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Richard Feynman on Teaching Math to Kids and the Lessons of Knowledge

Legendary scientist Richard Feynman was famous for his penetrating insight and clarity of thought. Famous for not only the work he did to garner a Nobel Prize, but also for the lucidity of explanations of ordinary things such as why trains stay on the tracks as they go around a curve, how we look for new laws of science, how rubber bands work, and the beauty of the natural world.

Feynman knew the difference between knowing the name of something and knowing something. And was often prone to telling the emperor they had no clothes as this illuminating example from James Gleick’s book Genius: The Life and Science of Richard Feynman shows.

Educating his children gave him pause as to how the elements of teaching should be employed. By the time his son Carl was four, Feynman was “actively lobbying against a first-grade science book proposed for California schools.”

It began with pictures of a mechanical wind-up dog, a real dog, and a motorcycle, and for each the same question: “What makes it move?” The proposed answer—“ Energy makes it move”— enraged him.

That was tautology, he argued—empty definition. Feynman, having made a career of understanding the deep abstractions of energy, said it would be better to begin a science course by taking apart a toy dog, revealing the cleverness of the gears and ratchets. To tell a first-grader that “energy makes it move” would be no more helpful, he said, than saying “God makes it move” or “moveability makes it move.”

Feynman proposed a simple test for whether one is teaching ideas or mere definitions: “Without using the new word which you have just learned, try to rephrase what you have just learned in your own language. Without using the word energy, tell me what you know now about the dog’s motion.”

The other standard explanations were equally horrible: gravity makes it fall, or friction makes it wear out. You didn’t get a pass on learning because you were a first-grader and Feynman’s explanations not only captured the attention of his audience—from Nobel winners to first-graders—but also offered true knowledge. “Shoe leather wears out because it rubs against the sidewalk and the little notches and bumps on the sidewalk grab pieces and pull them off.” That is knowledge. “To simply say, ‘It is because of friction,’ is sad, because it’s not science.”

Richard Feynman on Teaching

Choosing Textbooks for Grade Schools

In 1964 Feynman made the rare decision to serve on a public commission for choosing mathematics textbooks for California’s grade schools. As Gleick describes it:

Traditionally this commissionership was a sinecure that brought various small perquisites under the table from textbook publishers. Few commissioners— as Feynman discovered— read many textbooks, but he determined to read them all, and had scores of them delivered to his house.

This was the era of new math in children’s textbooks: introducing high-level concepts, such as set theory and non decimal number systems into grade school.

Feynman was skeptical of this approach but rather than simply let it go, he popped the balloon.

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.”

To Feynman this was a disease. It confused without adding precision to the normal sentence: “Find out if there are just enough lollipops for the girls.”

According to Feynman, specialized language should wait until it is needed. (In case you’re wondering, he argued the peculiar language of set theory is rarely, if ever, needed —only in understanding different degrees of infinity—which certainly wasn’t necessary at a grade-school level.)

Feynman convincingly argued this was knowledge of words without actual knowledge. He wrote:

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.

The point became philosophical.

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— pupil 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, adding:

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 absolute precision can never be reached and the search for degrees of precision that are not possible (but are desirable) causes a lot of folly.

Feynman has his own ideas for teaching children mathematics.

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Process vs. Outcome

Feynman 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 were flawed. The problem

29
+3

was considered a third-grade problem because it involved the concept of carrying. However, Feynman pointed out most first-graders could easily solve this problem by counting 30, 31, 32.

He proposed that kids be given simple algebra problems (2 times what plus 3 is 7) and be encouraged to solve them through the scientific method, which is tantamount to trial and error. This, he argued, is what real scientists do.

“We must,” Feynman said, “remove the rigidity of thought.” He continued “We must leave freedom for the mind to wander about in trying to solve the problems…. The successful user of mathematics is practically an inventor of new ways of obtaining answers in given situations. Even if the ways are well known, it is usually much easier for him to invent his own way— a new way or an old way— than it is to try to find it by looking it up.”

It was better in the end to have a bag of tricks at your disposal that could be used to solve problems than one orthodox method. Indeed, part of Feynman’s genius was his ability to solve problems that were baffling others because they were using the standard method to try and solve them. He would come along and approach the problem with a different tool, which often led to simple and beautiful solutions.

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If you give some thought to how Farnam Street helps you, one of the ways is by adding to your bag of tricks so that you can pull them out when you need them to solve problems. We call these tricks mental models and they work kinda like lego — interconnecting and reinforcing one another. The more pieces you have, the more things you can build.

Complement this post with Feynman’s excellent advice on how to learn anything.