Susan Addington wrote: <<stuff deleted>> > >My recent thinking on the subject of "covering the material" >(or uncovering it--thanks, Eileen) is that a spiral approach >is the best policy. I seem to remember a discussion on >the spiral approach on this list in the past year. >
While I think the spiral approach is an excellent idea, I recall the warning issued in _The Underachieving Curriculum_ (the Second International Mathematics Study Report, 1987) about the misapplication of the spiral curriculum in U.S. schools -- that the spiral curriculum had degenerated to a spiral of almost constant radius. The authors point to Bruner as setting forth the idea of spiraling but they go on to point out that the realities of schooling undermine this approach.
My interpretation of the problem documented in the report is the lack of articulation between teachers from year to year. It seems we can never be sure exactly what was covered (or uncovered) last year. In fact, our students guarantee that nothing was covered ("No, we didn't do that last year.") so we tend to re-teach a lot. I had the same high school teacher for 4 years and I think he was able to spiral because he could remind us that we saw a topic before -- most teachers don't have the luxury to meeting regularly with their colleagues and become aware, first-hand, of what is being taught in other classes.
The problem is even greater between high school and college. When I taught the one semester college precalculus class in college, I would sub-title it "Everything you already had in high school crammed into one semester." I don't think there is anything in a college precalculus (or college algebra) course that a student who earned 4 credits of college-prep mathematics in high school hasn't already seen once -- a spiral of constant radius.
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>Example: the subject of parametric functions (for example, describing >the position of an object as a function of time) is first introduced >in calculus. It comes up again in courses in differential equations, >and various flavors of geometry. Instead of yelling at the students, >"You were supposed to learn this in calculus", I give a fast >review, a problem or two, and explain how it fits into the current >subject.
This is a good example. In the Sept 93 issue of the _Mathematics Teacher_, Joe Cieply made a pretty decent argument for covering parametric equations in first-year algebra using graphing calculator technology. Again, the importance of articulation -- what would it benefit one group of students with Mr. Cieply as a teacher to see parametrics only to be mixed with other students in second year algebra (or third or college precalculus) and get a repeat of the same topic from scratch?