There are lots of students who aren’t interested in math, and lots of math teachers who want to motivate them to actually put effort into the subject. Most of this struggle seems to revolve around how useful math is, with kids saying (or at least perceived to be saying) something like “I’ll never need this” and teachers trying to convince them that math is everywhere.
For a variety of reasons, I’ve never been happy about this way of looking at why one learns math. Most importantly, I think, teachers are bound to fail at convincing students of the usefulness of math, even with the most applicable of topics. Part of the reason is that the subjects that apply the math are always taught after the math itself. Physics, economics, and so forth make the need for quadratic equations very clear, but you would never teach someone physics unless they had first mastered quadratic equations. This means that the math a student is currently learning is never being used in their other classes as they learn it. Some students see stuff they learned years before now being used routinely, and they learn to trust that the things they’re being taught now will turn out equally useful in years to come. That’s a healthy attitude, but even with an explicit explanation of the situation, many students will not adopt it.
It’s also worth mentioning that there are in fact topics taught in high school (especially in a school with a strong math curriculum) that aren’t likely to be used in the future by many of the students. Calculus is helpful in a variety of occupations, but far from all. Geometry, barring the straightforward basics, is only helpful in unusual circumstances. Imaginary numbers are unlikely to come up too frequently. A teacher is therefore bound to fail at showing the usefulness of these things directly.
I also think, though, that judging math by its usefulness is missing the point. Why do math teachers need to prove that their material will be vital for daily life in order to make it worth learning? Poetry would never be able to pass that test, nor would history or art. You don’t “use” Shakespeare very frequently. Of course, that’s obviously not the point. We fully recognize that sonnets aren’t “useful,” but we still learn them. We think it makes your life better to have the wider, deeper view of the world that comes with having studied art and literature. We think they’re part of being an educated person. We think that by studying them you build fundamental skills of critical thinking, imagination, and interpersonal relationships that are important, even if the actual material you’re learning is not. These things are all true about math as well, but people don’t think of math that way. They think of it as a prerequisite for other (important) things.
Math should be taught as something you learn because it’s interesting and enlightening. It should of course be mentioned that it’s also useful, but that should never be seen as the only reason why it’s being taught. To do this, it would help to change the math curriculum a bit. Anyone who’s ever taken proof-based math in college knows that mathematicians don’t spend their days doing the stuff that’s taught in high school. Engineers solve a lot more equations than mathematicians do. Mathematicians deal with abstract topics, proving new facts. This kind of open-ended, puzzle-like problem is a lot more fun to do, and a much better way to show students what math is really like, than the computational topics that take up the current standard curriculum.
If math can be taught as something that’s interesting, rather than as something that’s useful, it changes the way students look at it. It becomes the kind of thing that one would expect some people to really, deeply like. It can be fun and exciting. If it’s being done because it helps design bridges, it’s a chore. The application is cool sometimes, but the thing you’re learning never is. Math as useful calculation will never be appreciated by anyone who can’t see themselves going into science, engineering, or accounting, but math as clever puzzles that help us understand the wonder of pure reason can be something people really want to learn.
August 11th, 2008 at 10:48 pm
[...] 38th edition of the Carnival of Mathematics went up at Catsynth on Friday. It includes A’s recent post about typical math teachers’ efforts to make their classes seem relevant to students. [...]
August 14th, 2008 at 12:02 am
You took the words right out of my mouth. I have been drafting something on the topic for my blog, but your views are so similar (instead of Shakespeare I was using Chaucer, ha!) that the need isn’t there anymore.
Huzzah.
The one thing I think might be worth commenting on is this: “Anyone who’s ever taken proof-based math in college knows that mathematicians don’t spend their days doing the stuff that’s taught in high school.” Yes, two column geometry proofs are the best example.
However you also say “This kind of open-ended, puzzle-like problem is a lot more fun to do, and a much better way to show students what math is really like, than the computational topics that take up the current standard curriculum.” Which I agree with, to a point. I’m a new teacher, so maybe I’m totally off base, but there are a lot of students who need the structure of those computational topics. The drilling of the basics. Very specific types of learners find this open-ended environment fruitful. I think that some sort of thought-out mixture of the two is probably the way to go.
You might be interested in reading this (which I sort of disagree with, but it is on the theme of your post) if you haven’t yet: http://www.maa.org/devlin/LockhartsLament.pdf
August 16th, 2008 at 11:49 am
I’ve seen the Lockhart article before, and I also have very mixed feelings on it. I don’t really have any more teaching experience than you, but I do agree that some students like the more straightforward computation. I would never advocate totally removing it - it’s the part of math that has the most application to other fields and is definitely necessary. I just think we have to make clear what real math is. Those kids who enjoy the computation should be encouraged, but it’s the kids who enjoy the more abstract stuff that are the most promising math students. That stuff also earns math a lot more respect from those in other fields than the mindless computation does.
August 20th, 2008 at 10:21 am
Interesting article. I particularly agree with this:
If math can be taught as something that’s interesting, rather than as something that’s useful, it changes the way students look at it.
There is a big push in UK schools at the moment for a ‘functional mathematics‘ curriculum as an alternative for weaker students to standard GCSEs, which seems predicated on the belief that weak students will only be interested in maths that is useful in everyday life (see the link for an incredible amount of government nonsense-speak on this topic).
I think you’ve identified several ways in which this is a very bad idea.
An equally bad trend is the need to assert that ‘Maths is Cool’. As Rob Eastaway pointed out at in a keynote talk at a conference I attended recently, saying that something ‘is cool’ or ‘is fun’ does not automatically make it so. If we substitute ‘train spotting’ for ‘maths’ in the assertions, we can see how many students approach what we say!
August 20th, 2008 at 11:17 am
Yeah, I’ve never found those attempts at proving math is useful really even accomplish that very limited goal. The applications they give are always cartoonishly simplified.
Once, when I was teaching a tenth grade class and got the “How is this useful?” question, I gave the quick list of some places where the topic came up, but then said something to the effect of “Really, you can’t understand exactly how this will be applied right now. Remember when you were in 3rd grade, and they explained how fractions are useful because you can make change and count parts of pizzas? That was true, but you now use fractions all the time without thinking about it. Every time we graph a function, we assume it is continuous, and that makes no sense without fractional values.” They actually took this pretty well and seemed to find it a satisfying answer.
August 23rd, 2008 at 9:35 pm
I’ve been saying the exact same thing for years, sadly to people who teach Shakespeare and world history. There is a prejudice against math by many educators.