The Elementary School JournalVolume 108, Number 1! 2007 by The University of Chicago. All rights reserved.0013-5984/2007/10801-0003$10.00

Multiple Perspectiveson the Developmentof an Eighth-GradeMathematicalDiscourse Community

Edith Prentice MendezSonoma State University

Miriam Gamoran SherinNorthwestern University

David A. LouisThe Nueva School, Hillsborough, CA

Abstract

In this article we examine the development, over1 year, of mathematical discourse communities in2 eighth-grade mathematics classes in a suburbanpublic middle school. The curriculum topics in-cluded probability, functions, graphing, dataanalysis, and pre-algebra. The 50 students wereheterogeneously placed; most were from upper-middle-class families. Data included videotapedclassroom observations, field notes, and teacherreflections. We explored both the students’ grow-ing competencies with mathematical discourseand changes in how the teacher attended to stu-dents’ ideas. We present the teacher’s impressionsof the developing discourse community, and weapplied 2 research-based lenses, robust mathe-matical discussion to assess the strength of stu-dent discourse, and professional vision for class-room discourse to analyze the ways in which theteacher paid attention to, and reflected on, ideasstudents raised during discussion. Applying mul-tiple perspectives highlighted the complex natureof developing a discourse community and thechallenges facing the teacher as he worked to or-chestrate constructive dialogue for learning math-ematics and to become aware of what studentswere learning in this context. We also provide ananalytic tool, the robust mathematical discussionframework, that will be useful for teachers,teacher educators, and researchers to evaluate theevolving nature of classroom discourse.

Mathematical discourse has been the focusof many studies of U.S. reform mathematicsteaching (Lampert, 2001; Nathan & Knuth,2003; Silver & Smith, 1996). Such researchhas shown that developing a discourse com-munity can be challenging for both teachersand students. For example, supporting class-room discourse calls for teachers to attend tostudent thinking in ways that may be differ-ent from teachers’ typical practices (Heaton,2000). In addition, participating in class-room discourse often requires that students

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learn new ways to articulate and justify theirideas (Hufferd-Ackles, Fuson, & Sherin,2004).

Three Lenses for ExploringClassroom DiscourseAt the broadest level, research on classroomdiscourse has explored the ways in whichparticipants communicate with one another(Cazden, 1986; Pimm, 1996). Some research-ers, for example, have characterized pat-terns of classroom discourse (Mehan, 1979),whereas others have identified instructionaltechniques thought to foster effective dis-course (O’Connor & Michaels, 1996). Herewe take a different approach, drawing onthree perspectives, each of which illumi-nates, in a different way, the nature of dis-course in a mathematics classroom.

Teacher NarrativeIn the past, it was generally believed that

researchers generated important informa-tion about teaching and learning. More re-cently, however, it has become clear thatteachers are a critical source of knowledgeconcerning teaching and learning (Cochran-Smith & Lytle, 1993; Fenstermacher, 1994).Teacher narrative, in which teachers providea written account of their experiences, hasbeen found to be a valuable method forrevealing this “wisdom of practice” (Con-nelly & Clandinin, 1999; L. Shulman, 1987).Teacher narrative includes a variety of for-mats such as journal writing and case de-velopment. In both contexts, the goal is forthe teachers to provide a written accountthat simulates a reflective and reciprocal re-lationship between the teacher and his or herwork (Burnaford, Fischer, & Hobson, 2001).In the research reported in this article, weused teacher-written cases and journal en-tries as the source of the teacher’s perspec-tive on the mathematical discourse commu-nity that was developing in his classroom.

Robust Mathematical DiscussionA second perspective we used to ana-

lyze classroom discourse is a new analytic

tool called robust mathematical discussion(RMD), adapted from Mendez (1998). TheRMD lens provides a way to look closely atcomponents of student discourse to get asense of the “robustness” (mathematicaland discursive strength) of the discourse.Student verbal moves are categorized as re-lated to (a) the mathematical content, and(b) characteristics of a vigorous discussionforum. This distinction between the content(the substance of students’ comments) andthe process (how students participate in dis-course) has been discussed widely (Nathan& Knuth, 2003; Sherin, 2002; Williams &Baxter, 1996). In this study we extendedprior research by further elaborating keyfeatures of the content and process of class-room discourse. We identified these se-lected features from both policy and re-search documents that investigate the roleof discourse in student learning. Mendez(1998) contains a detailed discussion of thecriteria used to select these features.

Mathematics dimension. The mathe-matics dimension of RMD is comprised ofthree aspects of mathematical argumenta-tion: representations, generalization, andjustification (see App. A). Representationsrecognizes the value of alternate ways todescribe a concept. Research has shown thatrepresentations students develop provideevidence of their subject-matter knowledge(Wilson, Shulman, & Richert, 1987). Fur-thermore, the National Research Council(NRC) (2001) reported in Adding It Up thatusing different representations for mathe-matical situations and purposes indicatedconceptual understanding (p. 119). Thus, astudent’s ability to model an idea throughmathematical representations shows a depthof understanding that mere memorization oralgorithmic thinking misses.

Generalization involves the capacity toform connections among related ideas,which is central to solving problems (Polya,1945/1957) as well as to grasping the struc-ture of the domain (Bruner, 1960/1977). Asthe NRC (2001) noted, generalization allowsstudents to efficiently learn clusters of in-

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terrelated mathematical principles (p. 120).Certainly the coherence of a generalizationdispels the view of mathematics as a bag ofalgorithmic tricks.

The third mathematical aspect of RMDis justification. There is consensus amongtraditionalists and reformers alike that pre-senting a logical, warranted argument is abasic part of what it means to do mathe-matics. The NCTM Principles and Standards(2000) calls for regular practice in reasoningand proof (p. 56). Ball and Bass (2003) notethat reasoning in mathematics is as funda-mental as comprehension in reading (p. 29).The justification aspect is of particular con-cern in U.S. schools. As Hiebert et al. (2005)found in analyzing the Third InternationalMathematics and Science Study (TIMSS)1999 video study, which compared teachingin eighth-grade classes in the United Statesand six higher-performing countries, onlyin the United States were no instances ofmathematical justification or generalizationfound (p. 118). Further, only the Nether-lands and the United States had no lessonswith a proof.

Discussion dimension. RMD also ex-plores three aspects of the discussion pro-cess: engagement, intensity, and building.Engagement concerns the level of student in-volvement in the discourse. Research has in-dicated that student talk can promote learn-ing (Boaler & Greeno, 2000; Kazemi, 1998;Simon & Blume, 1996). For example, Kazemi(1998) highlighted discourse-related pro-cesses that supported student conceptualunderstanding. Boaler and Greeno (2000)found that students in discussion-orientedclasses expected to understand mathemati-cal relationships, not just to learn rituals (p.182). NCTM’s Principles and Standards forSchool Mathematics (2000) also emphasizesthe value of student participation, includingactive listening, for thinking critically (p. 63).

Intensity considers the ways in whichstudents enter the discussion. Research hasemphasized the importance of students in-terjecting key ideas into classroom discus-sions, both as an indication of student un-

derstanding of ongoing discourse and asan indication of students taking responsi-bility for their own learning (Cazden, 1986;Hufferd-Ackles et al., 2004). Thus, this as-pect of discussion emphasizes students vol-untarily entering a discussion rather thanresponding to a teacher request.

In discussing the value of classroomdiscourse, researchers have focused on theneed for students to build on each others’ideas, to comment on and critique oneanother’s mathematical ideas (Ball, 1993;Sherin, Louis, & Mendez, 2000). The build-ing aspect, therefore, concerns connectionsbetween student ideas. A student might saysimply, “I agree.” But in a robust discussion,students will add something germane thatconnects to earlier comments, whether insupport or disagreement, integrating theirremarks with those of other students. Insummary, the aspects of the mathematicsand the discussion dimensions of RMD pro-duce both a composite view of, and a toolfor analysis of, classroom mathematical dis-course.

Professional Vision for ClassroomDiscourseThe third lens we used to analyze the

evolving discourse community is that of pro-fessional vision (Goodwin, 1994; Sherin,2001). Members of any professional disci-pline become quite good at making sense ofa specific set of phenomena. For example,archeologists see patterns and textures insand, and meteorologists distinguish amongcloud shapes. This is their professional vi-sion.

For teachers, professional vision is con-cerned with attending to significant featuresof classroom interactions. In prior work, wehave described teachers’ professional visionas involving two key components (Sherin,2001, 2007). First, teachers draw on theirprofessional vision to decide which aspectsof instruction require their attention. In aclassroom, many things happen simulta-neously, so part of teachers’ expert knowl-edge is their ability to identify interactions

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that are significant and deserve a closerlook. Second, teachers’ professional visioninvolves reasoning about those selected fea-tures of instruction—that is, making senseof what they see. Thus, professional visionis also concerned with the kinds of strate-gies teachers use to interpret what they no-tice.

In this article, we examine part of oneteacher’s professional vision, what we callprofessional vision for classroom discourse(PVCD). This variable concerns how theteacher attends to, and reasons about, class-room discourse. We focus on the teacher’sPVCD as it relates to the ideas that studentsraise in discussion.

This focus on how and what teachers no-tice in classroom interactions, and on studentthinking in particular, is not new. Researchin mathematics education reform has, formany years, emphasized the need for teach-ers to pay close attention to the ideas stu-dents raise in class (Fennema & Nelson,1997; NCTM, 2000; Schifter, 1998). Several re-searchers have explored approaches forhelping teachers learn to pay attention to as-pects of student thinking. The CognitivelyGuided Instruction project, for example, pro-vided teachers with a research frameworkdescribing the ways in which students typi-cally understand and work through additionand subtraction word problems (Fennema,Franke, & Carpenter, 1993). Teachers thenuse this information to determine appropri-ate tasks for students and to decide what stu-dents understand about these concepts. Inrelated work, researchers investigated theways in which close examination of studentwork in study groups provided opportuni-ties for teachers to learn strategies for attend-ing to key features of student thinking (Ka-zemi & Franke, 2004). Building on suchresearch, in this article we consider how, inthe context of a discourse community, ateacher might develop new approaches forattending to the ideas students raise. Part ofour point, then, is that developing a dis-course community involves figuring out notjust what questions to ask students, or what

explanations to provide, but also what fea-tures of the student discourse the teachershould attend to.

MethodThis article draws on an intensive 3-yearcollaboration between the third author, ateacher-researcher, and the two other au-thors, university researchers.

ParticipantsDavid Louis was in his fourth year of

teaching when the project began. He had re-cently completed his master’s in educationat a local university and had expressed in-terest in conducting research in his ownclassroom. Thus, when researchers Mendezand Sherin sought a teacher-collaborator fora study of mathematics teaching, Louis wasan obvious choice.

Louis taught in a public middle school inthe San Francisco Bay area that housed ap-proximately 1,000 students. Most studentscame from upper-middle-class families, withthe student body 70% Caucasian, 20% Asian,5% African American, and 5% other ethnic-ities. The students in both eighth-gradeclasses were heterogeneously placed fromamong the 70% of students who were not inthe honors track. The curriculum was devel-oped to prepare these students for algebra.All student names are pseudonyms.

Data CollectionThe primary data for this study came

from two eighth-grade mathematics classesin a suburban middle school in the SanFrancisco Bay area. Across 1 calendar year(January–December), Louis worked to es-tablish a discourse community in his class-room. He focused his efforts initially in oneclass from January to June; the first episodein this article was from this class. Then, inthe fall of the same year, he resumed hisefforts with a new group of students, theclass discussed in the second two episodes.We (the first two authors) observed andvideotaped his classes regularly during thistime, for a total of 22 lessons from the first

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class and 44 lessons from the second class.We collected relevant classroom artifacts,took observation notes for an additional datasource, and transcribed pertinent video-tapes.

We used two additional data sources.First, Louis kept a reflective teaching jour-nal, generally writing twice each week afterschool about events that stood out to him inthe day’s lesson (Louis, 1997c). His entriestypically consisted of comments related tothe mathematics discussed in class, theideas students raised, and his pedagogicalmethods. Second, the teacher participatedin two case-writing workshops (J. Shulman,1992; L. Shulman, 1996) in the summers of1996 and 1997, both directed by Judith Shul-man, in which he reflected in detail onteaching dilemmas he encountered. NeitherMendez nor Sherin participated.

AnalysisOf the 66 videotaped observations we

collected, 56 involved extended whole-classdiscussions. From these, we selected threewhole-class discussions for analysis for thisarticle, one each from May, September, andDecember. We chose these lessons not onlybecause they occurred at different times, butalso because Louis considered them to bebenchmarks in his teaching and had writtenextensively about each one. Specifically, theteacher selected the May and December les-sons as the focus of his work in the case-writing workshops described below. Simi-larly, the September lesson was the subjectof an extended journal entry. Furthermore,related research (Sherin, 2002; Sherin, Men-dez, & Louis, 2004) suggested that the threeselected lessons were representative of theteaching and learning that took place in hisclassroom across the year. We now describethe three lenses we used to analyze each ofthe discussions in more detail.

Teacher narrative. Louis participated intwo case-writing workshops. Each tookplace over 3 days and involved conversa-tions with teachers and researchers con-cerning a “teaching dilemma” David iden-

tified, as well as time for the teacher to domuch of the narrative writing of the caseand to receive feedback. These cases (Louis,1997a, 1997b) and his reflective journal(Louis, 1997c) were the genesis of the teachernarrative presented in this article. We usedsegments and paraphrases of these docu-ments related to the teacher’s ideas about thedeveloping discourse community to con-struct the reflections presented here. Cita-tions from the documents are given forcomments from the cases or journal.

Coding robust mathematical discussion.We analyzed transcripts from the threeclassroom lessons using the RMD frame-work. Analysis focused on an episode ofwhole-class discussion delineated by its co-hesion in content (Sierpinska, 1997). Wecoded each of the RMD attributes sepa-rately, as described below (see App. B).

For the mathematics dimension, weevaluated representations by counting therepresentational forms for each mathemat-ical topic or solution in the lesson tran-scripts drawn from the classroom videos.For example, forms of representation froma discussion of percentages might includeratios, decimals, and fractions as well aspercentages. When students use more rep-resentations, a topic is examined and de-fined more completely, so greater numbersof forms of representation are associatedwith more robust discussion. The other twoaspects involve separate coding of individ-ual student speaking turns. For generaliza-tion, we coded each student turn as concrete(limited to one specific context), a specificcomparison between two contexts, or gen-eralization if the student recognized a pat-tern, categorized several cases in a generalway, or made a comparison over severalcontexts. These categories represent in-creasingly sophisticated understandings,so higher percentages of generalizationand specific comparison indicate a rich dis-cussion. For justification, we coded eachstudent turn as statement (an answer ormethod devoid of any explanation: “I agreewith Tina”), explain (an answer that in-

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cludes either why the answer holds or howit was obtained: “So I got 30 for the surfacearea just by counting all the squares”), orproof (a logical argument about why the so-lution is correct or a logical refutation orcounterexample: “It can’t be graph A be-cause that’s a constant, and the rate of fillingthe funnel is changing”). The discourse re-quired for the argument in a proof is betterdeveloped than that of an explain, which inturn demands more than a statement. Thus,the occurrence of these two more complextypes of student turns is prized in evaluat-ing justification.

For the engagement aspect of the dis-cussion dimension, students present in classare coded as either nonparticipant or par-ticipant in the discussion. Although this ig-nores nonverbal students, a high percentageof participants is a proxy for high involve-ment with the mathematics, and strong en-gagement. Intensity is observed by codingeach speaker’s entrance into the discussionas either elicited (the student is nominatedby the teacher) or volunteer (the student en-ters the discussion independently, withoutteacher mediation). This aspect values stu-dents who demonstrate their involvementby interjecting relevant comments into thediscourse. Thus, a high percentage of vol-unteer statements is associated with highintensity. Finally, to evaluate the buildingaspect, although each student speaking turnis coded separately, a sequence of theseturns determines whether students arebuilding on each other’s ideas. An unre-lated idea, which may be a direct answer tothe teacher or an unrelated new idea, showsno link to earlier student comments. A re-sponse shows some attention to other stu-dents’ thinking by agreeing or disagreeingwith a stated student position without anyelaboration or explanation. The highestvalue is placed on a turn that is codedbuild, in which the student elaborates onan idea another student has expressed,giving additional support for the view-point or developing a related result fromit. Teacher comments are not considered,

because RMD is a measure of student dis-cussion, not of teacher facilitation. Percent-ages of each type of statement are calcu-lated; high levels of build and responseindicate rich building.

We university researchers coded thethree selected lessons independently, usingtranscripts of the lessons as well as the videoas a reference. To do so, we assigned eachstudent turn a code for the attributes of gen-eralization, justification, intensity, and build-ing. The percentage of participants in the dis-cussion was calculated for engagement. Inaddition, we noted whether or not each stu-dent statement illustrated a new represen-tational form. We coded a total of 117 stu-dent turns. Initial agreement varied from86% to 100% across the six aspects. Consen-sus was reached through discussion.

Professional vision for classroom dis-course. Simultaneously with our investiga-tion of how students participated in themathematical discourse, we sought to char-acterize the role of the teacher. Toward thatend, we examined instructional strategiesthat the teacher used during whole-class dis-cussions (e.g., What kinds of questions didthe teacher ask? How did he encourage stu-dent participation?) as well as the teacher’smore metacognitive processes of reflectingon the classroom discourse. Elsewhere wehave reported our findings concerningchanges in the teacher’s instructional strate-gies. In Sherin, Mendez, and Louis (2000)and Sherin, Louis, and Mendez (2000) wediscussed prompts the teacher used to probestudents’ ideas and to encourage students tobuild on each other’s ideas. In Sherin (2002),we described the development of a particu-lar discourse pattern in the classroom inwhich the teacher contrasted two viewpointsstudents presented in an effort to enhancethe mathematical nature of students’ discus-sions. In contrast, here we consider howthe teacher reflected on the classroom dis-course, and, in particular, on the ideas stu-dents raised during discussion. We exam-ine both how the teacher paid attention tostudent thinking during instruction as well

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as which features of student talk were sa-lient in his reflections on the lessons.

We used three data sources to study theteacher’s professional vision in the contextof the selected lessons. First, we analyzedtranscripts and videotapes in order to char-acterize the features of each discussion thatthe teacher attended to most closely duringinstruction. His comments were categorizedas (a) general elicitations of student ideas,(b) requests for student elaboration, (c) syn-theses across student ideas, or (d) introduc-tion of new mathematical ideas. A smallnumber of teacher comments were coded as“other.” These primarily concerned class-room management. We independentlycoded each of the teacher’s spoken turns inthe three lessons along these lines. Initialinterrater reliability was above 90% forall three lessons. Consensus was reachedthrough discussion. This analysis suggesteda shift over time in the teacher’s attentionto more general versus more specific as-pects of student talk. To examine this fur-ther, we analyzed the teacher’s journal en-tries and narrative cases with respect to thethree focus lessons. To be clear, the teacher’snarrative cases and journal entries formedthe basis for the lens of his perspective thatis presented here and served as a focus ofanalysis of his PVCD. Specifically, we di-vided the three narratives into “idea units”(Jacobs & Morita, 2002), segments in whicha particular issue was discussed. We thenhighlighted those segments that includedthe ideas that students raised (or did notraise) in class for further analysis. We dis-tinguished between written comments thatconcerned specific aspects of student dis-course (“No one responded to Gary’s com-ment”) and those that focused on more gen-eral features of the discourse (“Studentsweren’t listening to each other”). Two re-searchers independently coded the journalentries and narrative cases. Initial agree-ment on which segments of the narrativesconsidered students’ ideas was 85%; agree-ment on whether the selected segments re-ferred to general or specific aspects of stu-

dent discourse was 92%. Consensus wasagain reached through discussion.

Results: Multiple Perspectives onThree Mathematical DiscussionsWe present our analyses of the three se-lected classroom excerpts. The first, fromspring of the initial school year of our col-laboration, features the first class in stu-dent presentations and discussion aboutthe surface area of figures. The second,from the fall of the following school year,concerns students’ attempts to representthe water level in a container being filled.The final episode is from the winter of thatsame year and focuses on expressing per-centage and, in particular, students’ effortsto make sense of the notion of “percentmore than.” Episodes 2 and 3 are from thesecond class.

Episodes ranged in length from 6 min-utes for the first, to 13 and 12 minutes forthe second and third discussions, respec-tively. We believe this difference reflectedthe infancy of whole-class discussion in thefirst episode and its development over time(Sherin, 2002). Interestingly, the ratio of stu-dent to teacher talk remained the sameacross the first two episodes, with a ratio ofapproximately 1:1. In the third episode, theratio of student to teacher talk approached3:2. The number of students present wasrelatively stable across the three lessons,with 24 students in the first excerpt and 25students present in the remaining two ex-cerpts.

Recognizing the Need for ClassroomDiscourse: Amy’s BabyIn the lesson we call Amy’s Baby, the

class was working with Cuisenaire rods tocreate figures shaped like people (see Fig.1). Then students were to devise a methodto find the surface area of their figures. Moststudents simply counted, using the unitsquare as having an area of one. Amy, how-ever, used a different approach to determinethe surface area of the “baby” she created.She first found the total surface area of each

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Fig. 1.—The baby Amy’s group built of Cuisenairerods.

rod comprising the body, the limbs, and thehead and then subtracted two for each pointon the figure where two rods met, hidingtheir surfaces. David circulated through theclassroom, asking students, including Amy,to explain to him their solution strategies.Following this, students presented their so-lutions to the whole class, and the teacherfirst called on Amy. There were no ques-tions from the class, and David then calledon the next presenter.

David’s reflections. I can still remembermy excitement at seeing Amy’s method. Fi-nally, one student had gone beyond the typ-ical counting method to come up with amore general solution of subtracting two foreach contact point. Amy’s method wouldwork for any contact point—a conceptualstep toward the abstraction needed for suc-cess in algebra. I envisioned other studentsbeing intrigued about her method when shepresented it to the class. Alas, her presen-tation was a real disappointment—here’sthe actual transcript:

Amy: Okay, what we did was, we disre-garded the fact that some of them couldbe seen—that some of the surfacescouldn’t be seen. So we added up 18 plus18 plus 20 and then—and we got 56. Andthen we went back and took the baby andwe counted all of the squares that youcouldn’t see that were covered up, thatwouldn’t count as surface area. And thenwe took away 10 and we got 46.

Aside from mentioning the squares thatcouldn’t be seen, she gave no sign of theelegance or generalized solution that I re-membered from her earlier group worktime. Nobody else showed any interest, andI dropped the ball by neither probing Amyfor details nor commenting on her unusualsolution myself.

This episode bothered me and got methinking—what had gone wrong? Therewas such potential in Amy’s idea, but shedidn’t know how to articulate her solution.I had helped develop a culture of inquiry—the students were used to exploring math-ematical ideas—but I had not preparedthem for sharing and listening. I had nothelped create a discourse culture. As I wrotelater in my case about this: “The teachablemoment slipped through my fingers. Imissed it in the moment, but in hindsightmaybe I missed “it” in a larger context.Maybe the “it” was not Amy’s presentation,but the preparation I gave Amy ahead oftime to share her methods with other stu-dents” [Louis, 1997a, p. 51]. What mighthave happened if I had spent time helpingmy students learn how to share their ideas?I like to think that Amy would have givena better explanation and that the other stu-dents would have joined in with clarifyingquestions and comments. I myself didn’thave the tools to facilitate the discussion Iwas imagining. I hadn’t thought about thekinds of questions I could ask my studentsto keep the discussion going, so things justended there, “not with a bang, but a whim-per” [Eliot, 1925/1957].

Robust mathematical discussion analy-sis. Analysis using RMD confirmed David’sdisappointment in the minimal discourse inthis episode. This example consisted more ofserial student presentations than of class-room dialogue, but common features of stu-dent voices were at the forefront in the dis-course. Further, this episode was importantin David’s understanding of his own teach-ing, so it is useful to compare the RMD anal-yses with the two later episodes.

We first examine representations. Two

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forms are used to represent an object’s sur-face area. Amy used whole rods, countingthe entire area of each before subtracting thecontact points, whereas Joey counted indi-vidual visible square units. The teacher hadhoped that Amy would present her gener-alized method of finding the surface area ofany rod shape, but instead her presentationwas specific and concrete. RMD is designedfor analyzing only whole-class discussion,so the earlier insights she described to hergroup and to David were not considered.The other students in the public discoursesimilarly gave concrete responses, limitedto their own solutions, and thus the episodelacked any indication of comparing or gen-eralizing. Concerning justification, therewere no student comments deserving thelabel of proof. We rated several studentturns as explain, such as that by Joey: “So Igot 30 for the surface area just by counting.And I just counted all the squares.” But 76%of the student contributions were rated atthe lowest level, statement, without any jus-tifying. After Amy gave her explanation,the following exchange took place.

David: Do you want to finish? The wayyou calculated the volume?

Amy: Yeah. For the volume we got 11.David: Good.

David’s comments fit the initiation-response-evaluation standard classroom dialogue se-quence (Mehan, 1979) and show acceptanceof Amy’s unsupported answer, an indica-tion of the norm for student justification.

The discussion dimension indicatedsome engagement and intensity, with 29%of students participating in the discussion,and 24% of the speaking students enteringas volunteers. In the excerpt below, thereare both an elicited speaker, Charlene, anda volunteer, Sally.

David: What kind of measurements do wemeasure surface area in? Charlene?

Charlene: Square centimeters.Sally (to David): Is that right?

Regarding building, however, only 12% ofturns were in response to earlier student

comments. Sally’s query was one, but, likethe other responses, this added nothingmathematically new to the discourse.Eighty-eight percent of the student com-ments were direct responses to the teacher,as in Charlene’s comment. No studentturns were coded as build with the intro-duction of new, related ideas drawing onearlier student comments (see Table 1).

Professional vision for classroom dis-course analysis. It is clear from David’s nar-rative case (Louis, 1997a) that he believedAmy’s method of finding surface area hadgreat potential mathematically and that hewas eager to hear what other studentsthought about the idea. He explained, “Iworked hard at holding back my smile ofexcitement . . . when Amy decided to shareher solution as the whole class reconvened”(p. 50). Thus, it appears that, even in thisearly stage, before David had focused ondeveloping a discourse community, he waspoised to attend to student thinking in thiscontext.

This attention to student thinking, how-ever, was manifest only at a general level.During the discussion, David did notprompt students to respond to Amy’s idea.His professional vision was not focused onindividual student thinking; therefore, hedid not push students to say what theythought about Amy’s idea. Instead, hiscomments to the class were general (e.g.,“It’s interesting, [our answers using differ-ent methods] are all the same.”). Moreover,David did not probe Amy’s understandingof the idea, either as he spoke with her in-dividually, or in the context of the class dis-cussion. He recognized Amy’s method asnoteworthy but did not investigate thedepth of her understanding of the method.

Instead, as David described in his nar-rative case, he had a general sense that “theclass wasn’t getting it.” He explained fur-ther that “the class listened but did not en-gage in any lively discussion. . . . No onecommented that Amy’s method was differ-ent or unusual” (Louis, 1997a, p. 50). Thisquote illustrates that the teacher was at-

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Table 1. RMD Ratings for Three Class Discussions

Discussion

RMD Variable “Amy’s Baby” “Plugged Funnel” “400% More”

Mathematics dimension:Representation:

Forms 2 6 4Generalization:

Concrete 100 50 54Specific comparison 0 39 13Generalization 0 11 33

Justification:Statement 76 36 70Explain 24 48 30Proof 0 16 0

Discussion dimension:Engagement:

Nonparticipant 71 32 64Participant 29 68 36

Intensity:Elicited 76 62 44Volunteer 24 38 56

Building:Unrelated idea 88 23 30Response 12 35 42Build 0 42 28

Note.—All ratings except forms are percentages.

tending generally to students’ understand-ing of Amy’s method. Analysis of the nar-rative supported this claim. For example,72% of David’s statements (13 out of 18)about student thinking were coded as gen-eral, compared with 28% (five statements)that considered Amy’s method more spe-cifically. Furthermore, four of those fivestatements were the teacher’s reflections onthe value and application of Amy’s methodrather than on his view of the depth ofAmy’s understanding.

The need to orchestrate discourse. OnceDavid recognized the need to work to de-velop a discourse community, he focusedon the next year’s class. Throughout thesummer, we university researchers workedwith David to adapt curricula and designpedagogy that would promote classroomdiscussion. Studying videotapes of DeborahBall’s teaching (1989), we discussed waysthat David could develop a discourse com-munity (Fish, 1980) in his classroom. Thethree of us researched and discussed class-room norms (e.g., explaining your results or

referring to others’ work) to promote dis-course and attempted to devise ways (e.g.,asking open questions such as “What doother people think about that?”) for theteacher to scaffold students’ developmentof discussion skills.

An Established Discourse Community:The Plugged FunnelDavid’s students had spent the previous

day pouring water in measured incrementsinto containers of various shapes, recordingthe water’s height after each addition. Theirhomework assignment was a worksheet(Joint Matriculation Board and Shell Centrefor Mathematics Education, 1985) in whichstudents were to match drawings of con-tainers with graphs depicting height as afunction of volume. So, for example, a cyl-inder would be filled at a constant rate, andthe corresponding graph would be a diag-onal line. But the rate of increase in heightof water in a bucket with outward-slopingsides slows as the height increases, so a de-creasing curve would be a more appropri-

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Fig. 2.—Graphs for the plugged funnel lesson

ate graphical model. In this discussion, theclass struggled to decide which graphmatched the plugged funnel container, witha 13-minute discussion occurring beforestudents agreed on the correct interpreta-tion, graph b. (See Fig. 2. The sides on thelower portion of the plugged funnel arestraight and would therefore fill at a con-stant rate. This would be illustrated by astraight line. The top section widens, filling

more slowly, so the corresponding graphwould have a decreasing slope.)

David’s reflections. This was the day Ibecame convinced that we really had a dis-course community in my classroom! I feltthat I was giving students time to think,pressing them for reasons, and asking themwhat they thought of other people’s ideas.I watched the mathematical talk happen,and the students really carried the day.

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There was a split of opinions over whichgraph was the best representation of theplugged funnel. Ben began by selectinggraph b because, he explained, as theplugged funnel widens it takes longer to fillup, so the corresponding graph shouldcurve. When I asked what others thoughtabout Ben’s idea, Tina explained that shedisagreed because there was a point wherethe funnel changed its shape, and graph chad such a point. Students lined up on thetwo sides and gave a variety of detailedexplanations concerning their positions.Not only were they agreeing or disagree-ing with Ben or Tina, but they gave manydifferent reasons. Several students agreed,incorrectly, with Tina, arguing that a graphcorresponding to the plugged funnel neededa point that showed where the funnelchanged shape. In contrast, Ben understoodthat the right graph would have to curve, sothat b was the better choice. Later Jeff madea correct suggestion of what a container thatrepresented Tina’s graph might look like,one very different from the plugged funnel.It was then that Tina seemed to concede. Ithink that she understood her own ideasmore clearly as a result of having to defendthem.

During this discussion, I was able to geta clear sense of what mathematical ideas mystudents were expressing. Tina and someothers were hung up on the transition pointbetween the two parts of the funnel and ofits graph. Ben, Robert, and others saw thatthe important part was the difference be-tween a straight segment and a curved oneto represent the filling of the upper part ofthe funnel. Here I felt that, with my help, thestudents were co-constructing their mathe-matical knowledge. I understood their ar-guments and saw where they needed assis-tance. Both the discussion and the mathunderstanding went well, and the studentswere all active learners that day!

Robust mathematical discussion anal-ysis. RMD analysis again confirmed Da-vid’s evaluation (see Table 1). This discus-sion showed strength on each aspect of both

mathematical content and the process ofdiscourse. The students used six forms ofrepresentation for the correct graph. For ex-ample, Robert considered the visual fea-tures of the graph, describing it as a diag-onal line that became curved. Lia’s formrelated to the underlying variable of timetaken to fill the funnel, and she noted thatat the funnel’s flaring, “It goes a little slowerthan before, but it’s still faster than when itends.” Each of these variations deepenedthe discussion by offering a different way toview the graph. Regarding generalization,we coded 50% of the student turns as con-crete. These included descriptive state-ments such as Sam’s: “It, like, well, in thebeginning it’s like all steep. And ‘cause itgoes straight up. And then at the end itstarts to level off.” But, because his com-ment referred only to one graph, it wascoded as concrete. Specific comparisonscomprised 39% of the utterances. Tina’scomment in support of her graph choicewas an instance; she compared the graph ofthe plugged funnel to that of the bucket:“Because there’s a point. It’s not like thebucket where it just goes like that,” as shemotioned with her hand.

The remaining 11% of student turns,such as Robert’s “It, um, gradually getswider and wider so it would be curved,”were coded as generalizations. This com-ment generalized from a specific instance tothe recognition that filling any container ofincreasing diameter will result in a graphthat is curved, not straight.

The justification aspect indicated a highpercentage of more complex warrants, withonly 36% bald, unsupported statementssuch as Jeff’s: “I don’t know who talkedabout what. I agree with . . . I think it’s b.”Forty-eight percent of the comments weresupported and coded as explain, such asSam’s reason for the graph’s steepness:“’cause it [the funnel] goes straight up.”Most important, several arguments showedthe logic and structure of proof. Jason gavetwo proofs, the first one supporting graphc: “There’s a point in the funnel where it

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stops . . . it doesn’t change gradually. Sothere’d have to be a point in the graph, andthat’s the only graph that has one point init. So that’s the only choice.” Later, after lis-tening to his classmates, he provided a ref-utation: “It can’t be c then, because the sec-ond half is a constant, right? And the funnelisn’t constant.”

The strong presence of complex studentideas about content was reflected in themathematics dimension. The students wereat the forefront of the mathematical dis-course, and this finding was echoed in thediscussion dimension.

David provided access to the discoursefor all students with prompts such as,“What do people think about Ben’s idea?”Over two-thirds (68%) of the students par-ticipated in this discussion as speakers, astrong measure of engagement. The normin the class was for David to direct the dis-cussion by calling on students, so it is notsurprising that only 38% of the studentturns were volunteer and nonelicited. Thisstill indicates considerable student inten-sity. Finally, on the build aspect we notedsubstantial percentages of responses toother students’ comments (e.g., “I agreewith Tina”) at 35%, as well as 42% of stu-dent turns coded at the build level. Jason’sproof contributions above are both exam-ples of build. He started from informationprovided by his classmates, then addednew content to the discourse in each case.These levels of response and build indicateda discourse community in which studentswere carefully listening to, and thinkingabout, the comments other students made.Considering both the discussion and math-ematical dimensions, this episode con-tained indicators of vigorous discussion,confirming David’s reflection.

Professional vision for classroom dis-course analysis. David’s reflections on theplugged funnel discussion, as well as theanalysis of the classroom transcript by theuniversity researchers, revealed a shift inhis professional vision. At this point, he wasattending to student thinking in a new way.

Throughout the discussion he tried to un-derstand the meaning of students’ ideas ata detailed level. Seventy percent of his com-ments in the discussion focused on the ideasof individual students. In contrast, only6.5% of his comments (three statements) fo-cused on student thinking at a more generallevel; 17% (eight comments) were instancesof David introducing mathematical ideasinto the conversation, and the remaining6.5% were coded as “other.” Recall that inAmy’s Baby, David made no statementsthat were coded as elicitations or probes ofindividual student thinking.

For example, he probed Robert andTina, asking them for further explanationsboth of their ideas and of their reasoning:“Can you explain why?” “How does itcurve?” “Do you have other reasons?” Inaddition, David consistently prompted stu-dents to comment on the ideas that otherstudents had already raised in class. “Whatdo you think about what Ben said?” “Whatdo people think about that?” “Jeff . . . doyou have an opinion on [her idea]?” In sum,throughout the discussion, the teacherworked to gather in-depth information con-cerning students’ thinking about the math-ematical ideas raised in class.

This focus on individual student think-ing is also evidenced in David’s journal re-flections. Seventy-five percent of the jour-nal statements for this day concerned thecomments of individual students (as com-pared to 28% on Amy’s Baby). The follow-ing quote illustrates David’s close attentionto Robert’s statements about the pluggedfunnel and the corresponding graph, aswell as his attempts to carefully interpretthe meaning behind these statements.

Robert commented that he thought thatthe graph that belonged to the “pluggedfunnel” was graph b. This graph wasstraight for a while, but then curved. Hethought this because the plugged funnelwas cylindrical for a while, then openedup like a “v.” I think Robert chose thisgraph because [the plugged funnel] hada consistent piece at the bottom which

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corresponded to the straight part of thegraph, and [then the plugged funnel]opened up, which corresponded to thecurved part [of the graph] that was lev-eling off [as the funnel] was filling.(Louis, 1997c, p. 5)

Complexities of Managing School Talk:400% MoreThe students had been studying data

representations and were to write an ad-vertising slogan to best present results of aclass survey on whether students preferredto watch TV or listen to the radio (Lappan,Fey, Fitzgerald, Friel, & Phillips, 1997).Twenty students preferred TV; five pre-ferred radio. One student, Jeff, suggestedthe following headline: “TV watchers out-numbered radio listeners by 400%.” Al-though it is correct that 400% of 5 is 20(4.00 " 5 # 20), 20 is not 400% more than5. The mathematical confusion expressed iscomplicated by the twin issues of (a) 20being 4 times 5, yet only 15 more than 5, and(b) the level of abstraction required by per-centage. Learning mathematics entails learn-ing its conventions as well as the precise lan-guage needed to express one’s thoughts. Alengthy discussion then ensued in which theclass considered whether Jeff’s statement ac-curately reflected the data that had been col-lected.

David’s reflections. I had worked hardto develop a discourse community wherestudents had discussion skills and knewthey were expected to explain and value thecomments of other students. Still, I was notprepared when they took over a whole dis-cussion, and that’s what happened on thisday. Things began simply enough. Jeff sug-gested the following headline: “TV watch-ers outnumbered radio listeners by 400%.”Julie then responded in agreement, pointingout that 20 was in fact 4 times 5, and thatwould convert to 400%. Based on this andother students’ comments, I could tell thatthere seemed to be some confusion between“percent of” and “percent more than.”

Gary then started on the right track bysuggesting that he disagreed with Jeff’s

statement. He explained, “100% is, thatmeans radio listeners are equal to theamount of TV listeners.” I thought this wasa promising line of reasoning and a “teach-able moment,” but I wasn’t able to capital-ize on it. I tried to use Gary’s idea and makecomparisons with five TV watchers and fiveradio listeners as an example of 0% more,and then 10 TV watchers and five radio lis-teners as an example of 100% more. Butwhen I tried to move to 15 TV watchers andfive radio listeners, rather than respond tomy line of thinking, students raised newquestions. “Could we use ‘outnumbering’with percents?” “Does ‘four times as much’mean the same as ‘400% more than?’” I wasuncomfortable because I felt I had lost con-trol of the discussion; the students did notfollow my lead. The students were defi-nitely talking about math, but was it trulymeaningful mathematical discourse?

Robust mathematical discussion anal-ysis. David’s anxiety about this episode wasreflected in varied findings on the RMD as-pects. Although intensity and generalizationshowed strong indicators, justification hadno proof, and only one-third of the studentsparticipated in the discussion. Overall, boththe mathematics and discussion dimensionsshowed mixed results (see Table 1).

There were four representational formsof percentages in the discussion: “5 goes into20 four times . . . so that’s 4 times as much. . . 400%”; “100%, that means that radio lis-teners are equal to the amount of TV listen-ers”; “In 5 over 5 it’s the same, so they’re notoutnumbering at all”; and “It’s just the sameas reducing it to 1/5 from 5/25.” These il-lustrate the diverse approaches that theclass considered in trying to make sense ofJeff’s statement. The generalization aspectrevealed that although the majority of stu-dents’ comments were concrete (54%), 33%were coded as generalizations. For exam-ple, in one turn, Jeff referred to the specificproblem at hand, coded concrete: “I wordedit as if there are five viewers, equal 100%‘cause radio had five [listeners]. Twenty TVviewers is 400%.” Nate included a specific

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comparison with a second possible answer:“So I think it should be 300% instead of400%.” Jeff’s final statement, “When I said400%, I meant that for every one radio lis-tener there’s four, there’s four TV listeners,so it would actually be if you said ‘outnum-ber,’ 300%. But you could mean it as 400%.I thought that would be a better advertise-ment,” was coded generalization becausehe offered a general rule for percentagesand outnumbering. The justification aspecthad no comments with enough logicalstructure to be considered proof, and only30% had an explanation such as Jeff’s. Moststudent turns were devoid of reasoning,more like Nate’s. Thus, with strong claimsoffered by students, but little elaboration ofthese statements, the mathematics dimen-sion for this episode was mixed.

In looking at the discussion dimension,it was clear that this episode involvedmany volunteers in the intensity aspect,and David’s intuition that the students hadtaken control of the discourse was borneout. Gary and Nate went back and forthwithout waiting for the teacher, and otherstudents commented without David’s inter-vention—56% of the turns. However, theoverall engagement indicated that only 36%of the students were participating. So, al-though student comments dominated thediscourse, they came from a minority of stu-dents. Building was perhaps the most inter-esting aspect of this dimension. A closeanalysis of the transcript showed that 30%of the student comments were new ideas,unrelated to those previously stated in theconversation (“I don’t think you can useoutnumber with percents”). Another 42%of turns were coded as response, indicatingthat students stated agreement or disagree-ment with an earlier statement but pro-vided no additional amplification (“I thinkthat’s right”). Finally, only 28% of the turnswere at the highest coding of build withelaboration of previous student ideas. Anexample was Julie’s statement that sheagreed with Jeff because 5 goes into 20 fourtimes. The discussion dimension, therefore,

also illustrated the mixed nature of the dis-course. Rather than functioning as an ex-ample of a uniformly robust discussion, the“400% more” discussion clearly portrayedthe complexity of classroom discourse. Inthis way, the RMD analysis affirmed someof David’s unease while also pointing to as-pects of the discourse that were more andless robust.

Professional vision for classroom dis-course. The discussion of Jeff’s idea con-cerning “400% more” revealed another shiftin David’s professional vision. First, as be-fore, the teacher continued to analyze indi-vidual students’ ideas in detail. For exam-ple, early in the discussion, he attendedline-by-line to Julie’s comments, working tounderstand her thinking at each turn.

David: Okay. TV watchers outnumber ra-dio listeners by 400%? [Is] that true?

Julie: It is, because 5 goes into 20 fourtimes, and then, so 100% # 1.

David: Wait, I got you so far. So 5 goesinto 20 four times.

Julie: And then, so that’s 4 times as much.But if you want to put it into percents,you have to say 400%.

David: Oh, so are you saying that fourconverted to a percent is 400%?

Julie: Yeah, if you use it as a whole.David: So this would be the decimal, and

this would be the percent? All right,what do people think about what Juliesaid?

This kind of close examination of a stu-dent’s idea was not uncommon during thediscussion of 400% more. There were nu-merous other examples. In each case, theteacher prompted the student to further ex-plain her or his ideas, often pointing to apart of the student’s idea or claim that hehoped the student would elaborate.

At the same time, however, David alsoattended to students’ ideas more broadly.For example, during class he commented,“There’s some really important mathemat-ics here, but . . . I don’t know if you guysare completely [getting it].” Overall, 41% ofhis comments about student thinking dur-

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ing class were coded as general, and 40%were specific.

David’s narrative case similarly re-vealed a dual focus on student thinking. Forinstance, he stated that the class as a wholewas not understanding his ideas during thediscussion: “Unfortunately, students werenot following my line of thinking,” and“Students did not understand my approachto understanding the ideas.” Note that inneither statement did David go into any de-tail about students’ understanding of hisapproach to the problem. Instead, he madea general claim about the class as a whole.Similarly, he wrote that students were “gen-erally not commenting on the mathematicsfrom each others’ ideas,” and that there was“a small number of students participatingin the discussion.” He also used the narra-tive case as a forum to reflect deeply onGary’s, Jeff’s, and Amy’s ideas about themeaning of “400% more.” Thus, at thispoint, David’s professional vision involvedboth a focus on individual student thinkingas well as attention to the class as a whole.This was an important development. On theone hand, David had learned to pay closeattention to the ideas that students raised,and he worked to make sense of these ideas.Yet, at the same time, he kept track of theclass, using other kinds of cues to noticewhether most students were keeping pacewith the evolving lesson.

DiscussionIn this article we analyze three whole-classdiscussions as a way to characterize thechanging nature of a discourse commun-ity in a middle school mathematics class-room. The study was limited to two classestaught by one teacher over the span of 1calendar year. Although only three briefclassroom episodes were analyzed in thearticle, the corpus of data collected covered66 class observations. The teacher had sup-portive collaboration with two universityresearchers, as well as instruction in casewriting.

For each discussion, we began with the

teacher’s perspective on what took place.The narratives reveal Louis’s “personal prac-tical knowledge” concerning this area of in-struction (Connelly & Clandinin, 1999). Fur-thermore, his impressions of the discourseexhibited in each lesson can be thought of as“case knowledge” (L. Shulman, 1987), thatis, as knowledge of a particular teaching sit-uation that becomes representative of a classof situations. For David, Amy’s Baby repre-sented the case of a lack of discourse;Plugged Funnel, just the opposite, a case ofeffective classroom discourse. The 400%more episode offered a third window intodiscourse by representing the complexity ofstudent talk. In fact, for David, these lessonshave continued to be important benchmarks.At the start of each school year, David sets agoal for when he hopes to reach what hecalls “the Plugged Funnel stage.” At thesame time, he is wary of “those darn 400%More lessons” in which students take own-ership of the discourse, but the discoursedoes not build on the mathematical ideasthat are raised.

Over the past 2 decades, the use of casemethods for educating teachers has receivedincreased attention both by researchers andteacher educators (e.g., Colbert, Desberg, &Trimble, 1996). Yet much research on casemethods has explored the use of publishedcases as a pedagogical tool. In David’s situ-ation, in contrast, the construction of casesthemselves, both in formal and informal set-tings, provided a valuable learning forum.Whitcomb (1997) explained this phenome-non by hypothesizing that narrating a di-lemma calls for teachers to draw on severaldomains of knowledge, thereby promotingself-awareness. Moreover, for David, his nar-ratives served as learning tools not only atthe time in which they were written but alsofor many years following, providing an im-portant means through which he would re-flect on and learn about his current teachingexperiences.

The second lens we used to explore thethree lessons was RMD. As we noted ear-lier, research has distinguished between

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the mathematical and discussion dimen-sions of classroom discourse (Nathan &Knuth, 2003; Sherin, 2002; Williams & Bax-ter, 1996). Robust mathematical discussionextends such work both by defining com-ponents of the mathematical and discussiondimensions and by providing a continuumfor each component, along which the dis-course becomes increasingly robust. We be-lieve such elaboration is critical because ateacher who is encouraged to “increase stu-dent discourse” may not know what thismeans. For example, David was aware ofhis attempts to increase student discourse.Yet his narratives give little indication ofwhich discourse components he thoughtneeded to be in place. The complete journalentries and narrative cases relating to thethree selected episodes provide no addi-tional discussion of specific components ofclassroom discourse than what is describedin the teacher’s narratives here (see Louis,1997a, 1997b, 1997c, for further details). TheRMD analysis of the three lessons, in con-trast, provides such a structure.

For all three discussions, RMD con-firmed David’s impressions of the nature ofthe classroom discourse. One would not ex-pect this to happen. In several reports ofteacher-researcher collaboration, research-ers describe working to turn teacher atten-tion away from some aspects of instructionand toward others (e.g., Nathan & Knuth,2003; Sherin & Han, 2004). In our study,however, RMD provided a mechanism forcarefully examining the teacher’s perspec-tive on the discourse. This potential—tobring into focus features of discourse thatmay already be somewhat salient to teach-ers—is an important contribution of RMD,particularly in light of research on teacherlearning that emphasizes the need to buildon teachers’ existing knowledge and exper-tise (Putnam & Borko, 2000).

Finally, we also use the lens of a profes-sional vision for classroom discourse to in-vestigate the three target lessons. Althoughextensive research has explored the role ofthe teacher in classroom discourse, most

studies have examined instructional strate-gies teachers use, for example, the kinds ofquestions asked or explanations provided.Such information is critical to understand-ing how teachers establish and maintain adiscourse community; however, we believeit is only part of the equation. Researchersmust also explore how teachers perceiveand make sense of classroom discourse. Asother researchers have also noted, a keycomponent of teaching expertise, and of ex-pertise in general, is the ability to recognizepatterns and perceive meaning in complexsituations (Berliner, 1994). Our contributionhere, therefore, is that we have begun to de-scribe what it means for teachers to developexpertise in attending to classroom dis-course. Moreover, we expect the dual focusDavid exhibited in the 400% More lesson tobe an important element of a discourse vi-sion in a range of classroom settings. Teach-ers cannot be expected to understand eachstudent’s thinking all the time; taking stockof the class as a whole, in addition to moni-toring individual students, is likely to be avaluable approach.

Thus far, we have discussed each of thethree perspectives individually. Yet we alsobelieve that these lenses are connected insignificant ways. As implied earlier, RMDallows us to examine in detail the teacher’simpressions of the discourse in a productiveway, one that highlights more clearly theways students do and do not participate.Professional vision similarly illuminates theteacher’s impressions of student discourse.Yet, although RMD does so by focusing onstudent talk, PVCD does so by focusing onthe ways in which the teacher pays atten-tion to, and reflects on, this student talk. Weclaim that developments in both of theseareas contribute to a mathematical dis-course community and that each influencesthe other. As students participate more ro-bustly, there is a greater need for the teacherto learn to attend to student ideas. Further-more, attending closely to student ideasmay allow the teacher to better support stu-dents’ developing ideas.

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ConclusionOur goal in discussing these three perspec-tives on classroom discourse has been bothto show the value that each brings indepen-dently, and to show the benefits of usingmultiple perspectives to analyze classroomdiscourse. We claim that together thesethree perspectives illuminate the complex-ity of classroom discourse while providingtools for researcher analysis and teacherlearning. David’s narrative cases offer anongoing educational experience for him, amodel for other reflective teachers, and awindow into his thinking for researchers.Robust discussion allows the researcher andthe teacher to focus a broad concern aboutclassroom discourse into an organizedstructure by separating the discourse intokey components. Moreover, the profes-sional vision analysis gives evidence thatthe teacher’s professional vision changedover time, offering a different view on thedeveloping discourse community.

We believe the lenses of RMD and PVCDnot only advance the understanding of stu-dent discourse and teacher cognition butcan also be useful tools for teachers andteacher educators in helping teachers to

achieve robust mathematical discussion.The call for increased discourse among stu-dents in the context of mathematics educa-tion reform is clear. Yet what is not clear iswhat this means for teachers and students.By outlining components of robust dis-course, RMD helps to clarify the goals thatstudents and teachers can work toward.Furthermore, teachers can use RMD to as-sess their classroom discourse, determiningwhether components of the discourse are asenvisioned. Considering the notion of pro-fessional vision adds to what is knownabout how teachers can implement class-room discourse. In particular, teachers needto learn to attend both to individual studentideas as well as to the understanding of theclass as a whole. Because a teacher who isleading classroom discourse must interpretboth individual understanding and that ofthe whole group, this is an important de-velopment. In addition, using the idea ofprofessional vision, teachers can choosewhich parts of a lesson deserve their atten-tion—for example, a mathematical concept,or an idea a student raised unexpectedly.Teachers can guide their noticing duringclass and reflection on what took place.

Appendix AAspects of Robust Mathematical Discussion

Mathematics Dimensions• Representations: How many different forms of representation do students present on a given

math topic?• Generalization: Does the discussion go beyond a specific, concrete situation?• Justification: What level of logical reasoning is evident in the discussion?

Discussion Dimensions• Engagement: How many students are actively involved in the discussion?• Intensity: How do students enter the discussion?• Building: Do students respond to, and build on, comments of other students?

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Appendix BTable B1. Coding for Robust Mathematical Discussion

Dimension/Coding Category Coding Rubric

Mathematics dimension:Representations:

Form Each form for representing an idea(e.g., graph, table, equation,diagram)

Count forms of representation for eachtopic within discussion; consider thehighest number attained

Generalization:Concrete Statement limited to one context Code each student speaking turn as

concrete, specific comparison, orgeneralization; find % of each type

Specific comparison Comparison to one other contextGeneralization Pattern recognition, categorization,

or comparison that applies tomany contexts

Justification:Statement Answer or method stated, but no

justification givenCode each student speaking turn as

statement, explain, or proof; find % ofeach type

Explain Why answer holds and/or howanswer was derived

Proof Logical argument for why solutionis correct, or logical refutation, orcounterexample

Discussion dimension:Engagement:

Nonparticipant Student does not join discussion Code each student as nonparticipant orparticipant; find % of each type

Participant Student speaks in discussionIntensity:

Elicited Student speaker is nominated byteacher

Code each student speaking turn aseither elicited or volunteer; find % ofeach type

Volunteer Student speaker joins discussionwithout teacher mediation

Building:Unrelated idea Unrelated idea or direct response to

teacherCode each student speaking turn as

unrelated idea, response, or build;find % of each type

Response Agreement or disagreement withother student(s) withoutelaboration

Build Elaboration on previously statedstudent idea

Note

This research was supported by a grant fromthe Andrew W. Mellon Foundation to StanfordUniversity and WestEd for the Fostering a Com-munity of Teachers as Learners project, Lee S.Shulman and Judith H. Shulman, principal in-vestigators; by a grant from Northwestern Uni-versity; and by the National Science Foundationunder grant REC-0133900. The opinions ex-pressed are ours and do not necessarily reflectthe views of the supporting agencies.

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