It seems I wasn’t the only one who read about OECD’s report. Others have been reporting on it, and comments have been coming in. Of interest was one in which the commenter interpreted “applied math” to mean “rote memorization”. How he made that stretch is beyond me, but in his view, it meant telling students the procedures without the reason why, and thus the students become “math zombies” to use a term invoked by some–able to “do” math but not “know” math, and lacking the flexibility to apply concepts and procedures to problems outside of those whose solutions they have apparently memorized. He goes on to talk about how he links proportion (no cross multiplication allowed in his classroom), ratios, to direct variation, and then finally to the concept of slope. And of course he focuses on point-slope rather than slope-intercept.
Everything in its time and order, I say. I like point- slope too, and really, the concept of slope is not that hard. But unlike others who insist that students must KNOW and UNDERSTAND why a linear equation produces a straight line, and why a straight line has a constant slope, I believe that you start simple, start small, and address more complex ideas later when students are ready and possess what used to be called “mathematical maturity”. I was introduced to slope in algebra, and got the proof of constant slope and straight lines in geometry.
Even in calculus, first year students learn an intuitive approach to limits and then proceed to the application of limits–learning the power and usefulness of derivatives and integrals. In upper level courses, if they stick, students learn the formal definitions and proofs of limits and continuity, after having a sufficient basis of what their use is and having gained mathematical maturity.
But the nay-sayers to OECD’s report are interpreting it in the old refrain: Applied math is rote memorization and pure math is “math with understanding”. So of course it only makes sense that OECD would come up with the conclusion it did. And a right good one it was they say, drawing in the circle of wagons tighter and doing a chorus line kick and recitation of Kumbya.
The issue of “rote understanding” doesn’t surface in their arguments, and somehow they sidestep the artificiality and limited usefulness of “real world” problems as well as “open-ended” and “ill posed” problems. The value of initially worked examples, and scaffolded problems that escalate in difficulty is ignored. Apparently the new watchword is “get the students to ask the questions that make the concept relevant”. Which is an aberration of what Dan WIllingham meant when he introduced this idea, and is nothing more than the same old constructivist approach that hasn’t been working for the past two and a half decades.