Transcript Document

Focusing on the Development of
Children’s Mathematical Thinking: CGI
Megan Loef Franke
UCLA
Algebra as focal point
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“Algebra for All”
(Edwards, 1990; Silver, 1997)
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“gatekeeper for citizenship”
(Moses & Cobb, 2001)
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Difficult transition from arithmetic
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Not move high school curriculum to
elementary school
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Engages teachers in a new way, new
content
Algebra as generalized arithmetic
and the study of relations
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Viewing the equal sign as a relation
57 + 36 =  + 34
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Using number relations to simplify
calculations
5 x 499 = 
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Making explicit general relations based
on fundamental properties of
arithmetic
768 + 39 = 39 + 
Equality
8+4=+5
Equality 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
2
76
21
2
1Falkner,
K., Levi, L., & Carpenter, T. (1999). Children’s understanding of
equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-6.
True/false number sentences:
from worksheets to index cards
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Shift from a focus on
answer to a focus on
reasoning
Shift from a focus on a
single problem to a
sequence
Shift from sharing a single
strategy to a conversation
around the reasoning
Sequence of Number Sentences
3+4=7
5 + 5 = 8*
7=3+4
6=6+0
6=6
6=3+3
4+2=3+3
* denotes false number sentence
Mathematical Content
Equality
Number Facts
Place Value
Number Sense
Mathematical Properties
Multiplication
Equivalence
7=7
5+5=4+6
250 + 150 =  +100
45 = 100 + 20 + 
5+6=6+
37=7+7+7
½=¼+¼
Relational Thinking
24 + 17 – 17 = 34 + 
1,000 – 395 = ___
999 – 395 + 1
Relational Thinking
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Solve:
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576 + 199 =
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576 + 200 - 1
1,000 – 637 =
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□
□
999 – 637 + 1
4 x 24 + 5 x 24 =
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10 x 24 - 24
□
Generating Conjectures
Making relational thinking explicit
Representing Conjectures
b+0 = b
c+d = d+c
Variables
k + k + 13 = k + 20
Experimental Study Design
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Volunteer, urban, low performing
elementary schools in one district (19)
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District working to improve
opportunities in mathematics
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Schools randomly assigned to year 1 or
year 2 professional development work
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School site based PD monthly
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On site support
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End of one year assessed teachers
(180) and students (3735)
Teacher Findings
Generating strategies for 8 + 4 =  + 5
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No differences in teachers’
perceptions on time spent on
algebraic thinking tasks in
classrooms
No differences on knowledge
of algebra
Differences in teachers’
knowledge of student thinkingstrategies and relational thinking
Number
of
strategies
Participating NonTeachers
Participating
Teachers
1
6%
44%
2
38%
41%
3
25%
12%
4 or more
31%
4%
Student Findings

Students in algebraic
thinking classrooms scored
significantly better on the
equality written assessment.

Students in 3rd and 5th
grades were twice as likely
to use relational thinking
Publications

Book for teachers:
Carpenter, T., Franke, M., & Levi, L. (2003). Thinking
mathematically Integrating arithmetic and algebra in
elementary school. Portsmouth, NH: Heinemann.
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Research article:
Jacobs, V., Franke, M., Carpenter, T., Levi, L. & Battey, D.
(in press). Exploring the impact of large scale
professional development focused on children’s
algebraic reasoning. Journal for Research in
Mathematics Education.
Conjectures
Is a focus on children’s thinking enough?
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Show what students are capable of
Counter narratives
Change what we consider basic skills
Create ways in schools to make room for
understanding
Watch for how the status quo limits
opportunities…find ways to challenge it