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Mathematizing and Identifying:A Study of Fourth Grade Students

Learning Mathematics

CEMELA Seminar

September 15, 2008

Marcy B. Wood

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A Central Problem of Mathematics Education Research:

How to account for

mathematical learning and

differences in mathematical learning?

Acquisitionist

Participationist / Sociocultural

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Progression of Lenses

Participationist / Sociocultural

– Account for

– But what participation to focus on?

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Progression of Lenses

learning by looking at participation differences in

learning by looking at differences in participation

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Narrowing Participation

2 Lenses

Lens #1: Commognition (Sfard 2008)

Thinking = Communication

Study discourseActivity = Mathematical Discourse

(Mathematizing)

Outcome of learning = Change in discourse

Participation in theCommognitive Framework

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learning by looking at

participation mathematizing

differences in learning

by looking

at

differences in participation

differences in mathematizing

The outcome of productive learning is a change in mathematizing

Participation in theCommognitive Framework

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learning by looking at

participation mathematizing

differences in learning

by looking

at

differences in participation

differences in mathematizing

But who students think they are seems to affect mathematizing…

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Narrowing Participation

2 Lenses

Lens # 2: IdentityEx: 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)

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Narrowing Participation

2 Lenses

Lens # 2: IdentityEx: 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)

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)

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Narrowing Participation

2 Lenses

Lens # 2: IdentityEx: 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)

Identity

The activity of identifying

is

Turning statements about activity into statements about a person

(Sfard 2007)

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Identity

The activity of identifying

is

Turning statements about activity into statements about a person

(Sfard 2007)

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Narrowing Participation

2 Lenses

Lens # 2: IdentityEx: 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)

learning by looking at

participation mathematizing and identifying

differences in

learning by looking

at

differences in participation

differences in mathematizing and identifying

Narrowing Participation

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The outcome of productive learning is a change in mathematizing and identifying

learning by looking at

participation mathematizing and identifying

differences in

learning by looking

at

differences in participation

differences in mathematizing and identifying

Narrowing Participation

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The outcome of productive learning is a change in mathematizing and identifying

learning by looking at

participation mathematizing and identifying

differences in

learning by looking

at

differences in participation

differences in mathematizing and identifying

Narrowing Participation

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Noticed patterns or constellations of mathematizing and identifying activity = KINDS OF LEARNING

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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?

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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?

What kinds of learning (constellation of mathematizing and identifying activity) are present and how do they connect to the development of mathematical discourse?

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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

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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

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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

Initial DiscourseTeacher 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 DiscourseTeacher 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

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Teacher What is it? Tell me what it is.Jakeel EightTeacher How is that eight? I can’t tell that’s eight.Jakeel 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.

Rebecca Can I tell him? The teacher leaves.

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Teacher What is it? Tell me what it is.Jakeel EightTeacher How is that eight? I can’t tell that’s eight.Jakeel 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.

Rebecca Can I tell him? The teacher leaves.

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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.

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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.

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.

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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?

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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?

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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

Engaged Learning

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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

Engaged Learning

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Teacher What is it? Tell me what it is.Jakeel EightTeacher How is that eight? I can’t tell that’s eight.Jakeel 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.

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

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Engaged Learner

• Mathematizing:– Interact– Contribute– Make own sense

• Adopt others’ discourse• Produce own discourse• Substantiate ideas yourself

– Use others to support work toward understanding

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Identifying

Engaged Learner

• Mathematizing:– Interact– Contribute– Make own sense

• Adopt others’ discourse• Produce own discourse• Substantiate ideas yourself

– Use others to support work toward understanding

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Identifying Do own work Discourse for

self Part of group

Engaged Directed

40Jakeel’s work on H and I

Engaged DirectedRebecca 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 becauseyou messed up. Jakeel picks up his pencil and starts writing. Rebecca starts cutting.

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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.

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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.

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Directed Learning

• Mathematizing– Limited– Focus on physical activity

• Not explanations or

mathematical concepts

– No adoption, production– Substantiation entirely

others

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Identifying Passive Reactive Others are

directors Discourse for

others

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Teacher What is it? Tell me what it is.Jakeel EightTeacher How is that eight? I can’t tell that’s eight.Jakeel 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.

Rebecca Can I tell him? The teacher leaves.

Directed Engaged

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

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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.

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!)

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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.

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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

Looking to the future…

CEMELA Teacher Study Group

Mathematizing and identifying as lenses for thinking about teacher learning

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Identifying

Mathematizing

But teachers do more than engage with mathematics….

Mathematical Discourse

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Identifying

MathematizingPedagogizing

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Discourse

Pedagogical Discourse

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?

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Questions?Suggestions?

Feedback?