Mathematics Reform

It would be nice if we had math teachers starting in the primary grades. Every other industrialized nation has math specialists by the third grade.

Elementary schools have PE and music specialists. What is needed is math and science specialists. At PSU there is a math resource center for teachers. A third grade teacher once came in because the text book had the wrong answers -- it didn't.

I could see that. It'd have to be done very carefully, and it wouldn't be easy, but it might be workable, especially with a selected class. I see several things to be resolved.

The inherent difficulty of the subjects might make things difficult. Math is abstraction, generalization, and for many kids at that age, the concept of a variable or even of a negative or nonintegral number is causing difficulty. Get into sets, functions, things, it might cause difficulty. 3rd may simply be too early from a cognitive development standpoint. All this stuff has to be built like a tower, remember. You can't teach Analysis without calculus, and you can't teach Calculus without algebraic manipulation, and you can't teach algebra without arithmetic. Our schools are mired down to where kids don't know arithmetic until 5th grade; analysis in 3rd would be nuts! Algebra is probablyu a little less dependent on, well, algebra, so that might be better. I can explain groups to my brother a little bit, integrals, hardly at all.

Obviously the countermeasure is to teach it well (you'd need new books and curricula to teach groups in 3rd grade!) and perhaps to take a certain group of kids, who can handle it better.

Also, the instructors are an issue. You'd need math specialist teachers, as the poster above mentioned. They'd need to be very good to teach that kind of thing to kids so young, and to be willing to go into elementary schools, which might be hard to find among doctorates in math.

Finally, of course, there's inertia and political opposition. Parents'd hate it; talk about being unable to do your kid's HW. Politically it might be difficult to sell spending money, time, energy all to teach kids some pie-in-the-sky, totally abstract, intellectually elite math class with no applications. We can enthuse about its benefits, but it's not playing in Peoria.

All that notwithstanding, it'd be cool, damn cool. I'd love to be talking serious math with elementary school kids.

Ari.

The problems you raise are the key ones, I think. Getting qualified instructors would be nearly impossible, for one thing. Political opposition would be tremendous. Sacrificing useful arithmetic and algebra for utterly unpractical pure math would not be pleasing to most people.

I think it's an interesting idea mostly because when you learn real analysis, you really start from ground zero. Technically, it requires little additional math background. You define sets, then build and build. Before you know it, you have numbers and derivatives and everything else.

I have no idea if it would work, but think of how awesome it would be if you were doing serious analysis by 5th grade...

See, as excited as you are about advanced math for elementary schoolers, I just don't think that learning analysis would benefit them, as much as learning practical arithmetic and algebra. Most kids don't grow up do be mathematicians, or any job where serious analysis would be useful. Most kids don't even end up getting bachelor's degrees.

Now, maybe teaching analysis to a TAG group, or something like that, would make a little more sense. Instead of simply accelerating them along the same path as everyone else, taking them on a completely different path could have benefits.

But all of this still depends on elementary school kids being able to handle analysis. I'm not sure what analysis for elementary-school kids would look like, but I'm sure it would have to be different from what we would learn in college. And the watered-down analysis that they would have to teach would be even less useful.

And I guess the pattern of super-short posts getting lots of comments sort of continues.

Providing the instructors and political will to do something like that would require a major scientific rennaissance in the US. The sheer quantity of willing and capable instructors combined with the political will and funding; it'd be too huge. Great, but too huge.

You do start from the beginning, technically, but it comes in from a background of mathematical maturity. My algebra text's teacher's preface notes that although almost everyone in algebra has had calc. and most have had LA, but that these things are not technically necessary and that an exceptional student may be able to get past it. Emphasis on exceptional, for undergrads. Abstracting that much is not in the standard lexicon for kids that young, and although much acceleration might be possible, it also may not be. Certainly you'd have to be careful.

I think the best way to begin might be to take an elemtary class or 5, possibly with selected students, and get serious mathematicians for a pullout exercise like Music instruction, to test it. If you could show soul-crushingly positive results, then you could sell it really good.

Maybe the laborotory schools K-12 in Chicago (or whatever you call them) would do for the purpose. Hold a seminar to write a curriculum and get some professors from the U to lecture, with undergrad TAs to help the kids individually. Once it works, sell it to the school disticts in the rest of the city, or however else to run it.

That would be exhilarating.

Ari.

Colin, your practicality point is well-taken. I agree that this would be impossible in most contexts. My neighbor tutors at a high school in the ghetto a few blocks south of our dorm. He works in a low-level math class and an algebra class. In the lower class, they were learning subtraction. He gave a few students the problem 13-5=? The student's responses ranged from 8, to 18, to 11, to 23. In the algebra class, they aren't required to do this simple story problems because they are "too hard."

So, I think the only way to consider this is as a thought-experiment, in an elementary school at least comparable to the ones we attended (or the lab schools... the name is so appropriate for this sort of thing!!). Given that, I have two things to say.

Firstly, you say that people would be ignorant of basic tools like arithmetic and algebra. I believe that a solid grounding in abstract mathematics would make it tremendously easier for a student to learn this concrete math.

Secondly, you and Ari have claimed that abstract thought is harder for children. You're right, obviously. Still, I can't help but wonder whether the difficulty with abstraction is because we emphasize the concrete so much in early education. To what extent is abstraction really foreign to young minds? I'm guessing the concreteness is part of human nature, but it's still an interesting question.

Nevertheless, I'm not convinced analysis is abstract. Set theory is extremely concrete. Sets are just groups of things, and operations on them can be represented by visualizations like venn diagrams. We drew venn diagrams on the first day of 4th grade. Maybe you can't use analysis to help farmer Bob count his cattle, but I think analysis is an extremely visual and tangible subject. Besides, everyone knows that meat consumption is inefficient and should be discouraged...

Well, as I'm not really sure what exactly analysis is anyways, nor set theory for that matter, I'm kind of at a disadvantage in this argument. So I really can't tell if abstract math would help with concrete math. But I do think that decades of child psychology and teaching research that have led us to focus on the concrete in elementary education can't simply be discounted. Maybe there's a reason the system works how it does: maybe it's a pretty good system.

Anyways, I think doing an experiment in your lab school would be a good idea. However, Ari's proposal of having U Chicago professors and grad students teaching wouldn't work. Teaching elementary school kids actually takes some specific skills that those people probably don't have. Which brings back that finding-enough-teachers problem: not only would we have to find teachers prepared to teach abstract math, but teachers prepared to teach abstract math to elementary schoolers. And to implement that on a national level, I think, would be insurmountable.

I don't think the usefulness of the process is in doubt, if it could be done. If you had a class of 6th graders through analysis, it's doubtful they'd stay ignorant of algebraic manipulation for long! They'd pick it up in a snap, a simple special case of what they've done. Just as someone who's learned to solve triangles will surely have no trouble with the pythagorean theorem. So it'd be great, it's just doing it and selling it.

Certainly it would be a huge leap from what we have now, which, with respect to Colin, is awful. There are adults out there, millions of them, who went to college and took algebra yet can't get percentages or even unit pricing. Math education needs to improve, however we do it. Analysis in elementary school? Maybe.

I don't know to what extent abstraction is inherently foriegn to all minds, but the way ours are set up would seem to be focused on the concrete. Note the difficulty even very intelligent people have with the simple abstractions like geometric optics (shiver) if they fall outside what they've been trained to do. Note also how abstraction is one of the notable characteristics of human mentality, as if it's foriegn to the mammalian brain and we've managed to claw our way into some.

Analysis really is abstraction, Adam. You can show set concept with venn diagrams, like disjoint, union, intersection, identical, maybe even cardinality, but what about cross product? Power sets? Functions? The whole idea of generalizing to prove things about all sets is abstracting a logical argument to apply to all things with certain properties. Even logic itself is abstract. We've gotten used to it, but in 5th grade I had trouble with the concept of a function in generality, as distinct from a set of points in the plane. Bananas, now those are concrete. You take 5 bananas and 3 bananas and you get 8 bananas. Now ask a class of 3rd graders whether the group bananaZ is cyclic?

Colin, don't worry about some mystical "license to comment." A terrible lot of this hinges on developmental psychology, of which I haven't an iota, but I'm not shrinking from speaking.

Now to the test. Adam, can you tell us a bit more about the lab schools? Are they voluntary, is the sample representative? Is there tuition? Would the entry contract the parents sign allow UofC to run something like this?

I think that it could work, and that Math profs and students would be perfect to run it. The difficulties of teaching small children have never struck me as insurmountable. Certainly undergraduate education majors at most schools are not a terribly impressive group (class? set?). Aside from crowd control and issues of experience, what is there really that they'd have to pick up? The experiment doesn't have to be perfect; as long as it gives us the information. If they get analysis early and crowd the halls of universities with brilliant math majors, we know it works.

The real worst case scenario is that they don't get it. Then all you do is teach them arithmetic in 6th grade when they'd pick it up faster, and the little buggers're hardly behind at all. No ethical problems here!

Christ, this post is long. Maybe listserve would be better for this discussion?