But that's math competition stuff, which is seriously more daunting, and practising mathematicians will often claim is even less like "real, proper math." It's occasionally suggested that some real, long-standing unsolved problems should be set as IMO questions, because there's a non-zero chance they'd get solved.
> What many -- perhaps even most -- kids really want is a
> course where they do only simple mechanical work, and
> get a boost to their GPA.
"Most people would sooner die than think. And most of them do." -- G.B.Shaw.
> ... not all teachers care.
And not all teachers are capable.
And largely all of your points are exactly right, except for your misconception about what kind of math we want kids to be doing.
Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible.
I'm not offering solutions here, I'm just helping define the problem, and explore directions.
You're pretty much right on the math competitions front, but the first few problems on the AMC are pretty simple IMO(pardon the pun). They are barely above the level of mechanical problem solving. It does get harder, though, and I agree that that is more daunting.
> Exeter does not teach math with traditional textbooks. Instead,
> math teachers assign problems from workbooks that have been
> written collectively by the Academy's math department. From these
> custom workbooks, students are assigned word problems as
> homework. In class, students then present their solutions at the
> blackboard. This means that in math class at Exeter, students are not
> given theorems, model problems, or principles beforehand. Instead,
> theorems and principles emerge more organically, as students work
> through the word problems."*
This is what we want. Discovering math makes it fun for everyone. Honesty and actual learning with that method(my whole would be not be too difficult, because everyone would inevitably have their own personal spin on the origins of the theorems et cetera. Plus, it'd be embarrassing to go to the blackboard and say that you haven't done the work.
----
By the way, expect an email regarding what you said about the IMO.
Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible.
I'm torn on this.
I personally got hooked on math while trying to figure out the probability of getting various scores after rolling 5 dice and taking the sum of the top 3. (I wound up suggesting this as a Project Euler problem - http://projecteuler.net/index.php?section=problems&id=24... was the result.) So I know full well the value of messing around.
However I wouldn't trust the educational establishment with a task like this. Right now we're caught in a tension between opposing groups. One is apt to recite drill and kill and learning to learn and then proceed to avoiding teaching things that they think are too hard. (Which is apparently everything.) The other is fond of standardized testing at every opportunity as a way to force the first group to actually teach something. (Usually with pretty ridiculous tests.) At the moment in the USA, the testers have the upper hand. But if history is an indication, this won't last forever.
I would trust neither group with what you are trying to do. The first group would be quick to agree with you, grab a slogan, and then head off to do the wrong thing. The latter group would not see the point, and would either start talking about the 3 Rs, or would add to their tests some random, out of context, facts.
I'm not just saying this out of pessimism. We've seen this particular movie before in the New Math movement. Early experiments, with actual mathematicians involved, went well. The mathematicians presented interesting material, and kids enjoyed it. But when the educational establishment tried to imitate that success and get not particularly mathematically inclined teachers to repeat that, it was a disaster. (I was after the main movement, but there was still some of it going on. And I experienced first hand how bad it was when a teacher who didn't understand the material taught his misunderstandings rather than the material.) In the end outraged parents forced teachers back to the 3 Rs, and New Math became nothing more than a bad memory.
I would highly recommend studying that particular episode with the goal of figuring out what went wrong, where. Because what you would like to do has the potential to do the same thing.
> One is apt to recite drill and kill and learning to learn and then proceed to avoiding teaching things that they think are too hard. (Which is apparently everything.)
is somehow comparable to
> The other is fond of standardized testing at every opportunity as a way to force the first group to actually teach something. (Usually with pretty ridiculous tests.)
I don't. The former is, crudely put, evil, while the latter is, at the same granularity, stupid.
These stupid are not a huge problem. They're skeptical of being conned because that's what everyone tries to do to them, but they're open to what works so long as it actually works.
> In the end outraged parents forced teachers back to the 3 Rs, and New Math became nothing more than a bad memory.
And they were absolutely correct to do so because New Math, as delivered, was a sham.
I'm very familiar with the "New Math" debacle/fiasco, and keep it in mind whenever I think of new ways to teach "proper" math. But equally, I give over 100 talks a year on math to kids, and I can see some of them light-up with enthusiasm at the idea that math isn't sums, equations, arithmetic, formula, and mindless manipulation.
But the delivery mechanism is part of the challenge - most high school teachers aren't equipped to deliver the sort of thing we "serious math graduates" would love to see delivered, and certainly most primary school teachers aren't. Getting the delivery right has to be part of the deal.
"Hurray for New Math" - "The idea's the important thing" ...
Yes, Tom Lehrer has something to teach us as well about the dangers of radical ideas.
What we want is like messing about with instruments to see what they do,
and getting hooked on the curious things that are possible.
In high school, I wanted girls to have sex with me and adults to leave me alone (so I could get better at football and maybe girls would have sex with me). I got good at math to effect the latter, and it so happens that I got quite good. Until calculus senior year, I don't ever remember having any notion of "the curious things that are possible".
Now I work in scientific computing.
Anyways, I don't know if student engagement is as important as you seem to think.
And largely all of your points are exactly right, except for your misconception about what kind of math we want kids to be doing.
Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible.
I'm not offering solutions here, I'm just helping define the problem, and explore directions.