Let’s look at a simple optimization problem. Bear with me, because the point is not the problem itself, but in what we have to know and do in order to solve it. Here we go:
Suppose you have a string 12 cm long. You form it into the shape of a rectangle. What shape gives you the maximum area?
Traditionally, how do we expect students to solve this in a calculus class? Here is one of several approaches, in excruciating detail:
- Let and be the lengths of the sides.
- Then we know
- The area is
- Solve step (2) for to get
- Substitute (4) into (3) to get
- Differentiate (6):
- Set the derivative equal to zero:
- Solve for to find the extremum:
- Substitute back into (2) to get
- Solve for to get
- Notice that and both equal 3, and that that means the shape is a square.
Ignoring all of the other ways you could solve this problem more humanely, notice that of the 12 steps in this solution, only two have anything to do with calculus. In (7), we actually take a derivative, and in (8) we use the important calculus concept that the derivative is zero at an extreme.
We could say, “the rest is just algebra.” True. But looking more deeply, I think it’s a combination of two things:
- Thinking strategically about algebra and functions. For example, you have to see that you want an expression in x and not y. This motivates how you combine the expressions in (2) and (3) to get the one in (6).
- Understanding the geometry of the situation and knowing how to express it symbolically. In step (1), for example, to specify the rectangle, you don’t need all four sides—or the angles—but you do need two distinct sides. Steps (2) and (3) are formulas for perimeter and area. It’s more than knowing the formulas, though; you have to recognize that you need them, that perimeter and area are relevant (and sufficient).
I claim that many calculus students—mine, anyway—have much more trouble with the 10 non-calculus steps than the two calculus ones. They have trouble getting started. They’re not sure how many variables to use. They don’t know how to express the geometrical relationships. They can’t see what relationships they need in order to get what they want. And at the end of the problem, they have a value for x, and can’t close the deal: they appear to lose sight of the original goal—the shape of the rectangle—and can’t find it in the algebra. In this case, they need to find y and then notice that the result is a square.
Aside: In calculus texts, we often see problems with cones. Typical: “A tank in the shape of an inverted right circular cone has a base radius of 8 cm and a height of 20 cm. It is filled to a depth of h cm. It is draining fluid at 15 cubic cm per second…” Just envisioning the setup requires a lot of geometry. And then, depending on the question, you probably have to visualize cross-sections of the cone and set up proportions that arise from similar triangles. Crikey. The actual derivative is a cakewalk in comparison.
This leads to another realization: In my own work, I rarely need calculus (except to teach calculus), but I need the lead-up to calculus and the wind-down—the mathematicization, the re-interpretation of results, the de-mathematicization, the modeling—all the time. That is, the geometry stuff, and the strategic thinking, are more important, and more useful, than the calculus itself.
So if students founder (or flounder 🙂 ) before they ever get to the calculus steps, and that’s the most useful math anyway, why don’t we spend more instructional effort on those parts of the problems?
I bet there are at least two reasons:
- There is so much to calculus that there isn’t enough time.
- The problem concepts are from geometry, intermediate algebra, and precalculus, which are all prerequisites to the calculus course. Students should already know this stuff.
Both of these reasons are terrible, and dangerously elitist. They lead to an attitude that justifies high failure rates.
A better explanation, I think, is that calculus is a deep study of change, as expressed using mathematical functions. It takes years to really understand functions, and calculus provides a universe of situations in which to practice. As you do this practice, you accumulate understanding of calculus itself, a kind of next-level-up functional understanding and, not incidentally, one of the crowning achievements of the human intellect. Even if you never do calculus later, having done calculus gives you mastery of simpler understandings and techniques, and a broader perspective. (Kind of like, when you study music theory, you analyze Bach chorales. You may never write a chorale after Music 101, but the underlying principles seem to govern a wide swath of the musical landscape.)
Sounds good, huh? And yet the paternalism shines through. And it doesn’t explain why we maintain this nutty focus on getting through so much calculus when the students have such trouble in more foundational mathematics.
I think we should instead try to figure out two things (yes, another list of two things):
- What are the elements of problem-solving, strategic thinking, and connecting geometry to symbols—and any other important category we discover—that students really need to know in their lives (even if their lives include being STEM graduate students)?
- How can we teach these topics? For example, what other settings, in additional to simple calculus, legitimately use these techniques? The same math might be accessible to more students.
Finally, why, when this is a blog mostly about teaching stats, am I writing about calculus? Two reasons:
- Like many of you, I wish we did not put calculus at the apex of our high-school-math-education pyramid, and I’m always looking for alternatives. For example, imagine a capstone course called Advanced Modeling, that has some calculus—not too much—and many other—possibly more interesting—topics.
- I think we have a shockingly parallel problem in teaching statistics, where for “calculus” substitute “frequentist inference.” In stats, however, we are even worse than traditional calculus instruction, because we usually present data that are already in the form students need in order to apply an orthodox technique. We don’t even let them (metaphorically) figure out the cone. Then, when we give them a project to do where they collect their own data, we are shocked—shocked!—to discover that they have no sense of how to set up their data, or even to choose a topic for which a paired t-test (or whatever) is appropriate.
The parallel to the calculus-substitute Advanced Modeling? Maybe Modeling with Data, or, dare we suggest, Introduction to Data Science.