TLDR: We can say "it's wrong" less and "it depends" more. (1/25)

#MTBoS @CmonMattTHINK @j_lanier @katemath @mrdardy

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Our expert blind spot: "Of course this means the MAXIMAL set of definition, so there's a right answer."

Reality: Unless it is tacit knowledge, 👏 domain 👏 is 👏 a choice 👏. (9/25)

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They're tacit knowledge to us who teach & use algebra daily. But only our most privileged & high-achieving students have fluency in "Mathematish." (10/25) researchgate.net/profile/Hakan_…

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But is it an identity? We can't decide until you/me/someone(!) says what are the domains of the functions on both sides! #WhoSaysMath (12/25)

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"The best math problems all have the same solution: 'It depends.'"

Is this an identity? It depends on domain chosen for f(x)=√x + √-x, and for g(x)=0. The former must be a subset of the maximal domain {0}. Latter cld be any subset of ℝ. (13/25)

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Always call a choice a choice. Too much dehumanization & anxiety arises in the math classroom from overuse of "THE" where "A" is appropriate. Give students agency to choose & knowledge to assess ramifications of their choice. (14/25)

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Category theory is a way to formalize that idea. (16/25)

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Common-sense asks: every object has an arrow to itself, we can follow two consecutive arrows to create a third, and following 3 consecutive arrows obeys associativity. (17/25)

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Or maybe only the polynomial functions (algebra 1), or differentiable functions (calculus)... (18/25)

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Why? Because every object has a (unique) arrow to itself, the so-called identity morphism 1ₓ : X → X. View these as "the objects!" (19/25)

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f(x) = x

doesn't tell us nearly enough about f. But it does, the moment you tell me its domain. Because it "is" its domain, in a unique way! (20/25)

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That's the role of "numbers" in our category. They're (bunches of) functions too, 1 for each subset X! (22/25)

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