theory into practice

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Megan LoefEranke Eham Kazemi Learning to Teach Mathematics: Focus on Student Thinking T HE FIELD OF MATHEMATICS EDU'CATION has miade great strides in developing theories and re- search-based evidence about how to teach elemen- tary school mathematics in a way that develops students' mathematical understanding. Much of this progress has grown out of research projects that engage teachers in learning to teach mathemiatics. These projects have not only shown what is possible for teachers and students but have also provided in- sight into how to support teachers in their own leamn- ing. We use one particular research and development project, Cognitively Guided Instruction (CGI), as an illustration of how theory and research inform the teaching and learning of mathematics. The key to CGI has been an explicit, consis- tent focus on the development of children's math- ematical thinking (Carpenter, Fennema, & Franke, 1996). This focus serves as the guide for under- standing CGI's contributions as well as its evolu- tion.' Initiated over 15 years ago by Carpenter, Fennema, and Peterson, CGI sought to bring to- gether research on the development of children's mathernatical thinking and research on teaching (Carpenter, Fennema, Peterson, Chiang, & Loef, 1989). This research project engaged first-grade teachers with the research-based knowledge about the development of children's mathematical think- Megan Loef Franke is associate proJessor of education at the University of Califoqrnia, Los Angeles: El.ham Kazemi is assistant professor of education at the Uni- versity oJ Washington, Seattle. ing. We studied the teachers' use of this knowl- edge within their classroom practice and examnined the changes in teachers' beliefs and knowledge. This work created a rich context for our own learning as well as for the teachers. We learned a great deal about teachers' use of children's think- ing, teacher learning, and professional development. Since the initial project, our understandings and our work with teachers and students have contin- ued to evolve. We now work with K-S teachers across a number of different content areas (Franke & Kazemi, in press; Kazemi, 1999). Our learning reflects much of the learning occurring in the field. 2 In this article, we divide our learning about CGI into two sections. These sections represent an ongoing shift in our thinking from a consistent cog- nitive paradigm to a more situated paradigm. These different notions of learning have influenced our views of student and teacher learninig andl how to maximize that learning. We tell this story by de- scribing our initial CGI work and what we learned about students and teachers. We then elaborate by describing and characterizing our current work and how that has influenced our learning about creat- ing a focus on students' mathematical thinking within professional development. A Cognitive Beginning CGI began by providing teachers with knowl- edge, derived from research, about the development of children's mathematical thinking and letting the THEORY INTO PRACTICE, Volume 40, Number 2, Spring 2001 Copyright (D 2001 College of Education, The Ohio State University 0040-5841/2001$1.50

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  • Megan LoefErankeEham Kazemi

    Learning to Teach Mathematics:Focus on Student Thinking

    T HE FIELD OF MATHEMATICS EDU'CATION has miadegreat strides in developing theories and re-

    search-based evidence about how to teach elemen-tary school mathematics in a way that developsstudents' mathematical understanding. Much of thisprogress has grown out of research projects thatengage teachers in learning to teach mathemiatics.These projects have not only shown what is possiblefor teachers and students but have also provided in-sight into how to support teachers in their own leamn-ing. We use one particular research and developmentproject, Cognitively Guided Instruction (CGI), as anillustration of how theory and research inform theteaching and learning of mathematics.

    The key to CGI has been an explicit, consis-tent focus on the development of children's math-ematical thinking (Carpenter, Fennema, & Franke,1996). This focus serves as the guide for under-standing CGI's contributions as well as its evolu-tion.' Initiated over 15 years ago by Carpenter,Fennema, and Peterson, CGI sought to bring to-gether research on the development of children'smathernatical thinking and research on teaching(Carpenter, Fennema, Peterson, Chiang, & Loef,1989). This research project engaged first-gradeteachers with the research-based knowledge aboutthe development of children's mathematical think-Megan Loef Franke is associate proJessor of educationat the University of Califoqrnia, Los Angeles: El.hamKazemi is assistant professor of education at the Uni-versity oJ Washington, Seattle.

    ing. We studied the teachers' use of this knowl-edge within their classroom practice and examninedthe changes in teachers' beliefs and knowledge.

    This work created a rich context for our ownlearning as well as for the teachers. We learned agreat deal about teachers' use of children's think-ing, teacher learning, and professional development.Since the initial project, our understandings andour work with teachers and students have contin-ued to evolve. We now work with K-S teachersacross a number of different content areas (Franke& Kazemi, in press; Kazemi, 1999). Our learningreflects much of the learning occurring in the field.2

    In this article, we divide our learning aboutCGI into two sections. These sections represent anongoing shift in our thinking from a consistent cog-nitive paradigm to a more situated paradigm. Thesedifferent notions of learning have influenced ourviews of student and teacher learninig andl how tomaximize that learning. We tell this story by de-scribing our initial CGI work and what we learnedabout students and teachers. We then elaborate bydescribing and characterizing our current work andhow that has influenced our learning about creat-ing a focus on students' mathematical thinkingwithin professional development.

    A Cognitive BeginningCGI began by providing teachers with knowl-

    edge, derived from research, about the developmentof children's mathematical thinking and letting the

    THEORY INTO PRACTICE, Volume 40, Number 2, Spring 2001Copyright (D 2001 College of Education, The Ohio State University0040-5841/2001$1.50

  • FrtanIke anad KazemiLearning to Teach Mathematics

    teachers decide how to make use of that knowl-edge in the conitext of their own teaching practice.We shared this knowledge with teachers in a rangeof mathematical content areas. However, what weshared was not a random set of ideas but rather anorganized set of frameworks that delineated thekey problems in the domain of mathematics andthe strategies children would use to solve them.The frameworks provided teachers the opportunityto understand how this knowledge about the devel-opment of children's thinking fits together so theteachers could make it their own (Carpenter, Fenne-rna, & Franke, 1996; Fennema et al., 1996).

    The teachers discuss CGI as a philosophy, away of thinking about the teaching and learning ofmathematics, not as a recipe, a prescription, or alimited set of knowledge. CGI teachers engage insense making around children's thinking. They con-tinually evaluate their understanding, adapt and buildon their knowledge, and figure out how to make useof it in the context of their ongoing practice.

    Although it fails to completely capture whatCGI is about, understanding the k:nowledge weshare with teachers and how it fits into organizedframeworks that are related to one other is criticalto understanding CGI. Constructing models of chil-dren's thinking entails focusing on organized, prin-cipled knowledge about problems within the variouscontent domains, along with the range of strate-gies children often use to solve the problems. Thestrategies discussed with the teachers are relatedto one another, both within and across problems.The strategies build within problems, in terms ofmathematical sophistication, while across problemsclasses of strategies exist. For example we can takea imnultiplication problem and detail (a) the strate-gies students will most likely use to solve it, (b)how those strategies build on each other, and (c)how the solutions relate to solutions of other group-ing problems and problems using other operations.

    To solve the grouping problem presented inFigure 1, a child could use a range of strategies.However, these strategies often fall into one offour classes. A child could initially solve the prob-lem by direct modeling the action in the problem.Here the child physically represents the number ofboxes and the number of crayons in each box. Thecounting strategy builds on the direct modelingstrategy as the child continues to represent the ac-

    tion in the problem but does so by physically keep-ing track of the number of groups and counting upthe nurmber in each group using skip counting. Thederived fact strategy no longer models the actionin the problem but rather makes use of what thechild already knows about multiplication facts.Recall shows the child knows the fact and doesnot need to represent the action.

    Grouping problem:Tessa has 6 boxes of crayons. Each box contains 5 cray-ons. How many crayons does Tessa have altogether?Possible solutions:Direct modeling: Draws out 6 boxes and miakes 5 miarksin each box. Counts the total number of marks.Counting strategy: Counts by 5's (5,10,15,20,25,30)keeping track of the 6 numbers in the counting se-quence on his or her fingers.Derived fact: Knows 5 times 5 is 25 so 6 times 5would be one more 5 that would be 30.Reca lied fact: Knows that 6 times 5 is equal to 30.Figure 1. A grouping problem and possible strate-gies for solving it.

    Knowing the sequence of strategies allowsteachers to interpret why a particular problem maybe difficult for a child. If we changed the numbers inthe problem to six boxes and seven crayons in eachbox, we would not create mnuch difficulty for a childwho was direct modeling. However, we might for achild using a counting strategy, as the child may notbe as efficient at counting by sevens and may use aless efficient counting strategy or direct model. Know-ing the sequence of strategies enables teachers to cre-ate problems that challenge their students' thinking.At the samie time, it avoids engaging students in strat-egies that do not make sense to them.

    The parallels in strategy development acrossproblem domains help teachers understand themathematical ideas associated with those patternsand interpret the mathernatical understanding of thechildren in their own classrooms (Carpenter et al,1996). For instance, teachers do not often thinkabout posing a problem such as the following:Keisha had some markers. Her grandmother went shop-ping and bought Keisha 6 more markers. When Keishacounted all of the markers she now had, she found thatshe had 14 markers. How manv markers did Keishahave before her grandmother gave her the new ones?

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    Knowing that direct modeling is the least so-phisticated strategy for solving miiany addition andsubtraction problems enables a teacher to see thatthe direct miodeling approach will not make it easyfor children to solve this particular problem. Thosewho use direct modeling will not know how nmanycounters to put in their initial set and thus may notknow how to get started. A child would need tobegin by putting out some counters (the number ofcouniters the child thinks represents the "somee" inthe problemn), add 6 to that set, and count to see ifthere are 14 counters altogether. If not, the childwould needl to adjust the initial set and begin theprocess again.

    Understanding the relationship between aninitial unknown quantity and the direct modelingstrategy allows the teacher to think beyond thisparticular problemi and about what can make prob-lems difficult for children to solve and why. Asteachers engage in listening to their students' think-ing, they learn more about possible problems topose, strategies to expect, and relationships thatexist between problems and strategies. This newknowledge is connected with their existing knowl-edge as they continue to elaborate and build theirframeworks. COI teachers have interacted withknowledge about children's thinking in the dormainsof addition and subtraction, miultiplication and divi-sion, place value, algebra, fractions, and geometry.

    Results of early CGI workOur goal in our initial CGI work was to share

    this research-based knowledge about the develop-ment of childreni's mathematical thinking withteachers, provide them an opportunity to thinkabout what it might mean for their practice, andtheni watch to see what teachers did with the knowl-edge and how this affected their students' learn-ing. The teachers surprised us. While we anticipatedteacher learning, changes in teachers' classroompractices, and resulti-ng student learning, we didnot anticipate the scope of the learning that oc-curred or the ownership the teachers took in dis-serninating CGT.

    Teachers' expectations of their students'mathematical understanding changed dramatically.First-grade teachers saw that their students, whetherin a midwestern city or a large urban center, couldsolve word problems often omitted fromr. the cur-

    riculum until third grade. Teachers recognized thattheir students solved the posed problems using arange of mathematical strategies, strategies theynever expected to see in their own students. Teachel-ers realized they needed to listen to their students'mathematical explanations, create strategies andquestions to elicit those explanations, and under-stand enough about children's thinking and thecontent to know what to do with what they heard.

    In all of our work with K-5 teachers, we havefound thei to be surprised by their students' math-ematical thinking and by how often the studentssaid what the research indicated they would say.Teachers began listeniing more and telling less, reg-ularly eliciting students' mathematical thinking andanticipating multiple strategies. While not all teach-ers found it easy to use what they heard from stu-dents to make instructional decisions, most teachersfound ways to encourage students to elaborate theirmathematical thinking.

    The students' learning paralleled that of theteachers. The students in CGI classrooms could solvea wider variety of word problems, used a wider rangeof strategies, and knew their number facts at a betterrecall level than their control group counterparts. Inour longitudinal study (Fennema et al., 1996), wefound that students on average made a grade levelgain in achievement: The second graders performedas the third graders had at the beginning of the study,and so on. Additionally, the students in CGI class-rooms reported being more confident and better ableto uLnderstand mathematics.

    Listening to students' mathematical thinkinghad another benefit. It transformed teachers intolearners. They learned in the context of their prac-tice about the teaching and learning of mathemat-ics and became engaged in what Richardson (1990,1994) termiis "practical inquiry." Teachers strug-gled to make sense of the development of theirstudents' mathematical thinking and how that re-lated to their instructional decisions. Professionaldevelopment happened inside and not just outsideteachers' classroom doors.

    Generative growthAs we st.udied teachers involved in our longitudi-

    nal study 4 years after the professional developmnentended, we had the opportunity to learn more aboutpractical inquiry and teacher growth. We were able

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    to track how teachers had sustained themselves orcontinued to grow professionally since our last datacollection in the longitudinal study. We termed thepatterns we observed generative growth.

    We drew from the work on the development ofstudent understanding in mathematics to conceptual-ize ongoing teacher learning. One distinguishing char-acteristi of learning with understanding is that it isgenerative (Carpenter & Lehrer, 1999; Greeno,1988; Hiebert & Carpenter, 1992). Generativityrefers to individuals' abilities to continue to add totheir understanding. When individuals learn withunderstanding, they can apply their knowledge tolearn new topics and solve new and unfamiliarproblems. When individuals do not learn with un-derstanding, each new topic is learned as an isolat-ed skill that can be used only to solve problemsexplicitly covered by instruction.

    A second defining characteristic of learning withunderstanding is that knowledge is rich in structureand connections. When knowledge is highly struc-tured, new knowledge can be related to and incorpo-rated into existing networks of knowledge. This isnot a matter of adding knowledge incrementally ele-ment by element. Learning with understanding goesbeyond connecting new knowledge to existing knowl-eclge and includes reorganizing knowledge to createrich integrated knowledge structures. A third factorin learning with understanding is that learners seelearning as driven by their own inquiry. Carpenterand Lehrer (i999), for example, propose that learn-ers must perceive their knowledge as their own, be-lieving that they can construct knowledge throughtheir own activity.

    Using this way of conceptualizinig generativegrowth allowed us to examine the degree to whichteachers were engaged in generative growth and toadd details to what generative growth looks like inCGI teachers. We found that indeed a number ofthe CGI teachers were engaged in generativegrowth. These teachers could detail their students'mathematical thinking, had a way of organizingthose details that highlighted the principles under-lying the thinking, and saw the knowledge as theirown to adapt and create.

    Our initial definition of generative learningfocused on characterizing the nature of the knowl-edge teachers were continually developing. It wasclear that teachers were redefining how they inter-

    acted with their students, with their colleagues, andwith mathematics. Yet in our initial analysis ofteacher learning, we did not explicitly theorizeabout the nature of teachers' practical inquiry orthe way it changed their way of being in the class-room and beyond.

    Considering a Situated ApproachAs we began to work with teachers and ex-

    plicitly support the development of generativity,we found ourselves limited by our cognitive ap-proach. Teachers were learning along with eachother, their students, and us. We began to accountfor and theorize about teachers' practice as occur-ring in their classrooms, professional developmentsessions, and conversations with each other, theirprincipal, and parents. Situated theory provided alens for beginning to understand how a focus onstudents' mrathematical thinking supported teacherand student learning and for creating professionalopportunities that would enhance that learning.

    Drawing on the work of Lave, Wenger, Cobb,Wertsch, Rogoff, and others, we now struggle torecast our thinking about teacher learning and gen-erativity to highlight learning within a comrnunityof practice (Cobb, 1999; Lave, 1996; Lave &Wenger, 1991; Rogoff, 1994, 1997; Wenger, 1998;Wertsch, 1991). Lave, Wenger, and Rogoff havedescribed learning as being evident in shifts in par-ticipation in communities of practice. A shift inparticipation does not rmerely mark a change in aparticipant's activity or behavior. It involves atransformation of roles and the crafting of a newidentity, one that is linked to new knowledge andskill.

    The situated perspective for us highlightsteachers' appropriation of knowledge as they par-ticipate in comnmunities of practice. The communi-ties of practice become critical to understandingteacher learning. The artifacts that are used as weengage together can afford or constrain participa-tion and support or not support teacher learning.3We became interested in understanding teachers'changing participation and identities as they en-gaged in a particular community of practice, wheremathematics is central, and the ways in which tools(in our case, student work) support or constrainthe practices and reasoning of those participating

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    (Saxe, 1991; Wertsch & Kanner, 1992). This viewof teacher learning pushed our initial notionis ofgenerativity.

    Rethinking professional developmentTaking on this more situated stance pressed us

    to rethink our engagement with teachers and createprofessional development opportunities that reflectedthis perspective. The professional developmentteachers engage in is a part of their practice. Pro-fessional developtnent sessions constitute a sitewhere teachers and professional developers engagein learning togethqer. Understanding the professionaldevelopment community of practice provides anopportunity to better understand teachers' learningand legitimizes the work of teaching occurringwithin the professional development.

    In applying our situated stance, we created anew community of practice with the teachers andadministrators from a single school site and ourselves(the professional development/research teatn). Thefocus of this particular community of practice re-mained on understanding the details of students'mathematical thinking; however, here we focusedon the mathematical thinking of teachers' own stu-dents and tracked the emergence of that thinkingwithin the context of the teachers' classroom prac-tice. We conjectured that a focus on teachers' ownstudents would facilitate a compelling link betweenthe classroom and professional development coI-munities (Franke & Kazemi, in press).

    Teacher work groupsWe began our work by getting to know the

    existing community. We spent time in the schooltalking with teachers, teaching a university mathe-matics m ethods course, and helping out when need-ed. A year later we approached the teachers andadministrators about working together on a consis-tent basis. We created four teacher work groups,each consisting of approximately 12 teachers frtmacross grades K-5 who worked together on a month-ly basis. We hoped to create conditions whereteachers could establish a new dimension of theirpractice centered on examining student work andtheir classroom practices.

    Each month the teachers posed a work-groutpmathematics problem to their students and brought

    their student work to the meeting. A typical work-group problem read as follows:

    Yvette collects baseball cards. She had 8 Dodgercards in her collection. How many miore Dodger cardsdoes Yvette need to collect so that she will have 15Dodger cards altogether?

    The teachers were given this problem and the sameproblem with the numibers 67 and 105. They couldadapt the numbers and the context but were askednot to change the structure of the problem. Thestudent work provided the basis for the work-groupdiscussion. The teachers shared their students'solutions to the problem, compared the mathemat-ical understanding of each strategy and how thestrategies may build on each other, and describedhow particular strategies were elicited and devel-oped. The teachers were asked to carefully detailthe student thinking, create ways to make sense ofthat thinking, and situate the students' thinkingwithin particular classroom interactions.

    As we facilitated the work-group meetings,our goal was to scaffold the conversation in a waythat allowed the teachers to create a common lan-guage and focus for talking about the teaching andlearning of mathematics. Initially the group dis-cussions were dialogues between the facilitator andone memnber of the work group. Over time the dis-cussions became conversations among the mem-bers of the group. We also engaged the teachersamnd administrators outside the context of the work-group mneeting. We spent time in teachers' class-rooms, walked the hallways, and visited thelunchroom. We wanted to get to know the teachersand their students. We wanted to be seen as a partof the school community.

    In visiting classrooms, we helped the teach-ers see that we wanted to get to know their stu-dents, not evaluate their teaching. We stopped induring math tirne to see what their students weredoing and get some sense of who the students wereand what they were thinking, rarely staying for anentire lesson. We also stopped by each month pri-or to the work-group meeting to give the teachersthe problem for the next meeting. While we couldhave provided the problem at the end of the previouswork-group meeting, we used this as another pointof contact and an opportunity to engage in conver-sation with the teachers outside the context of the

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    meeting itself. We often stopped in the hallway tochat with teachers about their students' mathemat-ical thinking. We found that teachers began talk-ing more with each other about what their studentswere doing and saying during math class.

    We also spent time with the principal. Sheunderstood our work with the teachers in terms ofboth content and process. She valued our partici-pation in the school community and paved the wayfor community building. She used school meetingtime for the work-group meetings so teachers wouldnot give up their own time or part of their schoolday with students. And she evaluated the teachersby having them bring student work to her threetimes during the year, rather than observing in theirclassrooms. So both the details of structuring themeetings and the substance of her work with theteachers supported the development of the com-munities of practice.

    Results from work groupsWorking with one school over a 4-year period

    enabled us to characterize the communities of prac-tice and shifts in participation within them. Theteachers in the work groups became much better atdetailing students' mathematical thinking. They notonly detailed the strategies the students used butexpanded the details to include the pedagogicalpractices that supported that student thinking. Thus,in detailing the student thinking for the group,teachers included rich descriptions of the questionsasked to elicit that thinking, the responses of otherstudents, and the work that came before the sharedinteraction.

    The teachers also developed ways of talkingwith each other about the relationships across thestrategies that highlighted the mathematical ideasbeing developed. Teachers could talk about the re-lationship between a strategy used in multiplica-tion and a strategy a child used in solving amulti-digit addition problem, and could describethe common, place value understanding demonstrat-ed across the strategies.

    Beyond the ways teachers talked about stu-dent thinking, we noted shifts in the stances teach-ers took in their participation. Teachers began to seethemselves as able to contribute to the group asteachers of mathematics with knowledge and skillsto share. One teacher in our initial conversation

    described herself as having nothing to offer herwork group. Her expertise was in literacy, and shefelt she knew little about the teaching of mathe-matics. She was there to listen, and she opted notto pose the initial work-group problem to her class.

    By the end of the year, however, she not onlyconsistently posed the work-group problem to herclass but she also used what she learned from thestudents to pose a series of new problems. Shebrought student work from all of her problemsposed to the work-group meeting, and shared anddiscussed insights with her colleagues. She nowleads the group as a facilitator. The teachers, likethe one described, developed a sense of themselvesas learners. They used the work group as a placeto talk about their experimentation with the prob-lems and describe and enrich their learning fromtheir experimentation.

    Teachers' participation in their classroomsalso changed. Teachers began to see the work-groupproblem as a part of their ongoing practice ratherthan something separate they had to get done forthe meeting. They developed ways to elicit andlisten to their students' mathematical thinking. Theyintegrated the work-group problem into their class-room practice and saw experimentation as a partof their teaching practice. Yet, the changes in theirpractice were not limited to their classrooms or thework groups themselves. Teachers began to talkwith each other outside of the work groups aboutstudents' thinking and their teaching practices.They took on leadership roles within the schooland the district. Four of the teachers joined thedistrict math committee, and proposed and changedthe district math standards. The teachers developedidentities as mathematics teachers with expertise tooffer. They reported they could no longer teach at aschool without the perspective on teaching and learn-ing of mathematics that existed at their school. Andmost importantly, the work groups continue, 2 yearsafter we discontinued our participation.

    We learned that generative growth is not abouta set of characteristics the teacher possesses; it isabout teachers' developing knowledge and skills andthe identities that evolve in relation to the knowledgeand skills. Teachers engaged in generative growthwithin our project all possessed substantial knowl-edge about the development of students' mathemati-cal thinking. They could detail the knowledge in a

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    way that included the context in which the strate-gy occurred, the mathematical ideas involved, andtheir beliefs and values about the teaching andlearning of mathe.matics. These new understand-ings provided the support for teachers' developingidentities. We noticed in our previous work thatthe teachers engaged in generative growth saw theirclassrooms as places for experimentation and theirown learning. We see in our current work that theway in which the teachers see theim-selves and ne-gotiate their identity becomes critical to their gen-erativity.

    Taking a situated approach to understandinggenerativity changed our interactions with teachersand enriched our understanding of the characteristicsof generative growth. We found ways to begin toaccount for teachers' different learning trajectoriesby tracking the interplay between their classroomand professional development experiences. We paidattention to how teachers' participation both con-tributed to the development of the classroom andwork-group communities and was shaped by it. Asituated perspective allowed us to deepen our under-standing of why collaborating in detailed analyses ofstudents' mathematical work and experimentation inthe classroom supported teachers' generative growth.

    ConclusionStudent thinking remains at the core of our

    CGI work. What has changed is how we conceptu-alize what it means to engage with student work,how we come to understand what teachers and stu-dents are learning, and how we create opportuni-ties for teacher and student learning.

    Focusing on students' mathematical thinkingremains a powerful imechanism for bringing pdcia-gogy, mathematics, and student understanding to-gether. As teachers struggle to make sense of theirstudents' thinking and engage in practical inquiry,they elaborate how problems are posed, questionsare asked, interactions occur, mathematical goalsare accomplished, and learning develops. The de-tails surrounding these issues are seen in supportof one another, not separate from one another.Teachers' experimentation around student thinkingbecomies part of their practice. Whether talking withother teachers or with their students, t eachers see aclear relationship between their learning and theirstudents' learning.

    Some researchers have suggested that CGIhas been too narrowly focused on the cognitiveaspects of prescribed mathematical domains. Wewould argue, however, that the focused CCI workforms the basis for our understanding of teacherand student learning within a situated perspective.'While our current work with teachers and our re-search mnethods may look quite different from whenwe began over 15 years ago, we draw heavily onthe details of student thinking elaborated in thatwork, the organization of those details, and thecharacterization of teacher learning. The knowl-edge developed through the earlier CGI work pro-vides the basis for how we structure our interactionswith teachers and guides our facilitation as we en-gage teachers in discussing the details and avoidsurface-level discussions of student thinking.

    Understanding the details elaborated throughthe early CGI work also enables us to look with aparticular focus at understanding the stories teach-ers tell and the identities the stories support. Whileour theoretical perspectives are shifting somewhat,those perspectives have illuminated our early workand provided a way to better understand ongoinglearninig within communities of practice.

    NotesThe research reported in this article was supported inpart by a arant from the Department of Education Of-fice of Educational Research and Improvement (OERI)to the National Center for Improving Student Learningand Achievement in Mathematics and Science(R305A60007-98). The opinions expressed in this arti-cle do not necessarily reflect the position, policy, orendorsement of the Department of Education, OERI,or the National Center.1. CGI is one of a number of projects drawing on stu-dent thinking to focus and define the teaching and larn-ing of mathematics (see, for example, Barnett & Sather,1992; Borko & Putnam, 1996; Brown & Campione,1996; Lehrer & Schauble, 1998; Schifter, 1997; Warren& Rosebery, 1995).2. Certainly, we do not want to treat CGI as an isolatedexample. CGI developed in the way that it has because ofour interactions with our colleagues (researchers andteachers), from our readings of the work of others in thefield, and the evolution of theoretical notions available.CGI occurred in a context of exciting developments inmathematics education and is a product of the variousprojects anid people doing parallel and non-parallel work.3. A number of researchers within mathematics educa-tion have developed this situated perspective as they char-acterize and study teachers, students, and classrooms(Boaler, 20)00; Cobb, 1999; Stein & Browin. 1997).

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    4. And here, clearly we have only begun to explorethese perspectives in relation to teacher and studentlearning of mathematics in elementary schools.

    ReferencesBoaler, J. (2000). Exploring situated insights into re-

    search and learning. Journalfor Research in Math-ematics Education, 31, 113-119.

    Borko, H., & Putnam, R. ( 1996). Learning to teach. In D).Berliner & R. Calfee (Eds.), Handbook of educationalpsychology (pp. 673-708). New York: Macmillan.

    Barnett, C.S., & Sather, S. (1992, April). Using casediscussionzs to promote changes in beliefs amongmathemnatics teachers. Paper presented at the an-nual meeting of the Amnerican Educational Re-search Association, San Francisco.

    Brown, A., & Campione, J. (1996). Psychological the-ory and the design of innovative learning environ-ments: On procedures, principles, and systems. InL. Schauble & R. Glaser (Eds.), Innovations inlearning (pp. 289-326). Hillsdale, NJ: Erlbaum.

    Carpenter, T.P., Fennema. E., & Franke, M.L. (1996).Cognitively Guided instruction: A knowledge basefor reformn in primnary mathematics instruction. Fl-emnentary School Journal, 97(1), 1-20.

    Carpenter, T.P., Fennema, E., Peterson, P.L., Chiang,C.P., & Loef, M. (1989). Using knowledge of chil-dren's mathematics thinking in classroom teach-ing: An experimental study. American EducationalResearch Journal, 26, 499-531.

    Carpenter, T., & Lehrer, R. (1999). Teaching and learn-ing mathematics with understanding. In E. Fenne-ma & T.A. Romberg (Eds.), Mathematicsclassroonms that promtote understanding (pp. 19-32). Mahwah, NJ: Erlbaurn.

    Cobb, P. (1999). Individual and collective mathemati-cal development: The case of statistical data anal-ysis. Mathematicail Thinking and Learning, 1, 5-43.

    Fennema, E., Carpenter, T.P., Franke, M.L., Levi. L.,Jacobs, V., & Empson, S. (1996). A longitudinalstudy of learning to use children's thinking inmathematics instruction. Journal for Research inMathematics Education, 27, 403-434.

    Franke, M.L., & Kazemi, E. (in press). Teaching as learn-ing within a community of practice: Characterizinggenerative growth. In. T. Wood & B. Nelson (Eds.),Beyond classical pedagogy in elementary mathe-matics: The nature of facilitative teaching. Mahwah,NJ: Erlbaum.

    Greeno, J. (1988). Situations, mental models and gen-erative knowledge. (Rep. No. IRI. 88-0005). PaloAlto, CA: Institute for Research on Learning.

    Hiebert, J., & Carpenter, T.P. (1992). Learning andteaching with understanding. In D. Grouws (Ed.),Handbook of' research on mathematics teachingand lear, ing (pp. 65-97). New York: Macmnillan.

    Kazemi, E. (1999). Teacher learning within communitiesof practice: Using students' mathematical thiinkingto guide teacher inquiry. Unpublished doctoral dis-sertation, University of California, Los Angeles.

    Lave, J. (1996). Teaching, as learning, in practice.Mind, Culture, and Activity, 3, 149-164.

    Lave, J., & Wenger, E. (1991). Situated learning: Le-gitimate peripheral participation. Cambridge, UK:Cambridge University Press.

    Lehrer, R., & Schauble, L. (1998). Modeling in mathe-matics and science. Unpublished manuscript, Wis-consin Center for Educational Research, Madison.

    Richardson, V. (1994). Conducting research on prac-tice. Educational Researcher, 23(5), 5-10.

    Richardson, V. (1990). Significant and worthwhilechange in teaching practice. Educational Research-er, 19(7), 10-18.

    Rogoff, B. (1994). Developing understanding of theidea of communities of learners. Mind, Culture,and Activity, 1, 209-229.

    RogoftF B. (1997). Evaluating development in the processof participation: Theory, methods, and practice build-ing on each other. In E. Amsel & A. Renninger (Eds.),C.hange and development: Issues of theory, applica-tion, and method (pp. 265-285). Mahwah, NJ: Eribauin.

    Saxe, G. (1991). Culture and cognitive development:Studies in mathematical understanding. Hillsdale,NJ: Erlbaum.

    Schifter, D. (1997, April). Developing operation senseas a foundation for algebra. Paper presented atthe annual meeting of the American EducationalResearch Association, Chicago.

    Stein, M.K., & Brown, C.A. (1997). Teacher learningin a social context: Integrating collaborative andinstitutional processes with the study of teacherchange. In E. Fennema & B.S. Nelson (Eds.),Marl ematics teachers in transition (pp. 155-191).Mahwah, NJ: Erlbaum.

    Warren, B., & Rosebery, A.S. (1995). Equity in thefuture tense: Redefining relationships among teach-ers, students and science in language minority class-rooms. In W. Secada, E. Fennema, & L. Adajian(Eds.), New directions for equity in mathematicsedt cation (pp. 298-328). New York: CambridgeUniversity Press.

    Wenger, E. (1998). Communities of practice: Learn-ing, meaning, and identity. Cambridge, UK: Catn-bridge University Press.

    Wertsch, J.V. (1991). Voices of the mind: A sociocul-tural approach to mediated action. Cambridge,MA: Harvard University Press.

    Wertsch, J.V., & Kanner, B.G. (1992). A socioculturalapproach to intellectual development. In R.J. Stern-berg & C.A. Berg (Eds.), Intellectual development(pp. 328-349). Cambridge, UK: Cambridge Uni-versity Press.

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