This Spring at Oregon State University I
taught an experimental Calc I class, using an (informal)
infinitesimals-based approach. I examine two notational
registers students developed for reasoning with definite
integrals: Adding-Up-Pieces and
Multiplicatively-Based-Summation. Research indicates that
these two registers are much more beneficial for modeling
with integrals and reasoning with the Fundamental Theorem
of Calculus than are the more common ``antiderivative" and
``area under curve" registers.
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