Almost 25 years ago, my friend and colleague Phil Lewis wrote an article for Kaleidoscopes called (if memory serves) “Subject to Gates and Bars.” If I were better organized, I would be able to find that article; Phil always describes me as mentally organized but physically disorganized, and it’s physical organization that’s needed in order to find a hard-copy back-issue of Kaleidoscopes. So I’ll have to go from memory here…
What brought Phil’s article back to mind was a recent discussion at Weston about students in precalculus whose algebraic skills are not up to the job. Our current curriculum rightly has a major emphasis on big ideas — on concepts rather than manipulations. This isn’t Tom Lehrer’s new math, where he said in his famous line, “The important thing is to understand what you’re doing rather than to get the right answer.” We do pay attention to right answers, and to the mathematical skills that are needed to get those answers.
Nevertheless, we subordinate skills to concepts, as we should. The problem lies in Lehrer’s italicized “rather than”: the line is amusing because we all know that it’s essential to understand what you’re doing and to get the right answer. This, of course, is what’s wrong with the so-called “Math Wars,” in which skills and concepts are portrayed as being opponents rather than partners. But what do we do with students whose understanding of concepts is sufficient for them to earn B’s but whose command of skills is insufficient to thrive in the next course? Having separate grades for skills and concepts would be an administrative and pedagogical nightmare, so don’t even bother suggesting that route. A more productive recent suggestion has been the use of placement tests: no matter what his or her grades might be, a student should not be able to move into precalculus without demonstrating a sufficient command of algebraic skills. (This proposal wouldn’t have to be limited to precalculus, of course, but that’s where the problem crops up most severely.)
What would be the effect of creating additional bars in the way of a student’s mathematical progress? An optimist might say that a placement test would actually be an additional gate to pass through, not a bar to further progress. The optimist would claim that a student who fails to demonstrate sufficient competence in algebraic skills would have the incentive to learn what s/he doesn’t know, would pass the test on a second or third try, and would then be qualified for precalculus.
Sounds a bit too much like No Child Left Behind and MCAS for my taste. Would it really work? I don’t believe the rosy scenario.
Admittedly, I’m usually an optimist, but in this case I see much more of a downside than an upside to the idea of putting additional bars in the way of students who want to take precalculus. The principal negative effect would be to widen the achievement gap at a time when we want more students of color to be taking calculus (and therefore precalculus before that). In a series of workshops last year, we were exhorted to find ways to narrow the gap — to get more black students, more Hispanics, and more girls into advanced placement math courses. I don’t see how we’re going to do that if we put more barriers in their way.
I’ve also heard a variant of the placement-test proposal: require a score above a certain cut-off in order to receive a grade of A in Algebra II (or other feeder course, if applied elsewhere than precalculus). In that case the test would not be a bar, since it would not be a ticket of admission to precalculus; it would merely be a gate that a student could pass through with eyes wide open, knowing that s/he does not yet have the recommended skills for precalculus. But I suspect that this variant might be the worst of both worlds: it would still discourage certain students, thereby widening the achievement gap; and it would let others who aren’t truly prepared for precalculus pass through anyway.
More thoughts to come…