A few days ago an interesting set of articles on mathematics education did the rounds on Hacker News. The first of those articles is “Problems versus Exercises” which talks about how a mathematical exercise (repetition to develop a skill) is different from a problem (something that is difficult to solve and often requires some creative skill). That articles is followed up by “Repetition and Practice“, “Resources about Learning Mathematics” and “Courage in the Face of Stupidity“. The last, especially, is a worthwhile read for people involved in any kind of intellectual, creative activity.

Personally, I had never quite realized the difference between exercises and problems, and I suspect neither have most people (even among the more mathematically inclined). Most of my mathematics education has been focused on exercises (however, I can distinctly remember examples of “problems” in physics). While I’ve never been afraid of math (and generally achieved respectable grades in mathematical subjects), I’ve never been particularly fond of it either. In fact, I’ve often said that I have a “grudging respect” for mathematics. I do wonder how much of that is due to my education, and how much is due to my having strong interests in the arts and humanities (then again, how much of *that* is due to my education, I wonder). All of this is a long winded way of saying that though I’ve never really articulated the difference between exercises and problems, I have been known for a while that the typical classroom exposition of mathematics is only provides a part of the picture.

Over the last few years I’ve been exposed to increasing amounts of mathematics. My main interest is in programming languages, which can be deeply mathematical (though it’s a breed of mathematics far different what most people would recognize as “math”). I recently started learning machine learning, which is yet another deeply interesting branch of applied mathematics. A lot of this exposure has been problem-centric, rather than exercise-centric (though there have been exercises along the way). While I certainly wouldn’t classify myself as a mathematician (I’m more comfortable building things than proving theorems), I’ve been developing an increasing respect for mathematical thought and tools.

A side-effect of this exposure is that my interests have been changing. I’ve been losing interest in things like web development, productivity tools and the like and becoming more interested in problems and ideas with an interesting formal foundation combined with practical applicability. In some ways, I’ve been losing interest in exercises (repeating questions to which I already know the answer) and gaining interest in problems (deep questions and fields of knowledge in which I have little to experience, and where the answers aren’t immediately obvious). I suppose this might just be a natural part of the passing of a few years since graduation.

For a long time now, I’ve been of the opinion that teaching mathematics is hard, and reading the above-mentioned articles has further cemented that notion. I think that mathematics is, by and large, an acquired taste. The dichotomy between exercises and problems (and to some extent, that between smartness and hard work) makes the acquisition of that taste particularly tricky and difficult. To acquire it, you need to be exposed to problem-oriented mathematics fairly early and regularly, but not so early that you don’t stand a chance of making progress. I think this is a balance that is very difficult to maintain, especially in large classrooms with students of varied skill levels.

I wish that I had understood the exercise/problem difference much earlier. Luckily for me, it’s not too late and I have plenty of opportunities to explore both sides. If your daily work involves mathematics (whether in terms of work, research or teaching) I’d love to hear further ideas on this difference (and how we can spread the word).

Thanks for posting this! You’re absolutely right that striking a balance between problem- and exercise-based instruction is one of the hardest parts of teaching. This isn’t exclusive to math either. Any skill-based subject – English included – requires just the right mix between problem-solving and exercises. As you point out, problem-based learning fosters engagement and reinforces learning by promoting higher-order thinking, but that’s only possible once a student has achieved the lower levels of competency with the skill in question. To do either type of practice without the other is to set students up for frustration and, consequently, disengagement with the subject.