Student difficulty in constructing proofs: The need for strategic knowledge

The ability to construct proofs is an important skill for all mathematicians. Despite its importance, students have great difficulty with this task. In this paper, I first demonstrate that undergraduates often are aware of and able to apply the facts required to prove a statement but still fail to prove it. They thus fail to construct a proof because they could not use the syntactic knowledge that they had. By comparing doctoral students and undergraduates constructing proofs in abstract algebra, I have hypothesized four types of `strategic knowledge' – knowledge of how to choose which facts and theorems to apply – which the doctoral students appeared to possess and undergraduates did not. The doctoral students appeared to know the powerful proof techniques in abstract algebra, which theorems are most important, when particular facts and theorems are likely to be useful, and when one should or should not try and prove theorems using symbol manipulation. This revised version was published online in July 2006 with corrections to the Cover Date.