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Newton etc.
Posted:
May 28, 1997 6:12 PM


Lately, visiting Rutgers University, I read the following book:
Sir Isaac Newton Universal arithmetics: or, A treatise of arithmetical composition 2d edition, very much corrected London, 1728
First, it was pleasant for me to see that Newton attached much importance to translation from English to Algebra. On p. 68 he formulates a problem:
A certain Merchant encreases his Estate yearly by a third Part, abating 100 l., which he spends in his Family; and after three Years he finds his Estate doubled. Query, What was he worth?
Then, on p.69, he explains how to translate it into algebra:
In English  Algebraically  A Merchant has an estate...  X  Out of which the first Year  X  100 he expemds 100 l.   And augments the rest  X  100 + (X  100)/3 or by one third  (4X  400)/3
and so on. There are several tables in his book, like this.
But my main point is the following.
On p. 75 Newton formulates what he calls Problem VII:
The Forces of several Agents being given, to determine X the Time, wherein they will jointly perform a given Effect d.
Then he solves it, denoting the Forces A, B, C and assuming that in the Times e, f, g they produce the Effects a, b, c respectively. Then the Time in question is
d X =  a b c  +  +  e f g
Then Newton gives what he calls an `example':
Three Workmen can do a Piece of Work in certain Times, viz. A once in 3 Weeks, B thrice in 8 Weeks, and C five times in 12 Weeks. It is desired to know in what Time they can finish it jointly? Here there are the Forces of the Agents A, B, C, which in the Times 3, 8, 12 can produce the Effects 1, 3, 5 respectively, and the Time is sought wherein they can do one Effect. Wherefore, for q, b, c; d; e, f, g write 1, 3, 5, 1, 3, 8, 12, and there will arize 1 8 X = , or  of a Week, that is, 1 3 5 9  +  +  3 8 12
[allowing 6 working Days to a Week, and 12 Hours to each Day] 5 Days and 4 Hours, the Time wherein they will jointly finish it.
Polya cites this `example' in his book `Mathematicsl Discovery', but calles it a `problem'. He does not notice or care that for a problem the numbers are rather cumbersome.
Also pay attention that what Newton calls `problem' is a `type' in the language of modern educators, to which they propose to decrease attention. Zalman Usiskin calles problems of this type phony and Morris Kline calles them hopelessly artificial.
Another observation. I mentioned that Newton included some fractional expressions into his textbook, e.g.
d X =  a b c  +  +  e f g
and
1 X =  1 3 5  +  +  3 8 12
That silly Newton did not realize that he was sending a wrong message to his students. Now we know that
``Teaching fractional expressions gives the inaccurate picture that there is harder manipulation to follow in later mathematics courses.''
This remarkable piece of modern educational wisdom is printed in large type on p. 160 of n. 2 of the jubilee volume 88 of `Mathematics Teacher' (February 1995).
Now I understand better why my American students are so much weaker that their foreign classmates in managing fractional expressions. In one of my classes a Chinese student evaluated that 1/(1/2  1/3) = 6 and others were shocked. When they needed to evaluate such an expression they needed calculators to turn 1/2 into 0.5, turn 1/3 into 0.33, then subtract, then divide and get something like 5.88 or worse. At the mathteach list there was a discussion whether common fractions should be taught. Ralph Raimi tried very patiently to explain why they should.
Andre Toom Department of Mathematics toom@universe.iwctx.edu University of the Incarnate Word Tel. 2106460500 (h) 4301 Broadway 2108293170 (o) San Antonio, Texas 782096318 Fax 2108293153



