Supporting the Participation of English Language Learners in ...

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Supporting the Participation of English Language Learners in Mathematical Discussions Author(s): Judit Moschkovich Reviewed work(s): Source: For the Learning of Mathematics, Vol. 19, No. 1 (Mar., 1999), pp. 11-19 Published by: FLM Publishing Association Stable URL: http://www.jstor.org/stable/40248284 . Accessed: 14/01/2013 17:20 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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of English Supportingthe Participation in Mathematical Discussions Language Learners JUDIT MOSCHKOVICH

The aim of this paper is to explore how teachers can supportEnglishlanguagelearnersin learningmathematicsand not only English. The focus of the analysis will be on one importantaspect of learningmathematics:participationin mathematicaldiscussions. I use a discourse perspective (Gee, 1996) on learningmathematicsto address two pairs of questionscentralto mathematicsinstructionfor students who are learning English (as well as for students who are nativeEnglishspeakers): • Whatcan a teacherdo to facilitate studentparticipation in a mathematicaldiscussion?How can a teacher supportstudentsin speakingmathematically? • Whatare the varietyof ways that studentstalk about mathematicalobjects?What are the different points of view studentsbringto mathematicalsituations? I examinea lesson froma thirdgrademathematicsdiscussion of the geometricshapes from a tangrampuzzle to illustrate how one teacher supportedmathematical discussion and describehow studentsparticipatedby talking about mathematicalsituationsin differentways. I use excerpts from the transcriptof this lesson to exemplify supportive teaching strategiesanddescribethe varietyof ways thatstudentscommunicatemathematically.In particular,the teacher did not focus primarilyon vocabularydevelopmentbut instead on mathematicalcontent and arguments, as he interpreted, clarifiedandrephrasedwhatstudentswere saying. Duringthis lesson, the teacher's instructionalstrategies included:1) using severalexpressionsfor the same concept; 2) using gesturesand objects to clarify meaning;3) accepting and building on student responses; 4) revoicing (O'Connorand Michaels, 1993) student statements using more technicalterms;and 5) focusing not only on vocabulary development but also on mathematical content and argumentationpractices.My analysis of student participation in the discussion shows that students brought several different ways of talking about mathematical objects and points of view of mathematicalsituations to the classroom discussion. Two importantfunctions of productive classroom discussions are uncoveringthe mathematicalcontent in studentcontributionsandbringingdifferentways of talking andpointsof view intocontact(Ballenger,1997;Warren and Rosebery,1996).

Frameworks for examining mathematical discussions One view of learningmathematicsis that English language learnersneed to focus primarilyon learning how to solve wordproblems,understandindividualvocabularytermsand translate from English to mathematical symbols (e.g. Mestre, 1988; Spanos, Rhodes, Dale, and Crandall,1988). This view is reflectedin many currentrecommendationsfor mathematicsinstructionfor English language learnersthat emphasize vocabularyand comprehensionskills (Olivares, 1996; Rubenstein, 1996; MacGregor and Moore, 1992). These recommendationsprovide a limited view of learning mathematicsand do not addressa currentincreasedemphasis on mathematicalcommunication. In contrast,in manymathematicsclassrooms,studentsare no longer primarily grappling with acquiring technical vocabulary, developing comprehension skills to read and understandmathematicstextbooksor solving standardword problems.Instead,studentsare now expected to participate in both verbalandwrittenpractices,such as explainingsolution processes, describingconjectures,provingconclusions and presentingarguments. I use a discourse perspective on what it means to learn mathematics to consider the participation of English language learners in mathematical discussions, and take a view of discourse as more than sequential speech or writing,using Gee's definitionof 'Discourses'[1]: Discoursesare ways of being in the world, or forms of life which integrate words, acts, values, beliefs, attitudes, social identities, as well as gestures, glances, body positions and clothes. (Gee, 1996, p. 127) Participatingin classroommathematicaldiscoursepractices can be understood in general as talking and acting in the ways that mathematically competent people talk and act when talking about mathematics in a classroom, and involves muchmorethanthe use of technicallanguage.Gee uses the example of a biker barto illustratethis. In orderto look and act like one belongs in a bikerbar,one has to learn much more thana vocabulary.While knowingthe names of motorcyclepartsor models may be helpful, it is clearly not enough to participatein a bikerbar community.In the same way, knowing a list of technical mathematicalterms is not sufficient for participatingin mathematicaldiscourse. Beyond technicalterms,mathematicaldiscourseincludes constructionsused to proveor explainstatementssuch as "If x, then y", "Let x be the case", "Assume ...", "This is the

For the Learning of Mathematics 19, 1 (March, 1999) FLM Publishing Association, Kingston, Ontario, Canada


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case because ...", "tomake comparisons,such as 'the higher ..., the smaller ...'" and to describespatial situations.There are also valued discursivepractices,such as abstractingand generalizing, making precise statements and ensuring certainty,and the need to acquirecontrolover the accompanying formswhich reflectthem(see Halliday,1978 for some instances). One important aspect of classroom mathematical discourse practices is participating in mathematical discussions, either as a whole class or with a group of students. For the purposes of this analysis, a mathematical discussionwill be takento be: purposeful talk on a mathematical subject in which thereare genuine pupil contributionsand interactions. (Pirie, 1991, p. 143) While duringa classroommathematicaldiscussion the conversation might involve some aspects of standard or canonicalmathematicaldiscourse,discussionin a classroom is not the same as the discourseof researchmathematicians. Studentsin mathematicsclassroomsshould not be expected to talk and act in the same ways that research mathematicians do. Nevertheless, in general, students participatein mathematicaldiscussions or communicatemathematically by makingconjectures,presentingexplanations,constructing arguments, etc. about mathematical objects, with mathematicalcontent, and towards a mathematical point (Brenner,1994). We can use several lenses to examine how English language learners(or other students) participatein mathematical discussions and how teachers support student participationin these discussions. Below I summarizethe instructionalstrategiesthe US NCTM standards(1989) and certainmathematicseducationclassroomresearch(e.g. Ball, 1991; Cobb et ai, 1993; Silver and Smith, 1996) suggest for orchestratingand supportingmathematicaldiscussions: • model desired participationand talk; support these when displayedby students; • encouragestudentconjecturesand explanations; • call for explanationsand evidence for students'statements; • focus on the process not only the product; • comparemethods,solutions, explanations; • engage students in argumentsfor or against a statement (move beyond "agree"or "disagree"); • encouragestudent-to-studenttalk; • ask studentsto paraphraseeach other's statements; • structureactivitiesso thatstudentshave to understand each other's methods. A comparisonof this discourse approachto mathematical discussions with recommended language development strategies for teaching mathematics to students who are learning English yields some reassuring similarities and some disturbingdifferences. Below is a brief summaryof some commonly recommended language development strategies:

• use realia, pictures, models, diagrams, gestures, charts,labels, dramatization,etc; • provide different forms of participation (orally, in writing,graphically)and interaction(with the teacher, with peers,alone); • adjustspeech (speed, enunciating,vocabulary); • check for comprehension; • providefeedback,appropriateresponseand adequate wait time; • provideresourceswheneverpossible and useful (dictionaries,translations,interpreters). I find these language development strategies disturbingin that there is little guidance on how to concentrate on the mathematical content of discussions. In general, the considerationof mathematicalcontentfor English language learners is apparently limited to providing exposure to content vocabulary and identifying differences in mathematicalconventions(such as the use of a comma or a period for decimals or the differentplacementof the long division sign). Another drawback of some language development approachesto mathematicsteaching is a focus on correction of vocabularyor grammaticalerrors(Moschkovich,in press), obscuringthe mathematicalcontentin what students say and the variety of ways that students who are learning English do, in fact, communicatemathematically.Instead, by focusing on mathematicaldiscourse,teacherscan move beyond focusing on errorsin English vocabularyor grammar to hear and supportthe mathematicalcontent of what studentsaresaying. Althoughthe strategiessummarizedabove can be helpful in orchestrating discussions, they are, however, only a beginning.These guidelines have some serious limitations. First, both the NCTM standardsand the language development strategies can be interpreted as presenting manipulatives and pictures as 'extra-linguistic clues', assuming that the meaning of an object can be communicated withoutusing language.Instructioncannotdependon using manipulatives as a way to deal with either comprehension or conceptual problems. Students need to use languagein context,participatein conversationsand clarify the meaning of the objects they are manipulating,looking at or pointing to. Second, neither set of recommendations provides enough guidance for how to listen to and understand the differentways that students talk mathematically or for how to bringtogetherthese differentways of talking in classroomdiscussions. The lesson excerpts presented below provide examples of strategies for supporting a mathematical discussion among English language learners and contrast the variety of ways that students talk about mathematical situations. They come from a third grade bilingual classroom in an urbanCaliforniaschool. In this classroom,there are thirtythreestudentswho have been identifiedas Limited English Proficiency. In general, this teacher introducesstudents to concepts and terms in Spanishfirst and then later conducts lessons in English. Studentsare surroundedby materialsin

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both Spanishand Englishanddesks arearrangedin tablesof fourso thatstudentscan work together. The studentshave been workingon a unit on two-dimensional geometricfigures. For several weeks, instructionhas includedtechnicalvocabularysuch as 'radius', 'diameter', 'congruent','hypotenuse'andthe namesof differentquadrilaterals in both Spanish and English. Students have been talking about shapes and have also been asked to point to, touchandidentifydifferentinstances.This lesson was identified by the teacher as an ESL mathematics lesson, one where students would be using English in the context of folding and cutting exercises to form tangrampieces (see Figure1). It also illustrateshow, when the goal is supporting studentparticipationin a mathematicaldiscussion,listening to the natureandqualityof students'mathematicaldiscourse is as importantas (if not moreso than)focusing on students' Englishlanguageproficiency.

How is the teacher supporting participation in a mathematical discussion? Excerpt1: Describinga rectangle 1. Teacher:Today we are going to have a very special lesson in which you really gonna have to listen. You're going to put on your best, best listening ears because I'm only going to speak in English. Nothingelse. Only English. Let's see how much we remembered from Monday.Hold up your rectangles. ... high as you can. [students hold up rectangles] Good, now. Who can describe a rectangle?Eric, can you describe it? [a rectangle] Can you tell me about it? 2. Eric: A rectangle has ... two ... short sides, and two ... long sides. 3. T: Two short sides and two long sides. Can somebody tell me something else about this rectangle? If somebody didn't know what it looked like, what, what ... how would you say it? 4. Julian:Parallel(o).[holding up a rectangle] 5. T: It's parallel.Veryinterestingword.Parallel,wow! Pretty interesting word, isn't it? Parallel. Can you describewhat thatis? 6. Julian:Never get together.They never get together, [runshis finger over the top length of the rectangle] 7. T: Whatnever gets together? 8. Julian:The parallela... the ... when they go, they go higher [runs two fingers parallel to each other first along the top and base of the rectangle and then continues along those lines] they never get together. 9. Antonio:Yeah!

Figure1 A tangrampuzzle The goal of my analysis of this lesson was two-fold. One was to identify teaching strategies during an actual classroom lesson in orderto providegenuineillustrations,while the other was to identify examples of the different ways Englishlanguagelearnerscommunicatemathematicallyand the different points of view they bring to a discussion, in orderto providea basis for hearingandunderstandingthese studentsbetterduringclassroomdiscussions. I presenta partialtranscriptof this lesson in fourexcerpts. The first segment involves descriptionsof a rectangle.The secondconcernscomparinga rectangleanda triangle,while the third involves comparisons of a parallelogram and a trapezoidin a small group and the fourthdetails a wholeclass discussion of whether a trapezoid is or is not a parallelogram.I have numbered conversational turns for ease of reference, and [] indicates non-verbalinformation to help with deictic references, such as 'this' or 'that'. In excerpt 3, () indicates an English translation of student remarksin Spanish.T is the teacherand Ss students.

10. T:Veryinteresting.The rectanglethenhas sides that will never meet. Those sides will be parallel. Good work. Excellent work. Anybody else have a different idea thatthey could tell me aboutrectangles? 11. Student:Anotherdifferent[unclear]parallelogram. 12. T: It's called a parallelogram, can you say that word? 13. Ss: Parallelogram. 14. T: Whatwere you going to say, Betsy? 15. Betsy: Also a parallelogramit calls a rectangle. 16. T: A parallelogramis also a rectangle?They can be both? 17. Betsy: Yeah. 13


18. T:Wow,very interesting.Canyou convince me that they can be both? 19. Betsy: Because a rectangle has four sides and a parallelogramhas foursides. 20. T: [unclear] 21. Eric:[unclear]a parallelogram. 22. T: You want to borrow one? [a tangram piece] I really wantto remindyou thatyou reallyhave to listen while your classmateis talking... 23. Eric:Becausethese sides [runshis fingersalong the widths of the rectangle] will never meet even though they get bigger,and these sides [runshis fingers along the lengths of the rectangle] will never meet even though they get bigger. And these sides [picks up a square]will never meet [runshis hand along two parallel sides] and these sides will never meet, [runs his handalong the othertwo parallelsides] 24. T: When you say get bigger you mean if we kept going with the line? [gesturesto the rightwith his hand] 25. Eric:Yeah. 26. T: Veryinteresting. During this lesson, the teacher employed some important strategiesto orchestrateand supportstudents'mathematical talk. [2] In general, he established and maintainednorms for discussions,askingstudentsto listen to otherstudents,to agree or disagree, to explain why they believed something and to convince the teacherof theirstatements.The teacher also used gestures and objects, such as the cardboardgeometric shapes, to clarify what he meant. For example, he pointed to vertices and sides when speaking about these partsof a figure. Although using objects to clarify meanings is an importantESL instructionalstrategy,it is crucialto understandthat these objects do not provide 'extra-linguistic clues'. The objects and their meaningsare not separatefrom language, but ratheracquiremeaningthroughbeing talked about and these meanings are negotiated throughtalk. Although the teacherandthe studentshad the geometricfiguresin frontof them, and it seemed helpful to use the objects and gestures for clarification,studentsstill needed to sort out what 'parallelogram' and 'parallel' meant by using language and negotiatingcommonmeanings. The teacherfocused not only on vocabularydevelopment but also on supportingstudents'participationin mathematical argumentsby using three other instructionalstrategies that focus more specifically on mathematical discourse. First, the teacher promptedthe students for clarification: for example, in turn 16 the teacherasked a studentto clarify the relationshipbetween two geometric figures, and in turn7 the teacherasked Julianto clarify what he meant by "they". Second, the teacher accepted and built on student responses,as can be seen in the above exchanges.In another

example (turns4-5), the teacheraccepted Julian'sresponse and probedwhat he meantby "parallel". Last, the teacher revoked student statements, by interpretingandrephrasingwhatstudentssaid (turn10 in relation to turn 8). Julian's "the parallela, they" becomes the teacher's "sides" and Julian's "they never get together" becomes "will nevermeet".The teachermoved pastJulian's unclear utterance and use of the term "parallela" and attempted to uncover the mathematical content in what Julian had said. He did not correct Julian's English, but insteadasked questionsto probewhat the studentmeant. Julian's utterance in turn 8 is difficult both to hear and interpret.He utteredthe word 'parallela'in a haltingmanner, sounding unsureof the choice of word or of its pronunciation. His voice trailedoff, so it is difficult to tell whetherhe said 'parallelo' or 'parallela'. His pronunciationcould be interpretedas a mixture of English and Spanish; the "11" sound being pronouncedin English and the additionof the "o" or "a"being pronouncedin Spanish. The grammatical structureis also intriguing.The apparentlysingular"parallela" is followed by a "the..." and then with a plural"when they go higher".[3] In any case, it seems clear thatthis was a difficult contributionfor Julian to make and that he was strugglingto communicatein a meaningfulway. Excerpt2: Comparinga rectangleand a triangle Students were folding the rectangle and cutting it into a folded triangle and a small rectangle. Holding these two pieces, the teacherasked the studentsto tell him how a triangle differs from a rectangle. 56. T: Anybodyelse can tell me somethingabouta rectangle that is different from a ... a triangle that's differentfrom a rectangle?Okay.Julian? 57. Julian: The rectangle has para ... parallelogram [runninghis fingersalong the lengths of the rectangle], and the triangledoes not have parallelogram. 58. T: He says that this [a triangle] is not a parallelogram.How do we know this is not a parallelogram? 59. Julian: Because when this gets ... When they get, when they go straight,they get together,[runshis fingers along the two sides of the triangle] 60. T: So, he's saying that if these two sides were to continuestraightout [runshis fingersalong the sides of the triangle],they would actually intersect,they would go througheach other.Very interesting.So, this is not a parallelogramand it is not a rectangle.OK. Duringthis shortexchange,the teacheronce againrevoked a studentstatement.In turn58, the teacherrestatedJulian's claim that"thetriangledoes not have parallelogram" as "this is not a parallelogram",and in turn60, the teacherrestated Julian'sclaim that"whenthey go straight,they get together" as "if these two sides were to continue straight out, they would actuallyintersect,they would go througheach other". Thereare at least two ways thata teacher'srevoking can supportstudentparticipationin a mathematicaldiscussion. The first is that it can facilitatestudentparticipationin gen-

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eral, by accepting the student's response, using it to make an inference and allowing the student the right to evaluate the correctnessof the teacher'sinterpretationof the student contribution.This move maintainsa space open for further studentcontributionsin a way that the standardclassroom initiation-response-evaluation(IRE) patterndoes not. The second way is thata revoking move serves to make and keep the discussion mathematical. In revoicing, a student statement is often reformulated in terms that are closer to the standarddiscourse practices of the discipline. For example, in the first excerpt the teacher uses the term "sides", which is more specific than Julian's phrase "the parallela",because it refers to the sides of a quadrilateral, ratherthan any two parallellines. This revoicing seems to have an impact on Julianwho later uses the term uside(s)" twice when talkingwith anotherstudent(see excerpt 3, turn 79). In the second excerpt,the teacherchanges Julian'scolloquial phrase "get together" to "meet", which is more formal and therefore more aligned with standard school mathematicaldiscourse.These two reformulationsserve to maintainthe mathematicalnatureof the discussion. Revoicingis not, however,as simple as I have presentedit so far. It is not always easy to understandwhat a student means.Sometimes,a teacherand a studentspeak from very different points of view about a mathematical situation. Even though in the second excerpt the teacher once again revoiced student statements, this time there seems to be more going on thanjust interpretingand re-stating what a studentsaid. To understandthis excerpt,it is importantfirst to considerJulian'suse of the phrases"has parallelogram" and "does not have parallelogram" and the teacher's responsesto Julian.It is not clearfrom this excerptwhether Julianwas using the expression"hasparallelogram"to mean 'hasparallelsides', as in "therectanglehas parallelsides"or to refer to the geometric figure "parallelogram",as in "the rectangle is a parallelogram"(i.e. has the parallelogram property).(Duringa laterconversationwith anotherstudent Julian (turn 85) seems to use the phrase "it's a parallelogram"to mean 'the lines areparallel'.) At one point (turn 58), the teacher interpretedJulian's utterance"does not have parallelogram"as "is not a parallelogram".However,in turn59, Julianseems to be referring to parallellines. By turn60, the teacherseems to be on the same trackas Julianwithouthaving focused directlyon the specific meaning of the phrase "does not have parallelogram"or having correctedJulian'sturn57 contribution. One way to comparethe teacher's and Julian'scontributions during the second excerpt is to consider the ways of talking and points of view each brings to the discussion. Julianseemed to be telling a story about the lines, saying "whenthey go higher, they never get together"and enacting the story by runninghis fingers along the parallellines. In contrast, the teacher was using the future tense, which sounds more predictive or hypothetical, saying for example, "willnever meet"and "willbe parallel". The teacherand Julianwere also talking aboutparallelogramsfrom differentpoints of view. While the teacherfirst referredto a categoryat turn58, in response Juliannarrated the situation once again, describing a property of the parallelogram(turn59). The teacherthen rephrasedJulian's

response in a more hypotheticalway using the subjunctive "if ... were ..., they would ...". He also explicitly markedboth revoicings,by using the tag "So, he's saying that ..." (in turn 60) and "Hesays that ..." (in turn58). While it is not completely clear what Julian originally meant in turn57 or what the consequencesof the revoicing in turn58 were, it does seem thatthe teacherand Julianwere bringingdifferentpointsof view to the discussion. This difference in the points of view of the situation (which reappearsin a laterdiscussion abouta trapezoidand a parallelogram in excerpt 4) may have contributed to the difficultyin interpretingand revoicingJulian'sstatement. There is one more way that this teacher supported student participation.Duringthe discussion in both excerpts 1 and 2, we see instantiated the teacher's general stance towards student contributions. The teacher moved past Julian'sconfusinguses of the term 'parallela'or the phrase "has parallelogram"to focus on the mathematicalcontent of Julian'scontribution.He did not correctJulian'sEnglish, but insteadaskedquestionsto probewhatthe studentmeant. This response is significant in that it represents a stance towards studentcontributionswhich can facilitate student participationin a mathematicaldiscussion:listen to students and try to figure out what they are saying. When teaching English language learners,this may mean moving beyond vocabulary or grammaticalerrorsto listen for the mathematicalcontentin studentcontributions.(Fora discussionof the tensionsbetween these two, see Adler, 1998.)

What is the mathematical content of this discussion? As studentsparticipatedin this discussion, they were grappling not only with the meaning of individual words and phrases, but also with some importantideas about quadrilaterals and lines. It may be easier to see the mathematical contentin this discussionby referringto the textbook'sdefinitions for some of the concepts invoked during the discussion. The definitionof parallellines is "straightlines in a plane which have no common point no matterhow far they are extended".The definition of a parallelogramis "a quadrilateralwith two pairsof parallelsides". One importantidea that studentsseemed to be grappling with was class inclusion, for example when students described a square (which is also a rectangle, a parallelogram and a trapezoid), a rectangle (which is also a parallelogramand a trapezoid)or a parallelogram(which can also be a trapezoid, depending on which definition is used [4]). Studentsalso seemed to be sorting out whether they were talking about a propertyof the figure, as in the phrase"ithas parallelsides", or a categoryfor the figure, as in the phrase "it is a parallelogram".These two points of view, one focusing on a propertyand the otheron a category, may be especially importantto sort out when talking about parallelograms, since the word "parallel", which describes a propertyof parallelograms,is partof the word "parallelogram",which describes a category of geometric figures. Studentswere also grapplingwith the concept of parallel lines. One importantaspect of parallelismis that one needs to imagine, hypothesize or predictwhat will happento the 15


two lines segments if they are extended indefinitely. Although two line segments may not meet here and now, what matterswhen deciding if two lines are parallelor not is whetherthe imaginedlines would ever meet. Different ways of talking about a mathematical situation Excerpt3: Comparinga parallelogram and a trapezoid in small groups 78. T [to whole class]: Whatdo we know abouta trapezoid. Is this a parallelogramor not? I wantyou to take a minute, and I want you at your tables, right at your tables. I want you to talk with each other and tell me when I call on you, tell me what your group decided. Is this a parallelogramor not? 79. Julian:[Julianand Andres have several shapes on theirtable:a rectangle,a trapezoidanda parallelogram] Porque si. Nomàs estas (Because ... Just these) sides get together[runshis fingersalong the two non-parallel sides of the trapezoid,see Figure2] pero de este (buton this side only), [runshis fingers along the base and top parallelsides of the trapezoid]

Figure2 Julian describinga trapezoid(turns79, 82, 85) The students in this classroom broughta varietyof ways of talking about mathematicalsituations to this discussion. In the three excerptspresentedabove, we have examples of students talking about mathematical objects in narrative, predictiveandargumentativeways. Forexample,Julian(see Figures 2 and 3) seemed to be telling a story aboutthe parallel lines and using his fingers to enact the story, an example of a narrativeapproach.

80. Mario:Yeste lado no. (And not this side) 81. Andres:No porque mira, aqui tiene un lado chico (No because, look, here it has a small side) [points to the two non-parallelsides of the trapezoid]y un lado grande y tiene cuatro esquinas (and a large side and it has fourcorners). 82. Julian: See? They get together, pero acd no (but not here), [runshis fingers along the base and top parallel sides of the trapezoid] 83. Andres:Acâ no. (Not here) 85. Julian:It's a parallelogram.Only this side, but this side meet, [runshis fingers along the non-parallelsides of the trapezoid][5]

Figure3 Julian describinga rectangle(turn8) Eric,in turn23 (see Figure4), was also telling a story.His use of the futuretense "will"and the phrase"even though" makes this story sound more predictive and hypothetical. His way of talking can be described as a combinationof a narrativeand a predictiveapproach.

86. T: [joins group]They would meet? 87. Julian:Yeah,but these sides, they won't, [runshis fingers along the base and top or parallel sides of the trapezoid] 88. T: OK. So one pairof sides meets, the otherdon't. So it is not, or it is? 89. Andres:No. It is. Figure4 Eric describinga rectangle(turn23)

90. T: It is? 91. Andres:Yeah. 92. T: Think about it now. I want you to talk about it. Remember,a parallelogramfrom what you said was, two sides, two pairsof sides.

In contrast,in turn81, Andres employed certain linguistic elements of the form of an argument,such as his use of "because"and "look",but neitherthe mathematicalcontent nor the mathematicalpoint of his argumentwere clear.

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Narrativeandpredictiveways of talkingarenot presented as a dichotomy.Notice that Eric's contributionwas a combinationof these two ways of talking.Althoughit may seem easy to identify hypotheticalways of talkingas mathematical, imagining and narratingcan also be a partof standard mathematicaldiscourse practices:for example, mathematical conversationsabout objects in four dimensions involve imaginingand narratinga situation. One way to thinkaboutthe relationshipbetweennarrative and formal mathematicalways of talking is that a mathematicaltext can be embeddedwithin a narrative.Solomon and O'Neill (1998) claim one of the characteristics of a mathematicaltext is thatthe mathematicalargumentis: atemporal[... and] achieves cohesion throughlogical ratherthantemporalorder,(p. 216) Seen in this manner,elements of both Julian'sand Eric's contributionsare similar to mathematicaltexts in that both contributionsare atemporalstatementsthat will always be true("thelines will never get together"or "thesesides will nevermeet")andare cohesive becauseof theirlogical rather thantemporalstructuring. The pointof the comparisonsabove is not to privilegeone way of talkingover anothernorto decide which way of talking is moremathematical.In a classroomdiscussion,various ways of talkingcan contributein theirown way to the mathematicaldiscussionandbringresourcesto the conversation. The point of these comparisonsis to diversify our view of the different ways that students talk about mathematical objectsand situations,to uncoverthe mathematicalaspects of whatstudentsare saying and to be able to hearbetterthe varietyof ways in which studentscan communicatemathematically. Different points of view of a mathematical situation The final excerptshows the differentpoints of view thatthe teacherandthe studentsbroughtto defininga parallelogram anda trapezoid.It also providesanotherexampleof how this teacher, ratherthan emphasizing language development, expectedthe studentsto focus on mathematicalcontent. Excerpt4: Deciding whethera trapezoidis a parallelogram or not 92. T: [to the whole class] OK. Raise your hand.I want one of the groupsto tell us whatthey do think.Is this a parallelogramor not, andtell us why. I'm going to take this grouprighthere. 93. Vincent:These two sides will never meet, but these two will. 94. T: How many agree with that.So, is this a parallelogramor not? 95. Ss: Half. 96. T: OK. If it is half, it is, or it isn't? 97.Ss:Is. 98. T Can we have a half of a parallelogram? 99. Ss: Yes.

100. T: Yes, but then, could we call it a parallelogram? 101. Ss: Yes. 102. T: What do you think? If we remember what Julian said. Would you repeat what you said at the beginning,a parallelogramis what? 103. Julian: It's a parallelogram's [meet] they never meet. 104. T: How many pairsof sides never meet? 105. Andres:Five. 106. Julian:Two. They have two sides .. to .. that .. a square has ... [takes a rectangle] these sides [runshis fingers along the lengths of the rectangle], they not meet, and these sides [runshis fingersalong the widths of the rectangle],never meet. That'sa parallelogram. 107. T: Did everybodyhear whathe said? 108. Ss: Yeah. 109. T: There have to be two pairsof sides that never meet. Let's see if this is or not. Wouldthese sides ever meet? [holdingup the trapezoid] 110. Ss: Yeah. 111. T: Wouldthey ever meet? 112.Ss:No. 113. T: No. Would these sides if we extend them up, would they ever meet? 114. Ss: Yeah. 115. T: OK. That'swhat you were saying. So, is this a parallelogram? 116.Ss:No. During this excerpt, the students and the teacher brought togethertwo points of view concerningdefining a parallelogram.They differ in termsof what types of definitionsare acceptable. While for the students "half a parallelogram" was an acceptable specification, this was not acceptableto the teacher. Rather than reflecting an error in students' reasoningor theirlack of Englishproficiency,this exchange uncovers two fundamentallydifferentviews of this mathematicalsituation. The standard definition of a trapezoid is "a quadrilateral with one pair of parallelsides" [6] and the standard definition of a parallelogramis "a quadrilateralwith two pairs of parallel sides". The teacher's initial question "Is this a parallelogramor not, and tell us why?"assumedthat this is an either/orsituation.The teacher'spoint of view was dichotomous: a given figure either is or is not a parallelogram. 17


In contrast, some students seemed to have a dynamic pointof view of this situation.Theirresponseto the question "Is this a parallelogramor not?"was "Half, implying that a trapezoidis half of a parallelogram.This is a reasonable definition if it is understoodto mean thatsince a parallelogramhas two pairsof parallelsides, and a trapezoidhas one pairof parallelsides, then a trapezoidis a half of a parallelogram because a trapezoidhas half as many parallel sides as a parallelogram (see Figure 5). The students' point of view is also differentin that they are focusing on whether and how these two figures possess the property of having pairs of parallel lines, ratherthan on whether the figures belong to one of two categories: "figureswith two pairs of parallel lines" or "figures with one or no pair of parallel lines".

as this teacherdid when saying that for a figureto be a parallelogram "therehave to be two pairs of sides that never meet". Anotherstrategymightbe to talk explicitlyaboutdifferent ways of talking, asking studentsto considerdifferentways of describing parallel lines and defining a trapezoid by contrastingthe teacher'scontributionsand the descriptions proposedby students.The teachercould explicitly compare descriptions focusing on the properties of a figure with descriptionsfocusing on the categoriesa figure belongs to, contrastingsaying "a figure has parallelsides" with saying "a figure is a parallelogram".The discussion could also focus on explicitly comparingdescriptionsof a trapezoidas "being a half a parallelogram"with "havinghalf as many parallel sides as a parallelogram", thereby working on grammaticalconstructions from within a content-focused discussion. A thirdway to talk explicitly abouttalk is to comparestudent ways of talking with textbook definitions,helping the studentto see theirdescriptionsin relationto the more regimented discourse of mathematics texts. During such a discussion, the teacher could explicitly compare student descriptions such as "the lines never get together" or "a trapezoidis half a parallelogram"with textbookdefinitions of parallellines, parallelogramsand trapezoids.

Figure5 A trapezoidis half a parallelogram

The excerptspresentedabove illustrateseveral instructional strategiesthatcan be useful in supportingstudentparticipation in mathematicaldiscussions, such as establishing and modeling consistent norms for discussions, revoking student contributions, building on what students say and probingwhat studentsmean. The teacherdid not focus primarily on vocabulary development but instead on mathematicalcontentand argumentsas he interpreted,clarified and rephrasedwhat studentswere saying. What does it mean to say that mathematicaldiscourseis more than vocabulary and technical terms? A discourse approachto learning mathematics means considering the different ways of talking about mathematicalobjects and pointsof view of mathematicalsituationsthatstudentsbring to classroom discussions. There are a variety of ways in which studentsargue,provideevidence or presentan argument. Sometimesthey predict,imagineor hypothesizewhat will happen to an object. Sometimes they focus on categories of objects and othertimes on the propertiesof these objects. Studentsmay have differentpoints of view of what an acceptable description or definition of a mathematical object is. A discourse approachto learning mathematicscan also help to shift the focus of mathematicsinstructionfor English language learners from language development to mathematical content. The lesson presented here shows that English language learnerscan and do participatein discussions where they grapple with important mathematical content. It is certainly difficult to consider carefully the mathematical content of student contributions in the moment.However, it is possible to take time after a discussion to reflect on the mathematical content of student contributionsand design subsequentlessons to addressthis

This interpretationof "a half parallelogram"would be consistentwith Julian'searlierusage of the phrase"is a parallelogram" to mean "has parallel sides". If "is a parallelogram"means "hasparallelsides", then a trapezoid has one half as many pairs of parallel sides as a parallelogram. Anotherway to see the students'pointof view as dynamic is to considera trapezoidas a parallelogramin transformation. We can take a trapezoid and transform it into a parallelogram.

Figure6 A dynamicpoint of view There are many possibilities for where this lesson might go next in a way that honors both mathematicaldiscourse and the mathematicalcontentof studentcontributions.One possibilityis to encouragemorestudent-to-student talk, asking students to consider and address each other's contributions.Anotherpossibility is for.theteacherto continuebuildingon studentcontributionswhile using ways of talkingwhich arecloser to standardmathematicaldiscourse,

Conclusions

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content. But, it is only possible to uncover the mathematical content in what students say if students have the opportunity to participate in a discussion and if this discussion is a mathematical one. This teacher provided this opportunity by moving past student grammatical or vocabulary errors, listening to students and trying to understand what students said. He kept the discussion mathematical by focusing on the mathematical content of student contributions, asking students for clarification, accepting and building on student responses and revoking student statements. Revoking can be difficult to carry out, perhaps especially when working with students who are learning English. It may not be easy or even possible to sort out what aspects of a student's utterance are a result of the student's conceptual understanding or of a student's English language proficiency. However, the analysis of this lesson suggests that, if the goal is to support student participation in a mathematical discussion, determining the origin of an error is not as important as listening to the students and uncovering the mathematical content in what they are saying.

Acknowledgments This work was supported by a grant from the US National Science Foundation (grant #REC-9896129). I would like to thank my colleagues at the Chèche Konnen Center at TERC for their contributions to the analysis of this lesson: several of the ideas in this article were developed in conversations with them and through my participation in the work of the Center. The videotape material came from the Math Discourse Project at Arizona State University (NSF grant # ESI-9454328).

Notes [1] While Gee distinguishesbetween'discourse'and 'Discourse',I will use lowercase 'discourse'hereto referto whatGee labelsuppercase 'Discourse*. [2] Although there are many valued instructionalstrategies focusing on eitherlanguagedevelopmentor mathematicalcontentthatthe teacherdid not use in this lesson (such as keeping track of student conjectures and conclusionson the blackboardor supportingmore talk between students), the goal of this analysis is not to focus on what the teacherdid not do, but ratheron how he did, in fact, supportparticipationin a mathematicaldiscussion. [3] Thereis a differencebetweenEnglishand Spanishin termsof how "parallel"is used as an adjective.While in English "parallel"can usually only be singular(one Uneis parallelto another;two Unesare parallel),in Spanish, "paralelo/d"can be singularor plural(estas dos lineas son paralelas; una linea esparalela a otra).This utteranceof Julian'smay be furthercomplicatedby the fact that sometimes Spanish speakersdo not pronouncea final us"sound, so that it may be difficult to tell whetherthis utterance was in fact singularor plural.While it may be interestingto considerthese differences,it seems unreasonableto jump to any simple conclusionsabout Julian'sdifficultieswith this word. [4] I have run into two definitions of a trapezoid. One is ua quadrilateral with exactly one pairof parallelsides", while the other is "a quadrilateral with at least one pairof parallelsides". [5] As mentionedearlier,in this line Julian seems to be usmg parallelogram"to mean 'parallel'. with no sides [6] The definitionsof trapeziumandtrapezoid(a quadrilateral parallel) are often interchanged. In Spanish, "... the word trapezoid is reservedfor a quadrilateral withoutany parallelsides, whereastrapeziumis used when thereis one pairof parallelsides. (This is oppositeto American Englishusage.)"(Hirigoyen,1997, p. 167).

References Adler, J. (1998) 'A language of teaching dilemmas: unlocking the complex multilingualsecondarymathematicsclassroom', For the Learning of Mathematics18(1), 24-33. Ball, D. L. (1991) 'What's all this talk about "Discourse"?', TheArithmetic Teacher39(3), 44-47. Ballenger, C. (1997) 'Social identities, moral narratives,scientific argumentation: science talk in a bilingual classroom', Language and Education11(1), 1-14. Brenner,M. (1994) 'A communicationframeworkfor mathematics:exemplary instructionfor culturally and linguistically diverse students', in McLeod, B. (ed), Language and Learning: Educating Linguistically Diverse Students,Albany,NY, SUNY Press,pp. 233-268. Cobb,P.,Wood,T. andYackel,E. (1993) 'Discourse,mathematicalthinking, andclassroompractice',in Forman,E., Minick,N. andStone,C. A. (eds), Contextsfor Learning:SocioculturalDynamicsin Childrens Development,New York,NY, OxfordUniversityPress, do. 91-1 19. Gee, J. (1996, 2nd edn) Social Linguistics and Literacies: Ideology in Discourses, Bristol, PA, Taylorand Francis. Halliday, M. (1978) 'Some aspects of sociolinguistics', in Interactions Between Linguistics and Mathematics Education, Copenhagen, UNESCO,pp. 64-73. Hingoyen,H. (1997) 'Dialecticalvariationsm the languageof mathematics: a sourceof multiculturalexperiences',in Trentacosta,J. andKenney,M J. (eds),Multiculturaland GenderEquityin theMathematicsClassroom:the Giftof Diversity- 1997 Yearbook,Reston,VA,NationalCouncilof Teachers of Mathematics,pp. 164-168. MacGregor,M. and Moore, R. (1992) TeachingMathematicsin the Multicultural Classroom, Melbourne, Instituteof Education,University of Melbourne. Mestre, J. (1988) 'The role of language comprehension in mathematics andproblemsolving', in Cocking,R. andMestre,J. (eds), Linguisticand CulturalInfluenceson LearningMathematics,Hillsdale, NJ, Lawrence ErlbaumAssociates, pp. 201-220. Moschkovich,J. N. (in press)'Learningmathematicsin two languages:moving from obstaclesto resources',in Secada,W. (ed), ChangingFaces of Mathematics (Vol. 1): Perspectives on Multiculturalismand Gender Equity,Reston,VA, NationalCouncilof Teachersof Mathematics. NCTM (1989) Curriculumand EvaluationStandardsfor School Mathematics, Reston, VA, NationalCouncilof Teachersof Mathematics. O'Connor,C. and Michaels, S. (1993) 'Aligning academictask andparticipation status through revoicing: analysis of a classroom discourse strategy',Anthropologyand EducationQuarterly24(4), 318-335. Olivares, R. A. (1996) 'Communicationin mathematicsfor studentswith limited English proficiency', in ElUott,P. C. and Kenney, M. J. (eds), Communicationin Mathematics:K-12 And Beyond - 1996 Yearbook, Reston,VA, NationalCouncilof Teachersof Mathematics,pp. 219-230. Pirie, S. (1991) 'Peer discussion in the context of mathematicalproblem solving', in Durkin,K. and Shire, B. (eds), Languagein Mathematical Education:Researchand Practice, Philadelphia,PA, Open University Press,pp. 143-161. Rubenstein,R. N. (1996) 'Strategiesto supportthe learningof the language of mathematics',in Elliott, P. C. and Kenney,M. J. (eds), Communication in Mathematics:K-12 and Beyond - 1996 Yearbook,Reston, VA, NationalCouncilof Teachersof Mathematics,do. 214-218. Silver, E. and Smith, M. (1996) 'Buildingdiscoursecommunitiesin mathematics classrooms: a worthwhilebut challengingjourney', in Elliott, P.C. and Kenney,M. J. (eds), Communicationin Mathematics:K-12and Beyond- 1996 Yearbook,Reston,VA, NationalCouncil of Teachersof Mathematics,pp. 20-28. Solomon,Y. and O'Neill, J. (1998) 'Mathematicsand narrative',Language and Education12(3), 210-221. Spanos,G., Rhodes, N. C, Dale, T. C. and Crandall,J. (1988) 'Linguistic featuresof mathematicalproblemsolving: insightsand applications',in Cocking, R. and Mestre,J. (eds), Linguisticand CulturalInfluenceson LearningMathematics,Hillsdale,NJ, LawrenceErlbaumAssociates,pp. 221-240. Warren,B. and Rosebery,A. (1996) 'This questionis just too, too easy', in Schauble,L. and Glazer,R. (eds), Innovationsin Learning:New Environmentsfor Education,Hillsdale, NJ, LawrenceErlbaumAssociates, pp. 97-125.

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