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 57 + 36 =  + 34  Using number relations to simplify calculations 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 12 17 12 & 17 1st & 2nd 5% 58 13 3rd & 4th 49 25 10 5th & 6th 76 21 Falkner, 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 Sentences 3+4=7 + = 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+ 3ì 7=7+7+7 ẵ=ẳ+ẳ Relational Thinking 24 + 17 – 17 = 34 +  1,000 – 395 = _ 999 – 395 + Relational Thinking  Solve:  576 + 199 = □   1,000 – 637 = □   576 + 200 - 999 – 637 + x 24 + 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 or year professional development work  School site based PD monthly  On site support  End of one year assessed teachers (180) and students (3735) Teacher Findings Generating strategies for + =  +    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 Numb er of strate gies Participat Noning Participati Teachers ng Teachers 6% 44% 38% 41% 25% 12% 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  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 ... 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... improve opportunities in mathematics  Schools randomly assigned to year or year professional development work  School site based PD monthly  On site support  End of one year assessed teachers... single strategy to a conversation around the reasoning Sequence of Number Sentences 3+4=7 + = 8* 7=3+4 6=6+0 6=6 6=3+3 4+2=3+3 * denotes false number sentence Mathematical Content Equality Number