02.16.08
In defense of fractions
A couple of people pointed out this article in USA Today about a mathematics professor who thinks fractions need not be taught in schools.
I know that I work for one of the most famous organizations in the world, and I’m well aware that the things I write on my blog could be misconstrued as coming from my employer or representing positions held by them. So I keep things pretty close to the vest. But I hope I’m not being too controversial when I say that I like fractions and hope they continue to be taught. I’m not quite sure how algebra can be done without fractions (what becomes of rational functions if you can’t abstractly divide polynomials?), and without the skills to algebraically manipulate expressions, calculus becomes very hard to do as well.
Prof. DeTurck compares fractions to roman numerals, which of course were abandoned once the arabic system became more widespread. The arabic system had place value, that is, the idea that the position of a digit within a number changed the value of the number. They also had one of the most important numbers: zero. With these ideas, finitely many symbols can be used to express infinitely many numbers.
The quickest to move to the arabic system from roman numerals were the accountants, and indeed, a big advantage of arabic numerals is the ease of computation. The same could be said of decimals versus factions: it’s usually easier to add decimals. Alexander Hamilton wanted a base ten system for the currency of the United States, rejecting the 12 schillings per pound, 20 pence per schilling system long before the UK themselves did. Computers are built for floating-point arithmetic rather than adding fractions, and so most stock exchanges list decimal prices for commodities now.
But fractions are for more than just arithmetic. And decimals are only useful ways to express numbers when the numbers themselves are expressed as points on a line. But there are other ways that numbers are used. If I show you a pie and offer you 0.25 of it, how would you cut the pie? If you said “I’d cut it in half, then cut one of the halves in half”, then you converted the 0.25 to 1/4 and used the fact that (1/2)(1/2) = 1/4. But if numbers are only lengths, I think the only way to do this is to cut a circular piece out of the center of the pie whose radius is 0.5 of the radius of the whole pie.
And don’t forget that many fractions aren’t expressible exactly in decimals. To get a sixth of the pie (a much more modest slice), you’d have to cut out a circle whose radius is the square root of 0.166666… which is about 0.408248.
So I say, vive les fractions. And mmm…pie….
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