Teaching is a demanding job. Maybe it works better if you learn with the students (try to go from their perspective), although obviously you need to have some idea where you're going. But you can still do something like that if you just pull out a random math problem you've never seen before to present to the class. You also get to have more fun!

A group is as strong as it's weakest member. Hence, it should be noted that if the weaker students are aided, and thus catch up to the smarter students, everyone will learn more. It is my belief that no student is too dumb to grasp any mathmatical concept taught in a basic elementary-, middle-, or high-school curriculum. America's schools are failing their students, and this needs to stop. I am currently not exposed to this problem, as TJ is an amazing school, but I was in 2nd and 6th grade. Basically, if the weaker students can be motivated to learn (especially during elementary school) then everybody wins.

In response to LordoftheMorons's post :

That would be nice, but isn't it a bit Utopian? The story of a large part of my life has been exhibiting patience or lack thereof with weaker students as they work to catch up. Much as I would like to believe otherwise, I do not think that we can motivate every student to learn, or even every student that does not have some disease in way of this, as much as it would be to eliminate crime completely. "America's schools are failing their students" is the wrong way to put it. Some students will always work harder, or have been working harder, than others, and they will draw ahead of the others. I agree that the mathematical concepts of grade schools are not too difficult for any student, but I do believe that they are too easy for many, particulary members of this forum. I feel that, if anything, too much emphasis is placed on aiding the weaker students. Indeed, I thought I was the only one who felt this way until the time when I went to national Mathcounts/discovered AoPS.

What little experience I have has taught me that students will adjust to the intellectual environment in which they were placed. I felt immensely smarter at and after Mathcounts nationals, despite falling much lower on the intellectual food chain than I do in everyday life. In my opinion, then, rather than worrying about the weaker students catching up, the teachers should constantly be increasing difficulty, as well as level, of mathematics. The stronger students will probably still have an easy time, and the weaker students will still be struggling, but (to make an overexaggeration) isn't struggling on an IMO problem 6 a huge improvement from struggling over a chapter Mathcounts countdown round question? Look at Mr. Fischer.

Please excuse me if this post was too blatantly depressing and pessimistic.

By the way, I completely agree with the stronger students working on their own.

I don't think so. Not everyone has to become good at math. Some people want to have careers in something else. While math class ought to be enjoyable (or at least not be dreaded) for everyone, you have a right to not really care. What's important is that everyone is spending their time productively, whether or not they reach a certain level of mathematical aptitude.

Hmmm...I had this same discussion with someone a few days ago. Sadly, we used chemical kinetics to describe it. He said that exact same thing and noted that the overall reaction rate was determined by slow reaction mechanisms. However, I noted that fast reaction mechanisms (the ones with the double arrows) could transform the slow mechanisms.

Top students (or any students for that matter) don't necessarily need a teacher to feed them all the materials/resources. Motivated students have a way of finding that on their own. Rather, students (of all levels) need mentors. The best teachers I've had, the ones that have made a greatest impact on my life, are the ones that weren't afraid to be mentors and friends as well. The one thing schools have going for them is that it's a great place to mentor students; we want someone who will encourage and motivate us to succeed in areas we've never considered or never thought possible, and we also need someone who will keep our ego in check whenever our head gets too big to fit through the door.

I was talking with Dr. Reiter yesterday, and he seemed to be under the impression that "math circles" in California, as well as math clubs in Charlotte, are a HUGE success. In SC, we have nothing like that. I suggest getting these clubs/circles started in every state. It would be a huge challenge, but well worth it.

MSRI at Berkeley is going to be trying to do that. We're helping those who are trying to do this on their own without grants, etc., by letting them use our Foundation as a fund-raising vehicle.

When I was at a summer math program, where all the students were among the best at the high schools they had come from, I was amazed to see that the best students were approximately six times faster than the slow students. Such a wide range of talent, even among this already narrowly selected group of students! Imagine comparing the best student at a math camp to an average high school student, or to a poor high school student. Some students are capable of learning much, much faster than other students.

I think it's a big mistake to group kids together based on age, or approximately based on age, which is what happens in most high schools. A system should be devised so that smart kids are free (and encouraged) to zip ahead as fast as they can. At any high school, it should be typical to have a few 13 year olds who are more advanced than many of the 18 year olds.

It may be a mistake to have homework due at a particular time, for example. Maybe it would be better if the policy were that you turn in the homework as soon as you can finish it. Future homework assignments could all be available immediately so that students can start them as soon as they are able too.

This kind of system would also unleash the power of competition, because students would race to advance the fastest. That competetive feeling was a huge motivator for me at the summer math program I went to.

These are probably common ideas among AoPSers...