How Should We Teach?

I'd like a short, pithy answer, but I don't have one. The best that I can do is talk around the question. I'll have as much to say about how we shouldn't teach as how we should. When I'm done, I'll quote from a Timothy Gowers' post on the failures of mathematics education. It's goes right to the heart of the matter.

I use the term 'recipe' quite a bit in what follows. By it I mean a mathematical algorithm that if faithfully followed will solve problems within a certain class. The Euclidean algorithm is a good example. The Power Rule is another.

School mathematics often takes this form:

1. Teacher presents mathematical recipe.
2. Teacher solves problems on board by application of recipe.
3. Teacher gives student two dozen or so problems like the one solved.
4. Teacher asks for questions the next day and then takes up students' work.
5. Repeat

I do admit that, at least in outward form, I often run my classes just like this. But where I differ from this sequence is in its first stage. I never (or at least try to never) simply give a recipe. For each new result that we encounter in class, either I prove it (sometimes), students prove it (sometimes), or we prove it together (most often). I never just give a recipe.

Why? Why not just give the recipe? (It's often what students want. It's what they've been trained to want.) First, recipes without explanations - and proofs are the form in which mathematicians give their explanations - aren't mathematics. This isn't ideology. It's history. Second, when an explanation can be given, and it always can, not to give it is intellectually lazy. Third, as Aristotle put it, human beings by nature desire to know, and if we give only the recipe, the better part of what can be known has been left out. Fourth, mathematics is not a loosely connected set of recipes (though if all you've ever know is school mathematics, that's likely just what you'll think). Instead, it is the most systematic of the sciences. What comes after is deduced from what comes before. This is the essence of proof. If we leave out the proofs, we miss the essence of mathematics. Fifth, students are unlikely to remember a recipe unless they know why it works, and if they how why it works, they can reconstruct it for themselves if need be.

I know of only three real objections to what I've said.

1. This demand to do genuine mathematics with students, to actually give the explanations that will satisfy their curiosity, is beyond most of them. It's just too hard.
2. Anyway, students aren't curious. Why give them something they don't want?
3. All that students will ever really need to know - either for standardized tests or for a later course - is mastery of the recipe.

I'll answer in reverse order.This claim about need is very narrow. What students really need is to be stimulated. They need to have good problems set before them and then given the space and time necessary to solve them. They need a teacher who can guide them in this pursuit. To do any less - to teach just recipes because "that's all they need to know" - is to abdicate the role of teacher.

Second, I know that it's false that students don't want to understand. I know this because I've taught thousands, and of those thousands most have been wanted to understand. It does often take some time for students to become accustomed to the idea that in mathematics, it's proper and indeed obligatory to demand explanations. Such is the state of mathematics education today that students often find it strange that there are such things as mathematical explanations. (How often I've been met with "It just is!" or "Because the teacher said so." when I ask for an explanation.) But once that hurdle has been cleared, students take to the mathematics like birds to the air.

Last, it's simply false that genuine mathematics is beyond most students. Again I know this from experience. I do not deny that some take to it more readily that do others. But all have some degree of ability, many a very high degree. (Every year I have students that could become research mathematicians.) The truth here, I suspect, is that to teach the recipe is much easier than to search out the explanation, and teachers often seek the easier path. (Another possibility - one that I think applies only to a minority - is that they believe, mistakenly of course, that the recipe is all there is.)

I'll let that suffice for now. No doubt I'll return to these issues later.

I end with a set of quotes from Timothy Gowers' post What maths A-level doesn't necessarily give you. I begin with Gowers himself. After I take a few lines from the comments. (If you haven't taken Calculus, you won't understand everything that's said. But I'm sure you'll get the points about pedagogy, which are the most important.)

Let’s suppose that your aim is simply to do well at maths A-level and that there are no questions that test your familiarity with the formula for the derivative of an arbitrary (nice) function at an arbitrary point. Which is better?

1. Don’t make any effort to learn and understand the formula, but simply learn a few basic examples of derivatives (polynomials, exponentials, logs, trig functions) and rules for differentiating combinations (linearity, product rule, quotient rule, chain rule) and you should be able to differentiate anything that comes up in the exams.
2.  Learn what the derivative means, derive the formula for the derivative of an arbitrary function at an arbitrary point, calculate a few derivatives from first principles, derive the product rule, quotient rule and chain rule, and then learn how to use them to differentiate combinations.

The answer is that if you are capable of doing 2, then 2 is far better. And the boy I was talking to was certainly capable of doing 2. Why is it better? Because (and this is something I plan to devote a blog post to at some point) memory works far better when you learn networks of facts rather than facts in isolation.

Gowers is right of course. 2 is far better. Indeed I would claim that if a student isn't capable of 2, there's no reason to have the student do 1. I do find it a bit strange, however, that the reason Gowers gives for the superiority of 2 is that it is an aid to memory. I don't doubt that this is true. (I said it myself above.) But of all the reasons for the superiority of 2, I'd think that that's the least important. Indeed, to give only it makes it seem as if the sole reason for 2 is that it serves the purpose of recipe memorization. I'm sure that Gowers doesn't believe that, but the passage seems to suggest it.

Now for a few responses that caught my attention.

Terence Tao: In general, I think fundamentals are too often given short shrift in order to advance prematurely to applications of said fundamentals. When I teach undergraduate real analysis, for instance, I like to spend a fair amount of time on construction of number systems, before getting to the limits and the deltas and epsilons. Unfortunately, the most basic topics are often the hardest to teach correctly…

Andreas: In Norway . . . the notion of proof has been completely removed from the school curriculum, except for some side remarks in the textbooks. In fact, it is perfectly possible to go through all of school (up to age 19) with top grades in the most advanced mathematics courses, without ever writing down a single proof. Even the formula for solutions of a quadratic equation is normally presented without proof, and to attempt a proof of the chain rule in class is just unthinkable.

As for making choices as a teacher, [the problem is that] we are extremely pressed for time, and choosing to include proofs and proper definitions would mean having to exclude other topics, which, unlike proofs, appear in the all-important end-of-year exam.

So I guess my point is that while it is possible that [Gowers'] young friend has been badly taught, it is more likely that the teacher did what she had to do in order to get the majority of ordinary students through the exam with decent grades. It all comes back to the politics determining the curriculum – who are the people actually making these decisions? Are there any mathematicians involved?

Greg Friedman: What seems to me to be the real problem is that students arrive with the idea that math isn’t something that you think about – it’s something where someone gives you a recipe and you follow it. I try explaining to them that, as pointed out in the post, it’s actually easier to learn something if you understand what’s going on than if you just try to blindly memorize formulas. I also try to instill the idea that if you don’t understand what it is you’re doing, there’s no point learning to do it since you’ll never be able to apply it (though of course to them, the application is the exam).

.