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Project Summary

Many consider mathematical reasoning to be a basic mathematical skill and inseparable from knowing mathematics with understanding. Accordingly, mathematical reasoning—proof, in particular—has been receiving increased attention in the mathematics education community. Many mathematics educators as well as reform initiatives advocate that proof should be a central part of the mathematics education of students at all grade levels. Yet despite its importance, mathematics education research continues to paint a bleak picture of students’ abilities to reason mathematically. In contrast, recent research in cognitive science has revealed surprising strengths in children’s abilities to reason in non-mathematical contexts, suggesting that children are capable of developing complex and abstract causal theories, and of using powerful strategies of inductive inference. Thus, this raises something of a paradox: Why are children so good at reasoning in non-mathematical contexts, yet so poor at reasoning in mathematical contexts?

The purpose of this research is to explore this seeming paradox. In particular, we seek to extend the cognitive science research into the domain of mathematics education and, more specifically, into the domain of middle school mathematics—a domain that marks a significant mathematical transition from the concrete, arithmetic reasoning of elementary school mathematics to the development of the increasingly complex, abstract reasoning required for high school mathematics and beyond. We believe, first, it is important to understand both the strengths and weaknesses of students’ reasoning in and out of mathematics and, second, that students’ ways of reasoning in non-mathematical domains may provide an important bridge to improving their ways of reasoning in mathematics.

The research has two inter-connected phases and includes the collection of written survey and interview data. In Phase I, we investigate middle school students’ inductive strategies in the domain of mathematics. This work draws on cognitive science literature describing inductive strategies in non-mathematical, causal contexts. A basic question is whether students use the same sorts of strategies in mathematical and in non-mathematical contexts. In Phase II, we explore students’ uses and evaluations of example-based justifications (the predominant form of justification among middle school students). We investigate how students assess degrees or qualities of justifications as well as the nature of the justifications students produce.

Intellectual merit. Although it is generally accepted that students’ mathematical reasoning abilities progress from inductive toward deductive and toward greater generality—and, indeed, various mathematical reasoning hierarchies reflect this expected progression—a theoretical (as well as pedagogical) model regarding the cognitive processes underlying this transition is lacking. By bringing together mathematics education and cognitive science researchers as well as their respective literatures, the research will support the development of a two-tiered theoretical model linking a) students’ ways of reasoning in non-mathematical domains to their mathematical ways of reasoning, and b) students’ ways of reasoning inductively with their abilities to begin to reason deductively. This coordination will offer deeper insights, both theoretical and pedagogical, into the connections between students’ ways of reasoning and, ultimately, will lead to improvement in their abilities to reason mathematically.

Broader impact. The knowledge gleaned from the study will contribute significantly to the knowledge base regarding the critical transition from informal, inductive reasoning to formal, deductive reasoning—reasoning that is fundamental to knowing and using mathematics. In practical terms, the research will serve to inform curricular and instructional efforts aimed at fostering the development of students’ abilities to reason mathematically.