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Topic: focus
Replies: 21   Last Post: Aug 1, 1996 12:07 PM

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Howard L. Hansen

Posts: 48
Registered: 12/6/04
Re: focus
Posted: Jul 29, 1996 2:23 PM
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Andre TOOM wrote:

> By Guidelines you probably mean the documents published by NCTM and
> often called `Standards'. I would rather call them misguidelines.
> Let us look at p.139 of ``Curriculum and Evaluation Standards'':
> ``Real-world problem situation. In a two-player game, one point is
> awarded at each toss of a fair coin. The player who first attains
> n points wins a pizza.Players A and B commence play: however, the
> game is interrupted at a point at which A and B have unequal scores.
> How should the pizza be divided fairly? (The intuitive division, that
> A should receive an amount in proportion to A's score divided by the
> sum of A's score and B's score, has been determined to be inequitable.)''
> Let us leave aside the mysterious phrase ``has been determined''.
> (This phrase alone tells any competent math educator that this is a hoax.)

> >From the text in the brackets we learn that the pizza should be
> divided equitably. But the most equitable division is half-and-half!
> What do the authors really want?
> Then they set a mathematical problem, which has nothing to do
> with equity, solve it and write:
> ``Validation in original real-world problem situation. Empirical evidence
> gained from actually playing out the game many times or, more easily,
> from computer simulation (using random numbers to represent coin tosses)
> confirms this solution.''
> Here the verb `confirms' is in the Present Indefinite tense,
> which is normally used in the English language to denote action
> which actually takes place in present. Does it mean that somebody
> actually played this game or simulated it using a computer when
> the book was written? I am sure that not. NOBODY PLAYED THIS
> GAME AND NOBODY SIMULATED IT. How do I know it? Because if
> somebody tried to do it, he or she would immediately notice
> that playing or simulation does not confirm anything, because
> there is nothing to confirm.
> So much about Guidelines, their authors, their concern about
> probability and solving problems and their competence
> in mathematics, education and English.

Contrived? Perhaps. But surely Professor Toom recognizes this classic
problem--The Problem of the Points--which by many accounts spawned the
modern era in probability. Here the Standards suggest using a geometric
approach to solving the problem, but certainly one could eventually make
the connection to Pascal's Triangle and extensions. I'm not sure I
would disguise the problem as was done here, but that's a matter of
I have played the game and simulated it many times on a computer. I do
not however recognize the term "Present Indefinite" tense. Perhaps
others will clarify.

Howard L. Hansen
Assistant Professor of Mathematics
Western Illinois University
Macomb, IL 61455
"Good mathematics is not how many answers you know,
but how you behave when you don't know the answer."

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