Why Are Books of Mathematics Bad at Explaining Math?
OK, so Shonk plus Salman Khan plus my math tutor have convinced me that I put things badly in my initial post on the topic of mathematical education: it is not mathematicians that are bad at explaining math, it is books on mathematics that are bad at explaining math.
An example: I have been going through a few of books on linear algebra, looking for one that I can connect with. They all immediately launch into ways of doing things with matrices: adding them, multiplying them, figuring out their determinants, ways of getting their minors, and so on. But none of them have told me what matrices are, or what they are for, other than brief mentions that we can solve systems of linear equations using them.
The whole exercise was as though I had been taken from deep in the Amazon rainforest, where I had never seen an automobile, and then placed in the front seat of one, with someone who started to explain to me how to manipulate each of the controls before me, without ever telling me what a car is or what it is for. I see I can make little lights blink on the dashboard, but know nothing about traffic or turning. I get that moving that stick in that direction makes that light come on, but I have no idea why I would want or need to know this.
Today I expressed this frustration to my math tutor, and after he spent about 20 minutes going over the topic, I said, "Ah, so matrices are one way of expressing linear transformations of vectors, and matrix multiplication is the way to compose new transformations out of existing ones!"
So, now my question is, is this a secret? Is no one allowed to reveal it in an introductory textbook on linear algebra? Is my tutor going to be punished by the secret brotherhood of linear algebraists for letting this slip out?
An example: I have been going through a few of books on linear algebra, looking for one that I can connect with. They all immediately launch into ways of doing things with matrices: adding them, multiplying them, figuring out their determinants, ways of getting their minors, and so on. But none of them have told me what matrices are, or what they are for, other than brief mentions that we can solve systems of linear equations using them.
The whole exercise was as though I had been taken from deep in the Amazon rainforest, where I had never seen an automobile, and then placed in the front seat of one, with someone who started to explain to me how to manipulate each of the controls before me, without ever telling me what a car is or what it is for. I see I can make little lights blink on the dashboard, but know nothing about traffic or turning. I get that moving that stick in that direction makes that light come on, but I have no idea why I would want or need to know this.
Today I expressed this frustration to my math tutor, and after he spent about 20 minutes going over the topic, I said, "Ah, so matrices are one way of expressing linear transformations of vectors, and matrix multiplication is the way to compose new transformations out of existing ones!"
So, now my question is, is this a secret? Is no one allowed to reveal it in an introductory textbook on linear algebra? Is my tutor going to be punished by the secret brotherhood of linear algebraists for letting this slip out?
Maybe try Linear Algebra Done Right?
ReplyDeleteAs for your question, most introductory textbooks on linear algebra avoid explaining the why of things because they're aimed at engineers, not mathematicians. Rightly or wrongly, I think the belief is that engineering students (who form the majority of introductory linear algebra students at most universities) just want to be given a formula or an algorithm for doing a computation, and are not interested in why things work or in the notion of a vector space as anything other than a collection of lists of real (or maybe complex) numbers. So the instinct is to try to translate everything immediately to matrices, because that's how you do computations.
In any case, I think a better characterization of your experience is the following: mathematicians are terrible at explaining mathematics when they're deliberately trying to sweep the actual mathematics under the rug because they believe (rightly or wrongly) that their audience is uninterested in or incapable of doing anything more than calculating.
It has over the years become my understanding that engineers are discouraged from approaching systems in a self-consciously abstract way. This is one of the great strengths of their approach -- it can be used to solve almost any problem. If they were in the habit of trying to rely on conceptual understanding to do things, they would be limiting themselves. By not relying on it, they are able to deal with situations that seem far too complex and/or inscrutible to the rest of us.
DeleteEngineering is something like the anti-rationalist version of science. Doing things like spending time on this site finally got me to understand that.
Gene, I agree 100% with your complaint about linear algebra. I'm one of those kids who learned linear algebra in high school, and when I got to the part about similarity and diagonalization, I kept thinking to myself "Why would anyone ever diagonalize a matrix, or care whether two matrices are similar or not?"
ReplyDeleteThen in college I took honors linear algebra, where we studied theorem after theorem all of the form "If M is a matrix representation of a linear transformation T, and T has such-and-such property, then M has such and such property." Finally I raised my hand and asked "We've come across so many theorems that allow you to deduce the properties of a matrix if you know it's a matrix representation of some linear transformation. But that means that if two matrices represent the same linear transformation (with respect to different bases), then we automatically know they have so many properties in common. So what's the relationship between two matrices that represent the same linear transformation?" My professor responded "We call such matrices 'similar'". That was the first time I really understood similarity.
This comment was written partially to support your point, and partially just to share another "secret" of linear algebra!
Thanks!
DeleteKeshav:
DeleteHad a similar experience, only it was with Calculus 2. Six of us doing independent study while a Calc 1 class was going on. Skipped the AP test, took the class at the local community college over the summer. Mad a lot more sense.
Gene:
ReplyDeleteCould it be that the textbooks themselves are designed primarily as aids for college classes? If that's the case, then the main reader is either thinking "Ooh, new math concepts!" or "Okay, this will help me out in my engineering/computer science/physics degree somehow . . . ".