ONE POINT OF VIEW: The new formalismAuthor(s): HENRY VAN ENGENSource: The Arithmetic Teacher, Vol. 18, No. 2 (FEBRUARY 1971), pp. 69-70Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187612 .

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ONE POINT OF VIEW

The new formalism

by HENRY VAN ENGEN

Once upon a time arithmetic as taught to elementary school children was

really bad. It was highly formal. There was no discovery of ideas, and it con- sisted almost entirely of rigid rules for computation.

Then about fifteen years ago a rich uncle, called Uncle Sam, made huge sums available to rescue arithmetic and the elementary^ school child. The rescuers and those who attached themselves to the rescuers' school of thought set out to make arithmetic make sense. (Of course, much of the change made sense for the rescuers. They didn't ask whether it made sense for the chil- dren.) Among the first changes that had to be made, of course, was to distin- guish between a number and a numeral. Certainly, the "new age" child would consider it nonsense if told to "write the number 2 on the chalkboard." And geometry must follow; so we don't put a point on the paper but all those little black circles are dots - not points.

The commutative law (or principle) must be taught in the new mathema- tics. But since this can follow as a result of set theoretic considerations, the child must be taught that

'a9b,clKJ lx,y) = {x,y' W {atb,c' and that

n{iatb,c' 'J {*,>>}) -«{{*, y } U {a,b9c'U n{a9b,c) + n{x,y) - /;{*, y' + n[a, b, c),

3 + 2 =2 + 3.

All of this is obviously true - as simple a. breathing! (For whom?) Geometry, too, needs reform - not so much in the ideas taught as in the

symbolism. Most certainly if we are precise in the symbolism, then it follows that the child "sees" a better set of geometric ideas. Hence, we must be care- ful to define an angle as the union of two rays (BA U Ì3C = ¿ABC); and, of course, the child must think of the rays as a set of points. Moreover, the

February 1971 69

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70 The Arithmetic Teacher

new formalism will not let us write ¿ABC = 60°; we must write m ¿ABC = 60°. (And the latter is not acceptable to the real purist!)

A^

Some reformers, in their zeal to be different and mathematically correct, forgot to consider the effects of the formalism being introduced on the atti- tudes and learning of the child. They also forgot that in initial learning it might be harmful to be precise in language and that pointing and general descriptions are frequently desirable.

So sometime later than the "once upon a time" beginning, teachers began to realize that being formalistic about ideas and symbolism is just as bad as the old computational formalism of the "bad" arithmetic of years ago. Actu- ally, there is a middle road. Children can intuitively grasp some of the proper- ties of a circle, square, or cube without being able to give a good definition of any of them. At some stage of mathematical development they should be able to define a square, but when?

The reformers have been preoccupied with "good" mathematics as a mature individual sees mathematics. As a result, in some programs the mathematics is good; but the pedagogical foundations are impossible. Once a teacher focuses exclusively on mathematics, then creativity, imagination, and language fly out the window. For the elementary school, informality and intuition may be better curricular guides than mathematics as the mathematician sees it.

Scrambled mathematics

Perhaps your students would like to unscramble each of the following to form words that deal with mathematics. Each expression represents one word. (Answers are on page 87.)

1. LEND PIERR A CUP 2. IDIOT DAN 3. GREEN TALC 4. IT BANS COURT 5. DIAL REAL QUART 6. CC : MICE RUN FREE : 7. OPEN NEXT 8. I RATE MITCH 9. PRIME TREE

10. GRUNT ONCE 11. ALL PEARL 12. CAUTION MIT PILL 13. note: sit nicer 14. GET ME ROY

- Lawrence P. Gross, Bialik School, Brooklyn, New York

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