| I feel like you're talking past what I'm saying to continue teaching the same math lesson you've taught hundreds of times before, which is exactly the kind of discontinuity in communication that I'm trying to highlight (and, evidently, failing). It's difficult to articulate, and I already feel like this reply is rambling quite a bit, but hear goes: > we are definitely not asserting that the two expressions are the same. Correct. Not the same, equal. Because that's definitionally what the equals sign means. "A=B" is a symbolic representation of "'The expression A' equals 'The expression B'". I hope we can agree on that? >if your view were correct we wouldn’t spend time teaching how to solve the equation. What I actually said implies the exact opposite. You teach how to solve equations because that's the use case for equations as tools. That's not a bad thing, it's an extremely useful thing to teach. But teaching how a tool is used is not the same as teaching the fundamentals of what a tool is. It can help in that goal, certainly, (and might even be required as a prerequisite) but it's not the same. It's exactly like you said: >We teach algorithms like long division and the quadratic formula because they are relatively easy computations to learn but they don’t in any way lead students to fully grasping a concept. It's not fair to blame students' "innumeracy" for not being able to derive "0=1" as "an equation without a solution", because they've successfully learned the thing that they were actually taught, that equations are "things with unknowns that we have to solve for". Of course generating a solution that has neither unknowns nor a solution is foreign, because everything they've learned about it as a tool goes against that. (It's worth noting that there's another reason that the teachers teach this, one that's perhaps even more important for the school system; it's an easy thing to evaluate student understanding of. You can easily test whether a student can "solve" an equation, and return the correct answer. It's something you can get immediate, iterative feedback on. You can't really test if they actually grok a definition, because they can just parrot a definition with no understanding.) Fundamentally, my argument is about language, not mathematics. You're saying that students aren't able to derive answers based on the definitions of terms, but not only are those definitions wildly divergent from their English meaning, they're divergent from the actual definitions the student are learning via practice. Take, for example: >There are values for which the two polynomials evaluate to the same number. Those are the solutions. Values of x, you mean but didn't say. Because it's so heavily implied in the existence of an "equation to solve" that unknown quantities you are solving for are the ones written in the equation itself, that it's not even worth mentioning. But it's precisely this linguistic assumption that obscures what an equation actually is to students. |