I’ve held these beliefs for a while, but kept them fairly private. Since a major theme of mine during the last year and a half has been the necessity of an appropriate educational system for nurturing great thinkers, I think I’ll state them in the open:
We really must, as a society, move beyond the “no child left behind” mentality. We cannot afford to slow classes down more and more in order to cater to the slowest learners. It does a great disservice to the normal children and a far greater one to accelerated learners. The manifestations of this are pretty plain: boredom, lack of focus on work (but not a general lack of focus), preference for self-devised side projects, frustration, detachment, etc. If these abound in a classroom, chances are the work is too slow.
The worst culprit, however, is not No Child Left Behind. It’s the concept of a grade. Even at young ages, mental ability is not uniform. It is absolutely unfair to group all children of a certain age into a single grade, learning all the same materials, and then to keep them there for a fixed amount of time. Sorry, but if a student learns multiplication halfway through 1st grade, for instance, he really should not need to keep learning it for the rest of the academic year – move the work up to something more appropriate for the student’s background.
The rest of this will concern math, because it’s the area I’ve tutored most extensively. However, it can be applied to any field in a similar manner, taking appropriate safety precautions in fields such as chemistry, of course. (“You know how to use a bunsen burner, now let’s work with some HCl!” is probably not a good idea).
From my own experience as a student and a tutor, I bet we’ll see students learning algebra in 3rd or 4th grade in such a system. I started learning it at 8, but I had no help in the matter (as usual; I’ve always had to do everything on my own), so I bet we could teach it even earlier, at least to the mathematically gifted. Taught properly, rules from arithmetic supply the intuition – things like multiplication and division being inverse operations, associativity, and even binomial multiplication (FOIL) on simple things that the student can verify conventionally, including showing how it emerges from the distributive property, such as (5 + 1) * (4 + 3) = 5(4 + 3) + 1(4 + 3) = 5*4 + 5*3 + 1*4 + 1*3 = 42. Anyone who knows simple addition and multiplication can of course verify this simple example by adding 5 and 1, 4 and 3, and then multiplying the resulting 6 and 7, which allows the student to verify for himself that the identity does indeed work on this example. After this is done, generalize by moving onto variables.
Once basic algebra is mastered (and it doesn’t need to take years), introduce the basics of calculus. If you’re feeling really adventurous, it’s probably an ideal time to introduce some abstract algebra as well – right after a student finishes generalizing numbers to variables, they’ll be in the right mindset to generalize variables to teach them to generalize algebra itself – to get rid of the whole “number” thing and just talk about systems.
Logarithms and things are all special cases in calculus and can be safely ignored until the students learn about them (their “special” properties really arise from the definition of the operators anyway). Limits are easy to teach – use a number line and terminology like “gets closer and closer”. And DO NOT tell me you can’t teach someone d/dx(x^n) = n x^(n-1) as soon as they know algebra, because I won’t buy it. If they can learn the quadratic formula, they can certainly learn how to differentiate a polynomial.
Integrals are taught pretty well as-is (Riemann sums and the Fundamental Theorem of Calculus give good intuitive foundations for the concept), but again, far too late. It should be 7th / 8th grade sort of stuff at the latest.
The idea is to teach students enough that they can learn the rest, then move on and let them learn the rest (with help, if necessary). Exactly how much is “enough to learn the rest” varies from student to student and must be recognized on an individual basis.
Forcing homework upon them isn’t going to help them learn the rest, either. They should be encouraged, but not coerced. Students start out wanting to learn – in my own experience, the younger my proteges, the more enthusiastic they were about learning science, engineering, and/or math. Associating boring and repetitive work with such subjects for years eventually turns most people off to it (or worse, makes them think they can’t do it when they really can). The high school and college students tended to be the least enthusiastic, because they had the choice of (a) rote study or (b) an active social life. Assignments force them to the choice while providing very little benefit. Einstein states that “it is a miracle that curiosity survives formal education”, but, scientist that he was, he hasn’t seen the casualties.
Ultimately, it is society that suffers.
Yes, I’m aware that Montessori came up with a similar idea, but I’ll stop saying it when I see people acting on it. It doesn’t matter whether the idea already exists if people don’t do anything with it. Further derivation just serves to underscore the need for an implementation.