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