Saturday, July 20, 2013

Making math fun

Plenty of the math exercises that are necessary for math mastery aren't much fun for anyone: namely, all those repetitive calculations that students go through in order to gain fluency in basic arithmetic. But once you've attained this, fun starts to emerge. Who (among those who have truly mastered basic arithmetic) hasn't enjoyed simplifying a messy sequence of fractions, reducing numerators and denominators and canceling things out? Who (among those who have truly mastered arithmetic and the beginnings of algebra) hasn't enjoyed simplifying a messy algebraic expression, combining like terms and grouping terms to fit abstract patterns like the difference of squares?

Some math problems, indeed, are appealing in exactly the same way the popular puzzles are. Consider, for example, KENKEN:


KENKEN, as the official directions explain, involves "filling in the grid with no repeats in any row or column using numbers from 1 to grid size" such that "the number in each heavily outlined set of squares must combine to produce the target number in the top corner using the mathematical operation indicated." For example, in this one, in row 1, you need to find two numbers between 1 and 5 that produce 2 when one is subtracted from the other and two other numbers between 1 and 5 that produce 2 when one is divided into the other, and then pick yet another number between 1 and 5 that will be one of three addends that sum to 9, etc., all the while ensuring that there are no repeats in any row or column.

This satisfaction of multiple operational constraints is just like what's involved in factoring a polynomial expression. Consider, say, 9x2 – 6x – 8: here you must find four numbers such that the product of two of them is 9, the product of the other two is eight, and when the product of one of the first pair and one of the second pair is added to the product of the other of the first pair and the other of the second pair the result is 8.

Then there are algebraic word problems, some of which are just like the verbal logic puzzles that plenty of people choose to do for fun. Here's one from the beginning of Wentworth's New School Algebra, published in 1898:
A man’s is four times as old as his won; in 20 years he will be only twice as old. Find the age of each.
Many Singapore Math bar modeling problems have a similar appeal, and they're similarly fun.

It seems to me that the best way to make math fun is not to have students dally around doing "playground math," but to help them master the rote aspects of arithmetic as quickly and efficiently as possible so they can move on to stuff like this. 

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