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Andre TOOM

Posts: 549
Registered: 12/3/04

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

Plain Text

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 math-teach 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. 210-646-0500 (h)
4301 Broadway 210-829-3170 (o)
San Antonio, Texas 78209-6318 Fax 210-829-3153