Learnin'

The Lumonator tackles the subject of education, and sort of seems to be going somewhere interesting until one of his fits takes him and he gets stuck cursing his many enemies. Learning, he thinks, is a bit like a wavelet decomposition - you start out with approximations and gradually refine them putting in more and finer detail. I can see this, to a degree. My toddler grandnephew used to refer to all four legged creatures as "dog" when he was little.

It seems that Lubos is doing some tutoring of one of his nephews - my sympathy to Kubo - who evidently needs to do a lot of simplification of rational polynomial expressions. Now that isn't a very useful skill, since most such don't admit much simplification, but it is an exercise in pattern recognition. The trick, of course, is to factor the polynomials.

Lubos also thinks that students don't get enough negative reinforcement, so I will offer some for him: your theory of progressive refinement is only the less important part of learning. Before you can refine, you must have generalizations that can be refined. Some of those are likely innate, like the concept of the four legged animal. Others, like the laws of physics, must be synthesized. One that seems to be difficult for the less mathematically inclined is the notion of equivalent fractions.

The idea that 1/3 and 27/81 are different ways of writing the same number, or that (x^2+4x+4)/(x^2-4) is the same as (x+2)/(x-2) doesn't seem to be reachable by refinement of some more primitive notion of number. Instead it is a generalization of the notion of number and the notion of equality.

Pattern recognition in general is a challenge to the idea of progress through cataloging of exceptions. The fundamental notion of pattern recognition is symmetry - the something that stays the same when other things are changed. This skill is very crucial for survival, so it's no surprise that we come equipped with a lot of pattern reconition hardware. A lot of education is adapting that hardware to slightly different tasks. There is a commonality in the recognition of the lion from his paw print and the recognition of a factorable polynomial.