A Note on Exercises and Problems

In my previous piece on writing problems, I brought up the frequently-made distinction between exercises and problems. The distinction is in what they require of the person (generally a student) to whom the task is set. An exercise requires the student to use what they have already learned or are learning in a straightforward way, and to execute techniques that they already know or have just been taught. A problem, on the other hand, requires creativity and ingenuity; to find its solution requires the solver to discover something genuinely new.

Problems are delightful and mind-expanding, and no education is complete without a healthy dose of them. Those who lament the stultifying effect of their absence from “school-math” are correct to do so. Most students need to experience more and better problems.

And yet, as bad as it is when assignment after assignment, class after class, is devoid of good problems, an educational diet of problems alone would be a disaster for almost all students. Direct instruction in specific techniques is necessary. Independent rote practice of those techniques — that is, doing exercises — is as well. As much as memorizing the multiplication table, or solving dozens of linear equations, or working through ten bog-standard proofs by induction, is a boring matter, it’s unavoidable if you want to actually learn. You can’t get very far in solving problems if you aren’t fluent in the language you need to think in to solve them.

This is why inquiry-based learning, as wonderful as it sounds to people who love discovery and problem-solving, so often fails. A student who is not fluent in the basics of a subject will not be in a position to make the imaginative leaps that are necessary for solving interesting problems. A student who has to slow down to think about each computational operation will struggle to see and think creatively about the bigger picture of the problem in which those computations are embedded.

Great problems are essential. But the less fun exercises are as well. Elementary school kids need to memorize their multiplication tables; algebra students need to learn to solve equations nearly automatically; math majors should be able to write a proof by induction or a standard ε-δ argument almost in their sleep. And then, once they can do these things, exciting new domains of fascinating problems become accessible to them.