two teachers' conceptions of a reform-oriented curriculum: implications for mathematics teacher...

GWENDOLYN M. LLOYD

TWO TEACHERS’ CONCEPTIONS OF A REFORM-ORIENTEDCURRICULUM: IMPLICATIONS FOR MATHEMATICS

TEACHER DEVELOPMENT?

ABSTRACT. This paper describes two high school teachers’ conceptions of the coopera-tion and exploration components of a reform-oriented mathematics curriculum. Althoughthe teachers appreciated the themes of cooperation and exploration in theory, their concep-tions of these themes with respect to their implementations of the curriculum differed. Oneteacher viewed the curriculum’s problems as open-ended and challenging for students,whereas the other teacher claimed that the problems were overly structured. Each teacherattributed difficulties with students’ cooperative work to the amount of structure and direc-tion (too little or too much) offered by the problems. Discussion of such similaritiesand differences in the teachers’ conceptions emphasizes the dynamic, humanistic natureof curriculum implementation and gives rise to important implications for mathematicsteacher development in the context of reform.

By demanding changes in both the content and activity of mathematicsinstruction, recent reform recommendations in the United States challengea lasting tradition (Gregg, 1995; Richards, 1991). In light of the impressivedurability of traditional teacher-centered and procedure-oriented mathe-matics instruction, how do veteran teachers deal with calls for reform? Thispaper describes two secondary teachers’ experiences with reform recom-mendations in the context of their implementations of a set of innovativecurriculum materials. Focus is on the teachers’ conceptions of the meaningand importance of certain mathematics classroom activities, in particularcooperation and exploratory problem-solving.

Cooperation and Exploration in Mathematics Teaching and Learning

Cooperation and exploration are prominent themes both in the curriculummaterials implemented by the teachers (Hirsch, Coxford, Fey & Schoen,1995) and in more general documents that promote mathematics educationreforms (Mathematical Sciences Education Board [MSEB] & National

? The research reported in this study was supported in part by the National ScienceFoundation (MDR-9255257). The views herein are those of the author and do notnecessarily reflect those of the National Science Foundation.

Journal of Mathematics Teacher Education2: 227–252, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

228 GWENDOLYN M. LLOYD

Research Council [NRC], 1989; National Council of Teachers of Mathe-matics [NCTM], 1989). Teachers are urged to establish mathematicsclassrooms in which students engage actively in exploration and cooper-ative work in order to help students develop rich understandings ofmathematics as a vibrant and useful subject.

Reform-oriented models of teaching and learning are supported bya broad base of empirical and theoretical literature about how studentsunderstand and learn mathematics. This literature makes a strong casefor classroom activities that give rise to genuine mathematical problemsfor students to resolve (Hiebert et al., 1996; Lampert, 1990; Schoenfeld,1992; P.W. Thompson, 1985; Yackel, Cobb, Wood, Wheatley & Merkel,1990). In contrast to traditional classroom activities that emphasize correctanswers and routinized solution methods, problem-centered instructioncapitalizes on opportunities for students to learn as they cooperate inthe solution process. This process can “include accounting for surprisingoutcomes, such as when two alternative methods lead to the same result,justifying a solution method, or explaining why an apparently erroneousmethod leads to a contradiction” (Yackel et al., 1990, p. 15). When studentswork in groups to communicate their ideas and questions, agree anddisagree among themselves, and negotiate joint theories and ideas, richmathematical learning can occur (Richards, 1991; Slavin, 1990; Voigt,1996).

Teachers’ Conceptions and Reform

How do teachers make sense of the themes of cooperation and explorationas they implement innovative curriculum materials? The role of teachers’conceptions and classroom experiences in mathematics reform cannotbe overemphasized. An extensive body of research provides consistentevidence that teachers’ conceptions strongly impact instructional prac-tice (Brophy, 1991; Fennema & Franke, 1992; A.G. Thompson, 1992).Moreover, teachers’ conceptions have profound effects on their inter-pretations and implementations of reform recommendations and reform-oriented mathematics curricula (Cohen, 1990; Lloyd & Wilson, 1998;Romberg, 1997; M. Wilson & Goldenberg, 1998; S.M. Wilson, 1990).For example, in S.M. Wilson’s (1990) case study of a teacher imple-menting a new curriculum as part of the California mathematics reforms,the teacher’s conceptions of appropriate instruction interfered with hisability to foster student inquiry in the ways intended by the reformcurriculum. Because reform-oriented pedagogies require that teachersreconceive their roles in mathematical activity and student learning,implementation of an innovative curriculum can pose significant chal-

TEACHERS’ CONCEPTIONS OF CURRICULUM 229

lenges even to the most committed teachers (Clarke, 1997; Smith,1996).

Reform documents and curriculum materials do not prescribe or definepractice for teachers, but rather offer visions “orienting individuals andinstitutions toward collectively valued goals” (Shulman, 1983, p. 501). Aricher understanding is needed of the relationships between teachers’ ownconceptions of mathematics teaching and the recommendations for changeoutlined in curriculum materials. The study reported in this paper inves-tigates how and why two teachers encountered particular successes anddifficulties as each implemented a set of novel curriculum materials for thefirst time. How did the teachers conceive of cooperation and exploration asthey implemented the curriculum materials in their classrooms?

RESEARCH METHODS

Curriculum Materials

The curriculum of the Core-Plus Mathematics Project attempts to supportteachers in enacting many recommendations of theStandards(NCTM,1989). Each year of the high school curriculum (Core-Plus Courses1–4) features algebra and functions, geometry and trigonometry, statis-tics and probability, and discrete mathematics. The curriculum’s units,each designed to guide approximately four weeks of student work, areorganized into several multi-day investigations that emphasize modelingreal-world situations, experimenting in order to develop and test theories,and debating with classmates. The materials direct teachers to organizethe classroom so that students can work cooperatively as they exploremathematical problems and ideas. Cooperative learning is complementedby whole-class discussions in which activities are introduced and summa-rized.

An example of a Core-Plus lesson may be helpful. The first lessonin a unit about probability,Simulating Chance Situations, begins witha description of Chinese government policies that restrict families toone child. Students are first asked to think of some alternative plans forcontrolling population growth. Then students are instructed to flip coinsto simulate an alternative plan, that each family may have two children,and to record their results in a frequency table. Students must decide howcoin flips can be used to simulate the births and to describe the possibleoutcomes for this particular plan. A variety of problems and questionsfollow, including: “Use your frequency table to estimate the probabilitythat a family of two children will haveat least oneboy” and “Estimate the


probability that a family of two children will have at least one boy usinga mathematical method other than simulation.” Next, students are askedto consider a different plan in which families continue having childrenuntil a boy is born, and to record data in frequency tables and histograms.This time, students predict responses to a set of questions prior to theirsimulation, and then compare predictions to actual results. Finally, studentsanalyze one of their own alternative plans proposed at the beginning of theinvestigation.

Although most Core-Plus activities are based in real-world contexts,such as the probability simulation described above, some engage studentsin exploring more abstract mathematical situations. For example, in a unitabout functions, an investigation titledThe Shape of Rulesasks studentsto generate tables and graphs associated with four sets of equations thatrepresent different types of relationships (linear, quadratic, exponential,etc.). Students are instructed to look for patterns across the three repre-sentations and summarize their ideas in statements such as “If we see arule like . . . , we expect to get a table [or graph] like . . . .” Further examplesand discussion of the Core-Plus curriculum can be found in a report by thematerials’ designers (Hirsch et al., 1995).

Participants, Site, and Context for the Study

This study investigated the conceptions of two veteran high school mathe-matics teachers, Mr. Allen and Ms. Fay, in a public school district in theNortheastern U.S. where the Core-Plus materials were being field-tested.Empirical study of Mr. Allen and Ms. Fay began in 1994 and 1996, respec-tively, when each teacher implemented the Core-Plus materials for the firsttime. These two teachers were selected for study because (1) in contrastto most other teachers at the same school, Mr. Allen and Ms. Fay imple-mented the materials voluntarily, (2) each teacher communicated a desireto integrate more cooperation and exploration into his or her instruction,and (3) the teachers’ individual experiences appeared to have importantcontrasting elements.

At the beginning of the study, Mr. Allen had been teaching mathematicsfor 14 years and, by his own description, had largely adhered to traditionalclassroom practices. In the spring of 1994, he was invited by the chair ofhis mathematics department to participate in the field-testing of the Core-Plus materials. He said, “If this is the way we’re going to go, I want tomake sure I have experience in it.” During the 1994–95 school year, Mr.Allen implemented the Core-Plus Course 1 materials for the first time inone class of 32 ninth grade students. In the 1995–96 school year, he againtaught one Core-Plus Course 1 class of 32 ninth grade students. In 1996–


97, he taught one ninth grade Core-Plus Course 1 class and two tenth gradeCore-Plus Course 2 classes.

In the Fall of 1996, due largely to her interest in the Core-Plusprogram, Ms. Fay joined the mathematics faculty at the high schoolwhere Mr. Allen taught. Her previous employment included both 10years of classroom teaching and, most recently, a state government posi-tion in which she visited schools with innovative mathematics educationprograms. According to Ms. Fay, in her previous teaching she had “alwaysused groups and. . . always had a project focus,” and she wished for theCore-Plus materials to support her in developing and extending thesecomponents of her practice. Her commitment to the Core-Plus innovationwas one of the primary reasons that she was selected as a participant in thisstudy. During the 1996–97 school year, Ms. Fay taught with the Core-PlusCourse 2 materials in two classes of tenth-grade students.

Data Collection and Analysis

Data sources consisted of teacher interviews, classroom observations, andfieldnotes. Because the two teachers joined the project at different times,the number and dates of interviews and observations varied considerably.Over a 3-year period between September 1994 and January 1997, Mr.Allen participated in 17 interviews (8 in Year 1, 7 in Year 2, and 2 inYear 3) and was observed 73 times teaching with the Core-Plus materials.This paper focuses primarily on data collected during his first two yearsof curriculum implementation. During the 1996–97 school year, Ms. Fayparticipated in five interviews and was observed 10 times in her Core-Plusclasses. Most interviews lasted approximately 1 hour, and all were audio-recorded and transcribed. All classroom observations were audio-recordedand approximately half were video-recorded. Fieldnotes were taken duringobservations, and written artifacts such as student work were collected andphotocopied.

In the interviews, teachers were invited to reflect on recent classroomevents, suggest goals for upcoming classroom activities, and respond toquestions about more general emerging themes. Data were analyzed duringand after collection (LeCompte & Preissle, 1993). Careful readings of tran-scripts and fieldnotes and creation of interview and observation summariesresulted in the identification of preliminary themes for subsequent focus ininterviews and observations. After the observation periods ended, morethorough review of data took place. The development of major themeswas aided by the use of taxonomic and thematic analytic strategies(Spradley, 1979). In the final stage of analysis, major themes were furthersynthesized within and across the data sources to illustrate important, and


often contrasting, aspects of the ways in which the teachers viewed theirimplementations of the Core-Plus curriculum.

THE CASE OF MR. ALLEN

Conceptions of Exploration

From the start of his teaching with the new curriculum, Mr. Allen posi-tively differentiated the Core-Plus problems from those found in traditionaltextbooks by pointing out ways in which they engaged students in sense-making activities. He described the Core-Plus activities as having anapproach that required students to develop “informal ideas based on theirexperiences” in contrast to “the teacher or book feeding it to them” (Int. 3,Yr. 1, 11/14/94). More specifically, in his second year using the materials,he indicated that the Core-Plus questions themselves required extensiveinterpretation by students:

The Core-Plus questions are a little more general – not vague, but not as specific and sosometimes the students have to figure out exactly what it is they want them to do. That’swhat they need to be able to work at and practice because that’s what they’ll use eventually– the problem solving skills and attacking a problem, reading it, and struggling with it. (Int.2, Yr. 2, 11/10/95)

One characteristic of the Core-Plus questions that required more analysisthan traditional exercises was that they were “different each time and theydon’t do the same thing over and over again.” He contrasted problemsthat challenged students to think actively about mathematics with a moretraditional approach that enticed students to have the attitude “Just tell methe steps I need to do and I’ll do those, but I don’t want to think about it toomuch” (Int. 5, Yr. 2, 12/8/95). Further, Mr. Allen suggested that with tradi-tional problems and instructional methods, student “understanding isn’tnearly what it is in Core-Plus.”

Another reason that Mr. Allen thought the Core-Plus materials engagedstudents in sense-making was that the activities centered on “realistic prob-lems and situations that drive the mathematics rather than having a bunchof made-up book-type problems” (Int. 1, Yr. 2, 10/17/95). For instance,he reflected on how the Core-Plus materials introduced the concept ofvariablewhen students explored the relationship between weight and cordstretch in a bungee jumping situation: “The kids understand what a variablecan represent based on some practical applications – give them situationswhere things are related and then they get a sense for what ‘variable’means” (Int. 5, Yr. 2, 12/8/95). In Mr. Allen’s view the overall benefit


of more open-ended, contextualized problem exploration was that Core-Plus students develop “deeper understanding of the material rather thanjust at the surface and knowing some algorithm to solve something and notreally know the background of it” (Int. 2, Yr. 1, 10/3/94). He suggestedthat, because Core-Plus students will “have a background in lots of mathe-matics andthinking,” when they need to develop procedures or algorithmsthey will be capable of deriving them: “If factoring is important, they’regoing to be able to do the algorithmic factor” (Int. 8, Yr. 1, 6/1/95).These comments illustrate Mr. Allen’s strong valuation of the Core-Plusemphasis on “thinking about situations but not necessarily having theanswer be the major outcome – it’s the process that’s important” (Int. 1,Yr. 2, 10/17/95).

His valuation of the “more investigative” approach of Core-Plus wasevidenced on numerous occasions in his classroom when he suggestedways that students could use their different solution strategies to yielddiscussion and further learning. For example, when some students labeledthe axes of coordinate graphs in the reverse order from the conventionalapproach, Mr. Allen invited groups to compare how the different axislabelings might affect the representation of the data in the graphs. Forinstance, when students switched variables on the horizontal and verticalaxes of a graph showing adistanceversustime relationship, he instructedseveral groups of students: “Compare the graphs withtime anddistanceswitched. See if they are the same.” He indicated in an interview (Int. 5,Yr. 1, 12/2/94) that he was hoping to elicit more discussion by havingstudents analyze these differences. Similarly, Mr. Allen often remindedstudents that their personal preferences (for instance, among differentrepresentations of data) would help them to solve problems effectively.As these examples show, Mr. Allen encouraged students to personalizetheir solution strategies and make sense of the problems and ideas forthemselves.

Although Mr. Allen valued the Core-Plus approach, he often worriedabout how to help students deal with the problems and questions in thecurriculum materials. He reported that while working on Core-Plus prob-lems, students repeatedly asked him questions about “what they needto do.” Because the Core-Plus questions engage students an analysisof complex mathematical issues with less guidance than is offered intraditional mathematics texts, students often “think that they have to dosomething completely different or they may read more into the questionthan what’s really there” (Int. 6, Yr. 1, 12/12/94). Mr. Allen attributedmuch of the students’ frustration to the novelty of these types of problemsand activities for them:


The students haven’t had a lot of practice with reading and understanding. . . a problem.They are used to reading a five- or six-word sentence and then simplifying the expressions.It doesn’t take a genius to figure out what you should be doing. While here, you have awhole story to read and in that story is some critical information. (Int. 1, Yr. 3, 9/20/96)

However, even when students understood the questions, they were unac-customed to attaining more than one correct final answer: “They want tomull over these questions like they’re right and wrong rather than thinkingabout them and giving a basis as to why they answer a question oneway or another” (Int. 6, Yr. 1, 12/12/94). Mr. Allen repeatedly remindedstudents to “just say what [they] think” as they worked on the Core-Plus activities. For example, in one lesson when students were asked topredict the relationship between the time and height of a ball thrown inthe air prior to modeling the situation, students expressed concerns suchas, “How are we supposed to figure out how high it goes?” In response,Mr. Allen encouraged students to “give a guess of how highyou thinkitwent.” In a subsequent interview, as he reflected on the students’ reac-tion to this question, Mr. Allen described, “Rather than ‘What do youthink about it?’ they wanted to know exactly how to do it” (Int. 6, Yr.1, 12/12/94). Because “getting them more into using their own thoughtsas being valid has been difficult,” Mr. Allen identified the need to buildstudents’ confidence in offering personal theories and opinions in responseto the Core-Plus problems.

Conceptions of Cooperation

As he began to implement the Core-Plus materials, Mr. Allen identifiedhis appreciation for “the philosophy of working in groups to get thingsdone” (Int. 2, Yr. 1, 10/3/94). Because “the teacher wasn’t going to be thefocus anymore,” the group work would give students “more ownership” ofthe mathematics under consideration. He communicated disappointmentabout the failure of traditional problems to “lend themselves to cooperativework”:

The traditional exercises are something that you can do by yourself. . . there’s not a lot ofdiscussion asked or describing asked. It’s more or less just get an answer or do some taskand get it done. You certainly can do that just by yourself. You might sit next to somebodyand ask them a question or watch how they did it or have them help you, but it’s not likeyou sit in a group and let one person do this part and one person do the other part. (Int. 5,Yr. 1, 12/2/94)

As this statement suggests, Mr. Allen wanted cooperative activities torequire explicit collaboration, a theme that is further illustrated by hisalteration of some Core-Plus activities to emphasize greater interde-pendence among group members. For example, in a Year 2 lesson that


involved students performing experiments with their graphing calculators,he explained to the class:

There is a lot of work to do so we are going to split up the work. I’m going to give onegroup Experiment #2 and there are six graphs to do in that group. We don’t have time to doall six ourselves so you want to split up the work . . . The idea is to use everybody in yourgroup.

To Mr. Allen, this lesson was “a perfect example” of how cooperationmakes most sense when the work involves more than could be reason-ably accomplished by one person. In other words, students must “relyon each other” while solving the problems (Int. 5, Yr. 2, 12/8/95). Inthis example, students analyzed and developed patterns based not only ontheir groupmates’ work, but also on the ideas of other groups. Because hechanged a recommended activity to encourage more extensive discussionand joint analysis by students, this example also illustrates Mr. Allen’ssincere interest in creating significant opportunities for students to worktogether, independently of him, on the Core-Plus activities.

However, Mr. Allen’s implementation of group work was not withoutconcerns or problems. In his view, struggles with tackling the more open-ended and complex problems of the curriculum particularly impactedstudents’ abilities to work cooperatively in their small groups. Because thematerial required more analysis and discussion on the part of the studentsthan did the traditional materials to which they were accustomed, Mr.Allen sensed that the students were “anxious about being out there a littlebit free and not really quite sure where this is all going” (Int. 6, Yr. 1,12/12/94). Unless students had “some perseverance and some stick-to-it-iveness,” they frequently stopped working on the mathematics and beganto socialize. During his second year implementing the materials, Mr. Allenexplained,

We have a ways to go with the groups really working at trying to understand things. If thematerial isn’t too difficult, the flow seems to go pretty well. As the material gets a littlemore difficult or more abstract, it isn’t in front of them real well, then they struggle withtrying to work at trying different things or . . . trying to figure something out as a groupwhen it’s not so obvious that it might take an error or two and not get frustrated and stop.(Int. 6, Yr. 2, 1/12/96)

As this statement reflects, Mr. Allen felt strongly that the success ofcooperative activities depended not only on student perseverance, but alsoon whether students understood what they needed to do and how to accom-plish it. Because, in Mr. Allen’s view, most Core-plus activities demandedthat students take the initiative in interpreting questions and situations,these factors frequently came into play in shaping his sense of the successof classroom activities.


Attempts to Address Concerns

Due to his concerns about students’ inclination to work productively andcooperatively on the Core-Plus activities, Mr. Allen often felt compelledto provide more direction than the curriculum materials recommend. Oneway that Mr. Allen added more direction to the Core-Plus lessons was thathe gave students review problems to complete at the end of each set ofinvestigations. As he explained when he developed the first set of reviewproblems, “It seems like a good way of having a limited task for all ofthem to get done. The task, more or less, summarizes the material. . . then Iknow that I have exposed them to the general ideas that I think they need toknow” (Int. 4, Yr. 1, 11/28/94). The review problems established “guidingposts of how and where we are supposed to be” (Int. 5, Yr. 1, 12/2/94).As these comments suggest, the review problems allowed Mr. Allen tocompensate for any weaknesses he perceived in students’ previous workwith the Core-Plus activities.

Notably, Mr. Allen made no changes to the problems printed in theCore-Plus curriculum materials. In other words, despite his concerns aboutthe open-ended problems, he did not re-write the problems or activities toinclude more structure or direction. This is consistent with Mr. Allen’ssense that to a greater extent than traditional textbooks, the Core-Plusmaterials restricted the teacher’s ability to personalize the curriculum:

The traditional textbook had this content that was just algorithmic type things and youcould then pick things to come into that class to make it come alive. You had that freedomto pull in an article or an experiment to liven up this idea we were going to do out ofthis algebra book. This [Core-Plus] is more –they are putting together the real-worldapplication that they want you to use so you are tied to that. It is hard to get away from it.(Int. 1, Yr. 3, 9/20/96)

Feeling tied to the mathematical situations presented in the materials mayhave contributed to Mr. Allen’s reluctance to modify the provided activitiesand to insteadappendreview problems.

The most prominent way that Mr. Allen’s concerns impacted hisinstruction was in the nature of his interactions with students and the wayshe chose to organize his classroom. Mr. Allen viewed teacher interventionas a way to improve students’ work on challenging problems and ques-tions: “To keep the flow of the class going I want to get in there and helpthem a little bit” (Int. 6, Yr. 1, 12/12/94). He indicated that providing direc-tion about certain concepts or activities was more efficient than havingstudents explore and develop ideas themselves, for example in the caseof solving algebraic equations: “Eventually they may come up with someunderstanding but I don’t know if it would be as clear that way as it is withme trying to make sure I give them some direction” (Int. 6, Yr. 2, 1/12/96).


In Mr. Allen’s first year, during a typical class session, he spent lessthan 5 minutes on whole-class instruction in order to introduce and reviewproblems. In his interactions with small groups, he attempted to “getthem going in the right direction.” As he circulated among the groupsprompting and directing students’ work, he often asked the same questionor made the same comment to each of the groups. Recognizing that hewas providing significant direction to students, he explained in an inter-view: “It takes spoonfeeding it a little bit at this point to let them knowwhat they really need to put down” (Int. 2, Yr. 1, 10/3/94). In his secondand third years of using the materials, Mr. Allen provided much lessdirection to groups than in his first year. Instead he decided to providemore whole-class instruction to help students acquire “a decent sense ofwhat they’re doing” and “the perception that they can be successful” inthe cooperative activities (Int. 3, Yr. 2, 11/21/95). Whole-class instruc-tion included an invitation to students “to be the teacher” so they wouldpresent problems to peers. It also included discussions of problems inwhich Mr. Allen attempted to make his classroom “more interactive.”However, more commonly, during his whole-class discussion he modeledproblems so that students would be prepared to work in a group withouthis help – otherwise, “You just turn them all loose and two minuteslater everyone’s got their hand up and it becomes chaos” (Int. 2, Yr. 2,11/10/95).

These comments and examples show Mr. Allen’s conception thatteacher-directed instruction can provide a sort of scaffolding or supportfor students’ small-group explorations. Although he believed that studentsshould discuss and learn important ideas in their small groups, he claimedthat teacher-directed instruction, particularly in whole-class formats, wasnecessary to insure that students had the opportunity to consider all of theimportant material. When he provided students with more structure anddirection, Mr. Allen felt better about his work in the classroom: “Eventhough I’m giving them more direction. . . it’s helpful for me to feel likeI’m covering the bases of what I think they should know” (Int. 6, Yr. 2,12/12/94). But he was also conscious of the potentially negative impact ofhis direction on students’ opportunities for mathematical exploration andcreativity. He suggested that his concerns about keeping students on taskand moving through the materials caused him “to be the opposite of what[Core-Plus is] really trying to emphasize” (Int. 5, Yr. 2, 12/2/94). In hissecond year, he became even more concerned that he “tended to do mostof it [him]self” (Int. 5, Yr. 2, 12/8/95). He pointed out that “You’ve got tobe able to balance it and not go overboard and start doing every problemfor them either” (Int. 2, Yr. 2, 11/10/95). The attempt to find this balance


was central to Mr. Allen’s goal of eventually developing a practice moreconsistent with “the Core way.” His sincere enthusiasm and interest inreforming his instruction is evidenced by his comment: “I am learning a lotabout myself as a teacher so I just enjoy it a lot more from my experienceof fifteen years . . . and I am finding it to be a lot more fun” (Int. 7, Yr. 2,5/17/96).

THE CASE OF MS. FAY

Conceptions of Exploration

As Ms. Fay began to teach with the Core-Plus materials, she commu-nicated her appreciation that the students were working on “really richproblems.” She identified real-life contexts such as distributions of studentshoe sizes or fast food nutrition as important problem features, and sheclaimed, “I never hear ‘When are we going to use this?’ because they allsee that it’s relevant. When the students saw the pure math, it didn’t makeany sense to them. It was just a neat strategy” (Int. 3, 11/5/96). The realisticcontexts of the Core-Plus problems seemed to Ms. Fay to be very effectivein engaging students in explorations of important underlying mathematics.

Despite her positive descriptions, she communicated disappointmentwith the opportunities for student exploration within the lessons. Herconcerns related primarily to the structure of the problems:

The kids are really led through the problem in Core-Plus. It’s like using the problem asa way to show them how to do this work instead of the kids saying “OK, let’s organizeourselves. What do we need?” Even though they are really good problems, it’s handed tothem how to solve it. (Int. 3, 11/5/96)

Ms. Fay viewed the activities in the Core-Plus materials as overly definedsets of questions that led students to particular answers or ideas throughpredetermined solution paths. She feared that the structured nature of theinvestigations might contribute to students attending more to the proce-dural aspects of problems than to the central concepts. For instance, sheexpressed that “Many times, students focus more on the steps for the bigideas than on the reasons behind them,” and she wished that students wouldask more “Why?” questions (Int. 1, 9/20/96). In her view, having studentswork on fewer problems with less guidance from the materials would allowstudents to take greater leadership in the solution process and, as a result,to develop more conceptually rich understandings.


Conceptions of Cooperation

Ms. Fay expressed many positive thoughts about students working cooper-atively. She believed that cooperative work allows students to articu-late and extend their understandings through give-and-take among groupmembers:

They are each articulating their own thoughts. If I just got up there and explained it, thenthey are only going to hear it one way. Instead they hear it from the kids . . . from othergroups, and it becomes a classroom of everybody teaching and everybody learning. (Int. 1,9/20/96)

But because the structured nature of the typical Core-Plus problemgave students few opportunities to “take responsibility for organizingand solving problems in their own ways” (Int. 2, 10/25/96), Ms. Fayalso expressed dissatisfaction with the cooperation component of thecurriculum.

Three weeks into the semester, she was pleased with her Core-Plusstudents’ abilities to work together, indicating that “they truly do listenand I think they have a lot better communication skills than some adultsI know” (Int. 1, 9/20/96). She attributed the students’ communicationstrengths to “their past experiences in the Core 1 classes” (the previousyear). However, at the end of October, she suggested that “the groups wereactually working better at the beginning of the year than they are now,”a phenomenon she related to the structure of the Core-Plus problems. Inparticular, she observed:

The groups are not discussing as much as they should be. They work apart until they havea question and then they hit their buddy and ask how he did it but then they go back toworking . . . or somebody says “I think this is the answer” and everybody says “Okay” andthey write it down. (Int. 2, 10/25/96)

This quote alludes to one of her main concerns – that many of the Core-Plus activities can be completed individually: “There is not a lot of workin these books that require a shared interest like a jigsaw puzzle whereeach person takes a piece and we will work together. Instead each personcan work alone” (Int. 4, 12/16/96). Although some investigations providedopportunities for Ms. Fay to have the whole class share the work, suchas when each student contributed to a large number of trials in a prob-ability experiment about Chinese birth policies, in general, she did notfeel that breaking up this work used everybody’s talents to contribute to asubstantial group product.


Attempts to Address Concerns

Despite her complaints about the over-structured problems, during myobservations, Ms. Fay did not change the Core-Plus investigations tomake them more open-ended by eliminating, altering, or adding prob-lems. Instead, during her whole-class instruction, which typically followedstudents’ group work, she adjusted the focus of some problems to attendto what she viewed as the two main negative implications of over-structured problems: They did not permit a conceptual emphasis, andthey did not encourage students to develop their own solution paths. Forexample, one Core-Plus investigation stresses several rules associated withgeometric transformations. In the whole-class discussions after studentshad completed the group activities, Ms. Fay emphasized repeatedly thatstudents should visually consider the graphs in order to perform therequested transformations, rather than rely solely on the rules. Whilemodeling exercise solutions, for instance, she stated to the class, “I findit easier to look at it visually” and “It doesn’t make sense to memorizeformulas.” Another way that Ms. Fay altered her review of problems thatfocused on procedures (such as multiplying matrices) was to ask studentsto explain verbally the steps involved, reminding them that “more than justthe answer is important.”

Ms. Fay’s whole-class instruction was primarily devoted to givingdetailed but very brief explanations of problem solutions. She empha-sized that the solutions were her own and that students might havedifferent approaches. For instance, she often prefaced her explanationswith comments such as “Here is howI did it . . . yours might be different.”These types of comments evidence that Ms. Fay wanted students to beaware that there could be multiple solution methods even within the limitedstructure of the Core-Plus problems. However, she did not capitalize onor elicit these potential differences from students. Her review of prob-lems typically consisted of writing her solutions on an overhead andexplaining them. When she asked students questions, she rarely pressedthem to justify their responses. Although she asked the class questions,such as “Do you agree with that?” or “Is that right?,”shegenerally decidedwhen answers were correct. Ms. Fay’s efforts to have students view prob-lems conceptually and to be aware of the possibility of multiple solutionmethods were restricted to discussion following, not during, students’group work. Most of her interactions with students in their groups involveddirect responses to their questions (e.g., “Is this right?” or “Am I doing thisright?”) and administrative tasks such as checking homework or discussinggrades. In fact, she expressed uncertainty about what to do while students


Figure 1. Two corresponding pieces of Ms. Fay’s dominoes game.

worked in groups, wondering, “Why do they need me here?” (Int. 1,9/20/96).

Ms. Fay did make some changes to the ways students worked on theCore-Plus problems in their groups. To address the individual nature ofproblems, she occasionally required students to work in smaller groupson the problems. Although not representative of a general approach to herconcern, one noteworthy example of an attempt to deal with her dissatis-faction with students’ cooperative work comes from her teaching of theinvestigation about transformation rules described above. At the end ofone class session, Ms. Fay divided the students into pairs for a game thatwas not part of the Core-Plus materials. She distributed sets of 14 domino-like pieces, each of which contained one exercise and one solution, andinstructed the teams to match up their pieces as quickly as possible. Todo so, students placed the domino containing the correct solution next tothe one with the corresponding exercise, as illustrated in Figure 1. Duringa subsequent interview, Ms. Fay pointed out that the students could “seethat if they work together then they will have a better chance of winningthan if we do it individually” (Int. 2, 10/25/96). In other words, successat this activityrequiredstudents to work jointly. This example illustrateshow Ms. Fay made an addition to compensate for what she viewed asa weakness of the curriculum materials. However, this type of additionwas not repeated during my observations, and Ms. Fay’s more typicalefforts to improve students’ group work involved verbally encouragingthem to work together, discuss problems in their groups, and value others’opinions.

Ms. Fay admitted that she did not develop many problems or projectsto supplement the suggested Core-Plus activities, and she explained thatshe felt very constrained in her ability to do so. One reason for notmaking more changes was that she did not know the materials wellenough to decide “which investigations and problems are crucial” (Int.5, 12/17/96) and moreover, her mathematical background was insuffi-cient. For example, she explained, “I don’t know if I had to come upwith a project on matrices that I could have . . . . I haven’t had matricessince Linear Algebra” (Int. 1, 9/20/96). In addition, her knowledge ofmeaningful mathematical problem contexts was lacking: “Many of theapplications my students are seeing for the first time, I am also seeing forthe first time.” She predicted that in her second year of implementation, she


would be more prepared to “eliminate some of the problems” and, basedon those problems, create “one big problem that students could organizeand solve” (Int. 4, 12/16/98).

Ms. Fay suggested that her inclination to develop and implementmore extensive projects than the Core-Plus materials provided was furtherrestricted by working in a department “where everybody tries to be at thesame place at the same time” (Int. 4, 12/16/96). Ms. Fay believed that“if everybody did their own thing then [she] would be teaching probablyvery differently” (Int. 5, 12/17/96). Her sense of obligation to colleagueswho preferred to maintain uniform movement through the materials wasmost problematic because she greatly valued giving students ample timeto explore and interact: “The more time that they have to work with thismaterial, the more time they have to interact with each other – time that isnot just taking notes and trying to figure out what the teacher wants” (Int.2, 10/25/96). During an interview, she referred to an occasion in which shesuggested to the other Core-Plus teachers that the students should work onone of the end-of-unit projects:

One of the Core-Plus activities. . . was to have students look through newspapers and findarticles or tables with cause-effect relationships. I said “Great, let’s break up the monotonygive them newspapers and each group do a presentation.” Everybody was like “Well wecan’t take a whole class hour to do that.” I’m like, “Why not?” (Int. 4, 12/16/96)

This example illustrates a struggle between Ms. Fay’s desire for moreexploratory activities and her colleagues’ concern for more efficientcontent coverage. This struggle contributed to Ms. Fay’s sense that shedid not fully own her Core-Plus classroom: “Inmyclassroom, we wouldbe doing some experiments or projects right now” (Int. 1, 9/20/96).Although the Core-Plus materials do suggest numerous in-depth projectsas unit summary activities, feeling rushed and behind some of the otherteachers contributed to Ms. Fay’s decisions at times to skip these morelengthy activities. Her preference was for these extended projects to bethe main components of the curriculum’s activities so that explorationand collaboration among students would be more integral to the program.Her disappointment with some aspects of her Core-Plus instruction aside,Ms. Fay felt that she was learning from the many challenges she facedwhile implementing the curriculum: “It’s really made me think and I willcontinue to think about what I feel is important in mathematics and what Iwant to stress in mathematics class” (Int. 4, 12/16/96).


DISCUSSION

The teachers in this study saw similar benefits of exploration and cooper-ation, but they interpreted the curriculum’s activities in terms of thosebenefits differently. Mr. Allen viewed the Core-Plus problems as chal-lenging and open to student interpretation – at timestoo open. On theother hand, Ms. Fay claimed that the problems were overly structured anddid not permit students to solve problems in their own ways or exploreconcepts sufficiently. Ms. Fay’s comments about Core-Plus activities, suchas “Students are led through the problem,” bear remarkable similarity toMr. Allen’s comments about the traditional curriculum’s exercises.

Both teachers expressed disappointment with the quality of theirstudents’ efforts to discuss and debate with one another as they workedin small groups. Moreover, both teachers attributed difficulties with groupwork in their classrooms to the nature of the mathematics problemsin the curriculum materials. However, the teachers’ contrasting viewsof the Core-Plus problems gave rise to different descriptions of theirspecific difficulties with students’ group work: Ms. Fay suggested thatthe overly structured problems could be completed individually and thuslimited students’ opportunities for discussion and cooperation, whereasMr. Allen believed weak group interactions resulted from open-endedactivities that sometimes placed too heavy demands on students to developtheir own solution strategies. These explanations illustrate the teachers’identification of different connections between the underlying structureof mathematics problems and the quality of students’ work with thoseproblems.

During their instruction with the materials, each teacher added occa-sional activities and experimented with group formats. For instance, eachteacher changed from groups to pairs at times and discussed group prob-lems as a whole class. However, neither teacher explicitly altered thewording, number, or sequence of problems on which students worked intheir groups. The primary way in which both teachers addressed concernswas through their own interactions with students. In Mr. Allen’s firstyear, these interactions aimed to communicate to students the import-ance of developing and exploring multiple approaches to mathematicalsituations, and occurred while students worked on problems in theirgroups. Like Mr. Allen in his second year, Ms. Fay typically savedher comments and questions for students until whole-class instruction,at which time she encouraged students to rely on their own under-standings rather than on rules. In other words, both teachers used theirinteractions with students to communicate messages about their concerns


with students’ work on the mathematics problems in the curriculummaterials.

As these results illustrate, curriculum implementation consists of adynamic relation between teachers and particular curricular features. Thisnotion is consistent with warnings that reform recommendations andassociated curriculum materials cannot and do not bring about changealone – educational change is a complex human endeavor (Cooney, 1988;Freudenthal, 1983). Although the Core-Plus materials were designed withthe intent of supporting student collaboration and exploration and theteachers were motivated to enact these themes, for the teachers to do sowas neither a straightforward process, nor always possible. Concurrentconsideration of Mr. Allen and Ms. Fay’s classroom experiences gives riseto many important themes and associated questions related to curriculumimplementation and classroom reform. Three of these themes are discussedbelow.

Teacher Control of Student Learning

One of the most prominent themes of Mr. Allen’s Core-Plus instruc-tion was his interest in structuring and controlling students’ engagementwith mathematics. Despite his belief that students benefit from exploringand discussing mathematics in their groups, he struggled with whetherstudents would learn appropriate mathematics without more direction fromeither himself or the curriculum materials and review sheets. Mr. Allen iscertainly not alone in his desire to structure or direct students’ learning.When experienced teachers, particularly those with a history of learningand teaching in traditional classroom settings, attempt to reform practicethrough the implementation of novel curricula, concerns about classroomauthority are prevalent (Cooney & Shealy, 1997; Romberg, 1997; M.Wilson & Goldenberg, 1998). Mr. Allen seemed most comfortable when heor the curriculum materials explicitly outlined and thus held the authorityfor what students should be learning.

Though not as immediately apparent in her case, Ms. Fay’s instruc-tion illustrates the issue of authority as well. Her view of the Core-Plusproblems indicates that to a large degree she believed students could learnsignificant mathematics without explicit direction from the curriculum orthe teacher. However, when students asked her questions or requestedher direction, she typically responded directly rather than press studentsto make decisions for themselves. In this sense, she seemed satisfied toallow students to view her as an authority for determining right and wronganswers. One factor related to this tendency might be Ms. Fay’s uncertaintyabout what to do when students worked in their groups. This uncertainty


highlights the need for teachers to rethink how their jobs in the classroommust change to coincide with novel types of student activities.

These issues suggest the value of continued investigation of howteachers involved with reform might renew their sense of efficacy. That is,as teachers change classroom practices, what changes occur in their senseof the ability to affect student learning? Teachers’ sense of efficacy canpowerfully impact student learning, but the conceptions that support manyteachers’ sense of efficacy may be rooted in models of learning that are notconsistent with current reforms. As Smith (1996) explained, teachers havetraditionally felt most self-efficacious when they tell students what theyneed to know. How then can teachers come to feel efficacious as they adoptnew forms of mathematics instruction? Smith identified four potential sitesfor re-establishing self-efficacy: choosing problems, predicting studentreasoning, generating and directing discourse, and judicious telling. Thefocus of the current study raises the question of the self-efficacy of teacherswho are implementing innovative curriculum materials: Can teachers moortheir sense of efficacy in the areas suggested by Smith when they imple-ment curriculum materials, and if so, how? Because the structure ofcurriculum materials may both support and constrain teachers’ efforts tochange instruction, as discussed below, the potential role of curriculumimplementation in teachers’ redevelopment of self-efficacy remains acritical question.

Opportunities to Personalize Instruction

Both Mr. Allen and Ms. Fay faced struggles with cooperation and explor-ation in their Core-Plus instruction, and both attributed their struggles tofeatures of the mathematics problems in the curriculum materials. Despitetheir criticisms of the materials, neither teacher significantly changedthe nature of the problems on which students worked in their groups.Rather, both teachers used their interactions with students to address theirconcerns. Why did the teachers not change the Core-Plus problems andactivities to better suit their personal goals? Ms. Fay frequently expressedthat she was notable to personalize her instruction to a greater extent.Although Mr. Allen never expressed this sentiment, he too may havebeen confined in his ability to make further changes to the curriculum’sactivities.

One constraint to altering the materials was suggested by Ms. Fay. Shelamented that the culture of the mathematics department, which encour-aged uniform implementation of the materials, confined teachers’ abilitiesto develop individual teaching styles. School culture shapes how teachersmake sense of their decisions and actions in relation to perceived shared


understandings of the faculty (Feiman-Nemser & Floden, 1986; Gregg,1995). Like many of Mr. Allen’s mathematics department colleagues, heviewed the curriculum as material to cover. Though he expressed littleconcern, he admitted that his decisions to teach in certain ways were ofteninfluenced by departmental pressures. When teachers feel restricted in theirattempts to make adjustments to the curriculum, they may face difficultchallenges in adapting the curriculum to suit the needs of their studentsand to best fit their own goals and strengths.

It may also be the case that the curriculum materials themselvespresented a form of constraint to the teachers. Recall that Mr. Allensuggested that the Core-Plus materials had accomplished the developmentof mathematical situations for students to explore, leaving the teacher tiedto those situations. He highlighted an interesting notion: When a reform-minded teacher uses traditional materials in the classroom, he or she maybe afforded more room for personalization because the goals of the mate-rials are so different from his or her own goals. Because reform-orientedcurriculum designers accomplish much of the alteration of mathematicalcontent and activity in their production of materials, teachers with strongand innovative visions may experience a profound loss of previously-heldopportunities to personalize their instruction.

The issue of constraints upon teachers who implement curriculummaterials is particularly interesting when related to Prawat’s (1992) iden-tification of one impediment to teacher change:

Instead of viewing students and curriculum interactively . . . teachers tend to regard themas similar factors that somehow must be reconciled . . . . Teachers focus on the packagingand delivery of content, instead of on more substantive issues of knowledge selection andconstruction. (p. 389)

This imperative begs the question of whether the detailed design ofcurriculum materials encourages teachers to focus primarily on pack-aging and delivery. That is, to what extent do curriculum materials permitteachers to view the learner and curriculum as interactive?

If teachers are to view students and curriculum dynamically, they needto learn to make classroom-based developments within the curriculumimplementation process. After all, curriculum developers may wish tocreate certain learning experiences for students, but they cannot fullyanticipate how particular students will interact with the mathematicalactivities. If teachers are to extend their focus beyond delivery, thedesign of curriculum materials may require substantial changes. Balland Cohen (1996) have suggested that curriculum materials designedwith the intention to engage teachers as well as students “could helpteachers to learn how to listen to and interpret what students say, and to


anticipate what learners may think about or do in response to instruc-tional activities” (p. 7). Given this potential for curriculum materials todraw teachers’ attention to details of student learning, curriculum imple-mentation may indeed offer first-hand experiences that can help teachersdevelop constructivist views of mathematics learning and enact associatedpractices.

Tensions Between a Curriculum’s Philosophy and Teachers’ Visions

Past research reports have described teachers whose traditional concep-tions interfered with their abilities to implement novel curricula (e.g.,Cohen, 1990; S.M. Wilson, 1990). These teachers’ struggles have beenattributed to their misunderstanding or lack of appreciation for the philo-sophies underlying the curriculum’s proposed activities. The present studyoffers an example of a distinctly different type of conflict betweenteachers’ conceptions and curriculum design. The two teachers in thisstudy seemed to desire problems that required more extensive cooperationamong students than they believed was provided in the materials. Duringtheir work on the Core-Plus investigations, some groups in both teachers’classes were observed working on problems individually even though theywere seated in groups. In addition, in his second year, Mr. Allen sometimesinstructedstudents to work individually on the investigations that had beenintended for groups. These observations support Ms. Fay’s concern thatwork on the Core-Plus problems did not always require substantial collab-oration among students. These results emphasize how tensions betweenteachers and curriculum may not necessarily be based on deficiencies inteachers’ conceptions, but may instead develop through complex interac-tions between teachers’ goals and specific characteristics of the curriculummaterials.

The dynamic interaction between teachers and curriculum has potentialto be highly educative for teachers. What conceptions of mathematics,teaching, and students are most helpful to teachers as they attempt touse curriculum materials to teach in reformed ways? What qualities ofcurriculum materials, and what underlying philosophies, most produc-tively help teachers change their practices? These questions suggestfruitful areas for continued research activity that will extend our under-standing of how teachers and curriculum materials interact in the processof reform.


IMPLICATIONS FOR MATHEMATICS TEACHERDEVELOPMENT

As illustrated by the findings of this study, teachers’ process of changedraws on their conceptions of mathematical content, students, pedagogy,and the department contexts in which they work. For curriculumdevelopers, these results have important implications: In the continueddevelopment of reform documents and materials, the visions and meaningsthat teachers attach to themmust be carefully considered. To better supportteachers in facilitating classroom activities that include student explorationand cooperation, curriculum materials must not only recognize but alsoexploit and profit by the teacher aspect of the teacher-curriculum dynamic.

This study’s results also present considerable challenges to teachereducators and others concerned with the professional development ofteachers. We bear responsibility to create opportunities for teachers, bothbeginning and experienced, to develop conceptions that will help them todeal effectively with reform themes in their mathematics classrooms. Likemany inservice teachers, preservice teachers often possess weak know-ledge and narrow views of mathematics and mathematics pedagogy –conceptions that are bolstered by years spent as students in traditionalclassrooms (Brown, Cooney & Jones, 1990; Lortie, 1975; A.G. Thompson,1992; Zeichner & Gore, 1990). Professional development activities muststrive to assist inservice and preservice teachers in making sense of themany discrepancies between current reform recommendations and theirown classroom experiences as students and teachers.

One area of mathematics teacher education that warrants increasedattention is the preparation of teachers to enter into a more dynamic rela-tionship with the curriculum. It is notable that in this study both teacherswere dissatisfied with the structure of activities presented in the Core-Plusmaterials, but neither teacher changed the problems on which studentsworked during class sessions. Ms. Fay and Mr. Allen’s treatment of thecurriculum as fixed suggests that teachers may struggle to conceive ofcurriculum as an adaptable guide that permits and encourages alterationswith respect to the demands of particular students. Teachers likely requiresupport not only in coming to recognize the need to adapt curriculum, butalso in learninghow to adapt it.

Of course, even if Ms. Fay and Mr. Allen had viewed curriculum moredynamically, it is not necessarily the case that student learning would havebeen more central to their thinking. In fact, neither teacher displayed evid-ence of recognizing student understanding as what Goldsmith and Schifter(1997) termed “both the guide and the goal of their practice” (p. 29). Both


Ms. Fay and Mr. Allen seemed to be more concerned with specific problemand lesson designs than with details of their students’ thinking. Ironically,had Mr. Allen made changes to the curriculum’s printed problems, hisinstruction may have less successfully enacted reform themes. Based onhis conception that students need to follow more structured paths towardmathematical understandings, his tailoring of the materials would mostlikely have involved more direction to the mathematics problems, limitingstudents’ opportunities for exploration. This example also suggests thatteacher education activities must aim to encourage teachers to conceive ofstudents as independent learners who are capable of and can benefit fromtheir own development of powerful mathematical strategies and theories.

There are likely many useful materials and contexts for extendingteachers’ conceptions in the areas for professional growth to which Mr.Allen and Ms. Fay’s experiences draw our attention. One potentially richcontext for extending teachers’ conceptions involves the use of reform-oriented K-12 curriculum materials in preservice teacher education andother professional development activities (Lloyd & Frykholm, 1998).First-hand experiences workingas studentson the investigations presentedin curriculum materials may prompt teachers to expand their conceptionsof teaching and learning. Although Mr. Allen and Ms. Fay were able toparticipate in some of these experiences second-hand by observing theirstudents’ work with the Core-Plus materials, they may have benefitedalso by engaging with the materials more personally. Such experiencesmay also allow teachers to develop new understandings of mathematicalconcepts and real-world contexts that they did not encounter in their ownmathematics education. Furthermore, use of teachers’ guides associatedwith both innovative and traditional materials to plan and facilitate lessonsmight engage teachers in recognizing and making pedagogical choices,thus explicitly initiating a more dynamic relationship with curriculum.

Innovative curriculum materials are certainly not the only source ofreform-oriented pedagogical themes. Videos and cases are particularlyappealing teacher education tools because they offer detailed images ofwhat reformed mathematics teaching and student learning can look like.Davenport and Sassi (1995) have suggested that images of classroomdiscourse, for instance, are highly valued by teachers at early stages in thechange process. Moreover, video explorations and case-based discussionsallow focused analysis of very specific issues, concepts, and practices.For instance, videos and cases can provide vivid representations of theclassroom successes and difficulties faced by teachers like Ms. Fay andMr. Allen. Through collaborative analysis of classroom vignettes, teacherscan consider and develop empathy for multiple perspectives on teaching


and understanding (Barnett, 1991). In doing so, teachers may learn tomore carefully observe and listen to students, and as a result, expand theirconceptions of students and how they learn mathematics.

Challenges to teachers’ conceptions about mathematics teaching mayalso occur in school settings, as was the case in this study. Implementationof the Core-Plus curriculum materials compelled both teachers to reflecton their past and present classroom instruction. Mr. Allen’s instructionwith the curriculum materials permitted him to view his traditional prac-tices more critically and to begin to identify and articulate his own desirefor new approaches to mathematics teaching. Despite her dissatisfactionwith some aspects of her implementation of the curriculum materials,Ms. Fay, too, valued the opportunity to contemplate different mathe-matics teaching practices. Because reform-oriented experiences demandthat teachers rethink what it means to teach, curriculum implementationoffers a promising way to initiate and support significant professionalgrowth in many areas related to mathematics, teaching, and learning.

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Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburg, VA 24061-0123USAe-mail: [email protected]

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