Somewhere in school I got the backward idea that solving math problems is hard but that formulating them is easy. I don’t know if anybody ever said that to me. Maybe it was just implied by years of solving problems someone else had formulated.

A related wrong idea that I also picked up was that formulating math problems was not a mathematician’s responsibility. Someone, probably an engineer, would formulate the problem and hand it over to a mathematician. That happens occasionally, but that’s not how it usually works.

Formulating problems is hard, and it’s usually the applied mathematician’s responsibility, ideally with generous input from a domain area expert.

There are a lot of ways to turn a real world problem into a math problem, and maybe several of them would be adequate for the task at hand. Then you might as well choose the easiest one to understand and compute. Knowing several ways to formulate a problem increases your chances of find one approach that’s tractable. Particularly when you can determine what problem *really* needs to be solved, not just the problem you first see, you might give yourself more options for how to go about it.

Applied mathematicians don’t need to be an expert in every area of application, and of course cannot be. But they do need to meet clients half way (or more). They need to know *something* about the problem domain. They need to listen well and need to ask good questions. The questions help the mathematician get going, andÂ they may also give the client something new to think about.

I think this is made even harder because formulating the problem is not practiced an awful lot in k through 16 mathematics. In fact, in my own experience, I don’t think I had one opportunity to formulate a problem for myself, every single problem was given to me by someone else.

It wasn’t until I started teaching that I started really formulating my own problems (some just for fun for myself, not for my students) and find out how difficult it is to formulate a well structured and solvable problem. In your line of work, it might be even more difficult because the problem and its solution have to math to some aspect of reality!

Funnily enough, this is often true in “pure” mathematics as well. Formulating the right problem is sometimes more than half the battle. It is unfortunate that this is so little emphasized in any of our education, even up through graduate school.

Agreed.