This link has been bookmarked by 86 people . It was first bookmarked on 27 Nov 2007, by Joel Liu.
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Textbooks rarely focus on understanding; it’s mostly solving problems with “plug and chug” formulas. It saddens me that beautiful ideas get such a rote treatment:
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Think we’re done? No way. In 1000 years we’ll have a system that makes decimal numbers look as quaint as Roman Numerals (“By George, how did they manage with such clumsy tools?”).
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Fractions (1/3), decimals (.234), and complex numbers (3 + 4i) are ways to express new relationships. They may not make sense right now, just like zero didn’t “make sense” to the Romans. We need new real-world relationships (like debt) for them to click.
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By the way, negative numbers weren’t accepted by many people, including Western mathematicians, until the 1700s. The idea of a negative was considered “absurd”. Negative numbers do seem strange unless you can see how they represent complex real-world relationships, like debt.
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- Factual knowledge is not understanding. Knowing “hammers drive nails” is not the same as the insight that any hard object (a rock, a wrench) can drive a nail.
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Don’t stop until it makes sense, or that mathematical gap will haunt you. Mental toughness is critical — we often give up too easily.
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Factual knowledge is not understanding.
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Marshall PreddyAn interesting approach to understanding how math works and why it works.
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ken .Anchoring the abstract in every day experience, embodied, making sense of why it makes sense, really - counting, naturally, physical things, accounting for things -> negative numbers model "relationships" - flows, information about inter-action :)
accounting communication complexity creativity economics flow information learning life math meaning systems thinking value review:nov
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Elegant, “a ha!” insights should be our focus, but we leave that for students to randomly stumble upon themselves. I hit an “a ha” moment after a hellish cram session in college; since then, I’ve wanted to find and share those epiphanies to spare others the same pain.
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