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Mathematizing and Identifying: A Study of Fourth Grade Students Learning Mathematics CEMELA Seminar September 15, 2008 Marcy B. Wood 1 A Central Problem of Mathematics Education Research: How to account for mathematical learning and differences in mathematical learning? 2 Progression of Lenses Acquisitionist Participationist / Sociocultural 3 Progression of Lenses Participationist / Sociocultural – Account for learning by looking at participation differences in by looking at differences in learning participation – But what participation to focus on? 4 Narrowing Participation 2 Lenses Lens #1: Commognition (Sfard 2008) Thinking = Communication Study discourse Activity = Mathematical Discourse (Mathematizing) Outcome of learning = Change in discourse 5 Participation in the Commognitive Framework learning differences in learning by looking at by looking at participation differences in participation mathematizing differences in mathematizing The outcome of productive learning is a change in mathematizing 6 Participation in the Commognitive Framework learning differences in learning by looking at by looking at participation differences in participation mathematizing differences in mathematizing But who students think they are seems to affect mathematizing… 7 Narrowing Participation 2 Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007) 8 Narrowing Participation 2 Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007) 9 Definition of Identity Narrative = an account of life events (Ochs & Capps, 2001) Significant = any change affects storyteller’s feelings about the identified Endorsable = “faithfully reflects the state of affairs” Reified = be, have, can, always, never, usually On many occasions when Josh answered math questions, his answer was correct and appropriate. Josh is smart at math. Josh is a talented math student. (Sfard & Prusak 2005, Sfard 2007) 10 Narrowing Participation 2 Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007) 11 Identity The activity of identifying is Turning statements about activity into statements about a person (Sfard 2007) 12 Identity The activity of identifying is Turning statements about activity into statements about a person (Sfard 2007) 13 Narrowing Participation 2 Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007) 14 Narrowing Participation learning differences in learning by looking at by looking at participation mathematizing and identifying differences in participation differences in mathematizing and identifying The outcome of productive learning is a change in mathematizing and identifying 15 Narrowing Participation learning differences in learning by looking at by looking at participation mathematizing and identifying differences in participation differences in mathematizing and identifying The outcome of productive learning is a change in mathematizing and identifying 16 Narrowing Participation learning differences in learning by looking at by looking at participation mathematizing and identifying differences in participation differences in mathematizing and identifying Noticed patterns or constellations of mathematizing and identifying activity = KINDS OF LEARNING 17 Research Question Initial Question: How to account for mathematical learning and differences in mathematical learning? Research Question How do the activities of mathematizing and identifying connect to the development of mathematical discourse? 18 Research Question What kinds of learning (constellation of mathematizing andQuestion: identifying Initial activity) are present and how do they How to account for mathematical learning and connect to the development of differences in mathematical learning? mathematical discourse? Research Question How do the activities of mathematizing and identifying connect to the development of mathematical discourse? 19 Methods • Setting – – – – 4th grade classroom Midwest school, midsize city Diverse SES, race/ethnicity 20 students total, 4 focal students • Data – Mathematics lessons (3 units, 34 lessons, 70 hours of video) – Student work – Interviews, conversations with teacher 20 Methods • Task – As a group, find which of the rugs covers more area or if they cover the same amount. • Groups: – Jakeel, Rebecca, Daren – Minerva, Jessica, Bonita • Analysis – change in discourse, mathematizing, identifying 21 Findings Focus on one learner: Jakeel – Initial discourse did not address triangular spaces – Change to more mathematically desirable discourse – Shift in kind of learning during the lesson, tied to changes in mathematical discourse 22 Initial Discourse Teacher Jakeel. What do you think? How can you prove this? …She picks up a paper with Figures H and I. Jakeel H and I. They both have squares. He points at the figures on the paper the teacher is holding. Teacher Okay but are they the same? Jakeel Yes … Teacher I mean Jakeel. How Jakeel Cause it’s a square. He points at the full square in Figure I. … Teacher But this isn’t a square She points at the two triangles in Figure I. Jakeel No, so that’s why it won’t cover the same. 23 Initial Discourse Teacher Jakeel. What do you think? How can you prove this? …She picks up a paper with Figures H and I. Jakeel H and I. They both have squares. He points at the figures on the paper the teacher is holding. Teacher Okay but are they the same? Jakeel Yes … Teacher I mean Jakeel. How Jakeel Cause it’s a square. He points at the full square in Figure I. … Teacher But this isn’t a square She points at the two triangles in Figure I. Jakeel No, so that’s why it won’t cover the same. 24 Teacher Jakeel Teacher Jakeel What is it? Tell me what it is. Eight How is that eight? I can’t tell that’s eight. Because one, two, three, four, five, six, seven, eight, nine, t-. Jakeel points to the spaces in Figure J as he counts. He points once at each space with the pinky of his right hand. Hold on. Jakeel points at each space in K with his pinky. This motion suggests that he is silently counting. 25 Rebecca Can I tell him? The teacher leaves. Teacher Jakeel Teacher Jakeel What is it? Tell me what it is. Eight How is that eight? I can’t tell that’s eight. Because one, two, three, four, five, six, seven, eight, nine, t-. Jakeel points to the spaces in Figure J as he counts. He points once at each space with the pinky of his right hand. Hold on. Jakeel points at each space in K with his pinky. This motion suggests that he is silently counting. 26 Rebecca Can I tell him? The teacher leaves. Final Discourse Jakeel: One. This word is elongated as Jakeel simultaneously puts his index and middle fingers on two triangles. Two, three, four, five, six, seven He counts each square, pointing with his index finger to each square. Eight He puts index and middle fingers on the last two triangles. 27 Final Discourse Jakeel: One. This word is elongated as Jakeel simultaneously puts his index and middle fingers on two triangles. Two, three, four, five, six, seven He counts each square, pointing with his index finger to each square. Eight He puts index and middle fingers on the last two triangles. 28 Kind of Learning: Engaged Learning Jakeel How we gonna get the answer if we didn’t even read the paper? He looks at Daren. Daren We know. Look. He picks up a paper with Figures H and I and points at the figures. This is half. If we put this together that’s two squares. He points to the triangles in Figure I. Jakeel That’s a obtuse. Jakeel points at Figure I. 29 Daren We don’t care about that! (?) Look, look. Jakeel has picked up his own copy of Figures H and I. You see how that Daren points at the paper Jakeel is holding. Jakeel This is I He points at his paper. Daren You see the one square and that’s got two. You put these together. That’s two squares. So they cover the same area. Daren is pointing at Jakeel’s paper. Jakeel Can you say that again? 30 Jakeel How we gonna get the answer if we didn’t even read the paper? He looks at Daren. Daren We know. Look. He picks up a paper with Figures H and I and points at the figures. This is half. If we put this together that’s two squares. He points to the triangles in Figure I. Jakeel That’s a obtuse. Jakeel points at Figure I. Daren We don’t care about that! (?) Look, look. Jakeel has picked up his own copy of Figures H and I. You see how that Daren points at the paper Jakeel is holding. Jakeel This is I He points at his paper. Daren You see the one square and that’s got two. You put these together. That’s two squares. So they cover the same area. Daren is pointing at Jakeel’s paper. Jakeel Can you say that again? 31 Engaged Learning Rebecca So let’s cut these out. Wait. We’ll write first and then we’ll cut. Okay Rebecca picks up her pencil. Daren and Jakeel also stop cutting and start writing. And then H. H has two squares. Hold on let me finish writing this and then I’ll show you what I’m writing. Jakeel is writing and erasing Like I Rebecca puts her paper in front of Jakeel See Jakeel. Jakeel, look. She points at her paper. Jakeel I don’t want to write that Jakeel pushes the paper away I know what= Rebecca You just won’t write 32 Engaged Learning Rebecca So let’s cut these out. Wait. We’ll write first and then we’ll cut. Okay Rebecca picks up her pencil. Daren and Jakeel also stop cutting and start writing. And then H. H has two squares. Hold on let me finish writing this and then I’ll show you what I’m writing. Jakeel is writing and erasing Like I Rebecca puts her paper in front of Jakeel See Jakeel. Jakeel, look. She points at her paper. Jakeel I don’t want to write that Jakeel pushes the paper away I know what= Rebecca You just won’t write 33 Teacher Jakeel Teacher Jakeel What is it? Tell me what it is. Eight How is that eight? I can’t tell that’s eight. Because one, two, three, four, five, six, seven, eight, nine, t-. Jakeel points to the spaces in Figure J as he counts. He points once at each space with the pinky of his right hand. Hold on. Jakeel points at each space in K with his pinky. This motion suggests that he is silently counting. 34 Rebecca Can I tell him? The teacher leaves. Rebecca Okay, Jakeel. You see those little triangles. She points to Figure J. Jakeel Yes Rebecca Those are half a squares. Look and if you put these together, those equal a square Rebecca points to Figure J on Jakeel’s paper. and so it’s eight, it’s eight and eight. Rebecca points to Figures J and K. Jakeel Oh. one= Jakeel is pointing to spaces in Figure J. Rebecca interrupts him. Rebecca So write that if you put the two half squares together it makes a square. If you put two half squares together, it makes a square. Write that on your paper. Write that. Then you’re done Jakeel. Jakeel points at the spaces in Figure J. He is silently counting. He points at squares with one finger and at triangles with two fingers. Jakeel Oh, yeah! 35 Rebecca Okay, Jakeel. You see those little triangles. She points to Figure J. Jakeel Yes Rebecca Those are half a squares. Look and if you put these together, those equal a square Rebecca points to Figure J on Jakeel’s paper. and so it’s eight, it’s eight and eight. Rebecca points to Figures J and K. Jakeel Oh. one= Jakeel is pointing to spaces in Figure J. Rebecca interrupts him. Rebecca So write that if you put the two half squares together it makes a square. If you put two half squares together, it makes a square. Write that on your paper. Write that. Then you’re done Jakeel. Jakeel points at the spaces in Figure J. He is silently counting. He points at squares with one finger and at triangles with two fingers. Jakeel Oh, yeah! 36 Engaged Learner • Mathematizing: – Interact – Contribute – Make own sense • Adopt others’ discourse • Produce own discourse • Substantiate ideas yourself – Use others to support work toward understanding 37 Engaged Learner • Mathematizing: – Interact – Contribute – Make own sense Identifying • Adopt others’ discourse • Produce own discourse • Substantiate ideas yourself – Use others to support work toward understanding 38 Engaged Learner • Mathematizing: – Interact – Contribute – Make own sense • Adopt others’ discourse • Produce own discourse • Substantiate ideas yourself Identifying Do own work Discourse for self Part of group – Use others to support work toward understanding 39 Engaged Directed Jakeel’s work on H and I 40 Engaged Directed Rebecca You wasn’t supposed to do that Jakeel. Jakeel. She takes his paper Jakeel what you was supposed to dooo is so this. What we did. She puts her paper so Jakeel can see it and she points to it. Jakeel Oh. Oh. He looks at Rebecca’s paper. Rebecca So you have to take it and rewrite it She tries to pull up Jakeel’s glued pieces. and I’ll cut these out for you. Rewrite what you wrote here on there. She points at an extra paper. Jakeel I don’t want to rewrite. Rebecca Well you have to because you messed up. Jakeel picks up his pencil and starts writing. Rebecca starts cutting. 41 Directed Learning Jakeel What am I supposed to write again? He talks to Rebecca. Rebecca You’re playing Jakeel What am I supposed to write? (Well you play too much) Rebecca They H and I cover the same amount of floor. … Rebecca I mean just cover the same amount Jakeel writes. … Rebecca H and I cover the same amount of floor. No just they cover the same amount. She watches Jakeel write. Same. Amount. Do you try to write small? All right. That’s all you gotta write. Rebecca reads as Jakeel writes. 42 Directed Learning Jakeel What am I supposed to write again? He talks to Rebecca. Rebecca You’re playing Jakeel What am I supposed to write? (Well you play too much) Rebecca They H and I cover the same amount of floor. … Rebecca I mean just cover the same amount Jakeel writes. … Rebecca H and I cover the same amount of floor. No just they cover the same amount. She watches Jakeel write. Same. Amount. Do you try to write small? All right. That’s all you gotta write. Rebecca reads as Jakeel writes. 43 Directed Learning • Mathematizing – Limited – Focus on physical activity • Not explanations or mathematical concepts – No adoption, production – Substantiation entirely others Identifying Passive Reactive Others are directors Discourse for others 44 Directed Engaged Teacher Jakeel Teacher Jakeel What is it? Tell me what it is. Eight How is that eight? I can’t tell that’s eight. Because one, two, three, four, five, six, seven, eight, nine, t-. Jakeel points to the spaces in Figure J as he counts. He points once at each space with the pinky of his right hand. Hold on. Jakeel points at each space in K with his pinky. This motion suggests that he is silently counting. 45 Rebecca Can I tell him? The teacher leaves. Rebecca Okay, Jakeel. You see those little triangles. She points to Figure J. Jakeel Yes Rebecca Those are half a squares. Look and if you put these together, those equal a square Rebecca points to Figure J on Jakeel’s paper. and so it’s eight, it’s eight and eight. Rebecca points to Figures J and K. Jakeel Oh. one= Jakeel is pointing to spaces in Figure J. Rebecca interrupts him. Rebecca So write that if you put the two half squares together it makes a square. If you put two half squares together, it makes a square. Write that on your paper. Write that. Then you’re done Jakeel. Jakeel points at the spaces in Figure J. He is silently counting. He points at squares with one finger and at triangles with two fingers. Jakeel Oh, yeah! 46 Final Discourse Jakeel: One. This word is elongated as Jakeel simultaneously puts his index and middle fingers on two triangles. Two, three, four, five, six, seven He counts each square, pointing with his index finger to each square. Eight He puts index and middle fingers on the last two triangles. 47 Summary • Jakeel enacts engaged learning – Exploring and talking about mathematical ideas – Identifying as capable of learning and understanding • Transitions to directed learning – Rebecca directs physical activity – Limited mathematizing – Identifying as needing to be told what to do • Back to engaged learning – Makes own sense of counting Figure J – Change in discourse (Learning Outcome!) 48 Discussion What kinds of learning are present and how do they connect to the development of mathematical discourse? • Engaged and Directed Learning • Differences in mathematizing and identifying connect to differences in mathematically desirable learning outcomes. 49 Implications • This close focus on participation – highlights students’ existing activities – demonstrates impact of interaction (student and teacher) on learning • Reform mathematics – Desirable learning can happen in groups – Need to emphasize engagement with mathematical ideas 50 Looking to the future… CEMELA Teacher Study Group Mathematizing and identifying as lenses for thinking about teacher learning 51 Identifying Mathematizing But teachers do more than engage with mathematics…. Mathematical Discourse 52 Identifying Pedagogizing Mathematizing Pedagogical Discourse Mathematical Discourse 53 Research Questions Initial Question: How to account for mathematical and pedagogical learning of teachers? Research Question How do the activities of mathematizing, pedagogizing and identifying connect to the development of mathematical and pedagogical discourse? 54 Questions? Suggestions? Feedback? 55