# Memorizing Mathematics: The Failure to Apply Mathematical Axioms Within Restrictive Models

- Rony Patel,
*Rutgers University*
- Jennifer Jacobs,
*Rutgers University*
- Rochel Gelman,
*Rutgers University*

## Abstract

Arithmetic, along with all mathematics, is built on axioms.
Mathematical education, however, favors mathematical models or algorithms without
appeal to the axioms they depend on. Our series of studies demonstrate adult
subjects’ inability to take advantage of the knowledge embodied in
arithmetic axioms. It is likely that students’ ability to master generative
proofs is related to the reliance on restrictive models.
Our first set of studies focused on subjects’ ability to apply the
addition-rule (mutually exclusive events) and the multiplication-rule
(independent events) with multiple rational number representations (percentages,
decimals, fractions, etc.). A second set of studies tested the understanding of
group theory properties, exploiting the effect of the order of numbers
(commutativity) and the effect of the digit zero (additive identity) on the long
multiplication model (LMM). All studies conducted revealed participants ability
to accurately perform arithmetic on chosen representation, but poor performance
on choosing representation.

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