ALCOCK, L. and INGLIS, M., 2008. Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69 (2), pp. 111-129.
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.
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