semantic structures.

Problem 1, belonging to “take away” problems, is the easiest subtraction word problemsfor first graders. Problem 2 belongs to comparison problems with the relational term “morethan”, and Problem 3 belongs to comparison problems with the relational term “fewer”.For first graders, solving Problem 3 with relational term “fewer than” is more difficult thanProblem 2 with relational term “more than” (Sue, Group discussion, 4/19/1999).

In addition to Sue’s own analysis, the researcher offered a theoreticalaccount from Riley et al. (1983) related to the distinction among variouscomparison type word problems during discussion of the case. The resultsof Riley and colleagues’ research suggests that the relational term “lessthan” of comparison type word problems for first-grade students is moredifficult than the relational term “more than”. Sue’s students’ performancein this lesson, observed by the teachers, reflected these results. The result

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of our study indicated that constructing the cases influenced teachers’conceptions of students’ learning; particularly, in this case, the teachersimproved their understanding of the levels of problems.

As another example, the data from Interview II, conducted earlier inthe middle of the study, shows that Huei became aware of the differencesamong subtraction word problems from discussing a case. Huei ranked thelevel of Problems 4, 5, 6, 7 as shown below and interpreted why she didso after she read a case entitled “What Are the Differences Among theSubtraction Problems?”

Problem 4: There are 5 caps for 8 children, one for each child. How manymore caps are needed?

Problem 5: There are 8 girls playing in the sand and 5 girls playing onswings. How many more girls playing in the sand are therethan girls playing on swings?

Problem 6: There are 8 girls playing in the sand and 5 girls playing onswings. How many fewer girls playing on swings are therethan girls playing in the sand?

Problem 7: There are 5 presents for 8 children, one for each. Are thesepresents enough? How do you know?

Huei ranked the four problems, from easy to difficult, as 4, 7, 5, and6. Her understanding was that comparing problems (Problems 5 and 6)are more difficult than matching problems (Problems 4 and 7). She alsooffered further interpretation of the differences among the problems basedon her experiences of teaching. She stated that

I remembered the case entitled ‘What Are the Differences Among the Subtraction Prob-lems?’ We have referred to various compare word problems. The unknown number inProblem 4 and 7 is the difference set. . . . Problems 5 and 6 belong to compare problemswith the relational terms “more than” and “fewer than”, respectively. Moreover, Problem 6is more difficult than Problem 5, since students are more familiar with the “more than”term in daily life (Huei, Interview, 2/10/1999).

Huei then applied this conceptual knowledge of subtraction word problemsin a later lesson and it served as a reference for categorizing students’solutions.

In observing the teachers’ teaching, a common error pattern was noticedfor first graders in resolving a problem in which the smaller number iswritten prior to the larger number in the text; for example, they wrote 8 –13 = 5. As a result of this new awareness, each teacher investigated the firstgraders’ difficulty with the problems using the relational term “less than”.The problems they used were:

Problem 8: There are 13 umbrellas and 8 students. Which number issmaller? What is the difference between the two numbers?


Problem 9: Hsin-Jen has 13 marbles. Pin-Huan has 8 marbles. Howmany fewer marbles does Pin-Huan have than Hsin-Jen?

Problem 10: Wei-Dert has 8 dollars and Ker-Ming has 13 dollars. Howmany fewer dollars does Wei-Dert have than Ker-Ming?

In the weekly meeting, the teachers analyzed individual students’ worksheets collected in the class and reported that there was a higher percentage(45%) of students using the incorrect expressions 8 – 13 = 5 for Problem 10than the other problems (3%, 6%). They realized that the smaller number(8) involved in Problem 10 was presented prior to the larger number (13).Therefore, the students without understanding the meaning of numbersentence but using a one-to-one correspondence strategy could not solveProblem 10 successfully. The teachers’ analysis showed that it needs onemore procedure for solving Problem 10 than for Problems 8 and 9, if firstgraders are to answer the problems successfully. The additional procedurefor solving Problem 10 is to transpose “How many fewer dollars does Wei-Dert have than Ker-Ming?” to “How many more dollars does Ker-Minghave than Wei-Dert?” Based on the analysis of the relationship betweenthe two sentences, the teachers understand that this type of problem isnot appropriate for first graders until they have successfully developed therelationship between subtraction and addition.

Teachers’ improving their awareness and competence in helping students’learning. In the initiation of the study, the teachers participating in thisstudy were not aware of making a distinction for the first graders in theorder of the three levels of representation, hands-on, picture, and symbolicrepresentation. The researcher, as a learning partner of the teachers,brought up this issue drawn from Jong’s lesson at the weekly meeting.5

The researcher used a student’s solution as an example to address the issue.

For the problem, There are 8 children and 5 presents. Each child is supposed to get onepresent. Are there enough presents? a student wrote the number sentence 8 – 5 = 3 on themarker board first, and then followed by drawing circles below it, like this�� – �� = ��Do you think the given problem can be represented as�� – �� = ��?Do you think those students who used the number sentence 8 – 5 = 3 first and then followedby drawing circles representing the problem make sense of 8 – 5 = 3? (The researcher,Group discussion, 11/23/1998).

Huei learned, from this group discussion, about students’ confusionwith order in using symbolic representation with pictorial representationand then reflected on her own lessons and what she had observed in Sue’slesson taught in October. She said:

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. . . As I observed Sue’s lesson, there were several students using either the number sentence3 – 8 = 5 or 8 – 3 = 5 to represent the problem, There are 3 red and 8 green chips. Howmany fewer red chips are there than green chips? These students were not able to givean explanation why they used the expressions, when Sue asked them . . . (Huei, Groupdiscussion, 11/23/1998).. . . The use of number sentences also frequently occurred in my earlier lessons. However,some of my students did not understand the meaning of the expression. When I asked themhow they learned the use of number sentences, one of the students said: “I learned thisfrom my parents and the teachers in cramming school”. Another student said: “I learnedthis when I was in kindergarten” (Huei, Group discussion, 11/23/1998).

Ling also paid more attention to the order of three levels of represen-tations in her later classroom practice. She stated in a weekly meetingthat

Since the professor told us the three levels of representations, I sat down with students inone group for a while and observed carefully the process of students’ solving the subtrac-tion word problem I gave them, to see if the circles were drawn prior to the use of numbersentences. According to my observation, some of my students used the number sentenceand circles in an inappropriate order. I can tell the circles drawn by those students whowrote the number sentence first merely for the requirement of the problem. I don’t thinkthose students understood the meaning of the number sentence even though they used it(Ling, Group discussion, 11/23/1998).

The issue of the order of use of the three levels of representationsbecame part of the Discussion Questions of a written case constructed inthe same manner as described in Phase 3. To avoid students skipping thepictorial learning from concrete experience to symbolic learning, the wayof helping first graders to use the pictorial representation to represent aword problem was also addressed in a weekly meeting.

The researcher, as one of the new curriculum reformers, suggesteda method of “Posing a problem with sub-steps” to teachers to resolvethe problem encountered in first-grader classroom practice. The additionor subtraction word problem is divided into several parts, one part ispresented at a time and students’ activity follows. For example, the addi-tion problem, “6 children are playing in the sand on the playground. 3 morechildren come to play. How many children are playing in the playgroundnow?” is typical as a complete statement. Alternatively, a solution of thisproblem can use three sub-steps. First, the teacher says, “6 children areplaying in the sand on the playground”; at this time, the students representthe situation using chips. Then the teacher says, “3 more children come toplay”, and students show them by placing three more chips. The teachersays finally, “How many are there now?” and students then count the totalnumber of chips.


The recommended method was examined in one of Jong’s lessons.The effect of “posing a problem with sub-steps” on students’ learning inaddition and subtraction problems was revealed in an interview with Jong:

The case developed in our previous discussion helped me to perceive my blind spotsin instruction. Reading it ahead of my teaching makes some difference to the effect ofinstruction. I re-taught the previously unsuccessful lessons and aimed at the use of pictorialrepresentation for subtraction and addition word problems by using the “posing a problemwith sub-steps” technique. I found that more of my students solved the word problem withcircles than with only numbers. Thus, if I can meet the objective of the lesson addressed inthe written cases, then I would not be frustrated with instruction; I could predict students’solutions in advance and know the way to deal with students’ difficulty in mathematicslearning (Jong, Interview, 02/08/1999).

Teachers becoming more reflective practitioners. Having teachers respondto a case seemed to reinforce insights in their thinking about mathema-tical learning. Teachers became active contributors to multiple perspectivesand became more reflective concerning their classroom practices. Thethird phase of constructing cases, designed to develop teachers’ abilityin writing cases, stimulated personal reflection. The preparation of casesseemed to help teachers develop skills central to reflective practice, forinstance, learning to focus on alternative approaches. One example waswhen Ling prepared the first version of a case conducted from her teachingwhich became the case “Is Binding Straws Important for Students?” Theobjective of Ling’s lesson was to enable students to learn to count by tensto 100. When first observed, Ling paid more attention to the differencesof students’ performance compared to Jong. Ling perceived that the differ-ences in strategies used by students in the two classes were because ofthe teachers’ different methods of teaching. Thus, Ling used the compar-isons of two instructional approaches as the main text of the case to bebuilt. However, Ling did not describe the key element in her teaching,which resulted in her students employing multiple strategies, until Jongaddressed the main difference between two approaches: “My students wereasked to bind ten straws into a bundle, while Ling’s students were not”.Thus, the case-development process internalized Ling’s insight into howto improve her teaching to enable students to count by tens. It is evidentthat writing using the context of Ling’s personal teaching enhanced her andother participants’ reflection on their teaching. Moreover, the case Lingconstructed effectively included her self-reports and other multiple pointsof view, such as Jong’s.

Teachers’ discussions resulted in an integration of their own teachingwith others’ multiple perspectives on teaching and hence became themain text of case writing. The discussion also made participants reflecton their teaching practices in-depth. A complete case was constructed

344 PI-JEN LIN

by the collaborative team and written by teachers throughout the processof observing a lesson, first weekly meeting, first version, second regularmeeting, second version, third regular meeting, third version, and finalversion. Without such revising activity and group member support,teacher-written cases seem unlikely to achieve such clarity and power.Therefore, discussing the issues of learning and teaching of mathe-matics with those who teach the same content is a beneficial vehicle forconstructing cases of teaching and also for development of thinking andreflection.

DISCUSSION

The study found that the use of cases enhanced teachers’ understanding ofstudents’ learning and improved their reflective thinking of teaching whencases were constructed consistently by a collaborative research team witha university professor and same-grade teachers. The result is consistentwith the literature research on the effect of cases on teachers’ knowledge(Schifter, 1996a, 1996b; Silver, 1999). However, the cases referred to inprevious research are created by researchers only (Fennema et al., 1993),by university faculty who also teach mathematics k-12 (Borasi, 1992), orby classroom teachers only (Barnett et al., 1994). The essential processesof constructing cases involved in this study include observing teachers’same grade instruction, discussing the pedagogical issues arising from theobservations in regular weekly meetings, and writing and rewriting thecases based on real teaching.

In group discussions, multiple perspectives and comments were sharedin the process of constructing a case. The issues discussed or debatedin group discussions became the candidates for Discussion Questions ofa written case. The issues could concern the structure of mathematicscontent, pedagogical problems, or students’ cognitive development. Theissues, initiated from first grade classroom teaching, are authentic pedago-gical problems and common experiences encountered in other first-gradeteachers’ teaching. Thus, the issues of each case, based on real same-gradeteaching as dialectic content, were centrally significant to teachers’ atten-tions and concerns and built toward a climax that exposes the dilemmasof teaching. The involvement of other teachers of the same grade helpsteachers to clarify the mathematical structure of teaching activities, thepresentation of the task, and students’ responses to the task. Teachers’pedagogical practice was therefore expanded and deepened in ways thatare in accord with Merseth’s study (1992, 1996).


In addition, the cases of the collaborative study, constructed by theresearcher and same-grade teachers, are more likely to be enrichingas exemplars. The intervention of the researcher placed more emphasison theoretical perspectives of children’s cognitive development, whilethe involvement of the teachers emphasized particularly the issues ofteaching in their classroom practices. During the process of constructinga case, the researcher’s theoretical perspectives were examined againstteachers’ classroom practice. For instance, Jong verified the researcher’sstrategy “posing a problem with sub-steps” for helping students’ pictoriallearning. Likewise, teachers’ perspectives of practice were examined byeach other and supported by the researcher’s theoretical perspectives. Forinstance, the teachers’ did not attend to students’ various strategies ofsolving subtraction word problems until the relationship between students’strategies and semantic structure of subtraction were addressed in groupdiscussion. Thus, case construction creates the possibility of connectingtheory with practice and makes questions about effective teaching morepublic. This is the second feature of case construction.

Case writing is part of constructing cases. The results of the studyindicate that case writing, involving the processes of observing teachingand several group discussions for revising four versions, improved thereflective ability of the teacher who wrote the case. Thus case writingalso involved the process of discussing a case. Through case discussion,each teacher’s cognitive awareness was enhanced from access to otherteachers’ perspectives. Thus, case discussion fostered personal reflectionthrough an external process. This is Vygotsky’s law of cultural develop-ment that individual development appears on the social plane followed bythe psychological plane (Vygotsky, 1978). Therefore, the case discussioninitiating social interaction is a potential factor in the source of the changesin the teachers’ ways of thinking and in the breadth and depth of pedago-gical content knowledge. The study found that case writing is an effectivetechnique for prompting teachers’ reflection on practice. The result canbe explained by Richert’s argument (1991). This argument claims thatcomponents of reflection are involved in writing a case as writing requiresinternal processes of choosing what to write, deciding the particular focus,perceiving an episode to write and developing it into a case, and learningwhat to accentuate. Although research on the use of cases suggests thatdiscussing a case for very experienced teachers seemed to be a catalyst forreflection and to promote metacognition (Levin, 1993), the result of thisstudy indicate that this is also true for case construction including bothaspects of case writing and case discussion. This study therefore recom-

346 PI-JEN LIN

mends that the use of cases, including case discussion and case writing, bea core component of the curriculum in teacher professional programs.

Compared to an individual’s case writing based on personal experience,such as in Shulman’s study taking a teacher’s self-report and turning itinto a teaching case (Shulman, 1992), the cases elicited by the same gradeteachers sharing homogeneous concerns of practicing become more avail-able for discussion and review than those based merely on particular orindividual experiences. Thus, the exemplar cases might prompt more easilythe rethinking of other school teachers who are not involved in the processof constructing cases.

The cases written in a form of a narrative or an episode can drawattention, lodge themselves in memory, and capture commitment. Caseconstruction offered authentic experiences of teaching for the teacherswho participating in the process of construction. The authentic experiencelinks others’ experiences to one’s own experience. Through discussion, theothers’ authentic experiences help one to understand the theory, concept,and knowledge readily. The cases which were based on teachers experi-ence and knowledge are real, so that they are a representation of teachers’knowledge. As teachers think about the teaching of others, they understandbetter the nature of teaching mathematics. Hence, it seems to be morelikely that such experiences will internally motivate teachers’ learning toteach.

ACKNOWLEDGEMENT

The author would like to thank Barbara Jaworski, University of Oxford,for her thoughtful comments on earlier drafts of the paper.

NOTES

1 An earlier version of this paper was presented at the 24th International Group for thePsychology of Mathematics Education which was held in Hiroshima, Japan. 23–27 July2000. The research reported in this paper was supported by the National Science Councilof Taiwan under Grant NSC89-2511-S134-001. The opinions expressed in this article arethose of the author and do not necessary reflect the view of funding agency.2 At that time, textbooks for only the first three grades had been developed completely,while textbooks for the rest of the grades were still being produced.3 Encouraging student’s various ways of thinking or solutions is one of the focuses ofthe learner-centered approach emphasized in the new curriculum reform of Taiwan. Itis a challenge for teachers to make a judgment between when teaching should focus onencouraging students’ multiple ways of thinking and when the teaching can move towardpromoting students’ thinking at a more advanced level. In some of the textbooks of Taiwan,


restricting the solution of a problem to a specific method which is proposed to students isa strategy for advancing students’ ways of thinking from a low level to a higher level.4 The designers of textbooks in Taiwan embed Bruner’s (1960) three levels of represen-tations (enactive, iconic, and symbolic) into learning activities for students to help firstgraders to understand the meaning of additive word problems. At the enactive level, firstgraders’ learning involves hands-on activity with materials like counters or chips. At theiconic level, learning is based on the use of visual media: films, pictures, diagrams, andthe like. For example, drawing 5 circles represents 5 children playing in the playground.Symbolic learning is the stage in which first graders use abstract symbols to representreality; for example, using a number sentence to represent a context in a word problem.Bruner contends that there is a common sense order implied by three levels because eachrequires familiarity with the earlier modes of representation (Bruner, 1960). However,most of the first graders had earlier learning of number sentences without learning theother representations when they were in kindergarten. Thus, the majority of first-gradershad earlier use of number sentences when they were learning the use of visual modes ofrepresentation.5 Following from Note 4 above: first, the textbook provides the first graders with concreteobjects. i.e., the first grader separates a set of 5 counters from a set of 8 counters anddetermines that there are 3 counters left. Secondly, the textbook provides the first graderwith a series of pictures; e.g., the first grader might draw a picture with 8 counters whichare separated from 5 counters in a second picture. The third picture shows 3 counters left,as shown in Figure 8. At the symbolic level, the textbook asks the first grader to use asymbolic representation to represent reality. For instance, 8 – 5 = 3 represents the problem:There are 8 children and 5 presents. Each child is supposed to get one present. Are thereenough presents?

Figure 8. Iconic level of a separating problem.

REFERENCES

Ball, D. L. & Cohen, D.K. (1999). Developing practice, developing practitioners: Towarda practice-based theory of professional education: In G. Sykes & L. Darling-Hammond(Eds.), Teaching as the Learning Profession: Handbook of Policy and Practice (pp. 3–32). San Francisco: Jossey Bass.

Barnett, C. S. (1991). Case Methods: A Promising Vehicle for Expanding the PedagogicalKnowledge Base in Mathematics. Paper presented at the annual meeting of the AmericanEducational Research Association, Chicago.

Barnett, C. S. (1998). Mathematics teaching cases as a catalyst for informed strategicinquiry. Teaching and Teacher Education, 14(1), 81–93.

Barnett, C. S. & Tyson, P. A. (1994). Facilitating mathematics case discussions whilepreserving shared authority. Paper presented at the annual meeting of the AmericanEducational Research Association, New Orleans.

Borasi, R. (1992). Learning Mathematics through Inquiry. Portsmouth, NH: Heinemann.Bruner, J. S. (1960). The Process of Education. Cambridge, MA: Harvard University Press.

348 PI-JEN LIN

Carpenter, T. P. & Fennema, E. (1988). Research and cognitively guided instructions. In E.Fennema, T. P. Carpenter & S. J. Lamon (Eds.), Integrating Research on Teaching andLearning Mathematics (pp. 2–19). Madison, WI: University of Wisconsin, WisconsinCenter for Education Research.

Cobb, P. & McClain, K. (1999). An approach for supporting teachers’ learning in social andinstructional context. In F. L. Lin (Ed.), Proceedings of the 1999 International Confer-ence on Mathematics Teacher Education (pp. 7–76). Taipei, Taiwan: National TaiwanNormal University.

Cooney, T. J. (1999). Conceptualizing teacher development. In F. L. Lin (Ed.), Proceedingsof the 1999 International Conference on Mathematics Teacher Education (pp. 1–34).Taipei, Taiwan: National Taiwan Normal University.

Doyle, W. (1990). Case methods in the education of teachers. Teacher EducationQuarterly, 17(1), 7–16.

Fennema, E. & Franke, M. L. (1992). Teachers’ knowledge and its impact. In D. A. Grouws(Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 147–164).NY: Macmillan.

Fennema, E. Carpenter, T. P., Franke, M. L. & Carely, D. A. (1993). Using children’smathematical knowledge in instruction. American Educational Research Journal, 30(3),555–583.

Harrington, H. L. (1995). Fostering reasoned decisions: Case-based pedagogy and theprofessional development of teachers. Teaching and Teacher Education, 11(1), 203–214.

Huang, H. M. (2000). Investigating of teachers’ mathematics conceptions and pedagogicalcontent knowledge in Mathematics. Paper presented at the 9th International Congresson Mathematics Education. Tokyo, Japan.

Jaworski, B. and Wood T. (1999). Themes and issues in inservice programmes. In B.Jaworski, T. Wood & S. Dawson (Eds.), Mathematics Teacher Education: CriticalInternational Perspectives (pp. 125–147). London: Falmer Press.

Krainer, K. (1999). Learning from Gisela-or: Finding a bridge between classroom devel-opment, school development, and the development of educational systems. In F. L.Lin (Ed.), Proceedings of the 1999 International Conference on Mathematics TeacherEducation (pp. 76–95). Taipei, Taiwan: National Taiwan Normal University.

Levin, B. B. (1993). Using the Case Method in Teacher Education: The Role of Discussionand Experience in Teachers’ Thinking about Cases. Unpublished doctoral dissertation.University of California, Berkeley.

Lin, P. J. (1999). Processes supporting teacher change within a school-based professionaldevelopment program. Proceedings of the National Science Council – Part D: Math-ematics, Science, and Technology Education, 9(3), 99–108. Taipei, Taiwan: NationalScience Council.

Lin, P. J. (2001). Developing Teachers’ Knowledge of Students’ Mathematical Learning:An Approach of Teachers’ Professional Development by Integrating Theory with Prac-tice. Taipei, Taiwan: Shu-Dah (in Chinese).

Merseth, K. K. (1992). Cases for decision making in teacher education. In J. H. Shulman(Ed.), Case Methods in Teacher Education (pp. 50–63). New York: Teachers CollegePress.

Merseth, K. K. (1996). Cases and case methods in teacher education. In J. Sikula, J. Buttery& E. Guyton (Eds.), Handbook of Research on Teacher Education (pp. 722–744). NewYork: Macmillan.

Miller, D. M. & Pine, G. J. (1990). Advancing professional inquiry for educationalimprovement thorough action research. Journal of Staff Development, 2(3), 56–61.


Ministry of Education of Taiwan (1993). Curriculum Standards of National ElementarySchools in Taiwan (pp. 91–132). Taipei: Tai-Jye.

National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Stand-ards for School Mathematics. Reston, Va: Author.

National Council of Teachers of Mathematics (2000). Principles and Standards for SchoolMathematics. Reston, Va: Author.

iaget, J. (1971). Biology and Knowledge. Chicago: The University of Chicago Press.Richardson, V. (1993). Use of cases in considering methods of motivating students. In H.

Harrington & M. Thompson (Eds.), Student Motivation and Case study Manual (pp.57–60). Boone, NC: Appalachian State University.

Richert, A. E. (1991). Using teacher cases for reflection and enhanced understanding. InA. Lieberman & L. Miller (Eds.), Staff Development for Education in the 90’s (pp. 112–132). New York: Teacher College Press.

Riley, M. S., Greeno, J. G. & Heller, J. I. (1983). Development of children’s problem-solving ability in arithmetic. In H. P. Ginsberg (Ed.), The Development of MathematicalThinking (pp. 153–196). San Diego, CA: Academic Press.

Schifter, D. (1996a). What’s Happening in Math Class: Envisioning New Practicesthorough Teacher Narratives. New York: Teachers College Press.

Schifter, D. (1996b). What’s Happening in Math Class: Reconstructing ProfessionalIdentities. New York: Teachers College Press.

Shulman, L. S. (1992). Toward a pedagogy of cases. In J. Shulman (Ed.), Case Methods inTeacher Education (pp. 1–30). New York: Teachers College Press.

Shulman, J. H. & Colbert, J. A. (1988). The Intern Teacher Casebook. Eugene, OR: ERICClearinghouse on Educational Management, Educational Research and Development.Washington, DC: ERIC Clearinghouse on Teacher Education.

Silver, E. A. (1999). Helping teachers learn from experience: Cases as a tool for profes-sional education. Proceedings of the 87th Annual Meeting of Curriculum for ElementarySchool (pp. 13–25). Taipei: San-Shia.

Strauss, A., & Corbin, J. (1994). Grounded theory methodology: An overview. In N.K. Denzin & Y. S. Lincoln (Eds.), Handbook of Qualitative Research (pp. 273–285).Thousand Oaks, CA: Sage.

Sullivan, P. & Mousley, J. (1999). Teachers as active decision makers: Acknowledgingthe complexity of teaching. Proceedings of the 86th Annual Meeting of Curriculum andResearch (pp. 1–7). Taipei, Taiwan.

Vygotsky, L. S. (1978). Mind in Society. Cambridge, MA: Harvard University Press.Wood, T., Scott-Nelson, B. & Warfield, J. (2001). Beyond Classical Pedagogy: Teaching

Elementary School Mathematics. New York: Lawrence Erlbaum.

Mathematics Education Department521, Nan-Dah RoadHsin-Chu City 30014Taiwan, R.O.C.E-mail: [email protected]

DESPINA POTARI and BARBARA JAWORSKI

TACKLING COMPLEXITY IN MATHEMATICS TEACHINGDEVELOPMENT: USING THE TEACHING TRIAD AS A

TOOL FOR REFLECTION AND ANALYSIS

(Accepted 6 August 2002)

ABSTRACT. This paper reports research that attempts to make sense of the complexity ofmathematics teaching and its development at secondary school level. The research wasconducted in partnership between two teachers and two educator/researchers over oneschool term in two U.K. schools. A theoretical construct, the teaching triad, was used asan analytical device (by the researchers) and as a reflective agent for teaching development(by the teachers). The focus of analysis was the interactions between teacher and studentsat whole class and small group level. Both micro- and macro-analyses were undertaken.We present details of the processes involved in examples from the teaching of one teacheras she translated theoretical aims into classroom practice. The use of the triad allowedaccess to complexity, involving both psychological and sociological elements, and to theposition of a sincere teacher with respect to competing forces in the educational system.The potential of the triad for teacher and teaching development is discussed.

KEY WORDS: analysis of teaching, classroom interaction, cognitive and affective sensi-tivity complexity of mathematics teaching, effective learning, teaching development,teaching triad

INTRODUCTION

This project involved a collaboration between two teachers and twoeducator/researchers to explore the contribution of a theoretical construct,the teaching triad (TT) to the construction and analysis of mathematicsteaching and its development in secondary school classrooms. It took placeover one school term (a third of the academic year) in which the teachers’lessons were observed by one or both researchers; the teachers were inter-viewed between lessons, and the team of four met periodically to examineissues arising from the research.

The study was designed to explore the use of the triad as a tool to

• analyse classroom data to provide insights into mathematics teaching,particularly its complexity and issues for teachers;

• encourage teachers’ reflection in all aspects of teaching.

The study shows that use of the triad in these ways revealed details ofcomplexity and allowed insights into teaching issues for both researchers


352 DESPINA POTARI AND BARBARA JAWORSKI

and teachers. We were able to go beyond simplistic judgments aboutlessons to search out cognitive and affective factors in learning and thewider social issues and concerns that impinge on classroom decisions.Such a revealing of factors and issues in a context of teacher/researchercollaboration provided a powerful environment for considering teachingdevelopment.

However, generality here is seen not in the particular issues that theresearch has raised, nor in the power of the triad to reveal particular issues;it is in the methodology of the use of the triad, the analytical process thatreveals issues, and the collaborative working of the research participantsin making such research possible. We take seriously the words of Cooney,who writes

But if we are to move beyond collecting interesting stories, theoretical perspectives needto be developed that allow us to see how those stories begin to tell a larger story. That is weshould be interested in how local theories about teachers can contribute to a more generaltheory about teacher education. (Cooney, 1994, p. 627)

We are interested in how our approach, using the teaching triad, cancontribute to a more general theory of describing and interpreting teachingpractice, and ultimately to indications for teacher education.

THEORETICAL BACKGROUND

The Teaching Triad Research

The teaching triad emerged from an ethnographic study of investigativemathematics teaching (Jaworski, 1994) of a small number of mathematicsteachers. Very briefly, this involved engaging students in open-endedand problem-solving tasks through which curriculum-designated mathe-matical topics would be approached and students’ mathematical thinkingand understanding fostered. The study led to identification of generalcharacteristics of investigative teaching and to a theoretical construct,the teaching triad, which linked the generalized characteristics to three‘domains’ of activity in which teachers had been seen to engage: manage-ment of learning (ML); sensitivity to students (SS) and mathematicalchallenge (MC). This triad attempted to provide a framework to captureessential elements of the complexity of the observed teaching and togeneralize these to mathematics teaching more widely.

Briefly, management of learning describes the teacher’s role in theconstitution of the classroom learning environment by the teacher andstudents. It includes classroom groupings; planning of tasks and activity;setting of norms and so on. Sensitivity to students describes the teacher’s

TACKLING COMPLEXITY IN MATHEMATICS TEACHING DEVELOPMENT 353

Figure 1. The teaching triad.

knowledge of students and attention to their needs; the ways in which theteacher interacts with individuals and guides group interactions. Mathe-matical challenge describes the challenges offered to students to engendermathematical thinking and activity. This includes tasks set, questionsposed and emphasis on metacognitive processing. These domains areclosely interlinked and interdependent as our current research shows. Onemajor aim of the research described in this paper was to explore theteaching triad beyond the situations from which it arose and to considerits applicability and uses for teachers and researchers more widely.

The following episode from our research (data extract: 4J/9: the fourthobserved lesson of a teacher, Jeanette, with her Year 9 class) illuminatesthe elements of the triad and provides initial insight into its use.

A lesson opening1

Jeanette’s class was divided into groups of four, five or six studentsaccording to students’ choice. She gave each group sheets of squaredpaper, some card, and a sheet describing a problem. Multilink cubes wereavailable for use. The problem was to design a box to contain 48 cubes,each of side 2 cm, using a minimum amount of card. Each group had towork on this task and provide a group solution. The lesson opened with aninteractive whole class session in which the task was clarified, describedbriefly as follows.

Jeanette invited someone to come to the board and draw a 2 cm cube.Michael accepted the invitation and drew a 2 cm square. Some studentssaid that it was a square (i.e. not a cube). Jeanette acknowledged it as “acube facing the front”. Another student came forward and drew a cube.Jeanette asked him “Can you write your dimensions as Michael has done?”The teacher then clarified the problem: “You need to make a box to fit 48


of those cubes. The squared paper is to try out some designs. It is a groupproject. Everyone needs to be busy.” She paused, waiting for silence in theroom. “The company will give a prize to the one who has used the leastcard. We want to find the best shape and size of box”.

In our analysis, in the category of management of learning, we recog-nized the teacher’s organization of the class in groups; her planning andorganization of the investigative task; her management of the whole classsituation in which she introduced the task; and her responses to the studentswho participated. All of these are part of the teacher’s organization of thelearning environment, her developing of social skills within the classroomsetting, and her creation of opportunity to engage in mathematics. In thecategory of sensitivity to students, we placed the teacher’s mode of intro-duction of the task, and her particular responses to students. She wanted allstudents to be able to make a start on the task; hence she offered a familiarpractical context, resources, and clarification of the problem. She involvedstudents in her introduction and was careful to value, and encourage othersto value, their contributions. As well as valuing Michael’s contribution,her words “a cube facing the front” acknowledged another perspective ofwhat students saw as a square. Thus she opened up mathematical challenge(MC), initially manifested in the problem itself, and enhanced throughcontextual emphasis on optimization in the box.

The teaching triad research, which is the basis of this paper, wasresearch into mathematics teaching in naturalistic settings – secondaryschool mathematics classrooms – aiming to study the full complexities ofteaching and reveal the tensions, issues and dilemmas that teachers face inconstructing effective learning situations for students. The research did nottry to promote any particular forms of teaching or approaches to teaching.It did not seek overtly to educate teachers or develop teaching. Its startingpoint was the teaching of the two teachers who were participants in theresearch. A study of their real lessons, in real circumstances was a sourceof opportunity to look at the potential of the triad to inform teaching, andto gain insights into teaching. We shall show in this paper how insightsgained were related to developments in a teacher’s thinking about teachingand potentially the development of teaching itself. This suggests the triadas a potential tool for teaching and teacher development.

Complexity

Three guiding assumptions in this research were:

• that the complexity of teaching-learning situations is such that anyattempt to provide simple descriptions is ‘rapidly shown to be hollow’(Bauersfeld, 1988);


• that the essence of teaching/learning processes lies in classroom inter-actions, the analysis of which provides the key to understanding thecomplexity acknowledged above;

• that teachers work from a position of sincerity and professionalism inseeking for students’ mathematical development.

The first two of these follow from the work of Bauersfeld andcolleagues in Germany (e.g. Bauersfeld, 1988) and from our own profes-sional experience before the research began. It is through a close scrutinyand analysis of interactions that we gain insights into teaching complexityand start to appreciate the myriad issues that lie beneath teaching decisions(Cooney, 1988; Cobb et al., 1990; Jaworski, 1994).

Without the third assumption, it is hard for effective collaborationbetween teachers and researchers to begin. Inevitably, in looking criticallyat teaching, one comes up against the apparent success or lack of successof a teacher’s approach in relation to students’ learning. It is important toget beyond such surface judgments. A strength of this project was that wewere able to address the roots of success, or, importantly, perceived lack ofsuccess, with the teachers through a common endeavour, a mutual respectand an appreciation of the problematic areas with which we were dealingjointly.

Values and Beliefs

Both teachers and researchers came to this project with strong views aboutthe nature of teaching that should lead to effective learning of mathematicsfor students in classrooms. Inevitably, such views were based with varyingdegrees of explicitness on theoretical perspectives that we can trace in theliterature.

At the root of the observed teaching were teachers’ aims for students’understanding of mathematics. This involves developing what variousscholars have called relational understanding (e.g., Skemp, 1976) orconceptual understanding (e.g., Brown, 1979). Teachers wanted studentsto go beyond ritualized knowledge (Edwards & Mercer, 1987) to appre-ciate reasons for procedures and rules, to relate apparently different areasof mathematics to each other, and to apply mathematical learning tosolving problems in mathematics and the wider world.

The words we use here are those of researchers and theoreticians.Teachers did not express their beliefs and intentions in such terms.Rather they spoke about the classroom processes that they were tryingto foster, the kinds of activities that would encourage thinking, andways in which they would encourage students to work together. Thus,dialogue was important: students were encouraged to articulate their


processes and perceptions; collaborative activity was encouraged: sharingand contrasting ideas, and reflecting critically; activities requiring explora-tion and inquiry were introduced with the negotiation of outcomes.Teachers believed that such classroom approaches and the resultinglearning environment would be conducive to the deep levels of mathe-matical understanding they sought in students. In such beliefs they were inaccordance with much of the mathematics education community currentlyin seeking to foster both individual and social forms of learning and leadingto a conceptual understanding of mathematics (e.g., Jaworski, 1994; Ball,1996; Lampert, 2001; Wood et al., 2001).

A Methodological Approach to Analyzing and Developing Teaching

The research here is related to other studies that explore mathematicsteaching in naturalistic settings, to describe and analyse in detail thenature of teaching that seeks students’ conceptual understandings ofmathematics. Such studies focus on students’ learning of mathematics,on the sociological construction of the classroom environment, and onthe roles of language and interactions in fostering mathematical learning(see for example, Lampert, 1985; Williams & Baxter, 1993; Jaworski,1994; Fennema et al. 1996; Wood et al., 2001). We see in such studiesa significant shift over the last decade towards a knowledge base relatedto a reflective approach to developing teaching. This knowledge basedocuments styles or approaches to teaching as well as dilemmas experi-enced by teachers in interpreting these styles or approaches. The teachingtriad research adds to this knowledge base. In addition, it develops amethodological approach to a study of teaching based on an analysis ofclassroom interactions. An assumption is that the teaching will result inclassroom interactions, the discourse of which can be analyzed by studyingepisodes in the dialogue between students and teacher (micro-analysis).This analysis can be reviewed critically alongside social factors in thewider educational system in which teaching takes place (macro-analysis).

A focus on such discourse and resulting dialogue fits with most ofthe projects referenced above. Whether the discourse is one of studentsarticulating their strategies for solving problems (Fennema et al., 1996),one of students challenging their peers’ arguments for a particular solution(Wood, 2001), or one of discussing the results of a classroom investiga-tion or experiment (Jaworski, 1994) it is possible to use the dialogue asthe basis for studying teaching and its outcomes, and learning about theissues and dilemmas for teachers. Most research reports from these variousresearch projects use dialogue to illustrate issues in teaching. However, therigorous use of dialogue actually to analyse teaching is less common. Such


analysis has been a hallmark of the work of a number of researchers inGermany (e.g. Voigt, 1985; Steinbring, 1998). Steinbring’s analyses haveinvolved an epistemological perspective that scrutinizes the mathematicson which teacher and students are focusing and explores ways in whichthe mathematical concepts are addressed. What is similar in our work isthe line by line micro-analysis which forms the basis of more detailedconceptualizations of the teaching situation.

These conceptualizations enable us to go beyond simplistic judgmentson the success, or otherwise, of teaching to the tensions and dilemmasthat affect teaching decisions at both micro and macro levels. The finedetails of a brief action or interaction can be related to the wider socialconcerns in which these actions and interactions reside. In earlier writingabout the teaching triad we identified the concept of ‘harmony’ (betweenthe elements of mathematical challenge and sensitivity to students) as a keyfactor in explaining the apparent success of a teaching/learning episode(Jaworski & Potari, 1998). In our writing here, we explore particularlya situation where there is a lack of harmony, where teaching would bejudged critically to fail in its objectives. Micro- and macro-analyses, usingthe triad as a tool, enable us to seek out the finer details of the complexinfluences on the teaching situation and its relation to teachers’ associ-ated beliefs. In these analyses we found ourselves drawing on constructsthat have been used and debated widely in considerations of teachingand its development; those of scaffolding (e.g. Bruner, 1985; Williams &Baxter, 1996) and funneling (Bauersfeld, 1988). Williams and Baxter talkabout analytical scaffolding and social scaffolding; the former relating tostudents cognitive development and the latter to their social development.These terms seemed to relate to concepts of cognitive and affective sensi-tivity. Goldin (in press) discusses the concept of ‘affect’ and its relationto beliefs and value systems, including how the cognitive and affectiverepresentational systems of individuals interact with socially or culturallyshared systems. Analysis enabled us to look critically at such constructs,and to see what factors in the micro and macro domains contributed tothinking and behaviour that could be described in those terms.

The teachers in this study used the teaching triad as a tool for theirplanning of lessons and reflections on students’ learning. This tool enabledteachers to be more explicit about their beliefs and intentions for teaching.In addition it allowed reflections on teaching to address occasions wherebeliefs and actions appeared not to coincide, where teaching did notseem to achieve its declared aims. Here we see an interface between theuses of the triad as a tool for thinking about teaching and for analyzingteaching. Beliefs, lying in the domain of theory, have to be interpreted into


classroom actions through the thinking and professional expertise of theteacher (Cooney et al., 1998). Analyses exposed factors in such interpreta-tion, revealing issues and tensions and encouraging alternative actions andinterpretations through an enhanced knowledge base. This means that theteaching triad was acting as a tool for teaching development through thisresearch and the activity of the teachers. Such activity and research fits intoa developing paradigm of collaboration between teachers and researchersin the development of mathematics teaching (Britt et al., 1993; Krainer,1993; Zack et al., 1997). Partnerships between teachers and researchers areseen as fruitful ways forward in the development of mathematics teachingmore widely (Jaworski, 2001).

METHODOLOGY

The Participants

The research was conducted by four participants, two mathematicsteachers and two university educator/researchers. It was a partnershipin the sense that teachers and researchers agreed to work together formutual interest, exploration and benefit, although towards differing goals.Researchers sought the nature of the practices observed, issues arisingand theoretical conceptualizations of the teaching, using and exploring theTT as an analytical device. Teachers considered how the TT contributedto their thinking and planning for teaching and evaluation of lessons inorder to improve teaching. They designed and taught lessons to a numberof classes of students; they discussed with the researchers their specificteaching objectives for each lesson before teaching and their reflections onpractices and outcomes after each lesson. They read and commented oninitial drafts of the analyses that the researchers produced. In this sectionwe provide a methodological account of our work with both teachers.However, to communicate key issues in the practice of any teacher requiresconsiderable detail and hence space in the paper, so we present detailedanalyses of the teaching of only one of the teachers (Jeanette). Wheredifferent issues arose from Sam’s teaching, these will be addressed in afurther paper.

The teachers work in secondary comprehensive2 schools (for ages 11–18 years) both of which are partners with the university in an initial teachereducation (ITE) programme in which the two teachers had each worked asa mentor to student teachers. Jeanette’s school is a boys’ school, designatedas a ‘Technology College’, for which extra funding is provided bringingwith it extra responsibilities for technology use within the school. Sam’s


is a mixed school. Both teachers are experienced mathematics teachers,well regarded by their peers and with positions of responsibility in theirschools. Jeanette is Head of Year in her school where she is responsible forstudents’ well-being within her year, their academic progress, behaviour,and self esteem. She liaises with parents to enable students to fulfil theirpotential during their time in school. Sam is the head of the mathematicsdepartment in his school, responsible for teaching and assessment of themathematics curriculum. Both had worked in an earlier project, the Mathe-matics Teachers Enquiry (MTE) Project, to study teachers’ classroomresearch into their own teaching (Jaworski, 1998). There they came toknow about the teaching triad and had expressed an interest in using itexplore their teaching. The language of the triad was a frequent part oftheir vocabulary in talking about teaching.

Both teachers’ chief aims for teaching involve the mathematical under-standing of their students and they have a ‘reform agenda’ as thebasis for developing their teaching such as is recommended in the UKCockcroft Report or the US Professional Standards for Teaching Mathe-matics (Department of Education and Science, 1982; National Councilof Teachers of Mathematics, 1991). In addition, in England, teachersare required by law to follow the National Curriculum which is testedthroughout primary and secondary schooling, satisfy the institutionalrequirements in their schools, and consider the current demands of societyfor the education of children. Fulfilling their academic aims and satisfyingthis multitude of requirements leads to issues and tensions some of whichthis research reveals.

The Data Collection

Data were collected

(1) through participant observation of lessons by the researchers(involving the researcher in sitting, observing, making notes, and occa-sionally talking with students) where field notes, audio recordings, andoccasionally video recordings, were produced;

(2) from interviews in the form of conversations between researchers andteachers before and after each lesson, audio recorded;

(3) from team meetings between all participants for a discussion ofresearch and teaching, audio recorded.

All recordings were transcribed.We studied classes in years seven to twelve (ages 11–17 years). Classes

were ‘sets’ in which students were grouped according to teachers’ percep-tions of their mathematical achievement – lower (L), middle, (M) or higher


(H). For Jeanette, for example, sets in years nine (M) and ten (H) wereobserved. Teaching varied according to the different characteristics of eachclass.

The conversations before and after each lesson were informal andconcerned issues about planning and teaching that were initiated eitherby the researchers or by the teachers. Three team meetings (each lastingabout three hours) were held during the period of the research and heremore general issues about teaching and learning were discussed. Conversa-tions and meetings enabled a development of mutual understanding withinthe team, nurturing trust and respect; offered further data to broaden theglobal characterization and contribute to macro-analyses, and acted in avalidatory role in which inferences and interpretations could be checkedthrough a discussion of issues.

Data Analysis

Our analyses have focused particularly on classes in which the teachers hadexplicit goals for changing their way of working with the class. In the caseof Jeanette, on whom we now focus particularly, it was her Year 9 classin which she had explicit goals for changing her way of working with thisclass. She had taught this class in the previous year and had an extremelygood relationship with the students. However, she was dissatisfied with thedegree of mathematical challenge she was offering: “Am I offering themenough challenge? . . . Am I pushing them enough?”

Analysis of data was in two phases: Firstly on-going analysis duringdata collection involved an inferential, descriptive and interpretativeprocess. Inferences about the teaching were made during classroom obser-vations and during the reading of notes and transcriptions. For each lessona summary was written describing briefly the various parts of a lesson(starting activity, group work, etc.) and including interpretations relatingto the inferences made. These were checked with teachers during conver-sations before or after subsequent lessons. From these summaries patternswere abstracted which described the teaching with a particular class, andsometimes, but not necessarily, across classes. These patterns form whatwe have called a global characterization of the observed teaching – adescription in broad detail, attending to styles of teaching and interactionwithout addressing the finer detail of students’ learning and understanding.Each characterization was related to the teacher’s expressed beliefs andintentions.

The second phase of analysis related to the teaching triad and wasdone by the researchers at micro and macro levels. In micro-analysisthe transcribed text from each lesson was read and divided into sections


according to the pattern of activity in the lesson. Each section was thensplit into smaller parts according to changes in teachers’ interactions. Forexample, in a section where the teacher was working with one particulargroup of students, a smaller part might involve her interaction with justone of the students in the group. The discourse of this subsection wasanalyzed line by line looking in fine detail at the language and meaningsof teacher and students in relation to the elements of the triad. At the endof each subsection inferences were made about relationships between theelements. During this process the researchers tested their inferences anddeveloped their conceptualization of the TT, both with reference to theempirical data and also in their own discussions of details of the analysis.Certain sections were analyzed separately by each researcher leading to acomparison of outcomes and a validation of inferences and their relationto the TT.

Following micro-analysis, a macro-analysis was done to attempt toexplain why a certain teaching behaviour occurred. This behaviour waslinked to relevant parts of the wider set of data including conversationsand team meetings, and also to lesson summaries from Phase 1 analyses.The macro-analyses allowed linkages to a wider sociological frame such asconstraints of curriculum and assessment, time pressure, parental expecta-tions, familiar patterns of classroom interaction, the particular aptitudes,experiences and expectations of students in any class and elements ofteachers’ views, beliefs and intentions.

Micro- and macro-analyses are complementary, and the analyticprocess involved a cycling between the two. For example, when widerissues emerged in the macro-analysis, a cycling back to further micro-analysis was done to see whether finer attention to detail might revealparticularities or consequences of the wider issues; when details of theinterrelationship of sensitivity to students and mathematical challengeemerged from micro-analysis, attention to the wider issues often served toroot such interrelationships in the realities of social concern. Moreover, theglobal analysis of the first phase contributed to linkages in the data. Thisprocess was successful in gaining deeper insights into the complexity ofthe teaching and explaining teaching tensions and issues as the followingsections will show.

AN ANALYSIS OF JEANETTE’S TEACHING

In this section we offer detailed examples of our analytical process toshow how the triad enabled key teaching issues to be revealed takingus beyond superficial judgments on teaching. First we provide a global


characterization of Jeanette’s teaching from our first phase of analysis.This characterization reveals general aspects of the teacher’s behaviour,deriving from observations in her classroom, and uses the teacher’s ownwords to connect behaviour with beliefs and intentions for teaching. Wethen go into the second phase of analysis to consider two separate cases.Case 1 refers to a situation where the teacher’s goals, to be sensitive tostudents needs and to challenge them mathematically, seem to be achieved:i.e., the teaching seems effective in terms of students’ mathematical devel-opment. The analytical process helps us to look critically at the meaningof the term ‘effective’ and to link it to the concept of harmony of thethree elements of the TT. In Case 2, in contrast, the teacher’s actions seemeither to neglect students’ thinking or not to challenge it further, so thatthe outcome seems ineffective in terms of students’ mathematical develop-ment. In this case it is tempting to say that there is a gap between what theteacher aims to achieve and what happens in her actual teaching. However,micro- and macro-analyses reveal a complexity of factors contributing tothe apparent discrepancy. As a result, we are able to appreciate betterthe issues that the teacher faces in making decisions about approachesto teaching. The first case was reported in Jaworski and Potari (1998)and therefore will be dealt with here more briefly than the second case.Following a discussion of the two cases, we look more closely at theteacher’s own thinking and development with reference to her use of thetriad.

Phase 1 Analysis: A Global Characterization of Jeanette’s Teaching

Jeanette’s teaching is characterized by an emphasis on individual andgroup work while whole class discussion takes place mainly for sharingideas from the group work, for reviewing a test or homework, for intro-ducing and clarifying a task and, relatively rarely, for introducing anew concept. She mostly coordinates whole class discussion, rather thanleading it. The group work is mainly on a task of investigative and practicalnature while the individual work is based on the textbook or on a structuredworksheet which describes a new topic and provides a number of prob-lems for the students to consider related to this specific topic. Jeanette’srole is mainly that of a facilitator of learning (c.f., Scott-Nelson, 2001),and includes a form of scaffolding (Bruner, 1985) which she describes as‘pushing’ or ‘pulling’ her students.

To push them I will ask an open ended question. What happens if we do this or have youthought about that, or why have you done this here? To pull them I will point to something,sometimes not even say anything, come out with a question and I will know the child wellenough . . . I will actually sit with them and, depending on their ability perhaps, will read


a whole question together, or will go through. I will say right lets draw this diagram, youdraw it, what do we know, what information have we got? And it’s much more guided:what do we need to find here, have we all the information we need, how are we going toset about doing this? Just the usual way through a problem. I will leave it there then, I willstart them on a next question and sometimes maybe sit there for five questions, graduallydrawing back, so that I am doing less and less of the question with them. (Data extract:TM2: team meeting 2)

In her interactions with the students she typically encourages reflec-tion and negotiation by asking them to explain and justify what they havedone, by praising their attempts and encouraging continuation and exten-sion of their work. She provides help by encouraging peer cooperation,by building links between current and previous work, by simplifying thechallenge, and by providing emotional support parallel to the cognitive. Insome cases her attempts to help a student result in closed questions leadingthe student towards an answer, or she provides an answer. However, sheconceives of teaching as process oriented rather than as a product orientedactivity and aims for the emergence of students’ ideas and strategies, andfor building their autonomy, self confidence and understanding as indi-cated in the following data extracts (Data extract: 1DJ9: discussion afterJeanette’s first observed lesson in Year 9):

In the whole class discussion, I would start with the right one [response from the groups],or the better one, and get the boys to discuss, to explain how they found that. Because thosewho had not got it right, it’s better for them to realize by themselves, instead of saying OKlet’s take that one. What is wrong with it?

I do always emphasize particularly with them that just because I am showing onemethod of doing things . . . there are other ways of doing things; certainly other waysof seeing how things come together . . . I think that they are quite happy with the idea thatthey can do things in different ways.

Phase 2 Analysis: The Two Cases

We focus, here, on a series of three lessons (of 70, 35, 70 minutes respec-tively) focusing on mathematical concepts of volume and surface area. Oneresearcher observed the first and last lessons and discussed the middleone with Jeanette. Case 1 is taken from the first lesson and Case 2 fromthe third. We have chosen the lessons to be representative of those withthe particular class and of the analytical process as a whole. The issuesemerging should be seen as indicative of the power of the analyticalprocess to reveal issues rather than representative of the issues it reveals.It is impossible to address all possible issues even for one lesson, as theemerging complexity shows.


Effective teaching: Interacting with a group of students (Case 1)The teacher wanted students to understand the nature of the least surfacearea for a cuboid of given volume. We have described earlier her openingactivity focusing on the construction of a box to fit 48 cubes each of side2 cm. Groups of students worked on this task. We focus first on an inter-action between Jeanette and two boys, Tom and Stewart. The boys hadtwo different organizations of the 48 cubes: Tom with 48 cubes in a line;Stewart with a 2×4×6 (4×8×12 cm) cuboid. They had each drawn nets oftheir solids to enable them to calculate surface area (respectively 776 cm2

and 352 cm2) for a volume of 384 cm3. The teacher looked, with the boys,at these two cuboids and through her questioning and their responses wegain evidence of Stewart’s appreciation of the key mathematical concept(T: teacher, S: Stewart).

Transcription 1. Dialogue from Volume and Area 1 (Data extract: 4J9).

1. T: Now I want you to think why Stewart’s is less2. S: Cause mine’s higher and wider and3. Tom: It’s easier to fit in the trolley. [He refers to a supermarket trolley]4. T: It is easier to fit in the trolley, yes.5. S: Because mine’s got more height and width than Tom’s.6. T: Right, so it will be (many voices) consequently its been made7. S: Shorter8. T: Shorter, (many voices in the background)9. S: More compact

10. T: Right. Compact. Good. Right now, is Stewart’s model the most compact model youcan come up with, or is there anything better? Well. I don’t know. Let’s look at Jamesand John’s to see if they’ve done better.

11. [Some interruption here from other students to whom the teacher responds]12. T: Stewart, you have done really well so far, OK?13. S: Yes.14. T: But you need to make sure you are listening in to the others’ designs as well.

A line by line analysis of this dialogue is given in Jaworski and Potari(1998). We therefore omit it here. We exemplify our line by line processof analysis in Case 2.

In this dialogue the students seem actively involved in a process ofconstruction of mathematical concepts; also in negotiation in which expla-nation, justification, and elaboration are essential features (c.f., Wood etal., 1993). The teacher’s questions lead to Stewart’s articulation of theword ‘compact’, his own word, that reflects an image that seems to fitwell with the concept of minimum area. It is an example of what Jeanettedescribed as ‘pushing’ her students and is characterized in line 1 by theteacher offering a mathematical challenge that fits with the current stageof students’ thinking (their construction of the two differing models). Theteacher’s actions might be seen to involve sensitivity to students in both


cognitive and affective domains: what Williams and Baxter (1996) havecalled ‘analytic scaffolding’ – scaffolding directed at cognitive develop-ment (e.g., lines 1, 6 & 10) and social scaffolding, directed at affectiveneeds, seen in her later encouragement of Stewart (lines 12–14).

This dialogue can be seen as a special case of a pattern the teacherdescribed as follows:

In an investigative lesson, I provide the stimulus for the initial problem and then givethem some time to explore. And so I now see a bit on a hillside, a bit rocky, fairly open,maybe a bit bleak. Some of them will stay very close to me, not physically, but close tome metaphorically, not close to the problem or their friends. Others will start perhaps to goround the problem, trying things, maybe coming back, making sure they are doing alrightand then go off again. One or two will skedaddle down the path and find something veryinteresting or get nowhere and come back again. And so my role will be making, I willmake sure that the ones who want to stay with me are walking with me round, mayberound or round about, encouraging them to go off. Sometimes I feel I push them and theywon’t go. And perhaps I will leave them and come back and they haven’t gone anywhereand then I will have to take them on with me a little bit further and then try and push themoff again and they will go. . . . When they’ve gone off on their own track and, and theydon’t think they’ve got anywhere, and they ask for help to come back that’s fine, it’s whenthey’ve gone off on their own track and I’m not sure whether they are getting anywhere ornot. . . . When do you give them a rope and pull them back? (Data extract: TM2)

For some of the students, the mathematical challenge can be seenmainly in the initial task as they take responsibility for guiding theirown learning with very little input from the teacher. For other studentsthe mathematical challenge is more tailored to their needs by the teacheroffering hints or asking questions to start them articulating the situation,or by extending the task to new possibilities for exploration. The manage-ment of learning works in two planes in this type of teaching: in the firstplane it is the actual interaction with the individual student; in a secondplane it is the coordination of different actions and decisions enabling theteacher to meet the different needs of all the students in the class.3 Wesee in this session a situation where the teacher values and at the sametime challenges students’ mathematical thinking by supporting them bothemotionally (praising, encouraging) and cognitively (questioning, seekingclearer conceptual articulation). We would claim, in this case, that the threeelements of the teaching triad are in harmony. The teacher through hermanagement of learning creates an environment where the sensitivity tostudents works in both affective and cognitive domains to make the mathe-matical challenge appropriate to students’ needs and thinking. In Case 1we see agreement between theory, the teacher’s perceptions about teachingand learning, and practice, the implementation of these beliefs in practice.


Figure 2. Summarizing students’ findings on the board.

Complexity in teaching: Reviewing the homework, forminggeneralizations (Case 2)This episode took place in the third lesson on the packing problem. Thestudents had completed the group work in this task and had reported someof their findings to the whole class. The teacher had asked them, for home-work, to explain the task and how their group set about it and to put downsome results about the characteristics of the boxes they constructed.

As part of this lesson the teacher led a plenary session to write ontothe board some of the results from the students’ work. She drew a table,as shown in Figure 2, and invited students to give her a set of dimensions.Two sets were given by students and written into the table by the teacher.In each case Jeanette asked how the box would look and the value of itsvolume. The third entry was added during subsequent discussion.

The following dialogue is from the whole class discussion betweenJeanette and the students in their attempts to calculate the surface area ofthe second box entered in the table (T: teacher, P: Peter, S: another student).

Transcription 2. Dialogue from Volume and Area 3 (Data extract: 6J9).

1. T: Right. Now, how am I going to find out the surface area?2. S: This is where I got stuck.3. T: Right. This is where people got stuck. Peter (the only one who puts his hand up)

you obviously got an answer. Could you tell us how you started that problem? Whatdid you do to work out the surface area?

4. P: I drew a net.5. T: Right. You actually drew a net. Can you describe for me roughly how it looks like?6. P: Is one . . . (he hesitates)7. T: (draws on the board a strip of 2 × 96) How many are like that? (Figure 3a)8. P: 49. T: Right. 4 like that. (She draws again. Figure 3b)


Figure 3. Building the net for the 48×1×1 cuboid.

10. P: Then one on this side and another one on the other11. T: (draws the two faces of 2 × 2) (Figure 3c) You need to write some dimensions. This

time you need to write some dimensions in cms. You need to work out the surface areain cms. Peter what is this length here? (length l in Figure 3c)

12. P: 9613. T: 96, good. What about all the way down there (she shows the total width of the four

faces (8 cms, d in Fig. 3c)14. P: . . . (cannot respond)15. T: No, ok. not to worry. What about this bit there (the width, w in Fig.3c)16. P: 217. T: 2 cms, up here18. P: that’s 219. T: that’s 2 as well. Down here is also 2 cms. What about these little end bits here (the

faces 2 x 2)?20. P: 2 by 221. T: The total in this case is (on the board she wrote: total = 192 + 192 + 192 + 192 + 4

+ 4 = ).22. T: Peter, what is the answer? Has everybody found the answer for this one?

According to the teacher’s metaphors, this situation might be describedas “standing on the platform, offering them a hand and pulling them up”.Here the teacher wants her students to have specific outcomes (solutionsto the questions about finding surface area) and uses questioning to try toachieve these outcomes.

We offer the following outline of our line by line micro-analysis, usingthe triad. SS/A and SS/C indicate sensitivity to students in the affectiveand cognitive domains respectively. Comments in brackets, with query,indicate interpretations regarding teacher intentions that are checked inmacro-analysis.

• 1–3. The teacher indicates a MC which the students had been askedto consider in their homework. A student admits that he had difficultyand the teacher responds at an affective level by stating that this wasdifficult for most students (SS/A). She asks Peter to give an answeras he is the only one offering to contribute (ML – time pressure,resulting in not asking other students?). She asks Peter to describe


his process of working (possibly to explore his thinking, SS/C/A, andto encourage the sharing of his ideas with the other students SS/A/C,through this ML).

• 4–10. Here, Peter says that he drew a net and the teacher asks himto describe how he did it (SS/A/C). She does not encourage Peterto show his working, for example by asking him to draw his net onthe board. (ML – time pressure?) Instead, she manages the situationherself, using Peter’s contributions to demonstrate the process to theother students. MC seems to be lacking here. It seems that in thispart of the dialogue there is mostly management of learning directedtowards the teacher’s mathematical objectives.

• 11–20. The teacher poses a new challenge of finding the dimensionsof the net (MC) and gives reasons why this is important. She reducesthe challenge by offering small steps in the form of short closedquestions. In line 14, Peter cannot respond to one of these questions,so (at 15) the teacher offers Peter emotional support and splits thequestion further so that in the end Peter gives a successful answer.The interaction here is similar to the previous one (4–10), lackingchallenge, but showing sensitivity to Peter at an affective level. Here,A appears to dominate over C which may be due largely to the timefactor.

Two issues seem to emerge from the above analysis: (i) the nature ofthe mathematical challenge and how it changes in relation to the othertwo elements of the triad, (ii) the dual nature of sensitivity to studentsand its relation to mathematical challenge. The mathematical challengestarts with a problem which seems to require both conceptual and factualconsiderations of finding the surface area of 3-D solids. From the way thatthe teacher manages the discussion, it seems that conceptual aspects aremainly implicit while the factual ones are revealed in small distinct steps.As a result, mathematical challenge seems to disappear, management oflearning dominates and does not provide access to students’ thinking,largely because sensitivity to students operates mainly in the affectivedomain. This is characterized by the teacher’s appreciation of students’difficulties and her reinforcement of classroom norms of respect and inclu-sion shown in her support and encouragement for Peter. However, theepisode seems to lack balance between the cognitive and affective dimen-sions of sensitivity to students. The missing cognitive dimension reflects acorresponding lack of challenge.

Micro-analysis reveals a three stage ‘elicitation pattern’ similar to thatdescribed by Voigt (1985, p. 80) which Bauersfeld (1988, p. 36) has calleda ‘funnel pattern’. The teacher posed an open task to offer different possi-


bilities for the students to explore; she then guided them towards a solution,and in the end produced a solution herself. Micro-analysis is (necessarily)limited to the dialogue in the episode to help us explain and interpretthe interactions. However, at a macro-level, other issues/concerns becomeevident. The teacher had found from the students’ homework that they haddifficulties with some parts of the task, so she wanted to make sure thatall the students could complete the task and develop some computationalskills on the topic of the area and volume of 3-D solids. At the same timeshe did not want to spend a lot of time in this phase of the lesson, as shewanted the students to move to her next activity – a text book exercisefor practice and consolidation of techniques in finding areas and volumes.This led her to reduce the challenge and guide the students towards hercontent oriented objectives. Time pressure, to achieve curriculum goals inthe available time, was evident. Her shift to the focus on skills neededto answer questions in the textbook resulted from this time pressure. Wecould only conjecture, in micro-analysis, that time was a factor, but ourconversations with the teacher, both in school after the lesson and in a laterteam meeting, confirmed this tension.

Before moving the class to working on the exercise, the teacher finallywound up the investigative activity by asking about the third column in thetable. Tom and Stewart contributed their understandings of compactness,giving a good explanation of how surface area can be minimized. Herethere seemed to be a possibility to re-enter a conceptual level of interac-tion, enabling the whole class to understand the meaning of compactnessthrough the presentation of Tom and Stewart. However, following theirexplanation, the teacher ended the whole class session. Why did she choosenot to build on these potentially fruitful contributions to work towards anabstracted conclusion?

To address this question, we need to look again to a macro-analysis. Indiscussion with Jeanette about her decision not to extend the challenge wesee that she perceived some students still to have difficulties and wanted tobe sensitive to the wider needs:

I felt that there were only a few that had actually got that idea and I think if I’d carriedit forward and said well what would be a more perfect number to use, that would havelost a lot of them. That’s why I left that; but I think it’s something to come back to whenwe come to look at something like this again; looking for a more perfect solution. (Dataextract: D5J/9)

She wanted the cubes activity to help students to address these conceptsin a more abstract context – the exercise in the textbook – which waspart of her planning to enable students to achieve curriculum objectives.


Moreover, it was Friday afternoon and many students were excitable andinattentive.

The funneling pattern discussed above is not one with which Jeanettefeels comfortable. Analysis shows two major factors motivating herapproach: her serious concern for students’ self-esteem, resulting often insensitivity that is effective in the affective but not always in the cognitivedomain, and a need to address her curriculum objectives under time andsocial pressures. As Head of Year in her school, students’ wellbeingand the school’s responsibilities to students with respect to parental andsocietal expectations are central to her views on education (we quote fromher own words later). The curriculum and time pressures are not just incon-venient impositions due to political forces, but also play a major role ininterpreting responsibility towards students’ wider educational fulfillmentand success. These factors acting together can result in a reduction inchallenge, and in leaving students with ill-formed concepts, and a depend-ency on instrumental skills. Jeanette’s recognition of a need for greaterchallenge is elaborated here in the particular detail of interactions: practicalwisdom adds to theoretical ideas and creates opportunity for the future torecognize patterns and develop alternatives, either in pre-planning or in theclassroom, as the next section shows.

The Teaching Triad as a Tool for Reflection and Development ofJeanette’s Thinking about Teaching

Jeanette used the teaching triad for planning her lessons and for reflectingon and analyzing her teaching actions. For the lessons above, Jeanette drewthe diagram in Figure 4 and tried to fit her planning to this diagram.

She conceived a dynamic interrelationship of the elements. First, shestarted to consider the kind of mathematical challenge she could offerto the students. Then she moved to management of learning where sheconsidered the resources she needed to offer, the classroom organiza-tion and the encouragement of communication in the class. Sensitivity tostudents was seen in providing students with the chance to explore, discuss,and argue but also to encourage the building of confidence in their ownideas. She also saw sensitivity to students as the basis for making mathe-matical challenge meaningful to students. The following extract indicateshow the teacher conceived the relationship between her planning and theTT.

When I’d written my lesson plan, I went on to reflect on the triad and it was the mathe-matical challenge that came out first. It comes out in my mind as well as to the top ofthe triangle. But with them [this particular group of students] . . . I don’t know, maybe it’sbecause it’s a practical lesson it’s sort of management of learning which is the next aspect


Figure 4. Jeanette’s fitting of her planning to the TT.

of the triad that I actually started to focus on. And they were more as prompts for me tohave the resources on hand, to work as a group and then to go around and see how well theyshare their ideas. And then, that was interesting, it was then that I considered sensitivityand so . . . I obviously, I wanted them to have a chance to explore and discuss and arguetheir ideas. But also, within that, I would encourage them to build their confidence to theirown ideas, so what I am looking for then is how . . . if they’ve got a chance to explore anddiscuss their ideas and to explore some things and put their own point of view forwardshow will that then extend the mathematical challenge. That’s what I am looking for in Year9. (Data extract: D4J9)

Later, at a meta-level, she reflected, “Now I am focusing much moreon it [the triad] and I am just starting, I am just interpreting my actions.”This shows Jeanette’s own overt recognition of her analytical activity inplanning and reflecting on her lessons. She is surprised that in her planningthe mathematical challenge has a priority even in this group where shethought initially that she did not offer enough challenge in her teaching.She has also started to see sensitivity and challenge (‘bits’ of the triad, asshe calls them) very much related to affective and cognitive interactions –notions we introduced to her – and this helps her to go more deeply intoanalysis of her teaching actions.

With Year 9, I can’t explain why but I feel I’m more in tune with their affective andcognitive and I wonder if that’s part of the movement between the challenge and thesensitivity bits. (Data extract: D4J9)


Jeanette feels that she should work next on management of learning.She has started questioning how this element is related to the manage-ment of the resources and to the flow of the teaching in moving from onetopic to the next, in order to make sure that she has organized an appro-priate environment in which students can work. In one discussion afterher teaching, she recognized differences between planning and teachingwhich shows her recognition of interactions between the three elements ofthe triad.

When I plan something, I think I know how I am going to challenge them mathematicallywith what I’m presenting. And I can make sure by managing resources and time that it’sgoing to work, that they’re at the right stage that they can move on. And I know which onesare going to need extra help. Now, then when you’re actually teaching and you’re actuallyin the lesson and you’re thinking on your feet the whole time, things will change, youhave to be prepared to change. Therefore then, within the lesson, maybe my perceptionof the mathematical challenge perhaps changes slightly which has a knock-on-effect tomanagement of learning and sensitivity towards students . . . you’ve planned it carefullybut now you’ve got to change so thinking quickly on your feet, the sensitivity bit – willthey be able to carry on with this for homework? How will they feel if they get stuck?Therefore, how much [homework] do I actually set them? Or should I just say OK, nohomework tonight? (Data extract: 4DJ9)

Through this description from Jeanette, we can see the dynamic natureof her use of the triad in the way that the three elements interact in thecomplexity of planning and interpreting her teaching. We also see how shehas started to elaborate the concept of management of learning and howthis is related to the process of teacher’s decision-making in classroomaction.

Towards the end of the project in the discussion at a group meeting,Jeanette showed a recognition of factors, other than those coming fromthe classroom culture, which influence her teaching. Thus, she developedfurther her conceptions of management of learning. In the followingextracts, Jeanette sees a difficulty in operating according to her teachingtheories as a number of different factors act in her decision makingprocess. Here, the management of learning is influenced by external pres-sures: the system of Examination (GCSE: General Certificate of SecondaryEducation4), the government’s educational policy, parents’ and students’attitudes, the school’s policy and its role as a technology college, and theteacher’s different roles in the school.

The caring side, the side of me that is sensitive to students, makes me think I still haveto build up their confidence. They find it difficult, so they stop. I have to work on theirconfidence, but the management of learning side – for those who are in five months timesitting their GCSEs, and there is so much riding on technology, on improving our standards.I’ve got a C group which is the grade Cs who are being targeted to improve and . . . all thatpressure piles in. And whilst one side of me says yes, keep building up their confidence,


give them easy questions, the other side is – I’m getting so frustrated because I know thatit is just years and years of apathy built up. And I’m positive, and the head of the year sideof me who meets a lot of parents and sees parents with these kids, I am convinced . . . thatparents are not allowing their kids to take risks they do far too much for them.

By the time they get to year 11 . . . they must be very confused because they have a dietof a lot of teachers who keep them very much pinned down, do this, do that, and then athome they are not thinking for themselves either because their parents are doing everythingfor them, or they will give in to them very easily. And there is a number of parents whocome to me and say ‘what can I do, he doesn’t want to come to school?’ What can I doto try to teach mathematics in the middle of all that? I’m actually finding it quite stressful,much more so than in any other school [I have worked] and I think it’s because there areso many, so many conflicts going on there. (Data extract: TM3)

Here we see Jeanette looking critically at her practice, questioning,acknowledging issues, recognizing tensions. It is clear that decisions, forher, go considerably beyond immediate issues of sensitivity and challenge,cognition or affect in particular mathematics lessons. However, it is useand consideration of the triad that have revealed the necessity to see thefiner details of mathematical learning within the complexity of these widerissues. We have not been able to discuss here the finer details of develop-ment of Sam’s teaching due to space considerations; however, these detailswere also considerable, and particularities which take us beyond the issuesaddressed here will be discussed in another paper. It would have beenvaluable to explore further the effects of Jeanette’s and Sam’s enhancedawareness on their subsequent classroom practice but the period of thisresearch limited what was possible.5

PERCEPTIONS OF THE TEACHING TRIAD EMERGINGFROM THIS RESEARCH

In striving to teach effectively (as teachers) or to talk sensibly abouteffective teaching (as teachers and educators) there are various principles,theories and beliefs to which we aspire. These are laid out at lengthin documents like the NCTM standards and the Cockcroft Report. Ulti-mately, however, these principles have to be translated into action for theclassroom. With whatever sincerity a teacher tries to follow such prin-ciples, there is a complex set of psychological and sociological factors thathave to be considered in constructing, managing and achieving classroomteaching and learning. The triad as an analytical tool helps to recog-nize, interpret, and understand processes of translation from principles tooutcomes. The triad as a developmental tool helps to make such translationmore effective.


The Triad as an Analytical Tool

Analyses here have shown that while mathematical challenge can be seento operate largely in the cognitive domain, sensitivity to students seemsto work in both the cognitive (C) and the affective (A) domains for themathematical learner. Cognitive sensitivity concerns the appreciation andrecognition of students’ thinking which can then be developed further byappropriate challenge. Affective sensitivity includes fostering students’personal beliefs in and valuing of their ability to do mathematics andthink mathematically; it concerns students’ wellbeing and positive attitudewithin the classroom setting, and does not always connect obviously witha cognitive dimension. It is possible to foster a good classroom atmospherein which students are happy, confident and well-motivated but do notachieve the quality of mathematical thinking of which they are capable.Jeanette suspected that this was the case for her Year 9 class: she hadworked hard on students’ self esteem, but felt that she had neglectedmathematical challenge. In trying to gain some access to what might beconsidered ‘effective’ teaching, we recognized that a balance betweensensitivity in the cognitive domain and mathematical challenge seemedcrucial. When such a balance seemed to be achieved we referred to it as aharmony between challenge and sensitivity. We asked about the extent towhich harmony depended also on affective sensitivity.

For example, we have seen that Jeanette’s teaching of her Year 9class included three main considerations: her concern for students’ selfesteem, an aim to offer a higher degree of mathematical challenge, and aneed to deal with factors of time, curriculum, examination requirements,parental expectations and so on. Self esteem and challenge are associatedwith principles of teaching: for example, principles that self esteem isnecessary for cognitive development; and that cognitive development isunlikely to happen without suitable challenge. Factors of time, curriculum,examinations, and parental issues are all a part of situational complexity.Analysis linked elements of the triad to elements of teaching principles andsituational complexity. Here, Jeanette’s concern for self esteem translatedinto classroom actions that we described as sensitivity to students of anaffective nature. In line by line analysis, we identified particular wordsand phrases as showing affective sensitivity – praising students, guidingtheir actions, listening to their explanations. Challenge was seen in tasksset by Jeanette and questions she offered. However, challenge needs to bejudged carefully to fit with students’ current levels of cognition. Also, evenif there is some degree of cognitive fit, challenge may not be taken up ifthe affective dimension is not also appropriate. Micro-analysis examinesrelationships between these dimensions mainly at a psychological level,


identifies actions and elements of dialogue that contribute to harmony ordisharmony, and leads to interpretations of motivating factors in a teacher’sdecision-making.

The struggle of a teacher to enable students to understand or to gainaccess to a concept was seen to involve pedagogic processes of scaffolding(Bruner, 1985) or funneling (Bauersfeld, 1988, p. 36). Such processesare linked to the teacher’s management of the learning situation (ML).We analyzed these processes further in terms of the TT. The followingsequence of stages in an interaction can be seen to develop into a funnelingprocess.

1) The teacher offers a challenge to which the student cannot respond.2) The teacher scaffolds in some way, reducing the challenge in order

to engage the student in the ideas. This indicates sensitivity in thecognitive domain.

3) Further scaffolding is offered, increasing sensitivity in either thecognitive or the affective domains, reducing challenge further toenable the student to engage.

4) (3) is repeated until either the student engages or the teacher tells thestudent what is required.

If at any stage in the above process, the student takes up the challenge,the process stops and the student engages with the challenge. Thus at stages(2) or (3) the scaffolding might be seen to have been successful, withsensitivity and challenge meeting harmoniously for the student to engageactively with the mathematics. However, if the student is not able to takeup the challenge at any of the earlier stages, the funneling process resultsin the teacher giving the answer, or in the challenge becoming negligiblefor the student or both teacher and student.

As educators we use words like scaffolding and funneling to refer topedagogic concepts in general terms. Scaffolding is often regarded posi-tively as a helpful process (e.g. Bruner, 1985; Williams & Baxter, 1996).However, it might lead, also, to encouragement of a student’s dependencyon the teacher – a ‘crutch’ (Jaworski, 1990). Funneling, on the other handis offered to describe an ineffective teaching approach (Bauersfeld, 1988).When we take the trouble to analyse closely particular interactions inwhich we perceive scaffolding or funneling, we recognize that the termsthemselves are insufficient to carry the complexity for the teacher of theinteractions with the student. Voigt (1994) writes critically of teacherscontinuing traditional patterns of interaction in the classroom perhapscontradicting the teachers’ beliefs or intentions, “teachers never realisethat traditional patterns of interaction are still alive in their classroomsnor that they contradict the teacher’s intentions” (p. 288). We believe that


our analyses show that this position may be more complex than it seemsat first sight. In the case of Jeanette, that was certainly revealed as westudied interactions at micro and macro levels. The triadic analysis goessome way to making evident this complexity and enabling us to address it,particularly when harmony seems lacking.

In the lack of harmony between challenge and sensitivity, we can seewhere alternative words or actions by the teacher might have allowedthe students to engage more effectively with the mathematical ideas. Forexample, in Case 2, we might ask why the teacher did not elicit morecarefully the thinking of students in building up the table of results. Herfunneling activity seemed contrary to her expressed aims for work withstudents, and different from her approach in small group work. Macro-analysis provided explanations for the perceived disharmony. Tensionsfor the teacher in navigating psychological and sociological factors led toineffective outcomes in students’ achievement.

It was in such considerations that we started to recognize the needto link a triadic analysis to the wider sociological concerns impingingon the classroom and the teachers. For example, we see in Case 2 thatJeanette’s decisions motivating interaction were related to wider socio-logical concerns as much as to cognitive factors. In a pragmatic immediacy,where decisions cannot wait on extensive reflection, the teacher recognizedthat further work needed to be done where students’ mathematical concep-tualization was concerned. There was not enough time, mood or cognitivereadiness to deal with her planned mathematical objectives at that time. Soshe decided to return to these at a later date.

The Triad as a Tool for Teaching Development

The teachers’ reasons for participating in the research was that it wouldgive them the opportunity to explore how the triad might be of use inenhancing their thinking, planning and classroom activity. We saw thatJeanette used the triad in planning her lessons and thinking about outcomesof lessons. Unsurprisingly, the language of the triad permeated our discus-sion about teaching and making sense of that teaching, particularly indiscussions between teacher and researcher about the lesson that was to betaught, or had just been observed. Such discussions took place in the imme-diacy of classroom decisions such as those discussed above, and before anydetailed analysis of the data was possible. In the timescale of this research,it was not possible to research the impact of analyses on the thinking ofthe teachers in the longer term. Much of the more detailed analysis tookplace after data collection ended. Another phase of research would havebeen necessary to test out the impact of analyses on teachers’ thinking


and future teaching. This was a limitation of the research describedhere.

However, during the timescale of this research, the triad served as astimulus for talking in depth about issues in teaching. The lengthy teammeetings provided evidence of the teachers’ thoughtful attention to issuesthat had been raised as a result of classroom activity and subsequent discus-sion, and provided considerable evidence of the wider social concerns thatmotivated or constrained teaching. Such raising of issues fed back intoteachers’ classroom activity.

Thus, we see that there was impact from use of the triad for the teachers’development of teaching during the timescale of the project. However,change in teaching is a lengthy process as we see, extensively, from theliterature in mathematics education and beyond (e.g. Fennema & ScottNelson, 1997; Fullan & Hargeaves, 1997). We need to ask what poten-tial there is for teaching development that builds on what this project hasachieved.

CONCLUDING THOUGHTS

The triad was a tool which provided access to complexity and helped us toperceive dualistic aspects of teaching (e.g. funneling/scaffolding, concep-tual/procedural) in a more integrated way in terms of both pupils’ cognitiveand affective development (SS) but also in terms of teachers’ obstacles anddecisions (ML). Its use led us to consider wider social concerns alongsidecognitively focused factors (such as harmony), and to realize the impossi-bility of separation of the cognitive and the social. It acted as a tool forour own learning and development by bringing these factors and relation-ships to a clear focus in both theoretical and practical domains, enabling adynamic linking of theory and practice.

These elements provide the basis of a theory of teaching by makingpossible explicit links between teaching principles, classroom interactionand teachers’ thinking about cognitive and social factors. We can see possi-bilities for a methodological applicability that goes beyond substantiveoutcomes. By this we mean that the practices we developed in analyzingteaching are available for wider use, with some confidence that theoutcomes will be fruitful in raising issues central to practice, enablingteachers’ serious engagement with these issues, and ultimately creatingpossibilities for modifying practice. For example, we ourselves are usingthe triad for analyzing teaching in other places and at other levels, suchas undergraduate level.6 The triad might also be used to address apparentinconsistencies that have been reported in the literature between teachers’


beliefs and their observed teaching (Raymond, 1997; Cooney, 1985; Voigt1994; Skott, 2001). Also, we see the triad entering the discourse in ourteacher education programmes and becoming a tool with which beginningteachers can start to analyse their own practices. Both teachers in ourproject are mentors in one of these programmes, and use the languageof the triad with their student teachers. In all of these ways we see thetriad contributing to the developing thinking and practices of teachers andteacher educators.

Finally, it needs to be said that the triad is no more than a tool. Thecritical dimension of mutual exploration developed between teachers andeducators through the use of the triad is in our view one key to successfulteaching development.

ACKNOWLEDGEMENTS

We should like to thank Tom Cooney and three reviewers for theirextremely helpful comments on a previous version of this paper.

NOTES

1 Some of the following discussion appeared in Jaworski and Potari (1998).2 ‘Comprehensive’ implies a full range of achievement of students within the school asmeasured according to standardised achievement levels and testing.3 These two planes have important connections with Vygotsky’s two planes: those of theinter- and intra-psychological in learning and development (Vygotsky, 1978, p. 57). Suchanalysis waits for another paper.4 The GCSE is the national examination across all subjects at 16+.5 This research was conducted during the sabbatical leave of the first author.6 Jaworski, B. (in review), Sensitivity and Challenge in University Mathematics Teaching.

REFERENCES

Ball, D.L. (1996). Teacher learning and the mathematics reforms: What we think we knowand what we need to learn. Phi Delta Kappan, 77(7), 500–508.

Bauersfeld, H.(1988). Interaction, construction, and knowledge: Alternative perspectivesfor mathematics education. In D.A. Grouws & T.J. Cooney (Eds.), Perspectives onresearch on effective mathematics teaching. Hillsdale, NJ: Lawrence Erlbaum.

Brown, M. (1979). Cognitive development and the learning of mathematics. In A. Floyd(Ed.), Cognitive development in the school years (pp. 351–373). London: Croom Helm.

Britt, M.S., Irwin, K.C., Ellis, J. & Ritchie, G. (1993). Teachers raising achievementin mathematics: Final report to the Ministry of Education. Auckland, NZ: Centre forMathematics Education, Auckland College of Education.


Bruner, J. (1985). Vygotsky: a historical and conceptual perspective. In J. Wertsch (Ed.),Culture, communication and cognition. Cambridge: Cambridge University Press.

Cobb, P., Wood, T. & Yackel, E. (1990). Classrooms as learning environments for teachersand researchers. In R.B. Davis, C.A. Maher & N. Noddings (Eds.), Journal for researchin mathematics education: Constructivist views on the teaching and learning of mathe-matics, Monograph no. 4 (pp. 125–146). Reston, VA: National Council of Teachers ofMathematics.

Cooney, T.J. (1985). A beginning teacher’s view of problem solving. Journal for Researchin Mathematics Education, 16(5), 324–336.

Cooney, T.J. (1988). Teachers’ decision making. In D. Pimm (Ed.), Mathematics teachersand children. London: Hodder & Stoughton.

Cooney, T.J. (1994). Research and teacher education: In search of common ground. Journalfor Research in Mathematics Education, 25(6), 608–636.

Cooney, T., Shealey, B.E. & Arvold, B. (1998). Conceptualizing belief structures ofpre-service secondary mathematics teachers. Journal for Research in MathematicsEducation, 29(3), 306–333.

Department of Education and Science (1982). Mathematics counts: Report of theCommittee of Enquiry into the Teaching of Mathematics in Schools (The CockcroftReport). London: HMSO.

Edwards, D. & Mercer, N. (1987). Common inowledge. London: MethuenFennema, E., Carpenter, T., Franke, M., Levi, L., Jacobs, V. & Empson, S. (1996). A

longitudinal study of learning to use children’s thinking in mathematics instruction.Journal for Research in Learning Mathematics, 27, 403–434.

Fennema, E. & Scott-Nelson, B. (1997). Mathematics teachers in transition. Mahwah, NJ:Lawrence Erlbaum.

Fullan, M. & Hargreaves, A. (1997). Tis the season. Learning, 26(1), 27–29.Goldin, G.A. (In press). Affect, meta-affect, and mathematical belief structures. In G.

Leder, E. Pehkonen & G. Toerner (Eds.), Beliefs: A hidden variable in mathematicseducation. Dordrecht: Kluwer Academic Publishers.

Jaworski, B. (1990). “Scaffolding” – a crutch or a support for pupils’ sense-making inlearning mathematics? In Proceedings of the 14th Conference for the Psychology ofMathematics Education (PME), Mexico, Volume III, pp. 91–98.


Jaworski, B. (1998). Mathematics teacher research: Process, practice and the developmentof teaching. Journal of Mathematics Teacher Education, 1, 3–31.

Jaworski, B. (2001). Developing mathematics teaching: Teachers, teacher-educators,and researchers as co-learners. In F-L. Lin & T.J. Cooney (Eds.), Making sense ofmathematics teacher education. Dordrecht, The Netherlands: Kluwer.

Jaworski B. & Potari, D. (1998). Characterising mathematics teaching using the teachingtriad. In A. Oliver & K. Newstead (Eds.), Proceedings of the 22nd Conference of theInternational Group for the Psychology of Mathematics Education. Stellenbosch, SouthAfrica: University of Stellenbosch.

Krainer, K. (1993). Understanding students’ understanding: On the importance of co-operation between teachers and researchers. In P. Boero (Ed.), Proceedings of the 3rdBratislava International Symposium on Mathematical Teacher Education. ComeniusUniversity: Bratislava.

Lampert, M. (1985). How do teachers manage to teach?: Perspectives on problems inpractice. Harvard Educational Review, 55, 178–194.


Lampert, M. (2001) Teaching problems and the problems of teaching. New Haven: YaleUniversity Press.

National Council of Teachers of Mathematics (1991) Professional standards for teachingmathematics. Reston, VA: Author.

Raymond, A.M. (1997). Inconsistency between a beginning elementary school teacher’smathematics beliefs and teaching practice. Journal for Research in Mathematics Educa-tion, 28(5), 550–576.

Scott-Nelson, B. (2001). Constructing facilitative teaching. In T. Wood, B.S. Nelson & J.Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics.Mahwah, NJ: Lawrence Erlbaum.

Skott, J. (2001). The emerging practices of a novice teacher: The roles of his high schoolmathematics images. Journal of Mathematics Teacher Education, 4(1), 3–28.

Skemp, R.R. (1976). The psychology of learning mathematics, 2nd edn. London: PenguinBooks.

Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.Journal of Mathematics Teacher Education, 1, 157–189.

Voigt, J. (1985). Patterns and routines in classroom interaction, Recherches en Didactiquedes Mathematiques, 6, 69–118.

Voigt, J. (1994). Negotiation of mathematics meaning and learning mathematics. In P. Cobb(Ed.), Learning mathematics: Constructivist and interactionist theories of mathematicaldevelopment (pp. 171–298). Dordrecht: Kluwer Academic Publishers.

Williams, S.R. & Baxter, J. (1993). Dilemmas of discourse-oriented teaching in one middleschool mathematics classroom. The Elementary School Journal, 97(1), 21–38.

Wood, T., Nelson, B.S. & Warfield, J. (2001). Beyond classical pedagogy: Teachingelementary school mathematics. Mahwan, NJ: Lawrence Erlbaum.

Wood, T., Cobb, P., Yackel, E. & Dillon, D. (1993). Rethinking elementary school mathe-matics: Insights and issues. Journal for research in mathematics education, MonographNumber 6. Reston, VA: National Council of Teachers of Mathematics.

Zack, V., Mousely, J. & Breen, C. (Eds.) (1997). Developing practice: Teachers’ inquiryand educational change in classrooms. Geelong, Australia: Centre for Studies inMathematics, Science and Environmental Education, Deakin University.

DESPINA POTARI

University of PatrasDepartment of Primary Education261 10 Patras, GreeceE-mail: [email protected]

BARBARA JAWORSKI

University of OxfordDepartment of Educational Studies15 Norham GardensOxford, OX2 6PY, UKE-mail: [email protected]

BOOK REVIEW

Haggarty, L. (Ed.) (2002). Teaching mathematics in secondary schools: Areader. The Open University, London. ISBN: 0-415-26069-8

OVERVIEW

This book presents wide ranging views of different authors which couldchallenge the beliefs of many beginning teachers. A strength of the book isthat it acknowledges at all times the complexity of learning, understanding,and effective teaching, and that these involve ongoing “learning”. Hence,it promotes teacher education as a professional endeavour, not a vocationalone best served only by some sort of apprenticeship. The inclusion ofpolitical aspects, national assessment and history of government prioritiesis to be commended as often students are limited to a few experiences oftheir own. The issues and international perspectives presented are relevantbeyond England which is where the book is primarily focused.

The book combines edited versions of previously published papersor articles with some new chapters. It is divided into seven sectionsallowing for a linear, systematic reading or a selection of chapters out ofsequence. The seven sections are: Student teacher learning; Mathematics(including a chapter on technology); Learning mathematics; Teachingmathematics; Assessing learning in mathematics; Social and contextualissues in mathematics; and International perspectives.

The section devoted to mathematics as a discipline is welcome. Thearbitrary nature of much mathematics (particularly definitions) and thehistory of how mathematics in schools (including the impact of tech-nology) and the students who study it have changed make interestingreading.

A strength of the book is that the material is pitched at an appro-priate level and so is accessible to the targeted audience: those studyingto become secondary mathematics teachers and those experienced mathe-matics teachers who want to develop further their understanding of issuesin mathematics education.


382 BOOK REVIEW

DISCUSSION AND REFLECTIONS

Even though the authors establish the need for social justice in educationand meaningful, constructivist-related teaching, they also present substan-tial practical influences on what is occurring in schools in England andthroughout the world. Hence, the book contains degrees of idealism (asit should), but also tempers them by recognising the external influenceswhich shape the curriculum and culture of schools. These external influ-ences and school cultures based on authoritarianism and closed methodsare what student teachers often confront head on when they enter schools.The issues of justice raised by Gates and Cooper, questions about “taken-for-granted assumptions and practices” by Hatch and Pepin, and theimpact of closed teaching methods and assessment (Boaler) may wellconflict with what the graduates encounter in their schools. Hence, theyare important inclusions and can go some way to breaking self-begettingcycles.

Another conflict graduates often face when entering school is apredominance of transmission, behaviourist-related teaching and evencontempt for those who may wish to pursue constructivist approaches.Sowder (2001) makes a similar point about the USA when she claimsmany teachers have little regard for research in mathematics education andthat often mathematicians who also do not value research in mathematicseducation are influential. Similar situations occur in Australia. Therefore,the theories of learning sections in this book play an important role inwidening student teachers’ knowledge of learning. Also, the chapters byJaworski, and Nickson present a more balanced view of constructivismthan was often the case ten years ago. For example, Ellerton and Clements(1992) agreed that much good had derived from the radical constructivistmovement, but raised concerns about “the missionary zeal” of some radicalconstructivists who accused those outside their ranks as Platonists advocat-ing transmission modes of teaching. These extreme views of radicalconstructivism belittled the practices of many committed teachers whileat the same time offered no real assistance for normal classrooms witha set curriculum. In this book, constructivism has moved on from thehard-line stances identified by Ellerton and Clements. Nickson’s two folddescription of constructivism - bottom-up (personal construction of mathe-matics) and top-down (personal reconstruction of mathematics that alreadyexists) (p. 231) is a useful perspective for teachers who are faced witha set curriculum. The latter, in particular, provides student teachers witha process approach to teaching which fits a normal curriculum and doesnot condemn those who love and enjoy the body of knowledge calledmathematics.

BOOK REVIEW 383

Supporting the notion of reconstructing mathematics is Jaworski’streatment of the abstract nature of subject: “the power of mathematicslies in its nature to generalize and to express unambiguously in abstractforms” (p. 77). Jaworski, like Nickson, situates social issues and construc-tivist ideas with mathematics as an existing body of invented knowledge.In so doing, the authors provide student teachers with the opportunity tore-examine their own beliefs and, it is hoped, when teaching, readers willseek to implement more socially interactive classrooms.

Abstraction and generalization, key aspects of mathematics, deserve thetreatment given. A closer look at generalization was given by Davidov(1990) who identified two types of generalization:

• empirical – a pattern-seeking behaviour activity;• theoretical – the consideration of deeper, structural similarities which

identify inner connections.

These two views of generalization help clarify Goldstein’s commentsabout pattern spotting (empirical) with computers alone not being enough(p. 151). He observes that time away from the computer is also needed (fortheoretical considerations). White (1999) supports Goldstein’s views thatusing dynamic geometry is useful for demonstrating a theorem (empiricalgeneralization), but offers no help with finding a proof of the theorem(theoretical generalization).

Another pervading issue is the use of contexts in mathematics learning,teaching and assessment. The use of “contexts” in mathematics is itselfpresented in a number of different contexts. For example, Noss (p. 35)makes the pertinent point that: “in trying to connect mathematics to whatis learnable, we have disconnected mathematics from what is genuinelyuseful”. Noss suggests that some school mathematics has become sodecontextualised that it is effectively useless. Boaler (p. 107) uses thecontext of house designs when arguing in favour of open investigations.Gates (pp. 224–225) advocates transforming traditional contexts into morerealistic ones which have social implications and examine social justiceissues. On the other hand, Jaworski states:

Theory has suggested that the use by teachers of everyday contexts in which to embedabstract mathematical ideas can be helpful to students in learning to cope with abstraction.However, research now shows us that students from working-class backgrounds find itmore difficult to see beyond the every day context to the mathematics on which they areexpected to focus. (p. 79)

This comment alludes to the later chapter by Cooper which looks at contextbased tasks used for pencil and paper assessments. Cooper’s argumentsabout the tasks he presents are persuasive. However, many would contest

384 BOOK REVIEW

that using contexts for teaching about abstract ideas is problematic isa valid inference from Cooper’s work. Other authors already mentionedargued in favour of using contexts. Irwin (2001) showed students from alow socio economic area exhibited fewer common misconceptions aboutdecimals when working in a familiar context than when working withnumbers alone. Perhaps the answer lies not so much in using or not usingcontexts, but in how they are used. Mitchelmore and White (2000) showedhow children do not see naturally the abstract features of angles in everyday examples, but by being directed to attend selectively to abstract anglefeatures (e.g.; where the lines are), the contexts provided a richer conceptof angle than occurred when students worked only with prompts likeabstract drawings.

By pertinently addressing key issues associated with the learning andteaching of mathematics, the book is an important stimulus for discussionand debate. It has much to offer mathematics educators in all parts of theworld.

REFERENCES

Davidov, V.V. (1990). Types of generalization in instruction: Logical and psychologicalproblems in the structuring of school curricula (Soviet studies in mathematics education,Vol. 2; J. Kilpatrick, Ed., J. Teller, Trans.). Reston, VA: National Council of Teachers ofMathematics (original work published 1972).

Ellerton, N.F. & Clements, M.A. (Ken) (1992). Some pluses and minuses of radicalconstructivism in mathematics education. Mathematics Education Research Journal,4(2), 1–22.

Irwin, K.C. (2001). Using everyday knowledge of decimals to enhance understanding.Journal for Research in Mathematics Education, 32(4), 399–420.

Mitchelmore, M.C. & White, P. (2000). Development of angle concepts by progressiveabstraction and generalization. Educational Studies in Mathematics, 41(3), 209–238.

Sowder, J. (2001). Connecting mathematics education research to practice. (Key noteaddress). In J. Bobis, B. Perry & M. Mitchelmore (Eds.), Proceedings of the 24th AnnualConference of the Mathematics Education Research Association of Australasia (pp. 1–8).Sydney.

White, P. (1999). Learning mathematics: A new look at generalization and abstrac-tion. Refereed proceedings of the annual conference (combined) of the Australianand New Zealand Associations of Research in Education. http://www.swin.edu.au/aare/99pap/whi99309.htm. Melbourne: Australia.

PAUL WHITE

Australian Catholic University

ACKNOWLEDGEMENT

The editor thanks the following colleagues who reviewed manuscripts forJMTE in 2002. We appreciate our reviewers’ thoughtful critiques of themanuscripts and their contributions to the field of mathematics teachereducation.

Sergei AbrahamovichJill AdlerJanet AinleyMike AskewDeborah Loewenberg BallLuciana BazziniMariolina Bartolini-BussiJeremy BeckerJoanne BeckerAlan BellSarah B. BerensonMaria L. BlantonGlen BlumeJo BoalerJanette BobisRaffaella BorasiChris BreenGeorge W. BrightCatherine A. BrownLeone BurtonThomas CarpenterJose Carrillo YanezOlive ChapmanDaniel ChazanMohan ChinnappanBarbara ClarkeDavid J. ClarkeCarne (Barnett) Clarke

Phil ClarksonAnne CockburnThomas J. CooneySandra CrespoBeatriz S. D’AmbrosioSandy DawsonThomas De FrancoHelen DoerrSusan EmpsonPaul ErnestRuhama EvenFrancis FennelMegan Loef FrankeJeffrey FrykholmToshiakira FujiiPeter GalbraithPeter GallinPeter GatesFred GoffreeEliot C. GottmanMaria GouldingAnna GraeberKaren GrahamBarbro GrevholmDouglas GrouwsLinda HaggartyAnjum HalaiKath Hart

Gillian HatchRuth M. HeatonStephen HegedusPatricio HerbstTony HerringtonJames HiebertMarj HorneCelia HoylesRobert HuntingKatherine IrwinBarbara JaworskiGraham A. JonesTony JonesJames KaputElham KazemiTom KeirenChristine KeitelDaphne KerslakeJeremy KilpatrickEric KnuthAndrea Peter-KoopEugenia KolezaMasataka KoyamaKonrad KrainerGoetz KrummheuerColette LabordeMagdalene LampertGlen Lappan

Journal of Mathematics Teacher Education 5: 385–386, 2002.

386 ACKNOWLEDGEMENT

Gilah C. LederNorman G. LedermannSteve LermanFou-lai LinMary LindquistRomulo Campos LinsCheryl A. LubinskiKay McClainAlistair McIntoshDouglas B. McLeodCarolyn MaherJoanna O. MasingiliaJohn MasonVilma M. MesaDenise MewbornL. Diane MillerCandia MorganJudith MousleyNel NoddingsJohn OliveErkki PehkonenBarbara PencePatricia PerksJoao Pedro da Ponte*

Despina PotariDave PrattNorma PresmegStephanie PrestageAlison PriceKristina ReissJanine T. RemillardBarbara ReysMelissa RoddTim RowlandHarallambos SakonidisMagdalena SantosDeborah SchifterAlan H. SchoenfeldLurdes SerrazinaMamakgethi SetatiAnna SfardMiriam SherinDiane SiemonMartin A. SimonJudith SowderLarry SowderKaye StaceyLeslie P. Steffe

Heinz SteinbringPeter SullivanRosamund SutherlandJulianna SzendraiPatrick ThompsonDina TiroshSteve TobiasRon TzurSuwattana UtairatLaura R. van ZoestLieven VerschaffelJanet WarfieldTad WatanabeAnne WatsonJane M. WatsonSteven WilliamsErich WittmannMelvin (Skip) WilsonTerry WoodErna YackelOrit ZaslavskyJudith ZawojewskiRose Mary ZbiekRobyn Zevenbergen

Journal of Mathematics Teacher Education

INSTRUCTIONS FOR AUTHORS

EDITOR-IN-CHIEF

Barbara Jaworski Editors: Konrad KrainerUniversity of Oxford University of Klagenfurt, AustriaDepartment of Educational Studies Peter Sullivan15 Norham Gardens La Trobe University, AustraliaOxford OX2 6PY Terry WoodUnited Kingdom Purdue University, U.S.A.

AIMS AND SCOPEThe Journal of Mathematics Teacher Education (JMTE) is devoted toresearch into the education of mathematics teachers and developmentof teaching that promotes students’ successful learning of mathematics.JMTE focuses on all stages of professional development of mathematicsteachers and teacher-educators and serves as a forum for consideringinstitutional, societal and cultural influences that impact on teachers’learning, and ultimately that of their students. Critical analyses ofparticular programmes, development initiatives, technology, assessment,teaching diverse populations and policy matters, as these topics relate tothe main focuses of the journal, are welcome. All papers are rigorouslyrefereed.

Papers may be submitted to one of three sections of JMTE as follows:

1. Research papers: these papers should reflect the main focuses of thejournal identified above and should be of more than local or nationalinterest.


388 INSTRUCTIONS FOR AUTHORS

2. Mathematics Teaching Around the World: these papers focus onprogrammes and issues of national significance that could be of widerinterest or influence.

3. Reader Commentary: these are short contributions; for example,offering a response to a paper published in JMTE or developing atheoretical idea.

Authors should state clearly the section to which they are submittinga paper. As general guidance, papers should not normally exceed thefollowing word lengths: (1) 10,000 words; (2) 5,000 words; (3) 3,000words.

All contributions to this journal are peer reviewed. Critiques of reports orbooks that relate to the main focuses of JMTE appear as appropriate.

MANUSCRIPT SUBMISSIONKluwer Academic Publishers prefer the submission of manuscripts andfigures in electronic form in addition to a hard-copy printout. The preferredstorage medium for your electronic manuscript is a 3 1/2 inch diskette.Please label your diskette properly, giving exact details on the name(s) ofthe file(s), the operating system and software used. Always save your elec-tronic manuscript in the wordprocessor format that you use; conversionsto other formats and versions tend to be imperfect. In general use as fewformatting codes as possible. For safety’s sake, you should always retaina backup copy of your file(s). After acceptance, please make absolutelysure that you send us the latest (i.e., revised) version of your manuscript,both as hard copy printout and on diskette.

Kluwer Academic Publishers prefer articles submitted in wordprocessingpackages such as MS Word, WordPerfect, etc. running under operatingsystems MS DOS, Windows or Apple Macintosh, or in the file formatLATEX. Articles submitted in other software programs, as well as articlesfor conventional typesetting, can also be accepted.

For submission in LaTeX, Kluwer Academic Publishers have devel-oped Kluwer LaTeX class files, which can be downloaded from:www.wkap.nl/kaphtml.htm/IFAHOME. Use of this class file is highlyrecommended. Do not use versions downloaded from other sites.Technical support is available at: [email protected]. If you are not familiarwith TeX/LaTeX, the class file will be of no use to you. In that case,submit your article in a common word processor format.

INSTRUCTIONS FOR AUTHORS 389

For the purpose of reviewing, articles for publication should initially besubmitted as hard-copy printout (4-fold: 2 with full author details and 3blind copies for review purposes) and on diskette to:

Kluwer Academic Publishers, P.O. Box 990, 3300 AZ Dordrecht, TheNetherlands.

MANUSCRIPT PRESENTATIONThe journal’s language is English. British English or American Englishspelling and terminology may be used, but either one should be followedconsistently throughout the article. Manuscripts should be printed ortypewritten on A4 or US Letter bond paper, one side only, leavingadequate margins on all sides to allow reviewers’ remarks. Please double-space all material, including notes and references. Quotations of more than40 words should be set off clearly, either by indenting the left-hand marginor by using a smaller typeface. Use double quotation marks for directquotations and single quotation marks for quotations within quotationsand for words or phrases used in a special sense.

Number the pages consecutively with the first page containing:

– running head (shortened title)– article type (if applicable)– title– author(s)– affiliation(s)– full address for correspondence, including telephone and faxnumber and e-mail address

AbstractPlease provide a short abstract of 100 to 250 words. The abstract shouldnot contain any undefined abbreviations or unspecified references.

Key WordsPlease provide 5 to 10 key words or short phrases in alphabetical order.

AbbreviationsIf abbreviations are used in the text, please provide a list of abbreviationsand their explanations.


Figures and Tables

Submission of electronic figures

In addition to hard-copy printouts of figures, authors are encouragedto supply the electronic versions of figures in either EncapsulatedPostScript (EPS) or TIFF format. Many other formats, e.g., MicrosoftPostscript, PiCT (Macintosh) and WMF (Windows), cannot be used andthe hard copy will be scanned instead.

AVOIDING PROBLEMS WITH EPS GRAPHICS

Please always check whether the figures print correctly to a PostScriptprinter in a reasonable amount of time. If they do not, simplify yourfigures or use a different graphics program.

If EPS export does not produce acceptable output, try to create anEPS file with the printer driver (see below). This option is unavailablewith the Microsoft driver for Windows NT, so if you run Windows NT, getthe Adobe driver from the Adobe site (www.adobe.com).

If EPS export is not an option, e.g., because you reply on OLE andcannot create separate files for your graphics, it may help us if you simplyprovide a PostScript dump of the entire document.

HOW TO SET UP FOR EPS AND POSTSCRIPT DUMPS UNDERWINDOWS

Create a printer entry specifically for this purpose: install the printer‘Apple Laserwriter Plus’ and specify ‘FILE’: as printer port. Each timeyou send something to the ‘printer’ you will be asked for a filename. Thisfile will be the EPS file or PostScript dump that we can use.

The EPS export option can be found under the PostScript tab. EPSexport should be used only for single-page documents. For printing adocument of several pages, select ‘Optimise for portability’ instead. Theoption ‘Download header with each job’ should be checked.

Submission of hard-copy figures

If no electronic versions of figures are available, submit only high-quality artwork that can be reproduced as is, i.e., without any part having


to be redrawn or retypeset. The letter size of any text in the figuresmust be large enough to allow for reduction. Photographs should bein black-and-white on glossy paper. If a figure contains colour, makeabsolutely clear whether it should be printed in black-and-white or incolour. Figures that are to be printed in black-and-white should not besubmitted in colour. Authors will be charged for reproducing figures incolour.

Each figure and table should be numbered and mentioned in the text.The approximate position of figures and tables should be indicated in themargin of the manuscript. On the reverse side of each figure, the name ofthe (first) author and the figure number should be written in pencil; thetop of the figure should be clearly indicated. Figures and tables should beplaced at the end of the manuscript following the Reference section. Eachfigure and table should be accompanied by an explanatory legend. Thefigure legend should be grouped and placed on a separate page. Figuresare not returned to the author unless specifically requested.

In tables, footnotes are preferable to long explanatory material ineither the heading or body of the table. Such explanatory footnotes,identified by superscript letters, should be placed immediately below thetable.

Section HeadingsFirst-, second-, third-, and fourth-order headings should be clearlydistinguishable. Headings should follow APA guidelines.

AppendicesSupplementary material should be collected in an Appendix and placedbefore the Notes and Reference sections.

NotesPlease use endnotes rather than footnotes. Notes should be indicated byconsecutive superscript numbers in the text and listed at the end of thearticle before the References. A source reference note should be indicatedby means of an asterisk after the title. This note should be placed at thebottom of the first page.

Cross-ReferencingPlease make optimal use of the cross-referencing features of your softwarepackage. Do not cross-reference page numbers. Cross-references should


refer to, for example, section numbers, equation numbers, figure and tablenumbers.

In the text, a reference identified by means of an author’s name should befollowed by the date of the reference in parentheses and page number(s)where appropriate. When there are more than two authors, only the firstauthor’s name should be mentioned, followed by ‘et al.’. In the event thatan author cited has had two or more works published during the same year,the reference, both in the text and in the reference list, should be identifiedby a lower case letter like ‘a’ and ‘b’ after the date to distinguish the works.

Examples:Winograd (1986, p. 204)(Winograd, 1986a; Winograd, 1986b)(Flores et al., 1988; Winograd, 1986)(Bullen & Bennett, 1990)

AcknowledgementsAcknowledgements of people, grants, funds, etc. should be placed in aseparate section before the References.

ReferencesReferences to books, journal articles, articles in collections and confer-ence or workshop proceedings, and technical reports should be listed atthe end of the article in alphabetical order following the APA style (seeexamples below). Articles in preparation or articles submitted for publica-tion, unpublished observations, personal communications, etc. should notbe included in the reference list but should only be mentioned in the articletext (e.g., T. Moore, personal communication).

References to books should include the author’s name; year of publication;title; page numbers where appropriate; publisher; place of publication, inthe order given in the example below.

Mason, R. (1995). Using communications in open and flexible learning.London: Kogan Page.

References to articles in an edited collection should include the author’sname; year of publication; article title; editor’s name; title of collection;first and last page numbers; publisher; place of publication, in the ordergiven in the example below.


McIntyre, D.J., Byrd, D.M. & Foxx, S.M. (1996). Field and laboratoryexperiences. In J. Sikula, T.J. Buttery & E. Guyton (Eds.), Handbook ofresearch on teacher education (3rd. Ed.; pp. 171–193) New York: Simon& Schuster.

References to articles in conference proceedings should include theauthor’s name; year of publication; article title; editor’s name (if any);title of proceedings; first and last page numbers; place and date ofconference; publisher and/or organization from which the proceedings canbe obtained; place of publication, in the order given in the examples below.

Yan, W., Anderson, A. & Nelson, J. (1994). Facilitating reflective thinkingin student teachers through electronic mail. In J. Willis (Ed.), Technologyand Teacher Education Annual, 1994: Proceedings of the Fifth AnnualConference of the Society of technology and teacher Education. VA:Association for the advancement of Computing in Education.

Blanton, M.L., Westbrook, S.L. & Carter, G. (2001). Using Valsiner’sZone Theory to Interpret a Pre-service Mathematics Teacher’s Zoneof Proximal Development. In M. Van den Heuvel-Panhuizen (Ed.),Proceedings of the 25th Conference of the International Group for thePsychology of Mathematics Education. The Netherlands: FreudenthalInstitute, Utrecht University.

References to articles in periodicals should include the author’s name;year of publication; article title; full or abbreviated title of periodical;volume number (issue number where appropriate); first and last pagenumbers, in the order given in the example below.

Thomas, L., Clift, R.T. & Sugimoto, T. (1996). Telecommunication,student teaching and methods instruction: An exploratory investigation.Journal of Teacher Education, 46, 165–174.

References to technical reports or doctoral dissertations should includethe author’s name; year of publication; title of report or dissertation;institution; location of institution, in the order given in the example below.

Hughes, M. (1975). Egocentrism in pre-school children, EdinburghUniversity, Edinburgh, unpublished PhD thesis.


PROOFSProofs will be sent to the corresponding author. One corrected proof,should be returned to the Publishers within three days of receipt by mail(airmail overseas).

OFFPRINTSTwenty-five offprints of each article will be provided free of charge.Additional offprints can be ordered by means of an offprint order formsupplied with the proofs.

PAGE CHARGES AND COLOUR FIGURESNo page charges are levied on authors or their institutions. Colour figuresare published at the author’s expense only.

COPYRIGHTAuthors will be asked, upon acceptance of an article, to transfer copyrightof the article to the Publisher. This will ensure the widest possibledissemination of information under copyright laws.

PERMISSIONSIt is the responsibility of the author to obtain written permission for aquotation from unpublished material, or for all quotations in excess of 250words in one extract or 500 words in total from any work still in copyright,and for the reprinting of figures, tables or poems from unpublished orcopyrighted material.

ADDITIONAL INFORMATIONAdditional information can be obtained from:

Journal of Mathematics Teacher Education, Kluwer AcademicPublishers, Attn Publishing Editor. Michel Lokhorst, P.O. Box 17,3300 AA Dordrecht, The Netherlands, tel.: +31-78-6576183; fax:+31-78-6576254. E-mail: [email protected]

CONTENTS OF VOLUME 5

Volume 5 No. 1 2002

EDITORIAL / JMTE: A Time to Reflect 1–5

JANINE T. REMILLARD and PAMELA KAYE GEIST /Supporting Teachers’ Professional Learning byNavigating Openings in the Curriculum 7–34

RUTH M. HEATON and WILLIAM T. MICKELSON / TheLearning and Teaching of Statistical Investigation inTeaching and Teacher Education 35–59

ERIC J. KNUTH / Teachers’ Conceptions of Proof in theContext of Secondary School Mathematics 61–88

Volume 5 No. 2 2002

BARBARA JAWORSKI / Editorial: Layers of Learning inInitial Teacher Education 89–92

JOÃO PEDRO DA PONTE, HÉLIA OLIVEIRA and JOSÉMANUEL VARANDAS / Development of Pre-Service Mathematics Teachers’ Professional Know-ledge and Identity in Working with Information andCommunication Technology 93–115

MARIA L. BLANTON / Using An Undergraduate GeometryCourse to Challenge Pre-Service Teachers’ Notionsof Discourse 117–152

BARBARA M. KINACH / Understanding and Learning-To-Explain by Representing Mathematics: Epistemolo-gical Dilemmas Facing Teacher Educators in theSecondary Mathematics “Methods” Course 153–186


396 CONTENTS OF VOLUME 5

Book Review

Lampert, Magdalene (2001). Teaching Problems and the prob-lems of teaching (DANIEL CHAZAN) 187–199

Volume 5 No. 3 2002

Editorial

TERRY WOOD / Demand for Complexity and Sophistication:Generating and Sharing Knowledge About Teaching 201–203

MIRIAM GAMORAN SHERIN / A Balancing Act: Devel-oping a Discourse Community in a MathematicsClassroom 205–233

JOANNA O. MASINGILA and HELEN M. DOERR / Under-standing Pre-Service Teachers’ Emerging PracticesThrough Their Analyses of a Multimedia Case Studyof Practice 235–263

LAURA R. VAN ZOEST and JEFFREY V. BOHL / The Roleof Reform Curricular Materials in an Internship: TheCase of Alice and Gregory 265–288

Volume 5 No. 4 2002

PETER SULLIVAN / Editorial: Using the Study of Practiceas a Learning Strategy within Mathematics TeacherEducation Programs 289–292

PATRICIA S. MOYER and ELIZABETH MILEWICZ /Learning to Question: Categories of QuestioningUsed by Preservice Teachers During DiagnosticMathematics Interviews 293–315

PI-JEN LIN / On Enhancing Teachers’ Knowledge byConstructing Cases in Classrooms 317–349

DESPINA POTARI and BARBARA JAWORSKI / TacklingComplexity In Mathematics Teaching Development:Using the Teaching Triad as a Tool for Reflection andAnalysis 351–380

CONTENTS OF VOLUME 5 397

Book Review

Haggarty, L. (Ed.) (2002). Teaching mathematics in secondaryschools: A reader (PAUL WHITE) 381–384

Acknowledgement 385–386

Instructions for Authors 387–394

Contents of Volume 5 395–397

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