# a defense of boring math

2011/08/15 9 Comments

There’s something wrong with the way we teach mathematics today.

So you will quickly hear if you talk to quite a few graduate students, professors, instructors, students, random people on the street… I do agree. There is something wrong with mathematical education (in the US, along with plenty of other places I would guess).

What exactly the problem is somewhat harder to say, but something that you will hear from graduate students and professors, people who have had exposure to “higher” mathematics, will often include the complaint what we teach is tedious, mundane, useless computations and manipulations. Questions have cookie-cutter forms that one presents multiple times throughout, demanding exacting details. Questions of any interest or requiring any independent though are not allowed in fear that students will complain or ignore them.

There is certainly truth to this. And I cannot say that I have anything resembling a solution to this problem or that I’m even a decent math instructor to have authority or experience to talk about what one should do. But despite this, I think that the demonizing of this mechanical understanding can be detrimental to the ultimate goal.

It’s been my understanding that mechanical understanding precedes deeper conceptual understanding. As far as I recall I memorized a list of numbers before I learned how to count things. When I first learned addition, I admit I sometimes leaped-frogged the number line, but with enough practice I just memorized what two arbitrary one-digit numbers added up to and then carried out an algorithm. Similarly I memorized the multiplication table without really caring about any deeper implication.

I think that in order to get a real grasp of calculus-track mathematics, one has to feel comfortable with numbers. And that one gets comfortable with numbers by familiarity with them, even in mundane and routine ways.

I think I generally agree that being able to apply mathematical concepts often precedes conceptual understanding, but not always.

Do you have any books that you could recommend that would help one get a better understanding of conceptual mathematics?

It somewhat depends on your background and interests. I’m going to assume that you have a minimal amount of mathematical background, so I apologize if some of these are below your level:

“Mathematics: A Very Short Introduction” is about what “higher” mathematics is, particularly in contrast to the type typically taught in school. I haven’t read it, but I hear good things and it’s written by a well-known mathematician. It’s probably the only reasonably priced one on this list.

“Conceptual Mathematics” is an accessible introduction to category theory, which codifies the notion of “mathematical structure.” Category theory proper can be extremely abstract (several mathematicians call it “abstract nonsense”), but the language of categories is useful in many areas of mathematics and explains a lot of what mathematicians study.

“Visual Group Theory” is a book that seems the closest to one that I wanted to write. Groups in some sense are the tool with which one measures symmetry. In my humble opinion, teaching groups with that emphasis is better than the axiomatic approach that is rather common.

If you know calculus (and maybe linear algebra), “Nonlinear Dynamics And Chaos” is a good book about differential equations and well, nonlinear dynamics and chaos theory. It emphasizes a more qualitative understanding than solving differential equations.

These are only the ones that came to mind. Do any of these pique your interest?

Yes, actually they all do. Thank you for the recommendations.

How do groups measure symmetry?

When I teach people about numbers and how to add and subtract, I start with the number line, and use arrows to show how the addition and subtraction works. I’m pretty sure that the concept should logically come before the algorithm, from a pedagogical view. Eventually, both should be understood in parallel.

Fair enough. I still think that mechanical understanding outpaces conceptual understanding at least in the beginning of the learning process.

Do you have evidence that mechanics precedes conceptual understanding aside from yourself? If not, I would suggest that you are not a good basis for such a statement, since you are one of the unusual few who has “survived” to graduate school in mathematics (I am unusual, too). Since you are such a special case, I would be very wary of suggesting that we design schools to do what worked for you precisely because it worked for so few people.

Fair enough. I certainly don’t pretend I have any evidence for what I claim above and was not meant to be prescriptive. I was more reacting to similarly unfounded claims that we shouldn’t be doing “boring” math on the grounds that it’s boring. If you know of a source that actually talks about math pedagogy I would be very much interested, since I have a very unscientific approach as of right now.

I would talk to some people in the mathematics education community. They have a boatload of evidence on what works and what doesn’t. Sadly, our two communities do not talk a whole lot.

A good place to start would be Tom Carpenter’s work on “Cognitively Guided Instruction” (http://en.wikipedia.org/wiki/Cognitively_Guided_Instruction).

In particular, a lot of people do spout unfounded opinions on pedagogy. But there are a lot of people (math educators) who have evidence to back up their opinions.