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Transcript of 1 Mathematizing and Identifying: A Study of Fourth Grade Students Learning Mathematics CEMELA...
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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
53Mathematical
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?