When I was young I played football (soccer). This was when tiki-taka football was all the rage. This tactic says: pass, pass, and pass, and then pass some more. Backwards, forwards, sideways, and back again––it doesn’t matter which way, just keep the ball in the team.
Our coach wanted to make us master passers so that we could play tiki-taka. So we spent hours and hours every week on one specific drill: playing football without goals. Without the “distraction” of trying to score goals, we could focus purely on passing and possession play. That was the idea.
In reality, what happened is that we got sick and tired of this boring drill that took all the fun and excitement out of the game and turned it into a pointless drudge. Instead of making us master passers it made us disinterested slackers who didn’t see any reason to put in our best effort.
Unfortunately much curriculum planning in mathematics is based on the same hare-brained logic. Again and again we see the same pattern: in order to do B you need A, but teaching A and B together in one course would be too much, so A is detached and drilled at length in a prerequisite course.
The problem is that B is the only reason anyone is interested in A in the first place, so now you are teaching an entire course on a topic A which serves no purpose whatsoever in and of itself. By severing A from B you are guaranteeing that your course has zero intrinsic motivation. You are not helping students by giving them “a good foundation in A” before moving on to B. Rather you are obliterating the meaning and purpose of mathematics and forcing your students to approach it as an empty chore.
A notable example is the plague of “intro to proofs” courses where the pedantry and mechanics of proof writing is detached from any context where these skills serve an actual purpose. In the same way we detach integration rules from their purpose, which is solving differential equations, and we detach rings and ideals from their purpose, which is number theory, and so on.
The fallacy is one of short-sighted, non-organic thinking. “I’m a teacher of B. I find that my students are lacking in A. Let’s solve my problem by having them do tons of A in a prerequisite course.” This is B-centered thinking that zooms in on one particular issue and wreaks havoc with the cohesion and integrity of the curriculum as a whole in order to “fix” it.
The A-course is rarely a success, because it’s hard to learn something well without knowing what it’s for, and it’s hard to stay motivated and excited when there is no purpose to what one is doing. But, unfortunately, the same blinders that led to this course in the first place also means that its proponents are blind to its failures. The disastrous outcomes only leads them to double-down on their short-sighted scheme. “Now they have a whole course on A and they still don’t get it! Obviously this proves how essential it is to drill and drill and drill A before going on to B.” Thus, as with medieval bloodletting, the failure of the treatment is taken as evidence that more of it is needed.
The naiveté is the same one that led to the myth that vitamin C cures and prevents the common cold. Compare: “Vitamin C is essential to the immune system. Therefore, taking tons of vitamin C pills will keep us from ever catching a cold.” “In higher mathematics it is essential to understand concepts and techniques relating to writing proofs. Therefore, making all our students cram these skills in a dedicated course detached from any content will have them flying through later courses without impediment.”
Just as a well-rounded diet gives us all the vitamins we need in a natural way, so also a well-rounded mathematics curriculum automatically incorporates any necessary material in an organic and natural manner. Unfortunately mathematical curriculum planners do not believe in teaching mathematics they way it grows naturally and organically. Instead they would rather play the role of a hubristic doctor in a sci-fi dystopia who thinks he can “improve” on nature by replacing all organic foods with artificial capsules.