# Doctoral students’ use of examples in evaluating and proving conjectures

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## Abstract

This paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and semantic reasoning strategies relative to a representation system of proof. We then use this conceptualisation as a basis for contrasting the behaviour of two successful mathematics research students whilst they evaluated and proved number theory conjectures. We observe that the students exhibited strikingly different degrees of example use, and argue that previously observed individual differences in reasoning strategies may exist at the expert level. We conclude by discussing implications for pedagogy and for future research.

## Keywords

Examples Generic examples Number theory Proof Reasoning Strategies## Notes

### Acknowledgements

We thank Pablo Mejia-Ramos, Dina Tirosh and Rina Zazkis for valuable comments on earlier drafts of this work.

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