Genius_ The Life and Science of Richard Feynman - James Gleick [240]
Feynman did not take the side of the modernizers. Instead, he poked a blade into the new-math bubble. He argued to his fellow commissioners that sets, as presented in the reformers’ textbooks, were an example of the most insidious pedantry: new definitions for the sake of definition, a perfect case of introducing words without introducing ideas. A proposed primer instructed first-graders: “Find out if the set of the lollipops is equal in number to the set of the girls.” Feynman described this as a disease. It removed clarity without adding any precision to the normal sentence: “Find out if there are just enough lollipops for the girls.” Specialized language should wait until it is needed, he said, and the peculiar language of set theory never is needed. He found that the new textbooks did not reach the areas in which set theory does begin to contribute content beyond the definitions: the understanding of different degrees of infinity, for example.
It is an example of the use of words, new definitions of new words, but in this particular case a most extreme example because no facts whatever are given… . It will perhaps surprise most people who have studied this textbook to discover that the symbol ? or ? representing union and intersection of sets … all the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical physics, in engineering, business, arithmetic, computer design, or other places where mathematics is being used.
Feynman could not make his real point without drifting into philosophy. It was crucial, he argued, to distinguish clear language from precise language. The textbooks placed a new emphasis on precise language: distinguishing “number” from “numeral,” for example, and separating the symbol from the real object in the modern critical fashion—pilpul for schoolchildren, it seemed to Feynman. He objected to a book that tried to teach a distinction between a ball and a picture of a ball—the book insisting on such language as “color the picture of the ball red.”
“I doubt that any child would make an error in this particular direction,” Feynman said dryly.
As a matter of fact, it is impossible to be precise … whereas before there was no difficulty. The picture of a ball includes a circle and includes a background. Should we color the entire square area in which the ball image appears all red? … Precision has only been pedantically increased in one particular corner when there was originally no doubt and no difficulty in the idea.
In the real world, he pointed out once again, absolute precision is an ideal that can never be reached. Nice distinctions should be reserved for the times when doubt arises.
Feynman had his own ideas for reforming the teaching of mathematics to children. He proposed that first-graders learn to add and subtract more or less the way he worked out complicated integrals—free to select any method that seems suitable for the problem at hand. A modern-sounding notion was, The answer isn’t what matters, so long as you use the right method. To Feynman no educational philosophy could have been more wrong. The answer is all that does matter, he said. He listed some of the techniques available to a child making the transition from being able to count to being able to add. A child can combine two groups into one and simply count the combined group: to add 5 ducks and 3 ducks, one counts 8 ducks. The child can use fingers or count mentally: 6, 7, 8. One can memorize the standard combinations. Larger numbers can be handled by making piles—one groups pennies into fives, for example—and counting the piles. One can mark numbers on a line and count off the spaces—a method that becomes useful, Feynman noted, in understanding measurement and fractions. One can write larger numbers in columns and carry sums larger than 10.
To Feynman the standard texts seemed too rigid. The problem 29 + 3 was considered a third-grade problem, because it required the advanced technique of carrying; yet Feynman pointed out that a first-grader could handle