On the relationships between (relatively) advanced mathematical knowledge and (relatively) advanced problem-solving behaviours

Abstract

This article discusses an issue of inserting mathematical knowledge within the problem-solving processes. Relatively advanced mathematical knowledge is defined in terms of three mathematical worlds; relatively advanced problem-solving behaviours are defined in terms of taxonomies of proof schemes and heuristic behaviours. The relationships between mathematical knowledge and problem-solving behaviours are analysed in the contexts of solving an insight geometry problem, posing algebraic problems and calculus exploration. A particularly knowledgeable and skilled university student was involved in all the episodes. The presented examples substantiate the claim that advanced mathematical knowledge and advanced problem-solving behaviours do not always support each other. More advanced behaviours were observed when the student worked within her conceptual-embodied mathematical world, and less advanced ones when she worked within her symbolic and formal-axiomatic worlds. Alternative explanations of the findings are discussed. It seems that the most comprehensive explanation is in terms of the Principle of Intellectual Parsimony. Implications for further research are drawn.

Keywords:

• advanced mathematical thinking
• problem solving
• problem posing
• exploring
• proof schemes
• three mathematical worlds
• intellectual parsimony

Acknowledgements

I am grateful to Miriam Farber for her valuable input in the first happy ending. I am also grateful to Kenneth Shilling for the second happy ending.