## You don’t suck at math (Part 1)

I want you to imagine that you’ve decided to play softball for your company’s softball team. Your coach, a guy from the accounting department, is a former math teacher, and has taken it upon himself train all of you. You spend an hour or so each day training. These training sessions take place inside one of those company meeting rooms. For the first 40 minutes, you listen to him talk about the different baseball games that he has been to; he draws various diagrams on the whiteboard, explaining in detail the dimensions of each ballpark. Then, for the final 20 minutes–in order to give you a more hands on experience–you watch some tapes of the most important moments in baseball history: Babe Ruth’s famous “called shot,” Bucky Dent’s home run in Fenway park during the division tie breaker, and the victory celebration of the 2004 world series, won by the Boston Red Sox.

Your teacher, who stresses the importance of some personal softball experience in learning the game, also assigns homework each week. Here is your first assignment: go home and make a video tape of yourself swinging the bat. Make sure the video is in focus, and that your whole body is in the frame. Take ten swings left handed, and ten swings right handed. Similar exercises are assigned throughout following weeks. Then your first game of the year comes along. Your team loses 23 to 0. The rest of the season doesn’t go very well either; you lose the first 10 games of the season. After a while, you manage to win a close one, and then a few more after that. But, for the most part, the season is a disappointment. You walk away from the experience believing that you and your teammates just aren’t the sort of people who can be good at softball. You don’t sign up for the team the next year.

This story, as unrealistic as it is, is not so different than the experience people go through when they are “taught” mathematics in school. In short, they are presented with a series of activities that superficially feature the subject matter, but in reality are absolutely unlike doing activity that generated that subject matter, and also quite unlike any secondary activity that allows one to really appreciate that subject matter. Watching baseball, and even mimicking the basic moves that baseball players do, is nothing like playing baseball–it won’t make you good at what they do. Similarly, watching someone drone on about the quadratic equation, or about the cosine and sine functions, or anti-derivatives, won’t make you good at math. Neither, really, will rote memorization of formulas or repeating procedures for doing problems (this may, however, at least get you to where you can pass the test). To get good at math, actual math, you have to sit and think, and explore, and even see things for yourself. Fortunately, this is easier, and more fun.

Many people tell me that they gave up on math when they could no longer follow what was going on in their math class. Well, I’m going to let you in on a secret: I almost never know what is going on in my math class. Granted, my classes are probably harder than the ones you were taking, but the difference isn’t so great as you might imagine. I don’t think I’ve ever been to a math lecture that I followed the entire way through. Losing track of what is going on in a math class is the norm. A good percentage of the time, it isn’t even your fault that you lose track of what is going on. Often, it is the fault of the lecturer. At some point along the way he may do something that actually makes it impossible for you to follow the rest of the lecture–for example, introducing a notation that you aren’t familiar with, and not explaining it. Or he could be assuming knowledge of a concept that he never taught. The lecturer isn’t necessarily being negligent in all of this. It’s just that explaining mathematics is actually quite difficult–you have to bring people to where you are, without knowing exactly what is missing. Knowing what information to supply so that your audience can see what *you* see can be even more difficult than the doing of mathematics itself. The format of a daily lecture, in which the students frantically write down everything that the lecturer is writing, while simultaneously attempting to follow the logic of what he is saying, simply doesn’t lend itself well to actual comprehension.

The sort of brain that is good at following a math lecture, in fact, may not be at all the sort of brain that is good at actually grasping mathematical structure. It is true that some brains are better than others at doing math, but I seriously doubt you’ve ever received reliable information on whether or not you’ve got one of the better ones in this regard. Think about this: there was a time when most people were illiterate, when reading and writing was reserved for the scholars. Now virtually everyone can read. This is because we, as a society, have figured out how to really teach reading, how to facilitate a person’s learning of what is actually quite an involved mental task. But I don’t think we’ve accomplished this with math yet.

A person said to me recently that she isn’t good at math because she just isn’t a right brain sort of person. I’ve heard similar comments in the past. My response is that you don’t do math with the right side of your brain, you do it with your whole brain! I’m sorry, my friend, but you’ve been duped by a false dichotomy. The logical and analytical are not opposed to the creative and fanciful (let alone separated neatly into distinct compartments in your physical brain. The things we believe these days!)–your mind is an organic whole, and I doubt any part of it is completely inactive in anything you do. If you are an artsy or creative type, that is probably going to help you do math, believe it or not. Does your mind have a tendency to go off on tangents? Can you not stay focused on what the teacher is saying? Good! This means that your brain is actually doing something. Don’t fight it! Quit paying attention, and pay attention to the thoughts that you are having (OK, sorry, now I’m sounding like Morpheus from *The Matrix*).

The main thing that makes math seem difficult is when someone is seeing something that you aren’t, but this isn’t communicated. They have a picture in your head, that you don’t see. Or there is an idea, or a whole train of thought, that is simply missing, and that makes everything seem arbitrary and complicated. Everything is difficult when you don’t know what it really is. For example, I bet you’ve heard of the quadratic formula. You may even remember part of it! You may even know that it’s used to “solve” quadratic equations. At some point in your math education, I bet you were presented with this monster:

What you are seeing is the result of moderately long process of reasoning that is not presented to you when you are given the formula. It’s easy to think that, since the result looks complicated, the process itself must have been so, and that it’s something you could never have come up with. But actually, it is merely the result of many steps that are, in themselves, straightforward. These steps constitute a perfectly logical and natural series of thoughts, a series of thoughts that maybe aren’t so different from the sorts of thoughts that you would start to have in your math class. You (or I) might not be able to come up with the quadratic formula on your own, but you can at least get a good idea of how someone did. In my next post, I’m going to give you a down to earth explanation of where this comes from, and how and why a person would come upon it naturally, and I’m going to try to get you to see it in the way that I see it. This, I hope, will give you an idea of how people think about mathematics. Moreover, I hope I can convey the idea that, when we see where these things come from, and see the kind of thinking that caused them, everything becomes a lot more interesting. The quadratic formula can actually be seen as a simple example of the sort of mathematics I study–Algebraic Geometry. I’m hoping, also, that I can (if it seems interesting enough), build up from there a more general description of what Algebraic Geometry is. We’ll have to see how far I get with that, though.