One thing that constantly surprises me is how often it’s my ‘dumb’ students who stun me with their mathematical insights. These are the students who describe themselves as bad at math, or whose parents describe them as bad at math, or whose teachers (in the form of bad grades) describe them as bad at math, but they are often the ones who see what’s really happening in the math, sometimes in ways I’ve never noticed. I suppose that’s probably a result of observer bias, at least in part. I’m told these students are terrible at math, and so I end up expecting that and I’m surprised when they turn out to have any level of insight. But I don’t think that’s the whole story…
For the purposes of this discussion, I think we can say that there are four different types of math students. At one extreme, there are the students who genuinely love (or at least like) math. They usually have both good mathematical insight (by which I mean the ability to step back and see patterns inside mathematical mumbo-jumbo) and are patient enough to do the sort of computational stuff that is typically required of them in math classes (usually, if you understand what’s really happening, the computation is easy), or they and their parents have the good sense to just not care too much about grades. As a tutor, I don’t see those kids much since they don’t need help, though I’ve occasionally done “enrichment” with them.
At the other extreme are the students who genuinely don’t “get it” when it comes to math. Before you accuse me of labeling some kids as dumb, let me point out that I’ve noticed two things about these students. The first is that they’re usually in classes that are way beyond the level they really should be working at. For whatever reason, they’ve muddled through a whole series of classes without ever really understanding, and their teachers should never have deemed them ready to move on to the next level. Regardless of what level you think students should be working at, it’s unfair to pretend that they are at that level if they’re not. The other thing I’ve noticed about these students is that they often struggle in ways that have nothing to do with math, e.g. they struggle so much to read (despite being in high school), that they can’t parse problems at all. As a tutor, I find I really can’t do much for these kids, because they needed help ages ago, and an hour a week with a tutor is just not enough to compensate for their dysfunctional families (it always seems to be these students whose parents pop in four or five times an hour, criticize their kids, and then try to get me to tell their children that they’ll never amount to anything if they don’t work harder), dysfunctional schools, and their dysfunctional self-images. For me, working with these students is an exercise in developing patience. For the students, it must be an exercise in humiliation.
But then there are the in-between kids, and they mostly come in two flavors: the ones who think they can do math (and whose parents and teachers think they can), and the ones who think they can’t. Here’s where things get interesting. Though it’s far from an absolute rule, the “good” kids in this group are often the highly conscientious students. They’re the ones who buy into the good-grades-will-make-me-a-good-and-successful-person hype of school and they have a lot to lose by doing badly, so they come to me as soon as things start getting a little difficult (which often means, the first time they discover they’re getting a B instead of an A). These are the students who often have the entire list of important trig functions memorized, who can recite the quadratic formula, and will unquestioningly do all the homework, no matter how offensively stupid the problems are…(even problems of the “use your calculator to find cos(72.63º)” variety).
Don’t get me wrong, I love these students. They’re usually kind, appreciative, attentive, and very easy to work with. But, they’re also usually content to just get the right answer. They don’t seem to care if they really know what’s going on or not, and they panic if they don’t have a nice set of rules to follow to get them to the answer. These are the students who look at me when I’m nuts when I start to show them ways to think about what’s actually going on underneath all those formulas. And that makes me sad. (On the flip side, helping one of these students who gets by on a starvation diet of memorization and conscientiousness come to a genuine mathematical insight is one of the highlights of my job.)
Then there are the students who think they’re bad at math. What I’ve noticed is that these students often eat up any explanations I can give them. And they often come up with insights that stun me. (Unfortunately, these insights usually make so much sense in the context that, even though I’m surprised that my student came up with it, I usually forget the insight itself. I can’t think of many good examples right now.) My “good” students might plod through computing all the important trig functions all the way around the unit circle, but it’s my “lazy,” “bad” students who will notice that once they’ve found the functions in the first quadrant, they can do all the rest easily.
All this has led me to believe that the students who think they are too dumb to do math are often the ones who aren’t satisfied by just having the right answer. They want to know why and understand how the math actually works. Given how obsessed our school culture tends to be with getting the right answer, and how little access most students have to the whys of math, it’s not surprising that when these students can’t figure out why the math is doing what it’s doing, but see the people around them (looking like they’re) doing just fine, they start to think that something must be wrong with them. What a tragedy.
Some students, of course, are lucky enough to have a decent curriculum, or have a teacher, or parent, or other role model who recognizes their need to understand. And some students are lucky enough to be able to figure it out on their own. But for the rest, I’m afraid they end up thinking that something is wrong with them, not with their textbooks or their tests or their schools.
By the way, I’m not in classrooms much, and teachers very a lot, so I can’t speak to how well or how poorly teachers help students genuinely understand the math they’re working on. From what I see, teachers mostly don’t do this well, but then, I typically see the students who are struggling to deal with incompetent or unreasonable teachers, so that’s not a balanced sample. What I have seen are lots and lots of textbooks, and from looking at those, I’m pretty convinced that anyone who tries to learn math from a textbook will end up having no idea what they’re really doing. I’ll rant about that in another post shortly.