Katie has a top-notch rant about the sorry state of public education in Canada. She explains what is exactly wrong with the teaching of most subjects – for example, in math, it’s the overemphasis on computation:
I would often ask my math teachers “Why?”. …I needed to know WHY we were to simply carry something over, WHERE a screwy/random number was coming from. I had to know. I needed to know the theory behind it – this is why I failed grade 10 math (that and not showing up…) because there was never any reason as to why we were doing the math. It didn’t help that the teachers didn’t care if we really knew the math – as long as we passed and they passed their grade quota.
Everyone can “get” math. People always tell me the just don’t “get” math. I was one of those people who didn’t “get” math. But really – its a matter of finding a way to make it work for you. The way it could have worked for me was the philosophical/theoretical approach to it. Reading Alon’s advanced math books, I understand more of some of those than I do of the grade 12 calculus book. But high school takes the easy way out, they prepare kids to be able to apply math to physics, or just give them formulas.
Having read the number theory book Katie’s talking about, I can add that it’s not a particularly comprehensible one. The same thing that happened to her is happening to me to some degree: I can do computations in Riemann surfaces, but I don’t understand the deeper meaning behind, say, sections of line bundles, or anything involving smooth functions, which means I do these computations worse than I do things in algebra, which I do understand.
More to the point, in the late 1950s, when Sputnik forced Americans to come to terms with the fact that their educational system stank, the US had to undergo serious educational reforms, a lot of which were a watered down version of what Katie’s recommending, at least with respect to having more theory in math and science. New Math was successful, in the sense that it improved math education in the US; the problems with it were that it went too far with set theory.
You don’t need to teach set theory to talk about the why of mathematics. The things in math that have reasons too deep for even the current high school curriculum can be skipped, since high school students don’t really need to know that e^pi*SQRT(163) is within 1/10^12 of an integer. Talking about her old math teacher’s daughter, Katie says, “They give her 13 bunnies, and tell her to make 3 equal piles. When she is left with one she understands why… that that bunny doesn’t fit into those piles so he has to start a new family.” Ideally it should be done way before high school, so that in high school they can explain why it is that quadratics where b^2 – 4ac < 0 have no real roots, but it illustrates the idea pretty well, I think.
The same applies to science teaching. The best way to make students hate science for the rest of their lives is to put them in high school science classes, where they learn all of biology is memorizing names of phyla, all of chemistry is memorizing the periodic table, and all of physics is memorizing names of particles. In contrast, Pharyngula and Cosmic Variance don’t attract tens of thousands of lay readers per day because they give you guides to how to memorize facts. Everything is easier when you only need to explain the general ideas – PZ wouldn’t attract so many readers if he structured his science posts as a textbook (for evidence, compare my traffic to Mark CC‘s) – but still, teaching the meat and potatoes of science should still be possible while offering a greater context.
One idea I’ve seen that is so farfetched it might actually work is hands-on learning. Mark Rosenfelder explains,
There’s more: every programmer knows, I think, that you can’t learn a programming language by reading a book. The book may give you a feel for it, but you don’t know it till you’ve written programs in it.
Now, I simply think that kids are the same way, if not more so. Illinois, where I grew up, requires classes on the Constitution in 8th grade. That’s utterly absurd— there’s zero practical use for that information at that age, so almost everyone is simply going to forget it.
The apparent success of schools is an illusion, due to several factors:
- The effectiveness of schools isn’t evaluated by kids (who are usually well aware of how poorly the system works), but by teachers and parents, who’ve bought into the system.
- A few kids, of a very abstract and systematic mindset, do learn something in school.
- Teachers tend to be of the same mindset. They love to see knowledge carefully set down in schematic order, and they assume everybody else does too.
- The system creates makework— i.e. tests— to focus on. The difficult larger goal (making kids learn things permanently) is replaced by a much easier one (making them learn things temporarily), and everybody pretends that they’re the same.
You might guess that I didn’t do well in school and resented it. Not true; I did very well. But 80% of it was a waste of time. The knowledge that’s stayed with me, from languages to geography to history to writing to programming, is all stuff I learned on my own.
How would I educate kids instead? Through learning by doing. School is supposed to prepare kids to live in a technological capitalist society. So a high school could be organized as a company to develop, produce, market, and sell real products. Accounting needs mathematics; marketing needs language skills; production uses all kinds of skills. Developing a video game, for instance, would require algebra and trigonometry.
Actually, on a somewhat personal note, I have a very abstract mindset, and learned nothing in school except how to get out of phys ed classes. But anyway, the idea of learning by doing is a really good one, although for the broadest and most theoretical applications, it appeals mostly to geeks.
The best way of learning about any social science is to have to write a fantasy or science fiction series that has world creation that wasn’t pulled out of Asimov’s ass. Mark himself notes that his fantasy setting required reading tons of historical linguistics, anthropology, and geography to be constructed right. The problem is that people who aren’t immersed in geekdom usually won’t care much for that kind of activity, while the people who are usually got that way by liking the dreck that passes for speculative fiction and that almost uniformly has substandard world creation.
Making math attractive to non-geeks that way is almost impossible, I think. Most people don’t have a business mentality, and at any rate, imitating businesses is a doomed project since the real world’s most powerful feedback, profitability, is absent (the same applies to model politics, by the way). It’s easier with science, because it is possible to imitate industrial production with a Soviet-style production quota replacing profitability.
But still, there are two ideas at odds here, which could work if combined. On the one hand, teaching people the why of things, especially math and science, will make them understand the concepts more fully. And on the other, devoting significant time not to drills and word problems but to more creative pursuits will make them concentrate on what interests them the most.
One of the problems of learning by doing is that when the curriculum gives totally free choice, it leads to overspecialization. In an environment where a significant number of people will go on to higher learning, where their interests might change, it’s not a good thing. It leads to a system that forces Katie to do social science not because she wants to but because she didn’t take enough science and math in high school. Ideally, making schools work better those ways should enable compressing 16 years of learning into 12, but in practice it’ll probably be a lot less spectacular than that.
Still, in a lot of contexts, learning by doing is the best way to go. I read a while ago about a freshman composition professor who taught his students syntax by making them design their own languages, complete with their own grammars. This is where the idea of learning by conworlding comes from: presumably, making people create their own worlds, and giving them more realistic feedback than what I had when I was 10, should stir them toward knowing enough about how societies in general work.
[…] Over at his excellent blog, Abstract Nonsense, Alon Levy has a hefty post with some rather scathing, yet overall very productive, criticism of math education. He gets right away to the root of the problem, which I know from relatively recent personal experience with the current system: a lack of emphasis on underlying concepts. Here’s what in my estimation is the meat, but read the whole thing: The same applies to science teaching. The best way to make students hate science for the rest of their lives is to put them in high school science classes, where they learn all of biology is memorizing names of phyla, all of chemistry is memorizing the periodic table, and all of physics is memorizing names of particles. In contrast, Pharyngula and Cosmic Variance don’t attract tens of thousands of lay readers per day because they give you guides to how to memorize facts. Everything is easier when you only need to explain the general ideas – PZ wouldn’t attract so many readers if he structured his science posts as a textbook (for evidence, compare my traffic to Mark CC’s) – but still, teaching the meat and potatoes of science should still be possible while offering a greater context. […]
Re Math teaching
I can only repeat my own experience in high school where I did much better in plain geometry then in algebra, probably because for former is a lot more fun. In plane geometry, one learns more about abstract ideas like axioms, propositions, and theorems and concepts of proof. Algebra was more rote memorization and is much harder to make interesting.
I feel in math it is important to learn by doing. Hands on experience is an excellent learning strategy to use to enhance students learning. Students learn by doing different projects that will help them understand math a lot better. Technology in the classroom, such as the computer guides students to higher learning and provides them many resources. Everyone will understand math, if children use resources that they use in everyday life. Technology has become an integral element in successful learning and especially in learning mathematics. The Internet can help students connect with their teachers and e-mail them any questions that they may have about an assignment. Technology-related training for teachers motivates teachers to incorporate videos into the classroom to share information with the students’, solve problems, increase communication, and decision making.
Thanks for sharing explanations and writing this article. Looking forward to more of your stuff. Hopefully you continually update your homepage often since you have found a loyal visitor .
I do agree with all the ideas you’ve presented in your post. They’re very convincing and will definitely work. Still, the posts are very short for starters. Could you please extend them a bit from next time? Thanks for the post.