Mathematics is the art of explanation.
Paul Lockhart
Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations.
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Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.
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A good problem is something you don't know how to solve. That's what makes it a good puzzle and a good opportunity.
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Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles?
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[Math] curriculum is obsessed with jargon and nomenclature seemingly for no other purpose than to provide teachers with something to test the students on.
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In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world.
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It is the story that matters not just the ending.
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Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved'.. Is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way.
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