Textbook problem sets aren’t great reading comprehension checks
Problem sets can help you assess your understanding of an explanation, but they aren’t great for identifying and remediating Reading comprehension issues.
Bad failure modes. At times I found myself stuck, but it wasn’t necessarily clear that I was stuck because I glossed over some important point in the text. When comprehension isn’t the issue, stubborn perseverance is usually appropriate, so I’d wonder if I just needed to try harder. So I’d flail at the problem, and the flailing wasn’t constructive, because I simply lacked some relevant information. The trouble is that I often couldn’t tell which situation I was in. The best remedy for a comprehension gap is usually to re-read some explanation in the text, but even with the solution manual in hand, it wasn’t necessarily clear where I should focus. In these cases, I’d end up flipping through the chapter again, looking for something that might be relevant. This seems like a more or less universal issue with problems as comprehension checks.
Biased coverage. In the course of a chapter, you’ll often learn that something is true, why it is true, and why that matters. Problems emphasize application, analysis, and synthesis, so they’ll mostly check if you comprehended that the thing is true, but not so much the other discussions. For example, the text tells me that the solution set of a linear system can be expressed as the sum of a particular solution and a linear combination of free variables which represent the solution set to the associated homogeneous system. Much of the chapter was spent discussing a proof of this property and some of its implications. But none of the problems checked my comprehension of the proof, and the interpretative remarks were only partially probed. I tested myself later to see if I could explain the proof, and I realized that I hadn’t understood a central move, though I had successfully completed the problem set.
You could argue that this is just a flaw in the textbook—that a problem set should exercise your understanding of everything the corresponding book section says. But I think problems are naturally predisposed to check comprehension of certain kinds of material and not others. If you push against that grain, you’ll end up with a different kind of activity, something that doesn’t feel like a problem.
Which problems cover new ideas? Some problems look very similar, but each has been cleverly constructed to exercise some different facet of the underlying material. But sometimes similar problems are just about repetition, to promote fluency. Often I’d feel like I didn’t need to, say, solve yet another system of linear equations: I felt fluent enough! But some of those problems hid subtle differences, novel comprehension checks. So I did every subproblem, and many of them felt unnecessary. Or, to put it another way: many redundant subproblems should have been “smeared” across the subsequent weeks, to support long-term memory. But I wouldn’t want to delay the comprehension checks.
Got the answer, missed the point. In a few cases, a problem was constructed so that it could be straightforwardly solved by applying some property that was discussed in the text. I’d solve it laboriously, and get the same answer, but I’d missed the point. To notice this happening, I needed to not just check my answers but to retrace the steps of every solution, comparing them to my own and watching out for important differences.