The following is from my husband David, industrial mathematician and professor at the Rochester Institute of Technology, as a follow-up to my post below on the Math Wars article as addressed recently by CNN. He was, needless to say, less than pleased with the way the article addressed how New Math is being taught in American schools. Here are his thoughts on the matter:Deb asked me to write a few comments on
CNN's recent article on the Math Wars.
A few years ago, I argued against the recent trends in mathematics education in an article called
The Math Wars, in which I present the thorough case against the approach promoted by the National Council of Teachers of Mathematics (NCTM). Please feel free to read that article, as I'm not going to rehearse the whole argument here.
I'll just summarize my thoughts on the
CNN piece, which offers a "black-and-white" summary of the Math Wars:
They call it the Math Wars: The debate, at times acrimonious, over which way is best to teach kids math. In its most black-and-white form, it pits schools hoping to prepare kids for a new world against reluctant parents who feel that the traditional way is best and that their kids are being shortchanged.
The kids
are being shortchanged. I say this as someone whose job it is to teach calculus to young adults who've just finished high school and are studying to become engineers or scientists, and as someone who has spent 20 years as an industrial scientist and an applied mathematician.
High school graduates these days do not know the math that they need to know. The reason that they don't know it is that
those who were responsible for teaching it to them didn't teach it to them.
The claim that schools that have done away with traditional mathematics instruction have done so because they want "to prepare kids for a new world" is ridiculous for many reasons. The most important is that the job of math educators is to get students to understand math, and the best way to understand math is to master the traditional algorithms.
That is: The best way to get students to understand math--
the best way to get them to grasp the conceptual structure of mathematics, the best way to get them to integrate the whole structure of modern mathematics at the abstract level--is to have them
learn, apply, master, and come to love precisely the traditional methods that the modern NCTM approach abhors. The position supported by Pat Cooney and Hank Kepner in the CNN article is not merely mistaken; it is diametrically opposed to the proper method for teaching students mathematical concepts.
My wife and are home schooling our daughters in large part to avoid their wasting time on this sort of nonsense.
The traditional approach has the wonderful, and useful, feature that students who have mastered the algorithms can actually do calculations. But its most important feature is that it
provides the tools that students need in order to get inside the abstract structure of mathematics. For a detailed argument for my claim here, please see
my Math Wars article. The CNN article discusses an admirable parent, Ms. Morey, who's teaching her son traditional math. She, bless her,
...feels no guilt. She says her son was relieved to learn long division. "He wants a quick and easy way to get the right answer,"
What this responsible parent probably doesn't know is that her son can not only divide numbers better than his teachers, he will probably grasp the nature of ratios and the operation of division and the structure of decimals--and, in the end, perhaps, how the solutions of certain differential equations reflect the quantitative facts of polymerization--because she taught him long division.
One problem, Cooney says in the CNN article, is that parents remember math as offering only one way to solve a problem.
"We're saying that there's more than one way," Cooney said. "The outcome will be the same, but how we get there will be different."
This what we call a
straw man argument. The traditional approach to mathematics never involved the claim, implicitly or explicitly, that there is only one way to solve a problem. Everyone understands that there are special ways to solve special problems, quirky methods of multiplying by nine using that finger trick, multiplying 199 by 7 quickly by doing 200x7 then subtracting 7, and on and on and on and on.
But, in a serious sense, there
is only one "way to solve a problem." That is, there is, e.g., exactly one universally applicable, perfectly reliable, optimally efficient, method for dividing one number by another: long division. And these qualities--universal applicability, reliability, efficiency--are precisely the qualities of good conceptual thought: They characterize what effective abstract thought is all about.
The author of the CNN article was provided with a typical sitting-duck example for the anti-traditional-algorithm approach:
Thus, when a parent is asked to multiply 88 by 5, we'll do it with pen and paper, multiplying 8 by 5 and carrying over the 4, etc. But a child today might reason that 5 is half of 10, and 88 times 10 is 880, so 88 times 5 is half of that, 440 - poof, no pen, no paper.
This, of course, is like the 199x7 example that I mentioned above. It's a special example. The propaganda for the NCTM approach is full of such examples. Let's amend it a smidgen:
Thus, when a parent is asked to multiply 838 by 37, we'll do it by multiplying 8 by 7, put down the 6 and carry the 5, and so on. But a child today might reason that...er...um...well...I'd better check with Ms. Cooney about this.
In the article, Pat Cooney is quoted as saying:
"The traditional way is really a shortcut,"..."We want kids to be so confident with numbers that it becomes intuitive."
Well, yes indeed. It's a shortcut. That's what conceptual thinking is all about. That's it! That's the reason that humans live the pleasant easy lives that we do and animals live, well, like animals:
We have the cognitive shortcuts of conceptual thought, and they don't. If education isn't about shortcuts, that is, about cognitive efficiency, about effective abstraction, about methods of organizing our awareness of the world into manageable units, what is it about?
Is it about, as Ms. Cooney suggests...intuition!? Please note carefully what Ms. Cooney is saying. The traditional method, because it's a shortcut--by which she seems to mean that it's a reliable, efficient, universally applicable method for solving a large class of problems--is bad. Intuition, whatever that is, is better.
In fairness, I want to recognize that math folks often use the term "intuitive" to refer to methods and facts that have become thoroughly automatic; effortless. And, this automatic quality is an important feature of conceptual thought. But such automatic, effortless, awareness can only follow from thorough mastery of the traditional methods.
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Please feel free to send questions or comments to me at
dsrsma@rit.edu.
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David S. Ross, PhDProfessor, School of Mathematical Science
Rochester Institute of Technology
