Understanding and Cultivating the Connections Between Students' Natural Ways of Reasoning and Mathematical Ways of Reasoning
It is generally accepted that students' mathematical reasoning abilities progress from inductive toward deductive and toward greater generality. Various mathematical reasoning hierarchies reflect this expected progression. Yet we lack theoretical and pedagogical models regarding the cognitive processes underlying this transition.
Recent research in cognitive science has revealed surprising strengths in children's natural abilities to reason in non-mathematical contexts. This suggests that children are capable of developing complex and abstract causal theories, and of using powerful strategies of inductive inference.
Why are children so good at reasoning in non-mathematical contexts, yet so poor at reasoning in mathematical contexts? The purpose of the proposed research is to explore this seeming paradox.
This research extends cognitive science research into the domain of middle school mathematics. This domain 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 it is important to understand both the strengths and weaknesses of students' reasoning in and out of mathematics. Students' natural ways of reasoning may provide an important bridge to improving their mathematical ways of reasoning.
This research will investigate
- middle school students' inductive strategies in the domain of mathematics,
- students' uses and evaluations of example-based justifications (the predominant form of justification among middle school students), and
- how curriculum and instruction might build on students' inductive strategies as a means to develop their deductive (proving) strategies.
This research will support the development of a two-tiered theoretical model
a) links students' natural ways of reasoning to their mathematical ways of reasoning, and
b) links students' ways of reasoning inductively with their abilities to begin to reason deductively.
This coordination will offer deeper insights into the connections between students' ways of reasoning and ultimately will lead to improvement in their abilities to reason mathematically.
The knowledge gleaned from the study will contribute to our knowledge of 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.