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> They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essen

by noahl 15y ago

> They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. It's not entirely clear what you mean by this, but in the most obvious interpretation this idea is correct. Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. The other interpretation, which you probably meant, was the question of mathematical style and interest: which axioms do we study, and why do we care about them, and why are some theorems important and others not important? Certainly the definition that 1/x^2 = x^(-2) was made for consistency, which is an aspect of style. You could make some other definition and do formal reasoning just as well (although you'd have to change some other things - maybe the definition of exponentiation, maybe your interpretation of symbols on paper). But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students.

yaks_hairbrush 15y ago
> Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. From one viewpoint, math deals with pure logic. It's something of a poor viewpoint from my perspective: Newton couldn't back up his calculus with logic, Euler and Riemann made numerous unfounded assumptions when looking at the zeta function, Heaviside built a telegraph across the Atlantic despite lacking a proper logical foundation. My personal viewpoint is that mathematics is a series of shortcuts for understanding and manipulating a wide variety of phenomena, and mathematical research is the development of further shortcuts. Often logic comes in, but it's usually after you get the result. Nothing wrong with logic (I sure do like computers, for example!), it's just not the panacea for anything mathematical. > But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students. You're absolutely right about this, and I was specifically thinking of two very bad high school teachers when I wrote my post. The fact is that I know very few HS math teachers, and so my opinion is quite clouded by these two.

by noahl 15y ago

yaks_hairbrush 15y ago

> Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet. From one viewpoint, math deals with pure logic. It's something of a poor viewpoint from my perspective: Newton couldn't back up his calculus with logic, Euler and Riemann made numerous unfounded assumptions when looking at the zeta function, Heaviside built a telegraph across the Atlantic despite lacking a proper logical foundation. My personal viewpoint is that mathematics is a series of shortcuts for understanding and manipulating a wide variety of phenomena, and mathematical research is the development of further shortcuts. Often logic comes in, but it's usually after you get the result. Nothing wrong with logic (I sure do like computers, for example!), it's just not the panacea for anything mathematical. > But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students. You're absolutely right about this, and I was specifically thinking of two very bad high school teachers when I wrote my post. The fact is that I know very few HS math teachers, and so my opinion is quite clouded by these two.