Over the vacation, I’ve had some thoughts which have just now bubbled to the surface:

When you call the Fountainhead or Atlas Shrugged a great book, please remember that the former was rejected by 12 different publishers before it saw the light of day.

When you use technologies developed around neural nets, think back on how much more developed the field (and consequent applications) would be if scientists did not neglect NNs for a decade because they misunderstood something Marvin Minsky said in his book.

When you speak of elliptical planetary orbits or heliocentricism, remember that your worldview would likely have you excommunicated and/or put to death 400 years ago. In fact, you can regard all of your thoughts that conflicted with anything Aristotle said – and a lot of what Aristotle said was wrong – in the same manner.

When you use technologies derived from genetics – when you speak of heritable traits – when you breed cats – when you are screened for diseases you have a high risk of getting because of family history – recall that evolution is still a contested idea that many would like to suppress.

The computer you view this on? Perhaps five of them would exist in the whole world if pundits had their way. The Internet? It would just be for universities and governments. Jesus himself was killed for expressing an unconventional idea.

So stop for a moment and think: do you take part in this collective? How many ideas have you suppressed? How can you be sure that an idea is “bad”? If you cannot, you lack the knowledge to make a rational decision.

Kant is an extremely underrated philosopher. He had some absolutely great ideas. Some of them are similar to principles of my own philosophy, such as his “categorical imperative” and stance as a nondeterminist, while others are similar to principles of Panidealism (which I guess is another facet of my own philosophy), such as the absolute inability to know what the nature of free will is and thus the lack of ability to make an objective moral judgment based on this inability. If you replace morals and free will with a general concept of an “idea”, you have a central Panidealist tenet. Actually, a quick review of his main works indicates that Kant may have hit on this in his “Critique of Pure Reason” (which, in my defense, took him 10 years to write) – the resulting philosophy is called “transcendental idealism”. However, it’s only one Panidealist principle, although an important one, and so my philosophy chugs onward.

Even where his ideas differ from mine, they remain intriguing.

So far I’ve been told my developing philosophy has similarities with Plato, Aristotle, Kant, Berkeley, and Russel, and, aside from the Republic, I’ve never studied any of their philosophy. I don’t study philosophy as a subject, as I believe that it’s foolish to let your own philosophy be influenced by the thoughts of others. I simply think, and the philosophy of others just… emerges.

Even if it bears similarities with other philosophers, my philosophy remains new at least in how I combine these principles. The best way to forge ahead in philosophy is to simply ignore everything that came before and think. Maybe you’ll reinvent many things. Maybe you’ll say the same thing in different ways. But somewhere in there, there will be novelty.

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.