My NSF-funded project Learning Algebra
and Methods of Proof (LAMP) demonstrated that teaching and
learning with and through viable argumentation (proof and
proving) improved Grade 8 students' mathematics
achievement. This talk presents the mechanisms that
caused the learning improvements. I illustrate a model for
how students can and should engage in mathematics, and in
particular, I illustrate cases where the argument students
create is the content to be learned. The framework offered
here offers opportunities to rethink mathematics education
at all levels, from kindergarten to post-secondary
mathematics education. In particular, I offer a useful
definition for what it means to know and comprehend a
mathematical concept.
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