Proof in school mathematics has received increased attention in the education community over the past decade, with researchers and policy makers arguing that it must be a central part of the education of all students at all grade levels. Middle school in particular has been identified as a critical time for students to develop the ability to reason deductively, resulting in recommendations for curricular and pedagogical changes emphasizing proof in beginning algebra classes. These recommendations pose serious challenges, however, given the documented difficulties faced by students in understanding proof and by teachers in supporting students’ proof competencies in algebra.

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SPARQ addresses these challenges by introducing into the ongoing investigation of the teaching and learning of proof a new strand of research—namely, research on quantitative reasoning. Reasoning with relationships between quantities (for example, length, area, volume, or speed) has been found to support students’ understanding of algebraic relationships, encouraging thinking that is flexible and general in character and thus conducive to deductive reasoning. SPARQ examines the ways in which reasoning with quantities can also support students’ developing proof skills in algebra. Our research activities provide a foundation for future work developing ways to support students’ deductive reasoning skills in whole-class settings, and will contribute to the education of pre-service teachers, in-service teachers, and mathematics education graduate students.
 

We have conducted four teaching experiments on linear, quadratic, and exponential functions and have shared our results with practicing middle school teachers, developing teachers’ abilities to promote their own students’ proof competencies through the use of quantitative situations. Scaling up our findings from small-group teaching experiments, we collected data in five teachers' classrooms in order to study their implementation of replacement units in whole-classroom settings. Our current analysis has two strands: We are developing research-based learning trajectories for middle school students' understanding of linear, quadratic, and exponential functions, and we are studying the teacher practices that have been effective in supporting students' abilities to reason algebraically and quantitatively in conceptually-oriented functions units.