Math Monday II: Common Sense From NPR's "Math Guy"
Stanford mathematics professor Keith Devlin (who is also NPR's "Math Guy") has some thoughts about the new emphasis on mathematics instruction in the public schools:
Saying the country needs more math teachers, as President Bush did in his recent State of the Union Address, is one thing. Deciding exactly what that entails is another. Although front-line reports from the so-called "math wars'' have largely disappeared from the nation's newspapers of late, there is still a significant disagreement among the experts as to exactly what it means -- or should mean -- to teach mathematics.I remain puzzled as to why our school systems continue to be advance kids to higher levels of math before acheiving mastery of basic concepts.
Broadly speaking (as with any generalization, there are exceptions), professional mathematicians, scientists and engineers, want the schools -- the pipeline that keeps those professions supplied with new personnel -- to ensure student mastery of numerical, algebraic and computational skills. "We don't want to spend our time having to reteach the incoming students how to add fractions!'' is a common refrain heard in university science and engineering departments.
Basic skills are not all they want, but they don't want them left out or de-emphasized.
Ranged against them (again, broadly speaking) is the mathematics education community, which argues that a focus on procedural skills is misplaced, and that the primary aim of school mathematics education should be to produce conceptual understanding. "It students understand the concepts, they can pick up any skills they need easily enough, as and when they need them.''
In my experience, when two groups of well-meaning, intelligent people seem to be in direct opposition, the "truth'' is most likely to be found somewhere in the middle.
While it sounds reasonable to suggest, as the math-ed community does, that understanding mathematical concepts should precede -- or at least go along hand-in-hand with -- the learning of procedural skills (such as adding fractions or solving equations) -- there is evidence to suggest that this is simply not possible. 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. Our brain does not find it easy to understand mathematical concepts, which are completely abstract. (I outline some of the evidence for this in my book "The Math Gene,'' published in 2000. And no, there is no "math gene''; the book's title is a metaphor.)
Ask experts at any activity what it took them to acquire their expertise and they'll tell you in one word: practice. Expertise does not come from understanding, it comes from practice. The part of our brain that provides conscious understanding did not evolve to control and direct our detailed actions, it developed to make sense of them -- after the fact. (The benefit of that sense making is that we can make use of our understanding to guide future action at a higher, more strategic level.)
We are not "natural-born mathematicians,'' but we are well equipped to learn new skills. 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. Anyone who has learned to play chess, play tennis, ski, drive a car, play a musical instrument, play a video game, etc. has experienced this progression from "following rules,'' through proficiency, on to eventual mastery and understanding. Mathematics is no different.
As a professional mathematician, I often have to learn a new part of my subject. Every time I have to go through the same process: Start by learning the rules, then practice using the rules, and keep practicing until understanding develops.
Practically every professional mathematician, scientist, or engineer I have spoken to has said more or less the same. Understanding follows experience.
Where the mathematics education community is absolutely right, however, is in saying that mastery of skills without understanding is shallow, brittle and subject to rapid decay. There is plenty of evidence to show that mastery of skills without (eventual) understanding is worthless. (At least it's worthless in terms of acquiring a skill that can be used. There may be some value in preparing the brain for future learning, just as doing push-ups makes the body better able to ski.) Understanding mathematical concepts is crucially important to mastering math. But because of the way the human brain works, that understanding can arise only as a consequence of practice -- lots of it.
And anyone who thinks that today's children don't have what it takes to practice for hours until they really "get it'' has not watched them playing a video game.
Update:(2/21) Writing over at Right on the Left Coast, Darren takes issue with a recent Washington Post piece by Richerd Cohen which states that Algebra isn't important.