Focusing on the Development of Children’s Mathematical Thinking: CGI

Megan Loef FrankeUCLA

Algebra as focal point

“Algebra for All” (Edwards, 1990; Silver, 1997)

“gatekeeper for citizenship” (Moses & Cobb, 2001)

Difficult transition from arithmetic

Not move high school curriculum to elementary school

Engages teachers in a new way, new content

Algebra as generalized arithmetic and the study of relations

Viewing the equal sign as a relation

57 + 36 = + 34

Using number relations to simplify calculations

5 x 499 =

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

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 Sentences3 + 4 = 75 + 5 = 8*7 = 3 + 46 = 6 + 06 = 66 = 3 + 34 + 2 = 3 + 3* denotes false number sentence

Mathematical Content

Equality 7 = 7Number Facts 5 + 5 = 4 + 6Place Value 250 + 150 = +100Number Sense 45 = 100 + 20 + Mathematical Properties 5 + 6 = 6 + Multiplication 3 7 = 7 + 7 + 7Equivalence ½ = ¼ + ¼

Relational Thinking

24 + 17 – 17 = 34 +

1,000 – 395 = ___999 – 395 + 1

Relational Thinking

Solve: 576 + 199 = □

576 + 200 - 1

1,000 – 637 = □ 999 – 637 + 1

4 x 24 + 5 x 24 = □ 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

Volunteer, urban, low performing elementary schools in one district (19)

District working to improve opportunities in mathematics

Schools randomly assigned to year 1 or year 2 professional development work

School site based PD monthly On site support End of one year assessed teachers

(180) and students (3735)

Teacher Findings

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 thinking- strategies and relational thinking

Number of strategies

Participating Teachers

Non-Participating Teachers

1 6% 44%

2 38% 41%

3 25% 12%

4 or more

31% 4%

Generating strategies for 8 + 4 = + 5

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.

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? 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