messy learning: preservice teachers’ lesson-study conversations about mathematics and students

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Teaching and Teacher Education 24 (2008) 1200–1216

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Messy learning: Preservice teachers’ lesson-study conversationsabout mathematics and students

Amy Noelle Parks�

College of Education, University of Georgia, 629 Aderhold Hall, Athens, GA 30602, USA

Received 1 June 2006; received in revised form 22 March 2007; accepted 17 April 2007

Abstract

Research about lesson study and other forms of collaborative, practice-based professional development has tended to

focus on identifying structures that support teacher learning. This study builds on this work by examining preservice

elementary teachers’ efforts to conduct lesson studies in mathematics. Analysis showed that the structures of the lesson

study project supported some groups in developing mathematical and equity lenses for looking at teaching; however, the

use of these lenses simultaneously led to productive and problematic learning. This finding suggests that future research

focus on challenges present in collaborative, practice-based work even when productive structures are in place.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Practice-based learning; Collaboration; Lesson study; Pre-service teachers; Teacher education

1. Introduction

Current thinking and theories support the notionof situating teacher learning in classroom practice.The benefits of contextual learning in both teachereducation and professional development have beentheorized widely (e.g., Ball & Cohen, 1999; Hawley& Valli, 1999; Hiebert, Morris & Glass, 2003;Little,1999; Wilson & Berne, 1999), and recently,the National Research Council [NRC] (2005), whichnoted that elementary children ‘‘continue to betaught by teachers who have at best a shaky graspof mathematics’’ (p. 4), suggested that practice-based learning play a role in developing teachers’mathematical knowledge.

ee front matter r 2007 Elsevier Ltd. All rights reserved

te.2007.04.003

7 333 2111.

ess: [email protected].

In particular, the NRC report highlighted thepotential of lesson study, a practice-based form ofteacher learning where teachers collaborate to plan,observe and analyze key lessons. Because the workis collaborative, student centered, and embedded inK-12 classrooms, researchers have argued thatlesson study, if done thoughtfully, could increaseteachers’ content knowledge, focus teachers’ atten-tion on students and help teachers transition towardreform-oriented teaching (Chokshi & Fernandez,2004; Lewis, 1998; Stigler & Hiebert, 1999; Takaha-shi & Yoshida, 2004). Similar claims have beenmade for other forms of teacher learning experi-ences that focus on interactions between studentsand teachers, including using multi-media tools(Van Es & Sherin, 2002), interviewing (Moyer &Milewicz, 2002) and letter writing (Crespo, 2003).

Much of the literature on practice-based pedago-gies argues that although the potential for teacher

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ARTICLE IN PRESSA.N. Parks / Teaching and Teacher Education 24 (2008) 1200–1216 1201

learning is great, opportunities are often missedbecause participants lack content knowledge, ex-perience with collaboration, or, in the case ofpreservice teachers, time in the classroom (e.g.,Fernandez, Cannon, & Chokshi, 2003). As a result,discussions about the challenges of learning inpractice-based contexts often involve strategies forovercoming the deficits of participating teachersthrough the involvement of knowledgeable othersor the development of structures that supportmeaningful collaboration. What often goes unsaidin these discussions is that the strength of practice-based learning is also an area for concern. As Balland Cohen (1999) point out, classroom teaching is‘‘complex, unpredictable and difficult to monitorand manage’’ (p. 10). The advantage of situatinglearning about mathematics or students in thecontext of the classroom is that complicatedinteractions can be captured, discussed and ana-lyzed. Opportunities for learning are rich becausesimple solutions to complex problems—like how toteach so that all students stay engaged and deepenunderstandings—are challenged through interac-tions with real children. However, because work inthe classroom is complicated and unpredictable, itcan be difficult for teacher educators to anticipatewhat their teacher education students will learnwhen they engage in practice-based experiences likelesson study.

In addition, collaboration itself can be proble-matic in terms of increasing subject matter knowl-edge and changing beliefs about teaching andstudents. Studies that discuss the difficulty oflearning from collaborative work (e.g., Fernandezet al., 2003; Grossman, Wineburg, & Woolworth,2001) lend support to Little’s (1990) cautionarystance toward collaboration.

Teachers’ collaborations sometimes serve thepurposes of well-conceived change, but theassumed link between increased collegial contactand improvement-oriented change does not seemto be warranted: Closely bound groups areinstruments both for promoting change andconserving the present. (Little, 1990, p. 510).

Because learning in context is complicated andincreasing opportunities for collaboration is any-thing but a certain cure, research that looks at theoutcomes of practices like lesson study in ways thatmake even productive collaborations problematic isnecessary. In other words, in addition to exploringways to make lesson studies more collaborative,

more content-focused, and more connected to theclassroom, research also needs to consider theproblems that teacher educators may face even ifthese goals are achieved. This study probes thecomplexities involved in practice-based, collabora-tive learning by looking at the intended andunintended learning of preservice teachers whoengaged in lesson study as part of graduate-levelelementary mathematics methods course I taught.

I incorporated a lesson study assignment into themethods class because I wanted to provide thepreservice teachers in my class opportunities todeepen their understanding of a particular area ofmathematics. I also wanted them to questionproblematic assumptions about children throughsystematic observation of the students in theirclassrooms and through collaborative analysis ofthose observations. To these ends, the lesson studyassignment directed the preservice teachers to focusnot only on the mathematical content of theirlessons but to also pay special attention to studentswho traditionally have not been successful inmathematics classes. As their instructor, I wantedto both deepen their knowledge of mathematics andtheir commitment toward equity-oriented teaching.

I expected the lesson study assignment to supportthese goals because lesson studies are both colla-borative and practice based—two conditions thatsupport teacher learning. Although virtually all ofthe published research on lesson study in US hasfocused on practicing teachers, rather than pre-service teachers (Fernandez et al., 2003; Fernandez,Chokski, Cannon, & Yoshida, 2001), lesson study isused frequently in preservice programs in Japan(Fernandez, 2002; Stigler & Hiebert, 1999). How-ever, because beginning teachers in US are unlikelyto be familiar with this practice, importing it intothe preservice program does present particularchallenges. For example, the amount of timeavailable for learning is constrained by the lengthof the methods course, so teacher educators may beunder increased pressure to introduce the structuresof lesson study quickly. In addition, beginningteachers often have not mastered many of thelogistical challenges that have become automatic fortheir more experienced colleagues. This may make itdifficult for beginning teachers to plan and teachlessons that allow them to learn about content orteaching practices. Little research has been done onthe particular challenges novice teachers face whenengaging in lesson study, although Chokshi andFernandez (2004) suggest that the practice may be

ARTICLE IN PRESSA.N. Parks / Teaching and Teacher Education 24 (2008) 1200–12161202

educative for beginning teachers if attention is paidto the key features of Japanese lesson study, such asthe focus on the research question and the collectionof rich data by observers during the teaching of thelesson. One of the goals of the current study is tocontribute to empirical descriptions of preservicelesson study outside of Japan as well as to exploresome of the challenges presented by this context.

As the instructor and facilitator of a preservicelesson study project, I was both pleased anddissatisfied with some of the things I saw and heardin the lesson study groups formed in my mathe-matics methods course. Seeking to better under-stand what aspects of this experience were and werenot productive, I asked the following researchquestions: how do beginning teachers talk aboutmathematics, teaching and students while partici-pating in lesson studies? What does their talk revealabout the challenges and possibilities for teacherlearning in such practice-based contexts?

2. Teachers learning in conversation

Like many other educational researchers, Ibelieve that learning can be described as participa-tion in practice and that learners produce knowl-edge in settings that are socially and culturallyconstituted (Boaler, 2000). This way of thinking isno longer uncommon in mathematics education.Many socio-cultural researchers cite Lave andWenger (1991), who describe learning as participa-tion in a community of practice, where novices takeon more and more central roles in the work of thecommunity over time. This theory allows research-ers to examine issues of collaboration by looking athow joint work is constructed by members of thecommunity and to examine the differences in thenature of the participation by expert and novicemembers. It also emphasizes the ways in whichlearning is contextual, by focusing on the localpractices and meanings of individual communities.For my project, Lave and Wenger’s theory helpedme to see lesson study as a practice that variedacross communities, even within my own classroom.In addition, this theoretical perspective seemedparticularly productive for framing this studybecause of the current emphasis on the role ofcommunity in teacher learning.

Recent research and commentary on teacherlearning has emphasized the creation of inquiry

communities as central to meaningful professionaldevelopment and preservice education (e.g., Ball &

Cohen, 1999; Greene, 2001; Grossman et al., 2001;Hawley & Valli, 1999; Hiebert et al., 2003; Lord,1994). In particular, teacher educators have arguedthat prospective and practicing teachers shouldengage in learning experiences that are embeddedin practice, geared toward collaboration, anddeveloped over time, rather than in ‘‘one-shotworkshops with advice and tips of things to try’’(Ball & Cohen, 1999, p. 4). In reviewing research onteacher learning, Wilson and Berne (1999, p. 174)described the lore about traditional workshop-stylelearning experiences as presentations of ‘‘outsideexperts’’ of ‘‘boring, pre-packaged information,’’resulting in experiences where ‘‘teachers learn little(or at least little of worth)’’. In contrast, theydescribed current images of meaningful teacherlearning as those where teachers had opportunitiesto work over time in inquiry communities aroundproblems that are close to teaching practice. Inaddition, this current image of meaningful teacherlearning tends to value the perspectives and ques-tions of participants. That is, rather than pursuingan agenda set entirely by the teacher educator, thesepedagogies are intended to involve prospective orpracticing teachers in deciding what and how theywill learn. For instance, in lesson studies, partici-pants decide what kind of lesson they want to teach,what research question they will pursue, and whatkinds of data they will collect in the classroom.Practices like lesson study, which provide opportu-nities for teachers to construct knowledge socially,seek to create a space for participants to shape theirown learning while also providing a structure thatallows teacher educators to meet learning goals,such as deepening content knowledge or developingmore equitable teaching practices. Research onlesson study and other inquiry-oriented forms ofteacher learning has tended to examine the difficul-ties involved in creating such spaces, such as thechallenges involved in supporting genuine colla-borations (e.g., Grossman et al., 2001) or in guidingteachers toward substantive conversations aboutsubject area content (e.g., Fernandez et al., 2003).Research has not tended to examine what learningopportunities may be lost in this shift away fromteacher education that emphasizes the recommenda-tions of experts (i.e., one-shot workshops or didacticteaching styles) and toward more inquiry-orientedpractices where teacher educators and workshoporganizers may have less control of the learningoutcomes. This study takes on this problem byexamining the necessarily unpredictable outcomes

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1Each district used one of the US reform curricula. The

development of these books of lesson plans, assignments, and

essays on mathematics and teaching were funded by the National

Science Foundation with the goal of helping teachers to teach in

ways described by the standards written by the National Council

of Teachers of Mathematics.

A.N. Parks / Teaching and Teacher Education 24 (2008) 1200–1216 1203

of an inquiry-oriented learning practice (in this case,lesson study) as opposed to the challenges involvedin initiating such a practice.

3. The participants and the lesson study project

I taught 27 preservice teachers (known as‘‘interns’’ at this point in their TE program) in a12-week, graduate-level math methods course.These interns had earned their bachelor’s degreesin education the previous spring and were in theprocess of completing the yearlong teaching intern-ship that the university requires before recommend-ing graduates for teacher certification. Four days aweek, the interns worked full-time in one of sixelementary schools, and on Thursdays, they at-tended my class in the morning and a literacymethods class in the afternoon. When planning thecourse, I sought to include as many opportunities aspossible for the interns to document and analyzeinstances of teaching, with the goal of connectingwhat they needed to learn about mathematics,pedagogy and children to the classrooms wherethey spent most of their time. In particular, I sawthe lesson study project as a way to connectteaching in elementary classrooms to the knowl-edge, skills and dispositions that I was trying todevelop in the methods course.

I introduced lesson study on the second day of thecourse with an article (Lewis, 1998) and a videotapeof a lesson study held in Japan (Lewis, 2000). I alsohanded out a project sheet, which described thethree major parts of the assignment. Each lessonstudy group was asked to write an introduction thatdescribed the group’s research goal and mathema-tical content, a detailed lesson plan, and an analysisof the data collected during the research lesson.Throughout the lesson study project, I providedstructures, including agendas, lesson goals andassignment requirements, with the intent to chal-lenge the interns to look more closely at bothmathematical content and issues of equity. Morespecifically, I hoped that through their participationin lesson studies, the interns would develop both amathematical lens and an equity lens, which theycould draw on to evaluate curricula and classroompractice. I expected that by paying attention tomathematics in the planning and the analysis of alesson, the interns would deepen their understand-ing of mathematical content as well their sense ofhow to present that content in reform-orientedways, such as through selecting meaningful tasks,

holding substantial mathematical discussions, andusing varied representations. I wanted them to beginto see the teacher’s role as one that required critiqueof content as well as pedagogy.

Although I encouraged the interns to choosegroups flexibly according to personality and inter-est, all the interns grouped themselves by theirelementary school placements. This resulted in threegroups of six interns and three groups of threeinterns. Table 1 shows a chart of the four focalgroups I studied for this project. I chose thesegroups to represent a range of group sizes, schooldistricts and grade levels.

Rather than providing time for my students todevelop their own research goals, I presented fourfrom which to choose. I did this because the lessonstudies needed to be conducted within the timeconstraints of the course and because I felt theinterns would need as much time as possible tobecome familiar with state standards and theirdistricts’ curriculum.1 The four research goals wereadapted from Cochran-Smith’s (1999) analysis ofpreservice teachers’ efforts to teach for social justice.I chose these goals to encourage the interns to haveexplicit conversations about special education,culture, language, gender, race and other issuesrelated to equity and social justice, in addition toconversations about mathematics. I asked thepreservice teachers to design lessons that would doone of the following: enable significant work for allchildren, offer assessments that allow all children toshow what they know, make inequity or activismexplicit, or draw on the cultural or linguisticstrengths of the children in the room.

Three 3-h class periods were devoted to theplanning of the research lessons, and most groupsalso met for some additional time outside of class.For each planning session, the groups were pro-vided an agenda that asked them to perform tasksrelated to their lesson study project, such asreviewing multiple curricula about their topic,addressing the state standards and locating articlesabout the mathematics they wanted to teach. Theagenda also included questions for the groups toaddress, such as ‘‘Which of the research goals doesit seem like teachers in your school are having the

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Table 1

The lesson study groups

Schools District Internsa Research goal Mathematics

content

Curriculum used

Oak Suburban 1 Sam, Noelle, Anna,

Caroline, Lucy, Eva

To diversify assessment

so all children can show

what they know

Graphing and

measurement

2nd grade

Investigations

Camden Hills Suburban 2 Lara, Mike, Gillian To enable significant

work for all children

Estimating and

rounding

5th Grade Everyday

Mathematics

Lakeside Suburban 3 Mia, Juliet, Sarah To enable significant


Multiplication with

arrays

4th Grade

Investigations

Jefferson Urban 1 Mara, Allie, Heidi,

Cate, Bethany,

Alyssa

To enable significant


through creativityb

Identifying and

drawing angles

4th Grade Math

Trailblazers

aLike me, all of the interns were white. The name of the intern who taught the research lesson is listed first. All names of interns and

schools are pseudonyms.bThis group of interns modified the goal I gave them of enabling significant work because they wanted to make creativity a central focus

of their lesson.

A.N. Parks / Teaching and Teacher Education 24 (2008) 1200–12161204

most trouble addressing?’’ and ‘‘What misconcep-tions might students have about your mathematicaltopic?’’ The groups had a week and a half to teachtheir lesson, and I was able to attend four of the sixresearch lessons. Each group met immediately afterthe research lesson to debrief with me or their fieldsupervisor. Following this, an additional period inthe methods class was devoted to analyzing the datacollected. Each group made a brief presentation tothe class and wrote a final report discussing whatthey learned about mathematics, children andteaching.

It would perhaps be more accurate to refer to thework described above as ‘‘lesson-study-like’’ ratherthan as ‘‘lesson study.’’ In designing this courseproject, I was unable to replicate the sort of lessonstudy work commonly done in Japan or being donein US by more expert practitioners. Differencesworth noting include the lack of attention to thedevelopment of research goals, the lack of revisionand reteaching of the research lessons, and theinexperience of the facilitator. (My own experiencewith lesson study comes from reviewing theliterature, attending a single workshop given byfacilitators with more expertise, and using theprocess in a previous course.)

Given these limitations, it would be possible todismiss the findings presented here because thisimplementation of lesson study deviated frommodel practice. However, these criticisms areunproductive for two reasons. First, many inexper-ienced practitioners are experimenting with lessonstudy in both universities and K-12 schools. Thus,

the experiences of lesson study novices should berepresented in the literature as a way of under-standing the process as it is enacted (as opposed toas it is idealized). Second, the practices described inthe study can be seen not only as a case of lessonstudy, but as a more general case of the kind ofpractice-based learning common in many teachereducation programs. Deviations from the lessonstudy paradigm do not prevent the viewing of thepractices described here as instances of collaborativeteacher learning.

4. Design and methodology

In this study, my goal was not to make broadgeneralizations about how lesson study ‘‘works’’ inmathematics or in preservice education, but tocarefully describe the participation of the teacherinterns and myself in these lesson studies, with theintention of helping others learn from my casebecause of (not in spite of) my attention to detailsthat are ‘‘quite small’’ and ‘‘radically local’’(Erickson, 1986, p. 129). The close examination oflesson studies by outsiders provides one way ofunderstanding the relationships between this prac-tice and teacher learning; however, the perspectivesof the insiders who engage lesson study are equallyimportant, because, as participants, they are in aunique position to describe the challenges involvedin attempting to lead lesson studies. When practi-tioners research their own teaching, they become‘‘ethnographers of their own situation’’ (Hymes,1972, p. xiv), and as practitioner-ethnographers,

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Table 2

Data summary

Whole-class discussions Fieldnotes from three classes

Transcript from whole-class

presentations

Small-group discussions

of lesson study

Transcripts from 3 different

discussions by the Oak, Jefferson,

Lakeside, and Camden Hills groups

Research lessons taught

in schools and

conversations

immediately afterward

Fieldnotes from four lessons

(Jefferson, Lakeside, Northview and

Eastside)

Written work Plans and analyses from all six

groups

Personal reflections by all 27 interns

Other communications Email to me about lesson study

project

Postings from electronic discussion

board

Teaching documents Syllabus

Assignment sheet for lesson study

project

Lesson plans from each class

Teaching journals from each class


they are able to apply the tools of qualitativeresearch to data and interpretations not available tooutsiders.

Throughout the project, I took a number of stepsto balance my roles of teacher and researcher. Forinstance, I graded all written work produced beforelistening to the conversations my students hadabout that work so my decisions as a teacher wouldnot be influenced by data that I collected as aresearcher. In addition, I only audio taped volun-teers in small-groups and I waited until the coursewas over and grades were filed to ask those who hadvolunteered for their consent to use the collecteddata for research purposes. My student-participantsdid not engage in data analysis of their own learningfor this research project, except to the extentrequired by course assignments such as writtenreflections and oral presentations.

4.1. Data collection

Throughout the lesson study project, I audiotaped all whole-class discussions and wrote fieldnotes after each class. In addition, I audio taped myfour focal groups each time they met in class. Itranscribed three small-group conversations foreach focal group: two planning sessions and theanalysis. For all transcripts, I recorded each speak-er’s comments verbatim, except when I was unableto make out each word because of the quality of thetape or when the conversation wandered off topic.In these cases, I summarized what was said in fieldnotes. I also collected a variety of other data,summarized in Table 2.

4.2. Data analysis

I focused the analysis on the written transcripts oftwo main events in the lesson study process: thethird planning session and the interns’ in-classanalysis of their lesson after it had been taught.(The interns had a brief conversation at their schoolsites immediately after the lesson and then engagedin a longer analysis in the methods class using datathey collected during the research lesson. Thissecond analysis was one of the focal events.)Because I was interested in the topics that internschose to discuss, the way they talked aboutmathematics, and the way that they describedstudents, I looked for events that would provideme the greatest insight into these issues both beforeand after the teaching of the research lesson. The

third planning session and the in-class analysisdiscussions offered me both the longest conversa-tions and the most focused conversations. I beganthe data analysis by reading through the transcriptsof the third planning session and the in-classanalysis. I then wrote open codes identifying majorthemes and explored these themes in analyticmemos (Dyson & Genishi, 2005; Emerson, Fretz,& Shaw, 1995; Erickson, 1986).

Initially, I identified instances of mathematicalconversations, pedagogical conversations and con-versations about students. I also broke out con-versations about the logistics of the lesson (e.g.,where to find materials), about my grading of andrequirements for the lesson study assignment, andabout unrelated topics. These latter conversationshelped contextualize the three types of conversa-tions I focused on but were not the subject ofintense data analysis. The conversations aboutmathematics, pedagogy and students were thensubjected to further fine-grain analysis. This con-sisted of breaking each conversation into thematicepisodes to look at turn-taking, wording, andpatterns of interaction. I defined a turn as a segmentof uninterrupted speech and an episode as a series ofturns about the same topic (Tannen, 1984).


Because this study was concerned with the extentto which the lesson study assignment could supportlearning about mathematics and equity, I examinedeach episode in relation to these two themes. Inmaking sense of these episodes I asked the followingquestions: What questions are being raised? Howare these questions being answered? What mathe-matical assertions are made? What assertions aremade about students? How are the participantsinterpreting their research goals in conversation?Because this study defines learning as participationin community practice, the transcripts of conversa-tion were considered the primary data to analyzelearning. After developing the codes with thetranscripts of my focal events, I coded all othertranscripts and then read the students’ written work,such as lesson plans and reflections, and analyzedthese documents using the codes I developed byanalyzing the transcripts. Table 3 shows a chart ofcodes used in analysis.

I used the codes under ‘‘participant structures’’ todetermine which topics most fully engaged theinterns during their lesson study conversations.For instance, some groups talked about mathe-matics only for several short episodes; whereas,other groups discussed mathematics for longerperiods of time. Similarly, conversations with moreopportunities for learning seemed to be those that

Table 3

Codes used In analysis

Thematic codes for content of

discussions

Codes for participation

structures

1. Students (ability, behavior,

mathematical knowledge,

personality)

2. Mathematics (definitions,

problems/tasks, big ideas,

procedures, relationships among

ideas)

3. Teaching (holding discussions,

assigning homework, structuring

group work, modes of

assessment)

4. Logistics (timing, materials,

supplies)

5. Lesson study assignment (due

dates, grading policy, assignment

requirements)

6. Unrelated conversations (job

search, gossip, planning for other

courses)

1. Asking questions

(open-ended vs.

closed)

2. Long episodes (more

than six turns)

3. Short episodes

(fewer than four

turns)

4. Making connections

to other themes

5. Drawing on

resources

(curriculum,

assignment sheet,

advice from

cooperating teacher)

drew on multiple resources and that made connec-tions to multiple themes. In the analysis below, Iargue that some groups engaged more deeply inmathematics and equity and much of this analysis isdependent on the participation structure codes. The‘‘content’’ codes allowed me to see which topicsinterns spent most of their conversational timeon–so that groups that spent much of their time onlogistical questions tended to have less time to spendtalking about mathematics and children.

4.2.1. Examining the lesson study work

To support the development of a mathematicallens, I asked each group of interns to reviewmultiple curricula, analyze the state standards andlocate articles about their area of mathematics. Ialso gave assignments that required them toconsider the pedagogical merits of various repre-sentations of mathematical ideas and to makeconnections across grade levels and mathematicalstrands. I hoped that the understandings the internsdeveloped during these tasks would encourage themto bring a mathematical eye to the more open tasks,such as the planning and analysis of the researchlessons.

In addition, I gave assignments designed tosupport the interns’ development of an equity lens.For instance, I asked groups to locate andsummarize two articles related to their researchgoals, which were chosen to support the learning ofall children. I also assigned course readings thatdescribed successful teachers working with studentswho traditionally have been seen as having lowability in mathematics. My goal in doing this was toencourage discussions that would bring assump-tions about children’s mathematical abilities outinto the open and to introduce the interns topractices that would help them meet the needs ofall students in their classrooms. As with themathematics-oriented assignments, I hoped thatthese conversations would shape the interns’ dis-cussions during planning and analysis and wouldhelp the interns to think about the ways thatcurricula and classroom practice might eitherexpand or constrain learning opportunities for allchildren.

During analysis, I found that the lesson studyassignment did seem to support the development ofa mathematical lens and an equity lens for at leastsome of the groups. However, in looking closely atthe data, I found that the attention some groupspaid to mathematics and equity resulted in learning


that I considered problematic as well as productive.The following two sections explore these findings. InSection 4.3, I describe the use of the mathematicallens and the equity lens and discuss the kinds ofconversations these lenses made possible. In thefollowing section, I discuss the ways that the use ofboth lenses led to learning that seemed to work bothfor and against my goals as an instructor.

4.3. Developing a mathematical lens

In analyzing my two focal events (the thirdplanning session and the analysis discussion), Iidentified 181 conversational episodes across thefour focal groups. I found that the fewest number ofepisodes—21—were related to mathematics. Ofthese, 16 episodes were evenly divided betweentwo groups: the Camden Hills interns who did alesson on estimation and the Lakeside interns whofocused on arrays. The majority of the episodeswere about students (38 episodes) and pedagogicalstrategies (39 episodes). Here I would like to explorethe differences between groups that seemed topursue mathematical topics in significant ways andthose that did not.

The Camden Hills group had the longest con-versations (with several 10-min episodes) aboutmathematics. In addition, they had the highestpercentage of conversational episodes about mathe-matics—about 22 percent. In the following episode,Lara pointed out a mathematical problem she sawin the curriculum, which suggested that studentsestimate by predicting the place value of theexpected answer.

Lara: However, say you have a number like 9.7and 9.8 and you multiply these. Well, the answeris going be, if you round it, it’s going to be in thehundreds. If you don’t round it, the actualanswer is not in the hundreds. It’s less than ahundred.

Mike: Did it always want to round like that orwas it looking for high rounds and low rounds?Because if you round down, your lower estimateis going to be 81. So it’s somewhere between 81and 100.

Lara: But this particular activity, see, is askingyou to round and then multiply. And you figureout, well, 200, it’s in the hundreds. But it’s not allthe time. That’s the thing.

Mike: You have to show them that if we’recircling the hundreds it doesn’t mean that it’sgoing to be 100. It’s going to be near 100.

Later in this episode, Lara continues her critique ofthe curriculum by pointing out that place valueestimates get less and less meaningful with largernumbers, saying that ‘‘there’s a big differencebetween 200 and 900’’ and asking what happenswhen the students start thinking about millions.Ultimately, this group decided to de-emphasize theplace value estimates suggested by the curriculum infavor of estimating techniques that they saw asmore meaningful for children, such as rounding to‘‘friendly numbers’’ before carrying out a calcula-tion.

The conversations of the Camden Hill internsrepresent one extreme of mathematical talk in myclassroom, with one other group (Lakeside) havingmathematical conversations similar to the CamdenHills interns. The other two groups (Oak andJefferson) had very few conversations about mathe-matics. Of the 106 episodes, I identified for thesetwo groups during the third planning session andthe analysis, only five were about mathematics.Table 4 shows a chart of all the mathematicaldiscussions for each group during both the thirdplanning session and the post-lesson analysis.

In addition to having fewer conversations aboutmathematics, the Jefferson and Oak groups also hadshorter conversations, and only one episode (theOak discussion about the purpose of non-standardmeasures) had real potential for enriching themathematical understanding of the group members.The kinds of questions asked during these discus-sions also tended to be different, with the Jeffersonand Oak interns typically asking factual questionslike ‘‘A right angle is 90 degrees?’’ or ‘‘What does aline graph show?’’ and the Camden Hills andLakeside interns typically asking more open-endedquestions, such as ‘‘What’s the difference betweentalking about a range for tens, hundreds andthousands?’’ and ‘‘What would we be teaching ifwe showed only one array for each number?’’

The groups that engaged in fewer mathematicalconversations tended to shift the discussion whenmathematical questions were raised toward peda-gogical issues, such as whether the task wouldprovide students with opportunities for discussionor creative thinking. For instance, the six internsfrom Jefferson Elementary discussed the mathe-matics of their lesson only twice during planning,

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Table 4

Mathematical topics discussed

Camden Hills (estimating and

rounding)

Lakeside (multiplication and arrays) Jefferson (identifying

angles)

Oak (linear measurement

and graphing)

Third planning session

1. Relationship between rounding

and place value (5 turns)

2. Concerns about the efficacy of

magnitude estimates and possible

alternatives (44 turns)

3. Why is it mathematically

reasonable to round up when the

rounded number is not a possible

answer in the real world (6 turns)

4. What math will studentshave to

do in estimating the students in

the school using totals of all

classrooms versus a sample of

classrooms (12 turns)

5. Analogies or descriptions of the

process of rounding (10 turns)

1. Qualities that make numbers easy

to find arrays for, such as size and

number of factors (19 turns)

2. Representing the commutative

property with arrays (10 turns)

3. Knowledge needed to find all

arrays for a number (8 turns)

4. Appropriate real world

representations of arrays (10

turns)

5. Appropriate role of prime

numbers in the lesson (5 turns)

6. Relationship of knowing

multiplication facts to

understanding the concept of

multiplication (12 turns)

7. Representations for

multiplication other than arrays

(4 turns)

1. Definition of right

angle (2 turns)

2. Right, acute and

obtuse angles can

be seen in real

world contexts (4

turns)

1. Difference between a

line graph and a bar

graph (8 turns)

2. Definitions of measures

of central tendency (4

turns)

3. Purpose of non-

standard measures (4

turns)

Analysis

6. Strategies for getting students to

estimate, rather than finding the

exact answer (5 turns)

7. Rules of rounding that would

support children’s ability to

accurately estimate (8 turns)

8. Interpreting the written work of

children in solving a math

problem (15 turns)

8. Relationship between arrays and

the concepts of multiplication and

area (7 turns)

A.N. Parks / Teaching and Teacher Education 24 (2008) 1200–12161208

and one of the conversations was initiated by theinstructor. The members of this group wereconcerned with giving children a chance to workcreatively in mathematics and with having mean-ingful discussions. They chose to design a lessonabout classifying and identifying acute, obtuse andright angles because the children had already had alesson on this with the cooperating teacher so theinterns thought they could develop previous under-standings. Their lesson involved having childrenidentify angles in geometric art and then makegeometric artwork of their own.

During the first planning session, I pushed thisgroup to consider the ways that they might developideas about angles in this lesson that wouldcontribute to other mathematical understandingsdown the road. I suggested thinking about therelationships of angles to degrees in circles and tothe identification of various polygons. After I left,

the group talked briefly about the requirements ofthe assignment, and then Heidi redirected the groupto my comments by saying, ‘‘So she said makingconnections to other things, like buildings andarchitecture.’’ My suggestion to look for mathema-tical relationships had been reframed as a sugges-tion to connect the lesson to ‘‘the real world.’’

After observing this group’s research lesson, Idrew their attention to a child who, when asked bythe teacher to show a right angle, pointed to a circleon his ruler. One of the interns hypothesized thatthis student had confused the degree symbol with aright angle. Despite this student’s apparent mis-understanding, he had been able to complete theprimary task in their lesson, drawing geometric artwith angles. Based on this evidence, I asked thegroup to think about whether it had been possiblefor students to participate in their lesson withoutdeveloping mathematical ideas about angles.

ARTICLE IN PRESS

2The chart describing the four lesson study groups shows which

curriculum was used by each group.


However, rather than looking back at the way thelesson did or did not develop mathematical ideas,the interns instead questioned the validity of myobservation.

Alyssa: But we don’t have any proof that theother kids didn’t know.Mara: I guess that surprised me from this studentbecause I thought that he was just being goofy.Or he actually knew and he was just nervous inthat situation even. I guess I would go back andactually ask it again.Cate: And if it’s just one kid too, you could kindof work with him individuallyAllie: It seemed to meCate: during a down time, instead of making awhole like, once youAllie: It seemed like everyone pretty much knewwhat the angles were, like that was a majortheme.

In this episode, the Jefferson interns turned mychallenge of explaining why a student who success-fully completed their task did not correctly identifya right angle into a statement that everyone knewwhat angles were. In addition, at no time during theplanning or analysis did they have a conversationwhere they considered what knowledge of anglesmight be important mathematically beyond theability to name and draw them.

In my analysis of why some groups were able todevelop a mathematical lens and others were not,two factors emerged as potentially important—theinterns’ dispositions towards mathematics and theway they interacted with the written curriculumprovided by the district. While it is possible that themathematical strand (i.e., geometry, multiplication,etc.) each group chose for the research lesson couldhave played a role in supporting and constrainingeach of the groups’ mathematical conversations,there was not enough data in this study to explorethis hypothesis. Multiple lesson studies would havemade such an investigation possible, whereas in thisstudy each group planned and carried only onelesson study.

Both the Camden Hills and Lakeside interns hada majority of members who reported positivedispositions toward mathematics. During the firstclass of the year, I asked all the interns to talk abouttheir experiences in mathematics. About a third ofmy students reported positive experiences. About ahalf reported negative experiences and the remain-ing students made comments that I would describe

as neutral, such as ‘‘I could take it or leave it’’ or ‘‘itdidn’t bother me; it was just school.’’ Both theCamden Hills and the Lakeside groups werecomposed of a majority of interns who reportedpositive experiences in mathematics. This was nottrue for the other two groups. Second, the ways inwhich the interns interacted with the curriculum intheir lesson studies seemed to play a significant rolein the development of a mathematical lens. Irequired all four groups to read at least one lessonfrom the school curriculum where their lessonswould be taught. All four groups taught in schoolsthat used reform curricula;2 however, the groupsused these curricula in different ways. Both theCamden Hills interns and the Lakeside internsbegan their discussions by reading and commentingon a particular lesson in their curricula. In bothcases, the intern teaching the lesson had been toldby her cooperating teacher to focus on a particularlesson during the lesson study. Throughout theirplanning conversations, both these groups had thelesson in front of them and referred to the writtentext. For instance, the Lakeside interns began theirconversations by criticizing the lesson described inthe Everyday Mathematics curriculum, saying thatthe student work page was ‘‘boring’’ and thatmagnitude estimates were not helpful because asnumbers grew larger, the range grew so wide as tobe useless. Although they were critical of the lesson,they used the lesson to structure their own work,giving similar problems to those posed in the textbut with different directions.

The Lakeside interns, who were using theInvestigations curriculum, were less critical of thelesson and adopted the proposed format suggestedin the text. They planned to model finding thearrays for one number and then assign othernumbers to pairs of children. Occasionally, thisgroup raised questions written in the text as part oftheir discussion, such as ‘‘How can you know whenyou have found all the arrays for one number?’’Although this group was less critical, some of theirmathematical discussions did arise from disagree-ments with the text. For instance, the text suggesteda movie theater as a possible context for arrays, butthe interns in this group thought the representationinaccurate because seats are staggered in theaters.This led to a conversation about logical representa-


tions of arrays in the real world as a way to frametheir opening problem.

Although all groups were required to read at leastone lesson in a published curriculum, I did notrequire that this be done first or that they use ideasfrom the curriculum in their teaching. The internwho taught the lesson for the Oak group openedtheir planning discussions by announcing that hiscooperating teacher had said he could teachwhatever he wanted for the lesson study. Thisgroup brought many resources to their planningsession and began by examining a book thatpromoted linking literature to mathematics. Earlyon, they decided to base their lesson on thechildren’s book How Big is a Foot? (Myller, 1962/1990), which describes the difficulty of building abed without a standard measure for a foot. Basedon this story, the interns decided to plan a lessonwhere children would measure the length of theplayground with their feet and graph the results.Most of the conversation during the planningstages—almost half the episodes—dealt with logis-tical issues, such as how to move the children fromthe playground to the classroom, which materials touse to record the measurements, and the organiza-tion of the room for the graphing. This groupseemed to read a lesson in the curriculum only as anafterthought. As the group was finishing theplanning of their lesson, one intern, who had beenreading the assignment sheet, said ‘‘Wait! We’resupposed to analyze a lesson from the curriculum.’’The group assigned two members to do this, whilethe other three finished planning. The result of thiswas that the mathematical ideas raised in thecurriculum had very little impact on the plannedlesson, although the analysis was included in thegroup’s write-up. The experiences of these groupssuggest that the way that the curriculum wasapproached—and perhaps the mathematical rich-ness of the curriculum—was an important factor inthe development of a mathematical lens.

4.4. Developing an equity lens

I provided fewer supports to help the internsthink about issues of equity than I did formathematics. Although I asked them to read articlesrelated to their research goal and engaged inconversations about equity outside of the lessonstudy project, none of the materials the interns drewon to plan their lessons—published curricula, statestandards and a mathematics methods text—pro-

vided much support for thinking about how tosupport students who have traditionally beenunsuccessful in math. Thus, it is unsurprising tofind that only one of my four focal groups seemed toraise the issue of teaching all students throughoutthe planning and analysis of their research lesson.

This group, the Lakeside interns, frequentlyreferred to their research goal while makinginstructional decisions during the planning of theirlesson, which was about drawing arrays. They, likethe Camden Hills and Jefferson interns, had chosenenabling significant learning for all students as theirgoal. When the Lakeside interns referred to theirgoal in conversation, they tended to emphasize thephrase ‘‘for all students.’’ This emphasis led thegroup to plan a lesson with multiple access points.The Lakeside interns chose to plan a lesson arounda problem that asked students to draw as manyarrays as possible for a given number. They decidedto group students by ability and assign a differentnumber to each group. After making this decision,Juliet redirected her group’s attention to theirresearch goal, saying: ‘‘If we look at our researchgoal, it’s supporting significant learning for allstudents. So what are we doing if we have a groupC lower or higher C that is like, done. What are wedoing? Like, how are we going to extend? Or if theydon’t get it, what are we doing? I think that we havekids who this is going to be super easy for. If theyreally get multiplication, I don’t, I mean I don’tknow.’’

Here, Juliet interprets the research goal asrequiring a plan to meet the needs of studentsworking at various levels of proficiency. In responseto this challenge, the Lakeside interns decided tomake supports such as multiplication tables andhundreds charts available for groups who werestruggling and to have more challenging numbersready for children who finished the assignmentquickly and without difficulty. The members of thisgroup used the language of the research goal torecognize that the children in the room would havedifferent needs in relation to the mathematics beingtaught and to launch conversations about how tomeet those needs. They made similar references totheir goal during the analysis conversation, withSara explicitly asking the others whether theythought that ‘‘all children had learned the contentof arrays.’’ This question resulted in a conversationthat drew on evidence from multiple sources,including Mia’s report that the homework assign-ment on arrays was the first that all students in the


room had completed and Juliet and Sara’s observa-tions of students they labeled as ADHD, gifted andstruggling.

The research goals did not seem to encourage theother lesson study groups to bring an equity lens totheir planning or analysis. In fact, the other threegroups of interns tended to refer to the research goalfar less often and when they did, they often droppedthe ‘‘for all children’’ from the goal. When Gillian,one of the Camden Hills interns, discussed sig-nificant learning during their analysis, she framedthe question as one about the kind of work studentswere being asked to do, saying, ‘‘significant workdoesn’t just mean paper and pencil. It means thatthe kids are engaged in what they’re doing and thatthey’re paying attention and they’re on task.’’ TheJefferson interns interpreted significant work insimilar ways, seeing creative, discussion-orientedwork as significant, as opposed to work that isteacher-directed and focused on the textbook.Similarly, the Oak interns, who had chosen thegoal of diversifying assessment so all children can

demonstrate what they know, never used the phrase‘‘all children’’ in their discussion or analysis andspoke of having diverse assessments in terms ofincluding individual, small-group and whole-classwork, rather than diversifying in response to theneeds of particular students.

The Lakeside interns had 18 conversations (out of42 total episodes) explicitly focused on respondingto differences among students in ways that wouldcause all students to learn. Even so, it is importantto note that in suggesting that this group used anequity lens to plan and analyze their lesson, I amdefining equity generously. This group did fre-quently talk about how they could adjust theirteaching so that all students could learn; however,they showed no awareness of larger social forces—such as attitudes about race, gender, class ordisability—that may have played roles in shapingwhich students were seen as ‘‘good at math.’’ Nordid they discuss any explicit desire to change thestatus quo. A fully formed equity lens wouldprobably include both these perspectives. Also, itseems worth noting here that none of the lessonstudy groups in my class, including those notdescribed in this paper, chose making activism or

inequity explicit or drawing on the cultural or

linguistic strengths of the children in the room asresearch goals, even though the Oak interns wereworking at a school with a significant number ofEnglish language learners. Although I have no

further data about these choices, this may suggestthat the interns were more comfortable when issuesaround equity were less explicit. Similarly, none ofthe interns in my focal groups had a singleconversation that explicitly addressed issues aroundculture, language, race or gender. So, while I wouldargue that being provided with equity-relatedresearch goals to choose from did provide anopportunity for at least the Lakeside interns todevelop an equity lens, it was not a highly focusedone.

5. Examining intended and unintended learning

Now, I explore the ways in which using mathe-matical and equity lenses during these lesson studiesresulted in learning that simultaneously workedboth for and against my stated learning goals,namely, to help the interns develop understandingsof mathematics that would support teaching inreform-oriented ways and to adopt practices thatincrease learning opportunities for all students. Toexplore this idea, I focus on the two focal groupsthat were ‘‘most successful’’: the Lakeside andCamden Hills interns. By ‘‘successful’’ I mean thatthey did have meaningful conversations aboutmathematics during planning and analysis. Becausethe Lakeside interns also paid significant attentionto the challenge of teaching all children, I use theirwork as the example for thinking about learningfrom the equity lens. I use the work of the CamdenHills interns for thinking about learning from themathematics lens. In doing this, I do not mean tosuggest that the work of these two groups was idealin reference to either mathematics or equity, or thatI could not improve my performance as a facilitatorin order to better meet both these goals. However, Ichose these relatively successful groups to make itpossible to consider the idea that improving therichness of discussions about content and studentswill not totally eliminate learning that is proble-matic, partial and messy.

5.1. The Camden Hills interns: learning from the

mathematical lens

By paying particular attention to mathematicsduring planning, the Camden Hills interns were ableto articulate a variety of definitions of estimationincluding, ‘‘an area, not a point,’’ ‘‘a ballpark,’’‘‘not a guess,’’ and ‘‘upper and lower limits.’’ Thediscussions that produced these definitions helped


to give them language to talk about estimation inthe classroom. Lara used many of these definitionswhile teaching the research lesson. In these con-versations, the interns also identified many troublespots in teaching estimation, such as children’sdesire to get exact answers and multiple under-standings of ‘‘friendly numbers.’’

The conversations about number and estimationalso offered a chance for Gillian, whose mathema-tical understandings did not seem to be as devel-oped, to learn from the interns with whom she wascollaborating. In the following exchange, Gillian isthinking out loud about the problem they are goingto set for students: to estimate the total number ofstudents in the school by adding up the number ofstudents in each classroom.

Gillian: Yeah, so you say to them: ‘Well what arewe going to do? How can we make this easier forourselves?’ So, if we said there’s 23 in one; there’s27 in another; there’s 24 in another; you couldsay well that’s about 20 students because on anygiven day some students are going to be absent.But that’s the thing, they might want to round upand you can’t say on any given day there’s goingto be 30 students. We’re never going to get there.Mike: No, but the 30 accounts for the 3 you justlost here (pointing to the 23). So as far asrounding goes, that’s okay. The answer stayssimilar because sometimes you’re rounding downand sometimes you’re rounding up.Gillian: Oh, yeah. Yeah. I see.

Here, Gillian expresses a misconception aboutrounding—that the reason it is okay to round downfrom 23 to 20 is because sometimes students will beabsent. Mike is able to correct this idea immedi-ately. It is unlikely this misunderstanding wouldhave come to light if conversations about pedagogyand students had dominated the Camden Hillsdiscussions, as they did in other groups of interns.

However, the same discussions that worked toexpand mathematical understandings also workedto simplify mathematical questions in ways thatwere unproductive for the development of strategiesfor reform-oriented teaching. For instance,throughout their planning and analysis, Lara, Mikeand Gillian used rounding and estimating assynonyms. Thus, the same conversations thatallowed them to develop their understandings ofthe content also reinforced a narrow perspective ofthe skills required by children for estimation. Theydecided to begin their lesson by asking students to

estimate the number of students in the school byusing a chart that showed the exact number ofstudents in each of the school’s homerooms. Whenthe group proposed this idea to me, I suggested thatthey consider taking a sample of just a fewhomerooms and encouraging students to make anestimate based on this information. I said that atask like this might promote more discussion sincestudents would have to consider whether sampleswere representative and how to account forpotentially bigger and smaller classes. When I leftthe group, the Camden Hills interns rejected thisidea, saying it would turn into a ‘‘basic multi-plication problem’’ because students could just picka number and multiply it by the number ofclassrooms in the building. For them, the taskof estimating lay only in the rounding of a series ofnumbers, not in making decisions about what tocount or in using context to judge the accuracy of afinal answer. Rather than challenging and extendingtheir mathematical ideas about this topic, theircollaboration around the mathematics of the lessonreinforced it.

During their research lesson, many of the childrenadded all 16 of the numbers posted on the boardand then rounded to get an ‘‘estimate,’’ and on theworksheet following big problem, several childrenrounded in ways that the three interns did notunderstand, such as changing 326–325. I encour-aged Mike and Gillian (Lara was absent during thein-class analysis) to think about what this workmight tell them about students’ sense of the purposeof estimation and to consider what sort of tasksmight help children to develop a more productivesense of the skill. However, Mike and Larainterpreted the student work differently. Based ontheir analysis, they wrote the following assertion:‘‘Rounding rules if not explicitly outlined can leadto misunderstandings.’’ They proposed modifyingtheir lesson, which had originally asked children tosolve the school estimation problem in any way thatseemed appropriate, to include extensive instructionon the rules of rounding, including the modeling ofsimilar problems, before allowing the students to getto work.

These three interns left the lesson study believingthat it was the open-ended nature of their problemthat caused mathematical difficulties for studentsand that the rules of rounding were the mostimportant content for students to learn related toestimation. They reinforced these beliefs for eachother in conversations about estimation before and


after the lesson. The problem here is different thanthe one posed by the case of the Jefferson internswho never discussed the mathematics of theirlesson. The Jefferson interns had almost noconversational opportunities to learn about mathe-matics. The Camden Hills interns had multiple ones.The defining of estimation in their own words andtheir critique of the way estimation was presented inthe curriculum almost certainly helped to developimportant understandings, including the belief thatit is part of a teacher’s job to evaluate themathematics presented in addition to the pedagogy.However, these same conversations also producedbeliefs about estimation and rounding that wereproblematic from my perspective. I wanted theinterns to think about ways that they might helpstudents see a meaningful purpose for estimationand rounding; however, they developed a belief thatexplicit instruction about rules was the best way toapproach rounding and estimation. In addition, thisnew belief has the potential to be tenacious, sincethe interns left the lesson study believing that theirsense of estimation was informed by evidence fromthe classroom.

5.2. The Lakeside interns: learning from the equity

lens

Now, I consider the ways in which the conversa-tions about equity held by the Lakeside internssimilarly worked in ways that I considered bothproductive and unproductive. Mia, Juliet and Saraspent a great deal of time during the planningsession talking about the types of students they sawin Mia’s classroom. For instance, they referred tostudents who ‘‘test out of math,’’ who are ‘‘better,’’who are ‘‘lower,’’ ‘‘middle,’’ and ‘‘higher.’’ Duringthe second planning session, they solidified thesecategories by using students’ performances on timedmultiplication tests to categorize them. They calledstudents who had passed tests on facts through thefours ‘‘low;’’ students who had passed tests throughthe sixes ‘‘middle;’’ and the other students ‘‘high.’’

This attention to student difference provided aspace where Mia, Juliet and Sara could considerwhat it would take to provide meaningful instruc-tion to each child in the class. Their decisions toprovide a variety of support, such as hundred’scharts, multiplication tables and textbooks, cameout of these conversations, as did their decision toprovide multiple access points to the problem byusing a variety of numbers. The interns were aware

that their decision to assign numbers based onability might have social consequences for children.At one point, Sara asked, ‘‘I wonder if you had thenumber 48 and you had all these examples and I hadthe number 10, where there’s only two differentexamples, or three or four different examples, thenam I going to feel dumb? Because my paper only hasfour and yours has a zillion different arrays?’’ Thisquestion led to a discussion about the role of primenumbers and a decision to emphasize during thediscussion that the number of possible arrays wasnot necessarily related to the size of the number.

During their analysis, Sara described two stu-dents who had been identified as low and who wereworking to find arrays for 18. Sara said they found2� 9 quickly and ‘‘then started skip counting bythrees. They kept saying you have to be able to getthere by three, but they missed 15.’’ Sara reportedthat eventually these students discovered theirmistake and drew the 6� 3 array. Later, Mia saidthat she felt their decision to use many numbers hadbeen successful because these two students wouldhave had great difficulty finding arrays if they hadneeded to skip count all the way to 48. ‘‘They wouldhave been counting forever,’’ she said.

As I discussed earlier, Mia, Juliet and Sara werethe only interns in my class to differentiateinstruction during their lesson; however, theseinterns’ attention to ability also worked to reinforcemany assumptions they held prior to their lessonstudy, especially ideas that students of similarability should work together, that ability is easilyrecognized by the teacher, and that students whostruggle with procedural aspects of mathematics areunlikely to contribute to the learning of their moreprocedurally able peers. Mia, Juliet and Saraseemed to hold on to these beliefs even in the faceof conflicting evidence. Because of an absence, theyended up pairing a ‘‘high’’ student and a ‘‘low’’student to work with a difficult number and decidedto tape the conversation because they were inter-ested in the results. In talking about this tape duringthe analysis, Mia said, ‘‘They were working reallywell together. If you listened to it, you wouldn’tknow who was who.’’ Another opportunity fordisequilibrium occurred when Sara described thework of two students she had been observing.

Sara: I could tell these students were right in themiddle of understanding multiplication because—

Mia: Actually those are low ones.

Sara: Are they the low ones?


Mia: Yeah. The highest they’ve gone to is four.

Sara: Wow. They were really y they weren’t justin there goofing around. They were really trying.

Despite these two episodes and my frequentchallenges to the idea of ability grouping both indiscussions and in writing, Mia, Juliet and Sarawrote in their final report that ability grouping‘‘allowed all students to gain understanding.’’ Theirreport did not mention either the mixed group thathad worked well together or Sara’s misidentificationof a pair of ‘‘low’’ students. My point in offeringthis example is to show how tightly linkedproductive and unproductive understandings canbe. If this group had not attended closely to theconstruct of ability, they would have been unlikelyto provide structures such as multiple resources andvaried numbers, which worked to open the lessonup to greater participation. At the same time, thisattention to ability served to reinforce rather than tochallenge their incoming assumptions about stu-dents.

6. Concluding comments

The structures of the lesson study assignmentdescribed here did provide some beginning teacherswith opportunities to develop mathematical- andequity-oriented ways of looking at the work ofteaching. However, the attention paid to issues ofmathematics and equity was problematic when itserved to reinforce assumptions in favor of rule-based mathematics or strict ability grouping. Thisfinding suggests new directions for research aboutthe use of lesson study and other forms ofcollaborative, practice-based learning. Previously,the conversation in the literature has largely focusedon developing or improving structures that supportproductive interactions among teachers (e.g., Fer-nandez, 2003; Fernandez et al., 2003; Grossman etal., 2001; Lewis, 2002; Wilson & Berne, 1999). Thisis valuable work, and it can help to solve theproblem exemplified by the Jefferson interns in thisstudy: that of collaborative work without learning.However, the focus on development of structurescan lead to an unspoken assumption that all of thedifficulties inherent in collaborative, practice-basedwork can be improved away. The problem exem-plified by the Camden Hills and Lakeside interns—the tight link between productive and unproductivelearning—suggests another area of exploration: onethat seeks to find pedagogical strategies for addres-

sing problematic learning, rather than assumingthat with the proper techniques these troublesomeunderstandings will disappear. In research, this willmean looking at not only whether the desiredlearning did or did not occur, but also looking atundesired learning. In teaching, this perspectiverequires that educators think about building assign-ments that offer opportunities for challengingproblematic understandings that may have devel-oped during previous work. For instance, when Iretaught the lesson study assignment in a subse-quent course, I planned a student work analysisassignment as a follow up to the lesson study. Idetermined the focus of the analysis based onlearning that I wanted to challenge from the lessonstudies.

As Wilson and Berne (1999) noted, pedagogicalpractices that are expert centered and didactic aregenerally considered to be ineffective within theteacher education community; whereas, inquiry-oriented practices are often described as creatingthe possibility for long-lasting and personally mean-ingful learning (Ball & Cohen, 1999; Cochran-Smith& Lytle, 1999; Putnam & Borko, 1997). One of myfears as an educator is that this is true. That is,Juliet, Mia and Sarah may have left my classroomwith a long-lasting belief in ability grouping,while Mike, Lara and Gillian may be committedto the explicit teaching of rounding rules. Had Ilectured, I would not have emphasized the benefitsof ability grouping. My lecture may have hadunintended consequences, but it is unlikely thatany changes in perspective caused by the lecturewould have been as deeply held as those developedin a practice-based, inquiry-oriented setting. Aseducators, we give up a certain amount of controlwhen we move toward inquiry-oriented pedagogies,and the more we locate teacher learning in K-12classrooms and the more we allow our students—bethey preservice or practicing teachers—to directtheir own learning, the greater the risk we assume.Our concern about the content of what our studentsmay learn should be all the more serious if, asresearch suggests, inquiry-oriented pedagogies arelikely to produce learning that is long lasting. Inaddition to examining strategies to launch lessonstudies and action research projects, teacher educa-tors may also find it productive to researchstrategies that they may use to guide the learningof others, even while turning over some control.

Finally, in considering the special context ofinquiry in preservice classrooms, this study suggests


that teacher educators may need to think aboutways to become more explicit in guiding beginningteachers’ work. For instance, teacher educatorsmight pay explicit attention to the development ofproductive lenses for examining curricula andteaching. In this article, I have focused on mathe-matics and equity, and previously Fernandez et al.(2003) have suggested others, including those ofresearcher and curriculum developer. It may be thatdiscussing more specific lenses with beginningteachers (such as place value, problem solving,assessment, or differentiation) might be more usefulin shaping their thinking and conversations. Inaddition, it will be important to look at both which

tools are most likely to support substantive learningabout students and content, and what kind ofinteractions with these tools are most educative. Inmathematics, it seems that work with reformcurriculum may have supported significant learning,but that this work needed to be carefully structuredby the instructor. It will be important to explore therole that written curricula might play in supportinglearning in other content areas and to look at theways that interaction with such curricula can bepromoted by instructors without constraining op-portunities for productive critique, which led tomathematical learning for some of the groups in thisstudy.

In doing this work as both practitioners andresearchers, we will need to move toward theoriesabout practice-based learning that not only help usdevelop structures that promote meaningful con-versations about content, students and teaching, butalso help us think about the unintended learningthat is almost certain to occur when practicing orpreservice teachers seek to make sense of theunpredictable world of the classroom. No doubt,there are many things I could have done as aninstructor to increase the likelihood that all of thegroups in my class talked about mathematics moremeaningfully, and things I could have done to helpmore groups think about how to teach all children.Research that looks at the ways that Japaneseteachers engage in lesson study or at particularlysuccessful instances of American teachers’ practice-based learning may help provide strategies to copewith similar problems. However, some parts of thechallenge of practice-based work cannot be im-proved away. No matter how well done, learningsituated in the classroom is likely to be messy, andour teaching and research will need to acknowledgethis.

References

Ball, D. L., & Cohen, D. K. (1999). Developing practice,

developing practitioners: Toward a practice-based theory of

professional education. In L. Darling-Hammond, & G. Sykes

(Eds.), Teaching as the learning profession: Handbook of policy

and practice (pp. 3–32). San Francisco: Jossey-Bass.

Boaler, J. (2000). Introduction: Intricacies of knowledge, practice

and theory. In J. Boaler (Ed.), Multiple perspectives on

mathematics teaching and learning (pp. 1–17). Westport, CT:

Ablex Publishing.

Chokshi, S., & Fernandez, C. (2004). Challenges to importing

Japanese lesson study: Concerns, misconceptions and nuan-

ces. Phi Delta Kappan, 85(7), 520–525.

Cochran-Smith, M. (1999). Learning to teach for social justice. In

G. Griffin (Ed.), The Education of Teachers: The Ninety-eighth

yearbook of the national society for the study of education

(pp. 114–144). Chicago: University of Chicago Press.

Cochran-Smith, M., & Lytle, S.L. (1999). Relationships of

knowledge and practice: Teacher learning in communities.

In A. Iran-Nejad, & P.D. Pearson (Eds.), Review of Research

in Education (pp. 249–305).

Crespo, S. (2003). Learning to pose mathematical problems:

Exploring changes in preservice teachers’ practices. Educa-

tional Studies in Mathematics, 52(3), 243–270.

Dyson, A. H., & Genishi, C. (2005). On the case: Approaches to

language and literacy research. New York: Teachers College

Press.

Emerson, R. M., Fretz, R. I., & Shaw, L. L. (1995). Writing

ethnographic fieldnotes. Chicago: University of Chicago Press.

Erickson, F. (1986). Qualitative methods in research on teaching.

In M. C. Wittrock (Ed.), Handbook of research on teaching

(pp. 119–160). New York: Macmillan.

Fernandez, C. (2002). A practical guide to translating lesson

study for a US setting. Phi Delta Kappan, 84(2), 128–134.

Fernandez, C. (2003). Learning from Japanese approaches to

professional development: The case of lesson study. Journal of

Teacher Education, 53(5), 393–405.

Fernandez, C., Cannon, J., & Chokshi, S. (2003). A US–Japan

lesson study collaboration reveals critical lenses for

examining practice. Teaching and Teacher Education, 19(2),

171–185.

Fernandez, C., Chokski, S., Cannon, J., & Yoshida, M. (2001).

Learning about lesson study in the United States. In M.

Beauchamp (Ed.), New and old voices on Japanese education.

Armonk, New York: M.E. Sharpe.

Greene, M. (2001). Educational purposes and teacher develop-

ment. In A. Lieberman, & L. Miller (Eds.), Teachers caught in

the action: Professional development that matters. New York:

Teachers College Press.

Grossman, P., Wineburg, S., & Woolworth, S. (2001). Toward a

theory of teacher community. Teachers College Record,

103(6), 942–1012.

Hawley, W., & Valli, L. (1999). The essentials of effective

professional development: A new consensus. In L. Darling-

Hammond, & G. Sykes (Eds.), Teaching as the learning

profession: Handbook of policy and practice (pp. 127–150). San

Francisco: Jossey-Bass.

Hiebert, J., Morris, A. K., & Glass, B. (2003). Learning to learn

to teach: An ‘‘experiment’’ model for teaching and teacher

preparation in mathematics. Journal of Mathematics Teacher

Education, 6(3), 201–222.


Hymes, D. (1972). Introduction. In C. Cazden, D. Hymes, &

John (Eds.), Functions of language in the classroom (pp.

xi–lvii). New York: Teachers College Press.

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral

participation. Cambridge: Cambridge University Press.

Lewis, C. (1998). Everywhere I looked—Levers and pendulums.

Journal of Staff Development, 23(3), 59–65.

Lewis, C. (2000). Can you lift 100 kilograms? (220 pounds): The

lesson research cycle. Oakland: Mills College Department of

Education Available at: www.lessonresearch.net.

Lewis, C. (2002). What are the essential elements of lesson study?

The California Science Project Connection, 2(6), 1 4.

Little, J. W. (1990). The persistence of privacy: Autonomy and

initiative in teachers’ professional relations. Teachers College

Record, 91(4), 509–536.

Little, J. W. (1999). Organizing schools for teacher learning. In L.

Darling-Hammond, & G. Sykes (Eds.), Teaching as the

learning profession: Handbook of policy and practice (pp.

233–262). San Francisco: Jossey-Bass.

Lord, B. (1994). Teachers’ professional development: Critical

colleagueship and the role of professional communities. In

The future of education: Perspectives on national standards in

America (pp. 175–204). New York: College Entrance Exam-

ination Board.

Moyer, P. S., & Milewicz, E. (2002). Learning to question:

Categories of questioning used by preservice teachers during

diagnostic mathematics interviews. Journal of Mathematics

Teacher Education, 5(4), 293–315.

National Research Council. (2005). Adding it up: Helping children

learn mathematics. Washington, DC: National Academy

Press.

Putnam, R. T., & Borko, H. (1997). Teacher learning: Implica-

tions of new views of cognition. In B. J. Biddle, et al. (Eds.),

International handbook of teachers and teaching (pp.

1223–1296). Netherlands: Kluwer Academic Publishers.

Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York:

The Free Press.

Takahashi, A., & Yoshida, M. (2004). Ideas for establishing

lesson-study communities. Teaching Children Mathematics(-

May), 436–443.

Tannen, D. (1984). Conversational style: Analyzing talk among

friends. Westport, CT: Albex Publishing.

Van Es, E. A., & Sherin, M. G. (2002). Learning to notice:

Scaffolding new teachers’ interpretations of classroom inter-

actions. Journal of Technology and Teacher Education, 10(4),

571–595.

Wilson, S. M., & Berne, J. (1999). Teacher learning and the

acquisition of professional knowledge: An examination of

research on contemporary professional development. In A.

Iran-Nejad, & P. D. Pearson (Eds.), Review of research in

education, Vol. 24 (pp. 249–305). Washington, DC: American

Educational Research Association.

http://www.lessonresearch.net

messy learning: preservice teachers’ lesson-study conversations about mathematics and students

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