4 ms·

Could you expand about absolute value? Is seems reasonable to me to think of it as a function.

lmm 12y ago
Maybe pedagogy recapitulates history of mathematics. "What is a function?" is kind of a wrong question, because "function" is just a word (what is it you need a "function" for?), but in a lot of early mathematics the idea tended to describe a sort of physical intuition, that a function was something like the shape that something physical (like a string under tension) - remember that in the early days mathematics was more a branch of physics than a discipline in its own right. In nature things tend to change smoothly - what we now call "analytic functions" - and you wouldn't ever have a sharp inflection point like there is for absolute value at zero. A more modern notion of a function only really came about after the discovery of the fourier transform, where it turns out that functions like absolute value (and sillier things like the dirac delta) have a fourier transform, and, worse, some perfectly ordinary functions like sin have fourier transforms that come out as these weird things. Eventually (and, I suspect, partially because the tools of fourier analysis were so useful) people started treating "something that can be fourier transformed" almost as the definition of a function. (The present set-theoretic definition of a function is even more recent; I would assume it comes from Russell/Hilbert/etc.'s efforts to give mathematics a formal foundation by expressing everything in terms of set theory)
MichaelGG 12y ago
Well it was introduced as "absolute value uses lines on each side. It means you drop the negative if there is one". I asked what the point of it was, cause it sounded sorta useless. Instead of saying "OK, well, how would you express distance, x1-x2?", she was just stuck. I suppose that's what happens when you have general teachers that don't really know maths or science, don't really feel that it's interesting and offers amazing ways to look at the world. In fact, I'd be surprised if she actually knew beyond what she was teaching. It's not like middle grade teachers have to take advanced courses, and they do have a wide workload.

lmm 12y ago

Maybe pedagogy recapitulates history of mathematics. "What is a function?" is kind of a wrong question, because "function" is just a word (what is it you need a "function" for?), but in a lot of early mathematics the idea tended to describe a sort of physical intuition, that a function was something like the shape that something physical (like a string under tension) - remember that in the early days mathematics was more a branch of physics than a discipline in its own right. In nature things tend to change smoothly - what we now call "analytic functions" - and you wouldn't ever have a sharp inflection point like there is for absolute value at zero. A more modern notion of a function only really came about after the discovery of the fourier transform, where it turns out that functions like absolute value (and sillier things like the dirac delta) have a fourier transform, and, worse, some perfectly ordinary functions like sin have fourier transforms that come out as these weird things. Eventually (and, I suspect, partially because the tools of fourier analysis were so useful) people started treating "something that can be fourier transformed" almost as the definition of a function. (The present set-theoretic definition of a function is even more recent; I would assume it comes from Russell/Hilbert/etc.'s efforts to give mathematics a formal foundation by expressing everything in terms of set theory)

MichaelGG 12y ago

Well it was introduced as "absolute value uses lines on each side. It means you drop the negative if there is one". I asked what the point of it was, cause it sounded sorta useless. Instead of saying "OK, well, how would you express distance, x1-x2?", she was just stuck. I suppose that's what happens when you have general teachers that don't really know maths or science, don't really feel that it's interesting and offers amazing ways to look at the world. In fact, I'd be surprised if she actually knew beyond what she was teaching. It's not like middle grade teachers have to take advanced courses, and they do have a wide workload.