> 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.
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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.
Your circle area formula is incorrect - you square the radius first, then multiply by pi. It's quite easy to see why if you substitute tau/2 for pi - 1/2 tau r^2 is clearly the integral of tau * r, where tau radians is a complete circle. Mixing diameter and radius in the same formula and hiding the resulting factor of two in a constant is a pedagogical disaster.
The mapping of the symbols onto reality was often missing.
Perhaps because it's not a part of math? Math is an art. It's totally unconcerned with things like "reality". If you're so concerned with reality, you've probably never done math.
If that's your definition of art, then I can only consider myself fortunate to have been doing something else. Though, it's hard to see why anyone would gain anything in studying it were that the case. They could just make totally random shit up and claim, with as good a justification as any other, that it was as worthwhile as the work of any renowned mathematician.
However, I don't agree that art is unconcerned with reality, nor that maths is. We learn to draw by looking at things in the world, we get our rules about anatomy and so on from there; we learn maths based - at least initially - on physical examples; we tell stories based upon common themes and situations. The basic rules of these things are drawn from their correspondence to reality; with what people have experience with; and form the meaningful grammar of the system. Art is always a language, and a language is always representative. Even when - as arts - we may bend a few rules and simplify certain aspects to lend emphasis or make it better suited to approaching a particular problem.
To hold otherwise seems to me to deprive them of a foundation to communicate any common meaning. It would be no better to learn maths, under that definition, than to spend your life insanely scraping crayons across a page. (Indeed, given that the symbols would also be arbitrary in their meaning, it would be hard to tell the difference.)
Is maths an art, is it a science?
It's a language of enquiry. To claim that it proceeds solely by a 1-1 correspondence to reality is to picture mathematicians off somewhere counting things and deriving pi from taking increasingly precise measurements of actual circles. (And, really, even physics proceeds with the help of a healthy dollop of imagination that forms the foundation of hypothesis.) To claim that it has no correspondence to reality is to reduce the whole exercise to nonsense.
I do not see how either position taken as an absolute is tenable.
Some do, others don't. There is an interesting dichotomy: much of mathematics does appear to be completely dissociated from reality, and yet in the other direction, reality appears to be entirely mathematics. I think to reach a unification, it is absolutely essential that we have people working in each direction. You aren't likely to develop a theory of (∞,n)-categories by working backward from reality, and yet it turns out to be useful to have done so once you start exploring e.g. topological quantum field theories.
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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.