Tuesday, January 25, 2005

Showing Your Work in Mathematics? Should We Insist?

We were recently talking to Jean Goerss, M.D. about this (she's setting up a school for the highly gifted in Arizona and an author of that wonderful book Misdiagnosis and Dual Diagnosis of Gifted Children and Adults)and we thought we'd share our ideas on this issue.

Although the neurobiology of mathematical ability is still early in its development, it is clear that different brain pathways are activated depending on pathway that is undertaken. In brain imaging studies of problem solving, researchers have been able to show that solving problems by deduction (step-wise, conscious) occurs by a very different route than sudden insight (Aha!, unconscious).

That means that in many cases, it may be difficult for a person solving mathematical problems by insight to know exactly how he or she arrived at an answer. As a general rule, it would therefore be best not to insist on a stepwise solution. But that being said, at the highest levels of mathematics, many gifted problem-solvers are very aware of how they may shift between conscious and 'sub-conscious' solutions, and a better understanding how all this brain stuff works only improves problem-solving for the future.

A number of mathematicians and physicists have written about their thought processes, and often the insight or inductive approaches to problem solving require some free association, manipulation of vague images (auditory or visual), and distraction (music, sleep, etc).

For our gifted mathematicians able to solve problems by insight then, although it may not be the best idea to take a hard line about showing every step of a solution, it might be beneficial to discuss different strategies for solving problems, and to discuss how certain problems may be more readily solved by particular method (e.g. visual, symbolic, verbal, mathematical...). The best situation, it would seem, is to have an arsenal of possible approaches, and flexibility and competence at undertaking different strategies to solve a problem most easily.

A great fairly easy read for topics such as this is James L Adam's Conceptual Blockbusting (A Guide to Better Ideas). We really enjoyed it. He addresses the various blocks which can occur in seeking alternate solutions to different problems (e.g. perceptual, emotional cultural and environmental, intellectual and expressive), and the need to thinking in alternative thinking languages to have the greatest flexibility and strength as a problem solver.

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