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Math Education

Math.

Math.

Here's the thing about math: People don't like it. I, for one, hated it in school. I remember math classes thusly:

1) Teacher writes some problems on the board and shows how to solve them.

2) Teacher gives an assignment of scores of similar problems, to be due the next day.

3) Teacher takes nap on desk while students start working on the assignment silently.

Is it any wonder that people don't like math or get "math anxiety" when it is "taught" like this? What was the point of that? Why was my youth squandered doing long division and endless algebra? We all knew that we'd never find a use for such skills, and for that knowledge, it seems, we were punished with hours of hunching over lined loose-leaf, scribbling with our pencils, watching the evening tick away into bed time. But that's the only way you learn, right? Right?

It wasn't until I was in college, taking my one required math class, that I had a different experience. I had a math teacher who was actually a teacher, as opposed to a coach who taught math. He did the craziest thing in math class: he lectured. He told us not just how to do these things, but reasons why we might want to do these things, and where to be careful, and where things really were as simple as they looked. He oftentimes had amusing anecdotes to go along with a new procedure, as a good teacher should (the most ineffective teacher is a boring one--if you can't get people to listen, they won't learn).

He also did something unheard-of: He only gave 4 or 5 problems for homework. He just asked that you did them by hand and thought about what you were doing. He gave the problems on half-page handouts. They were problems from the book, but he'd replaced all the numbers with numbers less than 10. "The point," he'd always say, "is that you learn how these things work, not to torture yourself with weird numbers and decimals." He said that, having worked on textbooks himself, he knew that when GTAs couldn't think of a harder problem, they just used really annoying values in them. "My goal is to let you do the arithmetic parts in your head, so you don't even need the calculator." I did all the assignments by hand, and learned tons. I had never gotten over a low B in a math class (despite having straight As everywhere else!), but I ended that college algebra class with a 98%. I spent a lot of time on the class, but not by doing repetitive and unenlightening busy work; I was learning all the math I should have learned in my math classes from age 12 on. More importantly, however, I learned the following:

1) Math is not really that hard.

2) Math is just a tool, like a hammer.

The second becomes more apparent as I work on ever more complex psychometric problems in my budding career as language tester. Problems in data can be solved with the right tools, just like problems in the home can be. Just as removing a toilet can be greatly facilitated by the use of a torque wrench instead of a crescent, the scores of a highly heterogeneous sample of examinees can be more easily understood by adding one or two additional IRT parameters. Maths are just tools, same as wrenches, drills, and hammers.

That fact, however, seems to elude the vast majority of math teachers, at least in my education. Just as a shop teacher would not assign a homework assignment of driving 100 nails into a board by tomorrow, a math teacher should not assign 100 quadratic equations. After you've done a few, you basically have the technique down, and should move on to how to use that skill to solve problems. Ideally, these should be actual problems that you actually want to solve, and, once again, you don't need a bunch of them (i.e. that half-page of "story problems" does not count).

There is an educational approach in law and med school these days called "problem-based instruction." What happens is the students are put into small groups and the class is given a quandary--a legal problem or some hypothetical patient's case file. The group then has a long time (days) to solve the problem. They can use the textbook, the library, their group members, other students, the internet, and of course the teacher. The idea is to make problems that will require them to put together a lot of different skills--both old and new--to solve the problem. There is no punishment for "cheating," because in real life you always cheat, only in real life it's called "research." You might end up using something much harder than you ever did before, or you might breeze right through with an ingenious workaround. It doesn't matter, because all that really matters is that you solve the problem--or not. When the time is up, the groups present what they did to solve the problem, and the teacher finally weighs in on what way is probably best. Groups that didn't solve it still learn as much as if they did. Everyone lays down some important cognitive trails to follow later in the next problem or in real life. Memories.

Can you imagine a math curriculum set up like this? Can you imagine how many people like me (people who are actually pretty good at math) it could save from getting degrees in bullshit just so they don't have to take any math? Instead of beating kids to death with homework, we could be giving them cognitive tools and skills and habits that would aid them the rest of their lives. True, in daily life we don't do a lot of long division. We don't do any quadratic equations. But the idea that any problem can be tackled with the right tool, and that that tool could just as likely be an idea as a physical object, is one that would aid them in every head-scratcher they'd ever come upon. Isn't that the point of education? To enable people to have a smoother and more successful life?

Math isn't hard; counting is. Accuracy over many iterations isn't important; precision in just a few is. And most importantly, kids aren't bad at doing math; teachers are bad at teaching it.

Comments

As one of those other kids who got a degree in bullshit and then later conquered his fear of math, you're damn right.

Of course, I was told all through childhood that math is just another tool, that didn't make anything easier. (I still had the same damned idiot teacher)

I was told that as well, but was never really clear on what kind of problems it could solve. "Story problems" don't really cut it, because no one cares when the trains collide, unless the idiot who wrote the problem is riding on one of them. I think math education should be focused more on application. It should be taught in science class, etc., not by itself. Just as you don't have a separate class for learning how to drive nails, a skill which will be later utilized in your "building a bookshelf" class, you shouldn't really have algebra II and trig separate from physics. Why? Past algebra I, you're never going to use math in every day settings.

Actually, increasingly, I've come to question the very idea of education. Why do we think we need people to know so many things? Most of the really useful stuff I learned, I had already learned by the end of jr. high. High school was just a prep for college. College was a prep for grad school. Grad school was finally an unabashed prep for a job. Grad school was the first time since jr. high that I didn't have an annoying voice in my head asking me if I wasn't just wasting my time and effort. If I had majored in something more reasonable as an undergrad, this wouldn't have been the case, of course.

But if you think about it, aside from a general grasp of history, general mathematics, the ability to read... What do people need? People are learning machines. If you give them the basic tools (literacy, etc.) and access to information (no problem with that in the modern world!), then they will bootstrap themselves into whatever it is they need to do. Intrinsic motivation is really the only predictor of academic (or vocational) success, so why beat people to death with "education?" I just don't know anymore. I think if I had a kid, I might do unschooling with him. Like, no school. The kids that do that don't actually perform any worse on the SAT, etc. Why waste people's time?

I am in complete agreement. I have long been a proponent of so-called problem based learning, and the use of more research, collaboration and applicability.

My first semester of college calculus (at Kenyon) was fantastic. Normally, calculus is a semester or more of rote learning of equations and formulae; then, at some high level, you get to apply them situationally. Calculus is actually incredibly cool, and is finally all about how math gets used in the real world. But most people just get freaked by the difficult math at the beginning. In THIS course, however, the very first day we were presented with a "briefing" by the CDC about an outbreak of influenza in a small town. Our job? Keep it from spreading. And the first half of the semester was on this one problem; at first, the vectors were very simple, and the equations we learned accordingly simple. But by the end of the semester we were all doing actual calculus and plotting disease vectors. It was amazing!