Transcript Document

Responding to Children’s Thinking and
Diversity: A Reflection on 20 years of Research
Megan Loef Franke
UCLA
Cognitively Guided Instruction
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Thomas Carpenter (University of
Wisconsin)
Elizabeth Fennema (University of
Wisconsin)
Linda Levi (University of Wisconsin)
Susan Empson (University of Texas)
Ellen Ansell (University of Pittsburgh)
Vicki Jacobs (San Diego State
University)
Elham Kazemi (University of
Washington)
Dan Battey (Arizona State Univ)
Annie, Mazie, Sue, Barb, Lilliam, Jo
Ann, Kim, Janet, and many, many other
teachers
Presentation Overview
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Why focus on Children’s
Mathematical Thinking
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Making use of the development
of children’s mathematical
thinking
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Adding it Up
Equity
Research findings
Supporting the development of
children’s mathematical thinking
in classrooms
Considering Understanding: Adding it Up
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Recognizing that no term captures completely all
aspects of expertise, competence, knowledge, and
facility in mathematics, we have chosen mathematical
proficiency
five interwoven, interdependent strands: conceptual
understanding, procedural fluency, strategic
competence, adaptive reasoning, productive disposition.
Mathematical proficiency is not a one-dimensional trait,
and it cannot be achieved by focusing on just one or
two of these strands
About children’s thinking
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Children come to school with
mathematical knowledge
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Children’s knowledge develops
through well documented
trajectories
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Development of children’s
thinking quite robust
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Development of children’s
thinking does not match the
way adults solve problems
Development of Children’s
Mathematical Thinking
Janelle has 7 trolls in her collection. How many
more trolls will Janelle need to buy to have 11 trolls
altogether?
How do you think children will solve this problem?
Watch what children can do…
Direct modeling
Counting strategy
Derived Fact
Recall
How can a focus on children’s thinking help?
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Notice what students’ can
do
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Make decisions based on
what students’ know
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Press for understanding
How can a focus on children’s thinking help?
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Create multiple ways to
participate
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Support the development of
mathematical identity—not
just one way, sense making,
question asking…
CGI Research and Development
Experimental
Longitudinal
School Case
Study
Study
Study
Teacher
Follow up
Experimental
Teacher/School
Case Studies
Study
Development of communities of inquiry
Development of tools to support learning in practice
Learning about the development of students’ mathematical thinking in classrooms
First Grade
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K- 3
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All school
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K-5
Algebraic thinking
Evidence that Attending to Student Thinking
Can Make a Difference
 CGI provides evidence that teachers’ classroom
practice that
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includes eliciting and making public student thinking,
involves eliciting multiple strategies,
focuses on solving word problems and
uses what is heard from students to make instructional
decisions
leads to the development of student understanding
Evidence that Attending to Student Thinking
Can Make a Difference
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Teachers who drew on
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detailed knowledge of the development of students’
mathematical thinking within a domain
an organization of student thinking in relation to the
mathematical content
notions that they could continue to learn from their
practice …identity
supported the development of student understanding
Supporting teachers to make use of
students’ mathematical thinking
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There is no single pattern or trajectory for teachers as
they come to make use of children’s thinking
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Can get teachers to ask students how they solved
problems
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Challenging to support teachers to make use of what
they hear from students, to engage students in
comparing strategies, to move forward in their
trajectories
Moving forward…learning more to
support teacher learning and practice
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Pushing on the research
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Learning through professional development
Moving towards understanding the details of
practice through research
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Listening to students talk makes it possible for the teachers (and
other students) to monitor students’ mathematical thinking
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The act of talking can itself help students develop improved
understanding
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Explaining to other students is positively related to achievement
outcomes, even when controlling for prior achievement
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(Brown & Palincsar, 1989; Fuchs, Fuchs, Hamlett, Phillips, Karns, & Dutka, 1997; King, 1992; Nattiv, 1994;
Peterson, Janicki, & Swing, 1981; Saxe, Gearhart, Note, & Paduano, 1993; Slavin, 1987; Webb, 1991; Yackel,
Cobb, Wood, Wheatley, & Merkel, 1990).
Less is known about teacher practices that are most effective for
producing high-level discourse in the classroom
Details: Supporting the development of students’
mathematical thinking
In classrooms where:
Students gave correct and complete explanations
Students scored the highest on the assessments
Teachers:
 Used a fairly coherent set of problems
 Asked questions very specific to what students said
 Engaged students in thinking and talking about important
mathematical ideas arising out of their suggestions
 All students participated in conversations about the
mathematics
Learning through professional development
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develop relationships: create a community where teachers can
learn together about the teaching and learning of mathematics
where the activities of the community were embedded in
teachers’ everyday work
make space for teachers to share their histories and make their
practice public
focus on the details and structures around students’
mathematical thinking
Focus on what students can do (Counter-storytelling)
attention to the artifacts and language that support the
development of students’ mathematical thinking in practice
Artifacts in our Professional Development work…
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Framework for the development
of student strategies within
mathematical domains
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• Problem types
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• Video of students and
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• Language
classrooms
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How did you get that?
Does that always work?
Strategy and problem names
Number sentence index cards
Join Change Unknown
Avita has 7 rocks. How many
more rocks does she need to
collect to have 11 rocks
altogether?
Join Result Unknown …
Artifact Travel
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Ongoing use across settings
Attention to and unpacking
of classroom use in PD
Trace where we are with
ideas around artifact
Helps to see teacher use of
artifact in both PD and
classrooms – raise questions,
note inconsistencies, conflict
etc..
How artifacts support learning and practice
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Focused on creating and negotiating meaning
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Focused conversation across and within communities of practice
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Supported the development of language and interaction that
could be used to support the development of new relationships
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Supported story telling across boundaries and be used to develop
counter stories
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Purposeful
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Challenged the existing cultural practices
Development of Children’s
Mathematical Thinking
Tom has 102 dog biscuits. His Dog Harmony
eats 12 biscuits a day. How many days will it
take Harmony to eat all of the dog biscuits?
Let’s see what children can do…
Attending to the Details of Children’s
Mathematical Thinking Allows Teachers to:
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Notice what students can do
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Make decisions that build on
what students know
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Create openings for varied
participation
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Support the development of
students who think of
themselves as capable of
making sense of
mathematics
Development of Children’s
Mathematical Thinking
Lucy had 38 dollars. One weekend she earned 25
making dollars raking leaves for her neighbors. How
much money did Lucy have then?
Watch what children can do…
What can you do?
Count by tens, solve problems using 20 and 30, take
numbers apart and put them back together.
Mathematical Proficiency
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conceptual understanding—comprehension of mathematical
concepts, operations, and relations
procedural fluency—skill in carrying out procedures flexibly,
accurately, efficiently, and appropriately
strategic competence—ability to formulate, represent, and solve
mathematical problems
adaptive reasoning—capacity for logical thought, reflection,
explanation, and justification
productive disposition—habitual inclination to see mathematics
as sensible, useful, and worthwhile, coupled with a belief in
diligence and one’s own efficacy.
Development of Children’s
Mathematical Thinking
8+4 = +5
Can students solve without computing each side?
Let’s watch David…
Equal Sign Data (8+4=  +5)
Student Responses1
Grade
7
12
17
12 & 17
1st & 2nd
5%
58
13
8
3rd & 4th
9
49
25
10
5th & 6th
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76
21
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1Falkner,
K., Levi, L., & Carpenter, T. (1999). Children’s understanding of
equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-6.
Evidence that Attending to Student Thinking
Can Make a Difference
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Students constantly surprise us…
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Kindergarten data
Fractions
Algebraic thinking
Focusing on Making Student Thinking Explicit
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need to be able to use student strategies as the center of the
workgroup conversation, as one of the tools teachers
interact with
expertise shared
change in power structures, teacher as expert
provides an explicit trace of the group’s thinking
extends to other communities of practice
centers the role of the professional developer
Moving Towards the Details of Practice
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Need to know more about student participation in mathematics
classrooms if we are to support teaching
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Often large scale studies focus on what occurs in public discourse
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Smaller scale studies document more specifically student
participation and what that means for student learning
Forman, et al, 1998; Lampert, 2001; Moschkovich, 2002; O’Connor & Michaels, 1996;
Palincsar & Brown, 1984, 1989; Yackel, Cobb, & Wood, 1991
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Want to look to the relationship between student participation,
teaching, the mathematics and student outcomes
Development of Children’s
Mathematical Thinking
19 Children are taking a mini-bus to the zoo. They
will have to sit either 2 or 3 to a seat. The bus has 7
seats. How many children will have to sit 3 to a seat
and how many can sit 2 to a seat?
How will children solve it?
How about a kindergartener?
59% had a correct strategy
51% correct answer, 33% 1st graders, 26% 2nd graders