Wu and I

Allison pointed me to some notes from a presentation by Hung-Hsi Wu at NCTM's 2007 meeting, and I've been meaning to write about them.

I gave a presentation at that conference, so during Wu's talk I was, if I remember correctly, holed up in my hotel room frantically reviewing my notes—a shame, considering that what appears to be the central point of Wu's presentation was, with fewer differences than similarities, the idea that I tried to flesh out for textbooks when I first started writing this blog three years ago. (A further shame is that many of my posts which directly addressed that idea are gone forever.)

Wu's idea below, minus the brilliant parallel to engineering, is essentially what I have referred to as the Boundary Principle (emphasis in original):

Engineering cannot cater to any human need no mater [sic] how scientifically absurd, any more than it should produce any product that is useless though scientifically correct.

Engineering must mediate between two extremes:

(1) inviolable scientific principles.
(2) user-friendliness of the final product.

Mathematics education, as Wu explains, is constrained by the same impossibility on the one hand and impracticality on the other. That is, it is not possible for mathematics education to serve every student desire, nor practical for it to attempt to do so. Rather, mathematics education (what Wu refers to as "mathematical engineering") must strike a principled compromise between its often conflicting fealties to both mathematics students and to mathematics itself.

Wu goes on to list five "inviolable scientific principles" for mathematical engineering, four of which fit reasonably well under my three for textbooks—accuracy, coherence, and language:

Precision: Mathematical statements are clear and unambiguous. At any moment, it is clear what is known and what is not known.

Definitions: Bedrock of mathematical structure (no definitions, no mathematics).

Reasoning: Lifeblood of mathematics; core of problem solving.

Coherence: Every concept and skill builds on previous knowledge and is part of an unfolding story.

Purposefulness: Mathematics is goal-oriented. It solves specific problems.

I would disagree that the last of these rises to the importance of being a principle, and Wu's contention that the damaging split (visit the links in Allison's post [linked above] for more details) is between mathematicians on the one hand and mathematics educators on the other is, I think, a bit simplistic. Nevertheless, I was heartened to see so many similarities between Wu's ideas and my own.

And here's just a fantastic section from Wu's article "How mathematicians can contribute to K-12 mathematics education" (again, see Allison), which, to my mind, connects with some ideas here, here, here, definitely here, and certainly here and here:

At present, what most children get from their classroom instruction on fractions is a fragmented picture of a fraction with all these different "personalities" lurking around and coming forward seemingly randomly. What a large part of this research does is to address this fragmentation by emphasizing the cognitive connections between these "personalities". It does so by helping children construct their intuitive knowledge of the different "personalities" of a fraction through the use of problems, hands-on activities, and contextual presentations.

This is a good first step, and yet, if we think through students' mathematical needs beyond grade 7, then we may come to the conclusion that establishing cognitive connections does not go far enough. What students need is an unambiguous definition of a fraction which tells them what a fraction really is. They also need to be exposed to direct, mathematical, connections between this definition and the other "personalities" of a fraction. They have to learn that mathematics is simple and understandable, in the sense that if they can hold onto one clear meaning of a fraction and can reason for themselves, then they can learn all about fractions without ever being surprised by any of these other "personalities".

Update: I promised to post this great quote from Allison on the Wu presentation/paper. It's a fantastic quote, and it's like a bottom line for people who read this post (or read this blog) and say, "Harumphh, he's so abstract":

The point is to simplify without lying, misleading, or otherwise undermining future growth, so that the presentation true for the 4th grader is still true for the 12th, but the 12th can handle more, and can see the connections to the stuff already in his mind. Mastery is then possible.

Update II: One reason I like Allison's statement so much--aside from the fact that it's pretty much spot on in interpretation--is that she uses the word lying. Exactly. I can go into detail later, but that's exactly what a lot of mathematics education is today--straight-up lying to kids' faces.

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