Amplifying our Op Ed piece and some personal experiences

A continued conversation based on the New York Times article: How to Fix Our Math Education, By SOL GARFUNKEL and DAVID MUMFORD

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DavidMumford
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Amplifying our Op Ed piece and some personal experiences

Postby DavidMumford » Sun Sep 25, 2011 5:02 pm

I want to amplify the Op Ed piece which Sol and I wrote, answer some of the criticisms and bring in some of my personal experiences which had no place there.

Many comments on our piece assumed we were advocating “dumbing down” math, going back to a discarded Voc Ed approach that presented rules without understanding, and throwing out the logic and critical thinking that traditional math courses seek to teach. This is absolutely not what we are thinking of. To put it simply, I believe students profit from knowing authentic examples of math in the real world and from learning a general “abstract” pattern in which multiple examples can be fit. This pattern may be an algebraic formula (like dist = speed × time) or it might be a geometric configuration (like similar triangles) or it might be an algorithm (like the rule for long division). At every level, increasingly sophisticated examples motivate more sophisticated and powerful mathematical patterns.

But I believe that, unless you are teaching math graduate students (and usually even then), you should always give multiple examples first and then explain the pattern. Except for a very tiny percentage of the population, this is how one learns. One reason this is not really obvious is that when a person has struggled to internalize some mathematical pattern, they forget how hard it had been to get there. So if they explain the idea to someone else, they tend to start with the general abstract situation, not with simple examples, thinking this cuts to the core. The abstraction has become the way they grasp that idea and the examples seem to be distractions. But if they remembered better, I’d bet that they needed concrete examples before they could build the conceptual mental structure embodying the abstract pattern.

Let me talk from personal experience. As a successful mathematician, people assume I must find it completely natural to grasp formal mathematical ideas which I haven’t seen before. There may be some who can do this (Grothendieck comes to mind) but not me. I could never have understood stochastic processes for example without studying Brownian motion first and, in fact, implementing a random walk on the computer. I could never have understood birational geometry of varieties without studying infinitely near points on the plane first and, in fact, making long lists of how various singular curves got resolved. I always get lost listening to a lecture on new research when the speaker introduces the subject matter by first laying out a string of definitions without any real examples. This way of lecturing to professional audiences has become the rule in the mathematical community and is another illustration of how basic pedagogical principles have been lost.

Those who wrote the CCSSM protest that they include applications in many places. But when you read it, you find that they have been added applications like window dressing, as supplements that help amplify the student’s understanding after a concept has been mastered. I would argue that instead each new concept has to be prepared for by laying extensive groundwork and that the real world uses of math must be placed front and center to engage students so that they have something to motivate difficult ideas like using algebraic formulas. As much time if not more must be devoted to applications as to the pure math.

I have another reason for believing that the present approach, especially to high school math instruction, is just not working. I have written a book (with two collaborators) and written and taught from extensive notes for a course, both of which were aimed at non-mathematicians. Both were intended for people who “could handle high school algebra with confidence” (as we said in the Preface to the book) but who were otherwise naïve in math. At the time I wrote these, I had the idea that this included most college graduates and that, building on this familiarity with algebra, I could explain in the simplest terms some quite exciting mathematical ideas. But to my dismay, a number of very intelligent friends stopped dead when they came to the first formula. They were having or had had fulfilling careers in the world while having absorbed absolutely nothing (but possibly fear) from their algebra classes. Ooph: time to rethink how to teach algebra especially.

To me the answer seems clear: base all K-12 math on authentic examples and draw generalizations and mathematical patterns from these only when students are prepared and motivated to learn them.

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