Wednesday, February 8, 2012

Know-How and Why

Science education needs to focus on the process, on mathematical modeling, on how to obtain results, instead of simply results themselves. There is some improvement in this regard, but the old approach and culture still persists. Even the greats of teaching were not immune to this problem; The Feynman Lectures on Physics is mostly about results rather than the process used to obtain those results. The real gold is in the process, the results can change. In Genius we read at a young age Feynman reached a point when he could model pretty much anything he wanted, he'd run around asking people for problems to model, and finally found one;
No one had ever analyzed the behavior of light passing through a parade of mostly transparent films thinner than a single wavelength [..] A few days later Feynman returned with the solution: a formula summing an infinite series of reflections back and forth from the inner surfaces of the coatings. He showed how the combinations of refraction and reflection would affect the phase of the light, changing its color. Using Feynman’s theory and many hours on the Marchant calculator, Cutler also found a way to make the color filters his professors wanted. Developing a theory for reflection by multiple-layer thin films was not so different for Feynman from math team in the now distant past of Far Rockaway [his childhood home]. He could see, or feel, the intertwined infinities of the problem, the beam of light resonating back and forth between the pair of surfaces, and then the next pair, and so on, and he had a giant mental kit bag of formulas to try out.
Representing functions are sums of infinite series is a deeply theoretical subject whose research was inspired by physics, beautiful mathematics. Tranmission of such "mental kit of formulas [or equalities, inequalities, theories, representations, etc]" should be the goal of education, not simply dropping a finished product / formula F = ma in a kid's lap, and have him / her solve mechanical problems one after another.

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