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
that
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.
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