One becomes a mathematician, even if only temporarily, if one becomes interested in an investigation into purely mathematical matters (e.g. in finding a rigorous proof of a statement) rather than in using mathematics as a computational tool in one's (original) area of interest, such as physics or biology. Even when one happens to invent a new mathematical method that works, one does not, in general, instantly become a mathematician - unless, of course, they lose the focus and turn their attention to making the method just discovered more efficient or better substantiated from the purely logical standpoint.

I'm thinking of an analogy: Musicians. The definition covers all levels of involvement and skill, but every "musician" pursues some amount of musical activity and learning, voluntarily.

Nobody has a choice about math in K-12 school, though there are kids who will choose the more advanced classes on their own. For instance my daughter proactively convinced her high school to let her skip a grade in math.

In college, you can choose to major in math, or in a math intensive subject like CS or physics. Within those disciplines, you can choose the more mathematical specialties. In the work world, you can volunteer for assignments that involve math, or get a reputation for being willing to solve hard math problems. That's me.

I don't see it as a "switch" because my interest in math was evident (so I'm told) before I could even talk.

I think you need to define your terms because I see at least a couple different classes of answers.

Speaking personally, a lot of people would probably say that I'm "good at math" in the sense that I got through a fairly rigorous engineering program and that I've never had issues with the quantitative side of business or other such pursuits. On the other hand, I've never personally considered myself a "mathematician" in the sense that anything approaching upper-level university math was something that came remotely naturally.

I'd say for me, it was when I properly learned how to write proofs. It just formalized everything I knew, and forced me to think about it logically.

In the context given in the article it sounds like non-mathematicians are students who choose to stop maths at age sixteen, presumably for the rest of their lives. This reflects the observation that children are born mathematicians and explorers and remain so until something kills their curiosity (often school is named as the culprit).

When you mentally translate theories and discussions from words to equations rather than equations to words.

I thought Terrence Tao had an interesting viewpoint on this:

https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

I think you switch to a non-mathematician as soon as you proclaim your hate for mathematics. I don't like seeing 'mathematics' as some kind of specialist activity that only a specific group of people is capable of.