I was just thinking about some of the difficulties that many high-school students have when attempting to learn math. Aside from those who face external obstacles — such as brain damage, severe emotional problems, or extremely inadequate teaching — we have those who don’t work hard enough and those who do. It’s easy to dismiss those who don’t work hard enough (“just work harder,” we advise, even though that might or not might help and they might or might have time), so let’s focus on those who have difficulty even though they have no visible external difficulties and they do enough work in math. “Enough” differs from student to student, of course, so we won’t try to quantify it. We’ll just say that if you do enough work, your problem is not that you need more practice.
What I’ve observed is two diametrically opposite subgroups of people who work hard but still have trouble with math: those who miss the trees and those who miss the forest.
- The first subgroup, which is overwhelmingly male (but by no means exclusively so), contains those who are apparently following Tom Lehrer’s satirical advice: “The important thing is to understand what you’re doing rather than to get the right answer.” These students get the big picture, or at least claim to do so, but they haven’t learned the details. They often say that they’ve made “dumb mistakes” and confidentally plan to do better on the retake with no additional studying. They correctly view math as a field with Big Ideas, but they incorrectly believe that that’s all there is. Because they can’t reliably execute the Big Ideas in the form of solving problems, they can’t actually apply what they know (or think they know). Usually they are deceiving themselves about their level of understanding; often they are not merely making “dumb mistakes” but actually are facing gaps in their knowledge. But it’s hard for them to improve, since they feel assured that they understand what’s important, and the rest is just details.
- The second subgroup, which has a slight preponderance of females (but not significantly so), contains those who focus on rote learning of algorithms. They view math as a bag of tricks, a collection of skills that can be mastered without context. They don’t see the big picture and often don’t believe there even is a big picture. If they have unimaginative teachers who always test them on problems that are exactly like the ones they’ve already seen with only the numbers changed, they are misled into thinking that they are successful learners of mathematics. And then they have a rude awakening when they eventually find that skills aren’t sufficient: understanding is also necessary. As John Holt put it, “The true test of intelligence is not how much we know how to do, but how we behave when we don’t know what to do.”