| > It seems that "solving algebra problems and doing two-column geometry proofs" is a necessary step on the road to "generating your own questions about whatever interests you and trying to answer them". That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics. --- I'd agree with you, if what we taught was an understanding of the concepts and established mechanisms. However, it seems to me that, most of what I saw in schools was just symbol manipulation. For example, people didn't actually seem to understand that to get the area of a circle you took the radius, multiplied it by the ratio of the diameter to the circumference and squared it. They understood that you took the radius, multiplied it by a magic number, and for unknown reasons squared that. The mapping of the symbols onto reality was often missing. It wasn't problem solving beyond the level of having a lookup table in your head that said 'When calculating an area do this, then this, then this.' All that said, there are things it makes sense to memorise after you understand them - low level components where the speed gained in doing so allows you to use them in higher level abstractions. My point isn't that it doesn't make sense to teach people tools. But that to just give them the tools without the understanding of how they function seems harmful to their ability to create and adapt their own tools down the line. |