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

1. Focusing on the Development of Children’s Mathematical Thinking: CGI Megan Loef Franke UCLA

2. 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

3. 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 + 

4. Equality 8 + 4 =  + 5

5. Equality Data (8+4=  +5) 1Falkner, K., Levi, L., & Carpenter, T. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-6.

7. 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

8. 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

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

10. Relational Thinking 24 + 17 – 17 = 34 + 1,000 – 395 = ___ 999 – 395 + 1

11. Relational Thinking • Solve: • 576 + 199 = □ • 576 + 200 - 1 • 1,000 – 637 = □ • 999 – 637 + 1 • 4 x 24 + 5 x 24 = □ • 10 x 24 - 24

12. Generating Conjectures Making relational thinking explicit Representing Conjectures b + 0 = b c + d = d + c

13. Variables k + k + 13 = k + 20

14. 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)

15. Teacher Findings Generating strategies for 8 + 4 =  + 5 • 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

16. 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

18. 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.

19. Conjectures

20. 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