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

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 =  + 34 Using number relations to simplify calculations 5 x 499 =  Making explicit general relations based on fundamental properties of arithmetic = 39 + 

Equality =  + 5

Equality Data (8+4=  +5) 1 Falkner, K., Levi, L., & Carpenter, T. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 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 Sentences = = 8* 7 = = = 6 6 = = * denotes false number sentence

Mathematical Content Equality7 = 7 Number Facts5 + 5 = Place Value =  +100 Number Sense45 =  Mathematical Properties = 6 +  Multiplication3  7 = Equivalence½ = ¼ + ¼

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

Relational Thinking Solve:  = □  1,000 – 637 = □ 999 –  4 x x 24 = □ 10 x

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 16%44% 238%41% 325%12% 4 or more31%4% Generating strategies for =  + 5

Student Findings  Students in algebraic thinking classrooms scored significantly better on the equality written assessment.  Students in 3 rd and 5 th 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