Why Do Those 'Clever Fools' Write Useless Texts?

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There has been a great deal of recent public discussion about the need to improve the teaching of mathematics and science in elementary and secondary schools. But it has concentrated almost entirely on ways to attract competent teachers, while overlooking another issue--the poor quality of many mathematics and science textbooks now in use.

An exceptionally good teacher can get good results even with a bad textbook, but the performance of an average teacher is likely to be much better with a good textbook than with a bad one.

I'm not a professional mathematician or scientist; in fact, my formal training in those subjects ended with my graduation from an English high school in the 1920's. But in looking at some of the math and science textbooks in the Montgomery County, Md., schools, which my grandchildren attend, I've made some disheartening observations.

Montgomery County's high-school mathematics textbooks place great emphasis on definitions and terminology and manage to make their subjects both dull and difficult, even though Euclidean geometry was one of the great early achievements of the human mind, showing that an elaborate system could be deduced from a few simple axioms. Euclid's Elements--written 2,000 years ago and found in translation in the local library--very clearly and simply explains theorems that were unintelligible in my grandchild's high-school geometry text.

Another feature of current textbooks is their emphasis on following prescribed procedures; this, they suggest, is as important as getting the right answer. Karl Friederich Gauss, a German mathematician, astronomer, and physicist of the 19th century, got his start in life at the age of seven when his teacher told the class to add all the numbers from one to a hundred. Gauss thought out the formula for the sum of an arithmetical progression and very quickly gave the right answer, while the rest of the class plodded through a laborious addition sum.

The teacher realized that he had a mathematical genius in his class and arranged the late-18th-century equivalent of a scholarship to send Gauss through high school and college. In a modern American school, Gauss might have been reprimanded and told to get the answer in the prescribed way.

There have been important developments in mathematics during the past century, but few, if any, are relevant to the kind of mathematics needed by children through the high-school level--applied mathematics. Nonetheless, half a semester's worth of work in my grandchild's calculus textbook deals with elaborate proofs of the obvious. One example: 5X-1 approaches 4 as X approaches 1.

Anyone who wants to make a career in science, or engineering, or economics needs calculus as a tool; proofs of what is intuitively obvious are really only of interest to those who wish to become pure mathematicians and could well be left for their courses at the university. There was almost nothing in my grandchild's textbook to show that calculus provides a way to solve many practical problems.

Sylvanus P. Thompson's Calculus Made Easy remains in use after 73 years because it fulfills the promise of its title. Its prologue says, "Some calculus tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics--and they are mostly clever fools--seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way." This book, that starts on the first page to explain what a differential is and goes on to relate calculus to actual problems, would be far better as an introduction to the subject than the textbook now used.

A science textbook used at a junior high school contained false statements and examples of gross carelessness. It listed lead as a naturally radioactive element. It described an experiment for finding the weight of air by weighing a balloon, first uninflated and then inflated (this would only show the very small difference in weight between the slightly compressed air in the balloon and the uncompressed air displaced by the inflated balloon). A drawing of a turboprop aircraft engine, showing the propeller, had a caption that read: "Jet engines have no propellers."

A junior-high-school mathematics textbook also contained errors. Some sections implied advocacy of the Pythagorean view that lines consist of a finite number of points. I explained to my grandchild the simple proof that the square root of two is irrational--a proof that destroyed the Pythagorean system. (However large the number of points in the sides of a square, no exact number of points can express the length of the diagonal.) My grandchild showed this proof to her teacher, who said that it was very interesting but that she was stuck with the textbook.

A general weakness of another grandchild's chemistry textbook was that it made chemistry a very abstract and formal subject because it said very little about things that would make chemistry interesting--the processes used to make important chemicals and some account of the development of chemistry.

Students get very distorted ideas of the way in which scientists work and of the nature of scientific knowledge from books that simply present the present state of scientific theories as if they were final and certain. With some understanding of the history of science, students would come to understand that what we learn from scientific theories is not final and certain truth, but only what is reasonable to believe on the available evidence, that though scientific knowledge is objective it is also uncertain because any theory may need revision in the light of new evidence. Teaching that gives students a real understanding of scientific method not only makes them technically competent but helps them think clearly in other fields.

Another weakness of American math textbooks is that they are continually being changed. If someone tries to produce a completely new textbook on an unchanged subject, he or she can only do so by putting in material that is really irrelevant and making the book more complicated and harder to understand.

Euclidean geometry goes back 2,000 years, and there has been no important development for more than a century in the parts of algebra, analytical geometry, and calculus that are appropriate for high-school students. Therefore, there is no real need for new mathematics textbooks. My own children, who attended school in Australia, were given slightly revised editions of the algebra textbooks I had been given in England 30 years earlier.

An official in the Montgomery County Department of Education explained to me that the school system could not find a high-quality textbook and continue to use it because publishers were continually producing new textbooks and allowing the old ones to go out of print. In effect, children are being exploited for the benefit of publishers and authors.

Instead, local education authorities should use the expertise of their staffs and produce their own textbooks; it would be a sound investment, for some texts could remain in use for decades. Local "talent" may, in fact, produce some very good textbooks, for mathematicians from leading universities might well be Sylvanus Thompson's "clever fools"--unqualified to write textbooks because they do not understand how to make fairly simple mathematics intelligible to school-age children.

Vol. 02, Issue 38, Page 18

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