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As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that
by wcarey 6y ago
As someone who has pushed pretty hard towards precise definition and rigor as a teacher, you're in the (happy!) minority being able to approach mathematics that way. For lots of students, the process of moving from definitions and simple rules to other statements is extraordinarily difficult.
I teach the properties of exponents to 14 year olds as a unit on logical necessity - the properties all flow necessarily from the definition of the exponent. About a third of the students say, "cool" and never miss anything on any assessment again because the answers flow necessarily from the givens.
About a third work their way through it fine.
About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 despite working many particular examples where that is manifestly false.
- 08-15 6y ago> About a third continue to maintain that, say, a^2 + b^2 = (a+b)^2 ...and I bet, their justification is "It looks right!" or "Why not?" I've seen this, too. A friend of mine had a habit of cancelling sums, such as (a+c)/(b+c) = a/b. Why? Because it looks right, and why not? And it isn't so different from (ac)/(bc) = a/b, is it? Of course, she never asked why a certain manipulation is legal. Arithmetic manipulation is legal because the teacher said so. I suspect, this isn't a property of their personality; rather it's a failure of their math teachers to convey the idea, that everything has a reason, and if you don't know the reason why some manipulation would be allowed, it very likely isn't. By the time you tried to teach exponentiation to these students, the damage was long done.