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Sunday, February 6, 2011

Quantum Field Theory For Toddlers

Last week, headlines in the newspapers and blogosphere blared:
American Students Struggle with Science
. Skimming the articles, we quickly learn that 99% of American fourth graders do not perform at an advanced level in science, according to a Department of Education report. The report also condemns (less laughably) the performance of our 8th graders and 12th graders, but fourth graders? How seriously can you take 'education experts' who worry about the science expertise of fourth graders? Do they have any concept of the nature of science?

Science in elementary school serves two purposes: (i) allay the concerns of pushy parents who erroneously believe elementary school science has content, and (ii) excite the interest of young children so that when they have the necessary mathematical tools, they are eager to study science. A bad elementary school science course is far worse than no science course at all. For, while neither imparts significant knowledge or lays the groundwork for subsequent courses, a bad course gives the young child the mistaken impression that science is dreary drudgery rather than an exciting pursuit.

Mathematicians often speak of the verticality of our discipline. By this, we mean that, for example, in order to understand partial differential equations, you need to understand functional analysis, real analysis, and complex analysis. In order to understand real analysis and complex analysis, you need to understand basic analysis. In order to understand basic analysis and functional analysis, you need to understand linear algebra and calculus. In order to understand linear algebra you need to understand algebra. Each subject rests on an edifice of one or more more basic subjects. Thus the earlier you learn mathematics the higher you can scale the tower of mathematical knowledge. Physics is similarly vertical. Fields which are significantly less vertical are sometimes referred to as "butterfly collecting." This is a disparaging term (often applied when the speaker does not know a field well enough to know its vertical structure). It refers to fields which are largely descriptive; the idea is that describing a swallowtail does not contribute significantly to the description of a sulphur.

At the elementary and middle school level all science must be taught as butterfly collecting. No real physics can be taught without calculus. A good deal of descriptive chemistry can be taught with a modicum of algebra, but a deeper understanding requires physics, which requires calculus. A great deal of descriptive biology can be taught, but a deeper understanding requires chemistry, etc. Teaching biology and chemistry at this level is useful; most students are not destined to become scientists or engineers. If they learn descriptive science, they are better able to understand their world and, as a bonus, are better able to evaluate the junk science pushed by Hollywood, Washington, and the media. There is nothing, however, that can be taught to a prealgebra (precalculus!) fourth grader that will not have to be relearned at a more advanced level by any student wishing to apply science in his work.

So, what are Washington's education experts telling us? Most importantly, they are telling us they have no concept of how best to spend our children's precious classroom hours (which were largely filled by Disney videos in my children's public schools). If you want to improve American's science performance, replace vacuous fourth grade science courses with more math to prepare them to learn real science while in high school. If more math does not fit the bill, foreign language instruction is a better use of their time at that age than playing with circuit boards.

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