>very few high school math teachers who have a good idea of what mathematicians do
(Professional mathematician here)
That's unfortunate, and it may be solvable. Indeed, professional mathematicians are usually the ones teaching prospective high school teachers during undergrad, so we are the ones to solve it.
You have define "what mathematicians do" properly, though. Here is a recent paper that I (and two others) wrote:
Nobody is going to understand that without a lot of specialized training. That shouldn't be the goal.
But problem solving comes close. For example, the "riddler" problems on fivethirtyeight.com are fantastic.
Here is an even better example:
- Write down all the integers which can be written as a sum of two integer squares: 0, 1, 2, 4, 5, 8, 9, 10, 13, ... Go up to at least 200.
- Look for patterns. At first, your descriptions will be vague. Keep refining them; eventually you get a nice exact description of which integers are on the list and which aren't.
- Why are these patterns true? You've made a conjecture, can you prove it?
- Are there similar such patterns with related questions? For example, integers which can be written as x^2 + 2y^2? As the sum of two cubes, or of three squares? What is the special sauce needed to make your pattern tick? Can you prove more cases, or a still more general theorem?
This problem is accessible to anyone who knows basic high school algebra, but it requires a lot of effort (and, probably, in most cases, a lot of mentorship to help the student through it). We should try to do more of this kind of thing with our students: it is exactly what mathematicians do.
> Indeed, professional mathematicians are usually the ones teaching prospective high school teachers during undergrad, so we are the ones to solve it.
The vast majority of people who are paid to use their knowledge of mathematics aren't doing math research. In fact, mathematics research is so uneconomical that most mathematics researchers spend 20%+ of their job on teaching.
The number of people who are paid a full 12 (or even 9) month salary to do mathematics research is vanishingly small.
> This problem is accessible to anyone who knows basic high school algebra, but it requires a lot of effort
That effort is really difficult to get out of students because few students have intrinsic motivation, and there's very little extrinsic motivation (unless 3 post-docs is your idea of a good way to spend your 30s...)
I'm not sure what the solution is, but showing more people what mathematics researchers in universities do with their non-teaching time seems like the wrong solution.
What I described is what I consider to be "doing mathematics". As distinct from applying mathematics to other problems (also a very worthwhile endeavor!)
I'm certainly not encouraging everyone to pursue a career as a math professor; the job market is poor, and is getting bleaker.
But I feel like prospective math high school teachers should have an experience like the one I described. Much more important, in my opinion, than learning how to row reduce matrices, do integration by parts, or prove the intermediate value theorem.
>This problem is accessible to anyone who knows basic high school algebra
Are you sure? Spotting the pattern isn't easy and I imagine a lot of high schoolers would give up in frustration while going down wrong roads. A good portion probably wouldn't even write down the numbers without errors. And even if you get past that, most high schoolers don't know how to do/write proofs. I'd be shocked if more than 10% could actually complete all your steps.
With challenging problem solving like that I'm afraid you're going to breed a lot of resentment.
I never said it was easy. In fact, in some sense, the problem is much harder than you might realize at first. This particular rabbit hole goes down very deep.
I am not urging this family of problem on all high school students. Rather, on all prospective high school math teachers. They should be able to communicate the joy of problem solving and discovery to any of their students who show an interest and aptitude -- which means these teachers should have done some of this themselves.
(Professional mathematician here)
That's unfortunate, and it may be solvable. Indeed, professional mathematicians are usually the ones teaching prospective high school teachers during undergrad, so we are the ones to solve it.
You have define "what mathematicians do" properly, though. Here is a recent paper that I (and two others) wrote:
https://arxiv.org/pdf/1708.00044.pdf
Nobody is going to understand that without a lot of specialized training. That shouldn't be the goal.
But problem solving comes close. For example, the "riddler" problems on fivethirtyeight.com are fantastic.
Here is an even better example:
- Write down all the integers which can be written as a sum of two integer squares: 0, 1, 2, 4, 5, 8, 9, 10, 13, ... Go up to at least 200.
- Look for patterns. At first, your descriptions will be vague. Keep refining them; eventually you get a nice exact description of which integers are on the list and which aren't.
- Why are these patterns true? You've made a conjecture, can you prove it?
- Are there similar such patterns with related questions? For example, integers which can be written as x^2 + 2y^2? As the sum of two cubes, or of three squares? What is the special sauce needed to make your pattern tick? Can you prove more cases, or a still more general theorem?
This problem is accessible to anyone who knows basic high school algebra, but it requires a lot of effort (and, probably, in most cases, a lot of mentorship to help the student through it). We should try to do more of this kind of thing with our students: it is exactly what mathematicians do.