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 just ways to get the answer. They aren't the important stuff. The important stuff is understanding mathematical operations and relationships.
A stress on algorithms is the bane of the mathematics classroom. A classroom in which algorithms reign does more harm than good. It destroys the natural mathematical intuition and curiosity children bring to the classroom; it cultivates disinterest and dislike. It substitutes symbol pushing for mathematical understanding. Children don't need algorithms, they need models and images that convey mathematical operations and relationships; they need to have their mathematical intuition honored and developed. Then, perhaps with a helpful hint or two from the teacher, they will develop their own algorithms that make sense to them. Even if an algorithm has theoretical significance, the thought that students appreciate this is absurd. In every math classroom I've been in—from arithmetic to calculus—where the emphasis is on algorithms, all students want to know is how to use them; they don't want to be bothered with the theory. I didn't as a schoolboy and I doubt if any of my classmates did.
I spent 17 years as a professor of mathematics in a research university and at one time or another I have been a member of the American Mathematical Society and the Mathematics Association of America, as well as the National Council of Teachers of Mathematics. In my experience, the typical research mathematician or scientist, Nobel laureate notwithstanding, has about as much expertise concerning the teaching of arithmetic as a Pulitzer prize-winning novelist has about teaching reading. Arithmetic is part of the inherent nature of a university mathematician—they never give it any thought. They are part of that small segment of the American public who breezed their way through school mathematics without much conscious effort. They have little idea of what's involved, much less any experience, in helping fifth graders make sense of fractions. or teaching algebra to a bunch of recalcitrant high school students. I have found, however, that a lot of them are pretty good at grousing about the sorry state of school mathematics, especially in the coffee room after teaching a class that didn't go well because, according to them, the schools did such a lousy job of educating their students. (I remember telling a colleague of mine who was ranting about how his calculus class didn't remember some elementary trigonometric property that, if he were any kind of teacher at all, he could teach them what they didn't know in less time than it took him to gripe about it.)
For those educators who wonder how to react to the open letter, I suggest they ignore it and assess the new programs in light of what the real experts on the teaching and learning of school math have to say—those folks such as Stanilas Dehaene and Brian Butterworth who have studied math cognition extensively (see, e.g., Dehaene's Number Sense: How the Mind Creates Mathematics and Butterworth's What Counts: How Every Brain is Hardwired for Math).
As for those research mathematicians and scientists who don't like what's going on in school mathematics, I suggest they hold their voices unless they're willing to commit time and energy to doing those things that earn them the right to be critics—such things as reading and reflecting on the effective teaching and learning of elementary mathematics, visiting schools and trying their hand at teaching fifth graders, becoming involved in mathematics courses for prospective teachers, developing and testing pre-calculus curriculum, teaching algebra to adults who didn't get it the first time around. Otherwise, I suggest they quit heckling Riley and get back to their blackboards and test tubes.