Many consider mathematical reasoning to be a basic mathematical skill and inseparable from knowing and using mathematics. Yet despite its importance, mathematics education research continues to paint a bleak picture of students' abilities to reason mathematically. In contrast, cognitive science research has revealed surprising strengths in children's abilities to reason in non-mathematical domains, 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 domains, yet so poor at reasoning in mathematical domains?
The purpose of this study is to explore this seeming paradox. In particular, our goal is 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 ina nd 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, casual contexts. A basic question is whether students use the same sorts of strategies in mathematical and 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.
The study was guided by the following research questions:
- What strategies do middle school students use to reason inferentially in both non-mathematics and mathematics contexts?
- How can instruction and curriculum build upon students’ inductive reasoning strategies in order to develop their deductive mathematical justifications?