The Life of Riley
By Dr. Eugene Maier |
The life of Riley is not without its bumps and bruises, especially if you are United States Secretary of Education Richard Riley. The latest missile headed Riley's way is an "open letter," made public through a paid advertisement in the Washington Post. The letter, signed by six mathematicians and endorsed by an additional 201 mathematicians and scientists, including four Nobel laureates, urges the United States Government to cease its promotion of 10 school mathematics programs developed with the support of the National Science foundation.
If you haven't seen the letter, it's reproduced at the end of this article. The stance taken by the signers is exemplified by a couple of excerpts. One bemoaning the "astonishing but true" fact that "the standard multiplication algorithm for numbers is not explained" in one of the programs. The other "that the standard algorithms of arithmetic are more than just 'ways to get the answer'--that is, they have theoretical and practical significance. For one thing," the statement continues, "all the algorithms of arithmetic are preparatory for algebra...."
I'm astonished that a mathematician would talk of the "standard" multiplication algorithm, as if one existed or even ought to exist. I suspect what the "standard algorithm" really means is "the algorithm I learned in school." (I remember when I accidentally discovered, early in my teaching career, that what I thought was the standard algorithm for subtraction was totally unknown to my students--I was taught an algorithm that was based on adding equal quantities to the subtrahend and minuend; they were all taught to borrow. And it wasn't until then that I thought about the theoretical basis for the algorithm I had been using for years. It's a rare student that pays any attention to the theoretical base of an algorithm they've been taught--I certainly never did.) There are lots of ways to carry out a multiplication and one of the problems for adults who have been taught traditionally is that they confuse the algorithm they learned with the process. One of the consequences is that most adults are terrible at mental arithmetic, which is the most efficient way to compute in that it doesn't require any external tools. Ask an adult to multiply 25 x 36 in their head and they are likely to try to recreate mentally the "standard" paper-and-pencil algorithm they have been taught, rather than recognizing say, that 25 x 36 is the same as 25 x 4 x 9, or 100 x 9.
It also astonishes me that a mathematician would imply that the "standard" way to carry out a multidigit multiplication these days is using some paper-and-pencil algorithm. From what I observe of the world outside of school these days is that the "standard" way to carry out such a calculation is with a calculator. I find it ironic that some 18 years ago Richard Anderson, then president of the Mathematical Association of American and professor of mathematics at Louisiana State University, stated that "calculators--fast, efficient, and nearly omnipresent" will eliminate the need for students to do laborious paper-and-pencil calculations, yet today it is members of his community that are resisting such a change. But then, Andersen foresaw such a resistance. "The arithmetic that people have studied tends to become the arithmetic they're attached to,'' he said, "if it was good enough for them, it's good enough for everyone."
All this emphasis on algorithms--in statements like those quoted above--I simply don't get. I sit here wondering of what great theoretical significance is a paper-and-pencil algorithm for long division--in what way is it preparatory for algebra? I can't think of a single arithmetical algorithm I know that I couldn't get along without. There's always another way to do it. Algorithms indeed are jus