> 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 taste. 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.
H^2 -- Howard L. Hansen Assistant Professor of Mathematics Western Illinois University Macomb, IL 61455 http://www.ECNet.Net/users/mfhlh/wiu/index.htm "Good mathematics is not how many answers you know, but how you behave when you don't know the answer."