(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.