Download A Century of Mathematics in America (History of Mathematics, by Peter Duren Richard A. Askey PDF

By Peter Duren Richard A. Askey

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Testing for understanding requires longer answers in which ideas must cooperate. Deriving one result from another, or applying ideas to novel situations demonstrates understanding. This is why most teachers value discursive, problem-solving exams over the true/false format. It is also why a lonely result with no derivation, even a correct result, is not a satisfactory answer when asked to solve a problem. ‘‘Show your work,’’ means show that you not only know the right answer, but that you genuinely understand what is going on.

Kuhn’s motivation for saying this is based on his survey of historical cases, but it could just as well be argued in principle, based on the essentially holistic nature of paradigms. The influence of a paradigm is pervasive in that it establishes the appropriate language of the science, the background knowledge against which new ideas are compared, and even the standards by 28 5 Scientific Change which new experimental results are judged. All aspects of a science are thus paradigm-dependent, including any guidelines for judging the acceptability of a paradigm.

But he did this before the use of astronomical telescopes. Subsequent measurements, made more precise with the use of telescopes, showed that the parallax motion did happen, as predicted by the new world system. The refined ability to observe nature allowed for this new, pivotal information. The case of stellar parallax as evidence against, or for, the hypothesis that the earth orbits the sun is a good place to demonstrate the important aspects of empirical testing developed in Chap. 3. The hypothesis that the earth moves is indeed about a phenomenon that could not be directly observed, at least not at the time of the Copernican Revolution.

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