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Multiple Methods for Analyzing Mathematics Classroom Discourse NCTM Research Presession April, 2009

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2 Overview of Presentation Project background, participants, & data collection Multiple concepts and tools for analyzing mathematics classroom discourse Teachers’ engagement with some of these ideas and tools Discussion

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3 Project Background 5-year project (2004-present) Broad goals of the project are to examine the nature of the discourse in eight middle grades (gr. 6-10) mathematics classrooms the ways in which teachers’ beliefs impact the discourse when working to enact reform-oriented mathematics teaching how this information can be used to incorporate action research using ideas from discourse analysis to improve mathematics instruction

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4 Overview of Project Phases Phase 1: Theoretical work; Recruitment of teacher- researchers; Beginning data collection for professed teacher beliefs Phase 2: Data collection for case studies of classroom discourse and data collection for understanding professed beliefs Phase 3: Data analysis for case studies of classroom discourse and for understanding professed beliefs; Reading & discussion groups on classroom discourse & action research Phase 4: Action research projects (Feb 07-May 09) Phase 5: PD material planning & proposal to NSF for instructional material development (Jan, 09) AY 04-05 AY 05-06 ONGOING May 06- Jan 07

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5 STUDIES OF CLASSROOM DISCOURSE: Quantitative (TCBQ, time spent on particular activity structures, turn- taking length) Mixed methods (lexical bundle analysis (Herbel-Eisenmann & Wagner, 2008), examination of specific words (Wagner & Herbel-Eisenmann, 2008) ) Qualitative (analysis of “activity structures” (Lemke, 1990), thematic analysis, longitudinal analysis (Cirillo, 2008), positioning (Herbel-Eisenmann, 2009 ; Wagner & Herbel-Eisenmann, in press) ) Study of teachers’ discussions of discourse readings and use of ideas from this literature (Herbel-Eisenmann, Drake, Cirillo, 2009) GLOBAL STUDY: (Jaworski, 1998) how action research on classroom discourse impacts teachers’ professed beliefs and enacted beliefs over time LOCAL STUDIES: teacher’s action research projects (Herbel- Eisenmann & Cirillo (Eds.), 2009)

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6 STUDIES OF CLASSROOM DISCOURSE: Quantitative (TCBQ, time spent on particular activity structures, turn- taking length) Mixed methods (lexical bundle analysis (Herbel-Eisenmann & Wagner, 2008), examination of specific words (Wagner & Herbel-Eisenmann, 2008) ) Qualitative (analysis of “activity structures” (Lemke, 1990), thematic analysis, longitudinal analysis (Cirillo, 2008), positioning (Herbel-Eisenmann, 2009 ; Wagner & Herbel-Eisenmann, in press) ) Study of teachers’ discussions of discourse readings and use of ideas from this literature (Herbel-Eisenmann, Drake, Cirillo, 2009) GLOBAL STUDY: (Jaworski, 1998) how action research on classroom discourse impacts teachers’ professed beliefs and enacted beliefs over time LOCAL STUDIES: teacher’s action research projects (Herbel- Eisenmann & Cirillo (Eds.), 2009)

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7 STUDIES OF CLASSROOM DISCOURSE: Quantitative (TCBQ, time spent on particular activity structures, turn-taking length) Mixed methods (lexical bundle analysis (Herbel-Eisenmann & Wagner, 2008), examination of specific words (Wagner & Herbel-Eisenmann, 2008) ) Qualitative (analysis of “activity structures” (Lemke, 1990), thematic analysis, longitudinal analysis (Cirillo, 2008), positioning (Herbel-Eisenmann, 2009; Wagner & Herbel- Eisenmann, in press)

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8 Data Collection for Analyses of Classroom Discourse Collaborators: 8 math teachers in 7 schools Purposefully selected to vary context, certification, gender, experience, perspective on teaching/learning Baseline data during 2005-06 school year Four full weeks (every day (6), modified block schedule (1), block schedule (1)) approximately every other month (September, November, January, March) = 148 observations Pre- & post- observation, daily reflections, classroom artifacts Some follow-up observations (three in 2006-07; add’l for Michelle’s dissertation) Teachers’ data collection during action research cycles

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9 Remainder of Presentation Offer some of the concepts and tools from discourse analysis literature that we have used to make sense of this data Lexical bundles (Beth Herbel-Eisenmann, MSU) Concordance tables & collocation charts (Lorraine Males, MSU) Thematic analysis (Sam Otten, MSU) Amalgamation of ideas to understand longitudinal change (Michelle Cirillo, ISU) Share observations of teacher’s engagement with some of these ideas Discussion

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10 Remainder of Presentation Offer some of the concepts and tools from discourse analysis literature that we have used to make sense of this data Lexical bundles (Beth Herbel-Eisenmann, MSU) Concordance tables & collocation charts (Lorraine Males, MSU) Thematic analysis (Sam Otten, MSU) Amalgamation of ideas to understand longitudinal change (Michelle Cirillo, ISU) Share observations of teacher’s engagement with some of these ideas Discussion WHAT THE IDEA/TOOL IS HOW TO DO/USE IT ILLUSTRATION WHAT IT HELPS ONE SEE

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Looking with Lexical Bundles Beth Herbel-Eisenmann Michigan State University bhe@msu.edu

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12 Lexical Bundles (with David Wagner and Viviana Cortes) Corpus: a large data set that includes a representative sample of the context being studied “large enough to adequately represent the occurrence of the features being studied” (Biber 2006, p. 251) “diverse enough to represent the variation in the kinds of texts being studied” (Biber 2006, p. 252) Particular focus: lexical bundle analysis Lexical bundle: groups of three or more words that frequently recur, as a multi-word grouping, in a particular register (Biber, Johansson, Leech, Conrad & Finegan, 1999; Cortes, 2004)

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13 Examples of lexical bundles from two seminal studies in linguistics Academic discourse (Biber, 2006; Biber, Conrad, Cortes, 2004) as a result of, on the other hand, the fact that the Everyday conversation (Biber et al., 2004) I don’t know why, what do you mean, I said to him

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14 Identifying Lexical Bundles dentified empirically rather than intuitively, using a specially designed computer program that works on a corpus of texts Merge and “clean” transcripts then import full set into Lexical Bundles program (designed by Cortes, using Borland Delphi Professional, 1998) Program identifies frequently recurring word combinations across a set of texts Our corpus was 679,987 words, large for oral corpora (but not a million words like some corpus linguistic studies) Conservative cut point: four-word combinations, occurring 40 times in at least 5 texts

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15 Types of lexical bundles in this corpus Most were “stance” bundles index “personal feelings, attitudes, value judgments, or assessments” (Biber, Conrad, & Cortes, 2004, p. 966). I want you to, I would like you, we have to do, you need to do Some seemed to be about encouraging mental and verbal processes what do you think, does that make sense, what do you mean A few specifically mathematical find the area of, the square root of, one and one half

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16 When compared to other corpora analyses More lexical bundles in this data set than in university classroom teaching Many stance bundles in this data set particular to this data set

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17 Stance bundles that were particular to this data set Language forms

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18 Lexical Bundle analysis Finds some patterns across large data sets- phrases that structure discourse Recurring fixed phrases can play an imperative role in fluent linguistic production of a register people are learning (Cortes, 2004) Uncovers mundane phrases that typically go unnoticed Forms only Hegemonic practices Does not attend intensely to context Lexical bundles do not define a discourse Once forms are identified, can use different theories of language to interpret them

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19 Stance bundles: Interpreting function Teacher’s personal authority Disciplinary authority More subtle disciplinary authority Personal Latitude Theories: systemic functional grammar, critical discourse analysis, positioning theory

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20 Exploring discourse forms further… Concordance tables & collocation charts…can be used once you know what language forms you want to examine…

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Lexical Analysis – Focusing on specific words Lorraine Males Michigan State University maleslor@msu.edu

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22 Lexical Analysis – Focusing on specific words Another technique from corpus linguistics is the use of concordance and collocation tables to examine the functions of specific words Unlike, lexical bundle analysis, this type of analysis requires that we pick specific words or phrases to examine.

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23 Method All transcripts of classroom observations were imported into Transana. Collections were created for each teachers uses of specific pronouns ( “it” and “they” in relation to curriculum materials) These collection reports (which include the transcripts snippets) were imported into WordSmith Tools.

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24 Using wordsmith WordSmith Tools is a lexical analysis program (Oxford University Press)for the PC used for looking at how words behave in texts. Wordsmith includes three tools: WordList – allows you to see a list of all the words or clusters of words in a text (alphabetically or by frequency) Concord – a concordancer that allows you to see any word or phrase in context (see the “company” that your word or phrase keeps) KeyWords – allows you to find key words in texts.

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25 Using Concord - Concordance This displays a concordance table that shows the search words (“it”, “they,” “book”) with some context around the word. These were sorted alphabetically by the word following the search word

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26 Using Concord - Collocates This displays a collocation table that allows us to see the position of words relative to our search words. L5 – L1 are the 5 words prior to our search word and R1 – R5 are the 5 words that follow our search word

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27 Collocates Table L5L4L3L2L1Cent re R1R2R3R4R5 totalkthroughthosenowitsaysonherereadboth tointerp ret andsayokayitsaysouthereminusone Of the 216 R1s of “it,” forms of the word “say” appeared 78 times ( ~36% )

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28 Allows us to reduce larger data sets to focus on specific words, while still attending to context Allows us to see patterns in word usage within and across classrooms May uncover how certain word function by providing many examples of the word in use Does not attend intensely to meaning Does not attend intensely to the words around the chosen word Doesn’t easily lend itself to look at the word use over time Lexical Analysis – Focusing on specific words

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29 Focusing on Semantic Patterns Thematic Analysis (Lemke, 1990) Billy’s Bicycle

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30 Semantic Relations Element/Set Subset/Set Part/Whole Possessor/Possessed Extent/Entity Synonym/Synonym, etc… WHY FOCUS ON SEMANTIC RELATIONS?

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31 Thematic Analysis - Method Transcript selection Highlighting content words “Clean” map Marginal analysis “Flow” (transcript) map

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32 Products of Thematic Analysis “Clean” map

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33 Products of Thematic Analysis Marginal analysis & “Flow” map

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34 How is this useful?

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Analyzing Discourse Through Discourse Moves Michelle Cirillo Iowa State University

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36 Motivation for Studying this Topic Larger project - Discourse analysis: A catalyst for reflective inquiry in mathematics classrooms (Beth Herbel-Eisenmann, PI) One of the eight teachers was a beginning teacher – teaching deductive reasoning and formal proof

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Setting and Participant Matt: taught 10 th Grade Geometry (middle level of three tracks) used a conventional geometry textbook participated in PD focused on discourse through Herbel-Eisenmann’s project

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38 Research Questions How did Matt’s teaching of geometry proof change across three years? To what did Matt attribute these changes?

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39 Findings How did Matt’s teaching of geometry proof change across three years? 1.Matt supplemented the written curriculum with additional proofs and activities. 2. Matt explicitly focused on “doing proofs” (Herbst et al., 2009). 3. Matt focused on students in more sophisticated ways.

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40 “Talk Moves” Chapin, O’Connor, and Anderson (2003) similarly wrote about “talk moves” that support mathematical thinking and productive mathematical talk.

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41 “Discourse moves” Krussel et al. (2004) defined a discourse move as “a deliberate action taken by a teacher to participate in or influence the discourse in the mathematics classroom” (p. 309, emphasis added). Despite the fact that the move may have been made deliberately, many of the consequences of a teacher’s discourse moves may be unintended and even potentially “at direct odds with the teacher’s purpose” (Krussel et al., 2004, p. 307) or beliefs.

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42 Alternate definition I use the term “discourse moves” defined as: actions taken by the teacher that influence the discourse in the classroom. These actions may or may not be deliberate, and I consider the absence of talk (i.e., silence or wait time) to be a discourse move.

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43 Four Discourse Moves Revoicing (O’Connor, 2009; Forman, Larreamendy-Joerns, Stein, & Brown, 1998a, p. 531; O'Connor & Michaels, 1993). Wait time (Cazden, 2001; Rowe, 1974) Pronouns (Fortanet, 2003; Pimm, 1984, 1987; Rounds, 1987a, 1987b; Rowland, 1999, 2000) Questioning (Dillon, 1983; Gall, 1984; Herbel-Eisenmann & Breyfogle, 2005; Martens, 1999; Perrot, 2002)

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44 Beginning Proof in Year 1

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45 Beginning Proof in Year 1 - Revoicing 1 Matt: They, what about [the line segments] equal each other? 2 S: The length. 3 Matt: The length, right? So the first step in my proof is going to be to say, ‘okay, I was told this.’ If I know that PQ is congruent is to XY, what do I know about PQ and XY? Because they’re congruent. 4 FS: They have the same measure. 5 Matt: They have the same measure. Why do we know that? Because they’re congruent. That’s what it means to be congruent, right? In other words, to be congruent, they have to have the same measure. In order to be congruent, they have to have the same measure. Okay? Now I want to end up with the other way. What’s the first thing, what do I need to do in between in order to state it the other way? I know that PQ is the same measure as XY or PQ equals XY. What’s my third step gonna be? 6 MS: XY equals PQ. 7 Matt: XY equals PQ. How do I know that?

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46 Beginning Proof in Year 1 – Pronouns 1 Matt: They, what about [the line segments] equal each other? 2 S: The length. 3 Matt: The length, right? So the first step in my proof is going to be to say, ‘okay, I was told this.’ If I know that PQ is congruent is to XY, what do I know about PQ and XY? Because they’re congruent. 4 FS: They have the same measure. 5 Matt: They have the same measure. Why do we know that? Because they’re congruent. That’s what it means to be congruent, right? In other words, to be congruent, they have to have the same measure. In order to be congruent, they have to have the same measure. Okay? Now I want to end up with the other way. What’s the first thing, what do I need to do in between in order to state it the other way? I know that PQ is the same measure as XY or PQ equals XY. What’s my third step gonna be? 6 MS: XY equals PQ. 7 Matt: XY equals PQ. How do I know that?

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47 Beginning Proof in Year 1 - Questioning 1 Matt: They, what about [the line segments are] equal each other? 2 S: The length. 3 Matt: The length, right? So the first step in my proof is going to be to say, ‘okay, I was told this.’ If I know that PQ is congruent is to XY, what do I know about PQ and XY? Because they’re congruent. 4 FS: They have the same measure. 5 Matt: They have the same measure. Why do we know that? Because they’re congruent. That’s what it means to be congruent, right? In other words, to be congruent, they have to have the same measure. In order to be congruent, they have to have the same measure. Okay? Now I want to end up with the other way. What’s the first thing, what do I need to do in between in order to state it the other way? I know that PQ is the same measure as XY or PQ equals XY. What’s my third step gonna be? 6 MS: XY equals PQ. 7 Matt: XY equals PQ. How do I know that?

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48 Beginning Proof in Year 3

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49 Beginning Proof in Year 3 - Revoicing 1 Matt: So how many people are convinced that this has to be true? [Students raise their hands.] …Why? 2 Sally: Because, uhm, since ST equals RN and IT equals RU, the difference from, from UN, from RN equals the same as the distance from SI to ST. 3 Matt: Does that make sense? What Sally said makes sense? 4 Ss: Yeah. 5 Matt: Right? If you start with equal lengths and you subtract off an equal amount, it should be equal to the same thing. Now how are we gonna say that so that we're sure completely, completely sure that we have to be correct? That's what we're trying to write the proof for. So where do we want to start? 6 Jim: Uh, RN minus RU equals ST minus. 7 Matt: Why? Where'd the minus come from? 8 Jim: Because take off the IT and the RU and it just goes. 9 Matt: But how can you, how can we justify that? 10 Jim: Because it's given that IT and RU is the same. 11 Matt: Okay, so let's start with that. So IT, okay, so, the two major kinds of proof, on a two-column proof, you're going to write down a statement and then a reason, okay? So IT equals RU. Why?

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50 Beginning Proof in Year 3 - Pronouns 1 Matt: So how many people are convinced that this has to be true? [Students raise their hands.] …Why? 2 Sally: Because, uhm, since ST equals RN and IT equals RU, the difference from, from UN, from RN equals the same as the distance from SI to ST. 3 Matt: Does that make sense? What Sally said makes sense? 4 Ss: Yeah. 5 Matt: Right? If you start with equal lengths and you subtract off an equal amount, it should be equal to the same thing. Now how are we gonna say that so that we're sure completely, completely sure that we have to be correct? That's what we're trying to write the proof for. So where do we want to start? 6 Jim: Uh, RN minus RU equals ST minus. 7 Matt: Why? Where'd the minus come from? 8 Jim: Because take off the IT and the RU and it just goes. 9 Matt: But how can you, how can we justify that? 10 Jim: Because it's given that IT and RU is the same. 11 Matt: Okay, so let's start with that. So IT, okay, so, the two major kinds of proof, on a two-column proof, you're going to write down a statement and then a reason, okay? So IT equals RU. Why?

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51 Beginning Proof in Year 3 - Questioning 1 Matt: So how many people are convinced that this has to be true? [Students raise their hands.] …Why? 2 Sally: Because, uhm, since ST equals RN and IT equals RU, the difference from, from UN, from RN equals the same as the distance from SI to ST. 3 Matt: Does that make sense? What Sally said makes sense? 4 Ss: Yeah. 5 Matt: Right? If you start with equal lengths and you subtract off an equal amount, it should be equal to the same thing. Now how are we gonna say that so that we're sure completely, completely sure that we have to be correct? That's what we're trying to write the proof for. So where do we want to start? 6 Jim: Uh, RN minus RU equals ST minus. 7 Matt: Why? Where'd the minus come from? 8 Jim: Because take off the IT and the RU and it just goes. 9 Matt: But how can you, how can we justify that? 10 Jim: Because it's given that IT and RU is the same. 11 Matt: Okay, so let's start with that. So IT, okay, so, the two major kinds of proof, on a two-column proof, you're going to write down a statement and then a reason, okay? So IT equals RU. Why?

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52 Conclusion A combination of methods helped me to see how Matt created more space for student participation through discourse moves. Across time, a more open discourse was observed.

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Acknowledgment I would like to thank Matt for opening up his classroom to me for the last three years. I would like to thank my committee members who helped me shape this study: Beth Herbel-Eisenmann Corey Drake Katherine R. Bruna Leslie Bloom Heather Thompson This study was partially supported by the National Science Foundation under Grant No. 0347906 (Beth Herbel-Eisenmann, PI). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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54 Teacher’s Engagement Ideas from corpus linguistics Read the concordance tables and talked about how they thought the lexical bundles were functioning

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55 Discussion What are some affordances of the various approaches to discourse analysis? What are some constraints of the various approaches to discourse analysis? What do these methods highlight about the discipline of mathematics? What do these methods highlight about mathematics classrooms? How might the different methods be useful to mathematics education researchers? How might the findings from the different methods be useful to teacher educators and classroom teachers?

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