Saturday, February 6, 2016

Feynman on the disease of increased precision

We want to be careful about arguing from authority, but it is also important to consider credibility.

Richard Feynman was, to put it mildly, a big deal in 20th century physics. His opinions on scientific and mathematical reasoning carry a tremendous amount of weight. On top of that, he spent a great deal of time and serious thought to digging into the state of elementary and secondary math textbooks. For all these reasons, when he argues forcefully against a certain trend or practice in mathematics education, we should certainly take his arguments seriously.

by Richard P. Feynman

March 1965, Vol. XXVIII, No. 6
Words and definitions
When we come to consider the words and definitions which children ought to learn, we should be careful not to teach "just" words. It is possible to give an illusion of knowledge by teaching the technical words which someone uses in a field (which sound unusual to ordinary ears ) without at the same time teaching any ideas or facts using these words. Many of the math books that are suggested now are full of such nonsense - of carefully and precisely defined special words that are used by pure mathematicians in their most subtle and difficult analyses, and are used by nobody else.

Secondly, the words which are used should be as close as possible to those in our everyday language; or, as a minimum requirement, they should be the very same words used, at least, by the users of mathematics in the sciences, and in engineering.


Pure mathematics is just such an abstraction from the real world, and pure mathematics does have a special precise language for dealing with its own special and technical subjects. But this precise language is not precise in any sense if you deal with the real objects of the world, and it is overly pedantic and quite confusing to use it unless there are some special subtleties which have to be carefully distinguished

A fine distinction
For example, one of the books pedantically insists on pointing out that a picture of a ball and a ball are not the same thing. I doubt that any child would make an error in this particular direction. It is there- fore unnecessary to be precise in the language and to say in each case, "Color the picture of the ball red," whereas the ordinary book would say, "Color the ball red."

As a matter of fact, it is impossible to be precise; the increase in precision to "color the picture of the ball" begins to produce doubts, whereas, before that, 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 or just the part inside the circle of the ball? Coloring the ball red is clear. Coloring the picture of the ball red has become somewhat more confused.

Although this sounds like a trivial example, this disease of increased precision rises in many of the textbooks to such a pitch that there are almost incomprehensibly complex sentences to say the very simplest thing. In a first-grade book (a primer, in fact) I find a sentence of the type: "Find out if the set of the lollypops is equal in number to the set of girls" - whereas what is meant is: "Find out if there are just enough lollypops for the girls."

The parent will be frightened by this language. It says no more, and it says what it says in no more precise fashion than does the question: "Find out if there are just enough lollypops for the girls" - a perfectly understandable phrase to every child and every parent. There is no need for this nonsense of extra-special language, simply because that type of language is used by pure mathematicians. One does not learn a subject by using the words that people who know the subject use in discussing it. One must learn how to handle the ideas and then, when the subtleties arise which require special language, that special language can be used and developed easily. In the meantime, clarity is the desire.

I believe that all of the exercises in all of the books, from the first to the eighth year, ought to be understandable to any ordinary adult - that is, the question of what one is trying to find out should be clear to every person. It may be that every adult is not able to solve all of the problems; perhaps they have forgotten their arithmetic, and they cannot readily obtain 2/3 of 1/4 of 1-1/3, but they at least should understand that that product is what one is trying to obtain.

By putting the special language into the books, one appears to be learning a different subject and the parent (including highly trained engineers) is unable to help the child or to understand what the thing is all about. Yet such a lack of understanding is completely unnecessary and no gain whatsoever can be claimed for using unusual words when usual words are available, generally understood, and equally clear (usually, in fact, far clearer).

Feynman was, of course, writing in the Sixties, but we have a pretty good idea what he would say about this Common Core aligned handout for an eighth grade algebra class.

Dilation: A transformation of the plane with center O and scale factor r(r > 0). If
D(O) = O and if P ≠ O, then the point D(P), to be denoted by Q, is the point on the ray OP so that |OQ| = r|OP|. If the scale factor r ≠ 1, then a dilation in the coordinate plane is a transformation that shrinks or magnifies a figure by multiplying each coordinate of the figure by the scale factor.

Congruence: A finite composition of basic rigid motions—reflections, rotations,
translations—of the plane. Two figures in a plane are congruent if there is a congruence that maps one figure onto the other figure.

Similar: Two figures in the plane are similar if a similarity transformation exists, taking one figure to the other.

Similarity Transformation: A similarity transformation, or similarity, is a composition of a finite number of basic rigid motions or dilations. The scale factor of a similarity transformation is the product of the scale factors of the dilations in the composition; if there are no dilations in the composition, the scale factor is defined to be 1.

Similarity: A similarity is an example of a transformation.
This is not an isolated case. Both the standards themselves and the materials associated with them are filled with obtuse, overly precise language often used in an alarmingly imprecise way.

No comments:

Post a Comment