The metaphor of the assembly line seems to me to express best, and for good reason, a very troublesome problem facing education today. For economic reasons, we have had no choice but to use strategies of mass production, including the notion of interchangeable parts: Interchangeable faculty, interchangeable courses, interchangeable curricula, interchangeable textbooks, interchangeable students. The use of standardized tests reinforces the metaphor; they are the measuring engines that enforce interchangeability. We--faculty, students, the general public alike--perceive educational institutions as nothing more and nothing less than monstrous assembly lines. That's why the metaphor expresses the problem so well: Metaphor has become reality. The metaphor is the problem.
We have adopted the metaphor whole-heartedly. I think that many students (and faculty) believe that what one is doing at college or university is being processed. Students often seem to presume that simply going to college guarantees one the degree one seeks, provided only that one serve one's time. By the same token, many students seem to have the notion that merely having taken the prerequisite course should automatically qualify them for the current course. Some of them believe it to the extent that they think they are qualified for the current course even when they earned an "F" in the prerequisite course--I took it, didn't I?
I submit that herein lie the reasons underlying the need for calculus reform (as well as the full-fledged curriculum reform that real calculus reform will engender). Our mathematics curriculum is moribund because we have not paid attention to the dangers that attend application of techniques of mass production to education of human beings. We have tried to reduce human beings to hardware capable of retaining processed characteristics indefinitely while awaiting further processing. We may have to use techniques of mass production; let us use them humanely, remembering that neither we nor our students are interchangeable parts. A re-formed curriculum, no matter how lean and lively, that ignores this issue will be no reform at all.
I suggest that we should rely less on the results of standardized testing (i.e., how closely the student matches the criteria for the part that fits here in the machine) and on the number and nature of credit hours a student has accumulated (i.e., the amount of processing that a student has endured) when we place that student. We must find ways to determine more accurately what courses students are prepared to profit from, and we should place students into our courses on these new bases.
But the metaphor fails here, and we must be wary: I do not mean that we should invent better standardized tests in order to be sure that the part fits into its intended place, for there can be no guarantees that students--especially students who see themselves as passive receivers of processing--will have the capabilities tomorrow that they had yesterday or even today. We must continually assess how students fare in their courses, and we should be willing to advise them to review prerequisite work when they aren't prepared for current work. Sometimes we will have to advise students to re-take courses in spite of the good grades they got the last time around. And we must make that advice stick.
Finally, I would like to suggest a metaphor I think likely to serve us better than does that of the assembly line. We can conceive of mathematics as a poorly explored mountain range. We are experienced mountaineers. As researchers, we are explorers entering the unknown heart of the range. As teachers, we are mountain guides through more or less well explored parts of the range. Students come to the mountains seeking many things: recreation, natural resources (including beauty), to become explorers themselves. They must develop not only knowledge of the terrain itself, but the skills necessary for safe travel in mountainous country. Ultimately, it is the latter which are the more important--not the former.
I can't emphasize this point enough: The skills are much more important than familiarity with any particular route or piece of terrain. I should say that in our entire curriculum, as it stands today, we concentrate the lion's share of our efforts on describing trails that lead from one point to another, instead of on inculcating the skills for traveling. We must find ways to free our curriculum from its dependence on fixed course content so that we can attend to more important matters.
I think we can learn much from this metaphor and its ramifications. For
example, one would not ask a tenderfoot to lead even an easy technical
climb, and this suggests that one should not ask a freshman to invent an
epsilon-delta proof. Further, an experienced mountain guide does not
expect his charges to know the territory--even given that they've been over
it before--and this observation begins to address our problems with
prerequisites. I will leave it to the reader to extend the metaphor