Much of my rationale for believing that the way forward in math ed is to be found in the DMZ of the war comes from my recognition that on both sides you find significant numbers of smart, well-educated, well-meaning people, who genuinely care about mathematics education. Unless you are of the simplistic George W. mindset, that the world divides cleanly into righteous individuals on the one side and "evil doers" on the other, it follows, surely, that both sides have something valuable to say. ("Evil doers" always stuck me as a strange phrase to hear uttered in public by a grown-up, by the way. It sounds more like the box-copy description of a band of orcs in a fantasy video game.) The challenge, then, is to reconcile the two views.
In brief, the gist of my Mercury News opinion piece was this. While it sounds reasonable to suggest that understanding mathematical concepts should precede (or go along hand-in-hand with) the learning of procedural skills (such as adding fractions or solving equations), this may be (in practical terms, given the time available) impossible. The human brain evolved into its present state long before mathematics came onto the scene, and did so primarily to negotiate and survive in the physical world. As a consequence, our brains do not find it easy to understand mathematical concepts, which are completely abstract. (This is part of the theme I pursued in my book The Math Gene, published in 2000.)
However, although we are not "natural born mathematicians," we do have three remarkable abilities that, taken together, provide the key to learning math. One is our language ability - our capacity to use symbols to represent things and to manipulate those symbols independently of what they represent. The second is our ability to ascribe meaning to our experiences - to make sense of the world, if you like. And the third is our capacity to learn new skills.
When we learn a new skill, initially we simply follow the rules in a mechanical fashion. Then, with practice, we gradually become better, and as our performance improves, our understanding grows. Think, for example, of the progression involved in learning to play chess, to play tennis, to ski, to drive a car, to play a musical instrument, to play a video game, etc. We start by following rules in a fairly mechanical fashion. Then, after a while, we are able to follow the rules proficiently. Then, some time later, we are able to apply the rules automatically and fluently. And eventually we achieve mastery and understanding. The same progression works for mathematics, only in this case, as mathematics is constructed and carried out using our language capacity, the initial rule-following stuff is primarily cognitive-linguistic.
Of course, there is plenty of evidence to show that mastery of skills without understanding is shallow, brittle, and subject to rapid decay. Understanding mathematical concepts is crucially important to mastering math. The question is: What does it take to achieve the necessary conceptual understanding, and when can it be acquired? Certainly my own experience is that conceptual understanding in mathematics comes only after a considerable amount of procedural practice (much of which therefore is of necessity carried out without understanding). How many of us professional mathematicians aced our high school or college calculus exams but only understood what a derivative is after we had our Ph.D.s and found ourselves teaching the stuff?
In fact, I can't imagine how one could possibly understand what calculus is and how and why it works without first using its rules and methods to solve a lot of problems. Likewise for most other areas of mathematics. In fact, the only parts of mathematics that I find sufficiently close to the physical and social world our brains developed to handle that there are innate meanings we could tap into, are positive integer addition and subtraction for fairly small numbers, and perhaps also some fairly simple cases of division for small positive integers.
Interestingly, those were the only examples cited by the readers of my Mercury News article who argued against my suggestion that understanding comes only after a lot of procedural practice. Now it may be that in those particular areas, understanding can precede, or accompany the acquisition of, procedural mastery. Personally I doubt it, but I have yet to see convincing evidence either way. But, leaving those special (albeit important) cases aside, what about the rest of mathematics? Here I see no uncertainty. Understanding can come only after procedural mastery.
For example, physics and engineering faculty at universities continually stress that what they want their incoming students to have above all is procedural mastery of mathematics as a language - it is, after all, the language of science, as Galileo observed - and the ability to use various mathematical tools and methods to solve problems that arise in physics and engineering. Since even first-year physics and engineering involve use of tools such as partial differential equations, there is no hope that incoming students can have conceptual understanding of those tools and methods. But by a remarkable feature of the human brain, we can achieve procedural mastery without understanding. All it takes is practice. One of the great achievements of mathematics over the past few centuries has been the reduction of conceptually difficult issues to collections of rule-based symbolic procedures (such as calculus).
Thus, one of the things that high school mathematics education should definitely produce is the ability to learn and be able to apply rule-based symbolic processes without understanding them. Without that ability, progress into the sciences and engineering is at the very least severely hampered, and for many people may be cut off. (This, by the way, is the only rationale I can think of for teaching calculus in high school. Calculus is a supreme example of a set of rule-based procedures that can be mastered and applied without any hope of anything but the most superficial understanding until relatively late in the game. Basic probability theory and statistics are clearly far more relevant to everyday life in terms of content.)
Is mastery of rule-based symbolic procedures the only goal of school mathematics education? Of course not. The reason I am not focusing on conceptual issues is that much has been written on that issue - of particular note the National Research Council's excellent volume Adding it Up: Helping Children Learn Mathematics, published by the National Academy Press in 2001 (a book I have read from cover to cover on three separate occasions). My intention here is to shine as bright a light as possible on a mathematical skill that I think has, in recent years, been overlooked - and on occasion actively derided - by some in the math ed community. Life in today's society requires that we acquire many skills without associated understanding - driving a car, operating a computer, using a VCR, etc. Becoming a better driver, computer user, etc. often requires understanding the technology (and perhaps also the science behind it). But from society's perspective (and in many cases the perspective of the individual), the most important thing is the initial mastery of use. If something has been so well designed or developed that proficient use can be acquired without conceptual understanding, then the rapid acquisition of that skillful use is often the most efficient - and sometimes the only - way for an individual to move ahead. I think this is definitely the case with mathematics. I believe we owe it to our students to prepare them well for life in the highly technological world they will live in. In the case of mathematics, that means that one ability we should equip them with (not the only one by any means - Adding It Up lists several others, for instance) is being able to learn and apply rule-based symbolic processes without understanding them. That does not mean we should not provide explanations. Indeed, as a matter of intellectual courtesy, I think we should. But we need to acknowledge, both to ourselves and our students, that understanding can come only later, as an emergent consequence of use. (No shame in that. It took 300 years from Newton's invention of calculus to a properly worked out conceptual basis for its rules and methods.)