Understanding Proofs in Pure Mathematics How a step is derived Why the step was chosen How parts of premise are used.

By understanding mathematics I mean the understanding of the nature of a proof in pure mathematics. This involves at least three levels of understanding: 1. Understanding how each step of the proof is derived. 2. Understanding why the author of the proof chose this step. 3. Understanding how each part of the premise of the proof is used. In this research, MU models level 1, HAL models level 2, and level 3 is beyong the scope of the present research. Level 2 involves both the identification of heuristics underlying the proof and the plan of the proof. The former is modelled by HAL, whereas the latter has been investigated by Bundy. Problem solving is a good test of understanding.