# How do mathematicians learn math?: resources and acts for constructing and understanding mathematics

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## Abstract

In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a *network of mathematical resources* by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics graduate students that were asked to read and make sense of an unfamiliar, but accessible, mathematical proof in the domain of geometric topology. We find that experts are more likely to refer to definitions when questioning or explaining some aspect of the focal mathematical idea and more likely to refer to specific examples or instantiations when making sense of an unknown aspect of that idea. However, in general, they employ a variety of types of mathematical resources simultaneously. Often, these combinations are used to deconstruct the mathematical idea in order to isolate, identify, and explore its subcomponents. Some common patterns in the ways experts combined these resources are presented, and we consider implications for education.

### Keywords

Expert mathematicians Topology Proof Reasoning Knowledge resources### References

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