How is mathematics made?

Being at the very beginning of my journey to becoming a big and important mathematics professor as well as working on my thesis for the last couple of months, I have often thought about how complex mathematics comes into existence. Take my thesis topic for example; the theory behind understanding and solving the moment problem spans multiple different fields, like complex analysis, operator theory, functional analysis, and somehow connects all of them into this elegant framework, which guides us to the solution. So, while I am given this piece of artwork to ingest and understand as much as I can, I sometimes wonder how the connections between all of these seemingly unrelated facts were made. What questions were asked? What things were tried?

Coming from a contest math background, I have had exposure to this thinking process before, but it was always pretty limited to the problem I was given. The connections have never spanned multiple layers of theorems, propositions, and lemmas in order to get to the final answer. Looking at all of the elegance, complexity, and interconnectivity of mathematical fields, one gets overwhelmed simply because very few understand how a human mind can come up with all of this and structure it into a coherent tree of knowledge.

I believe the reason for this is the way mathematics is taught to us in schools, colleges, and universities. What we see at the very beginning are frameworks that were polished and consolidated to be taught in the most effective way possible. Even when motivations for theorems are introduced, they serve merely as a pause for students to catch up on their notes during the lecture. I would also argue that most classes assign homework in ways that fail to develop the exploration skills crucial for mathematicians. Learning how to master freshly introduced tools is important, but it is more important to learn how to recognize them in settings seemingly unrelated to the field that those tools come from. In the US education system, the first real experience of wobbling around and trying to engage your knowledge to create new mathematics comes from participating in REUs.

My humble opinion is that this should happen much earlier. If not in the form of creating new mathematics, then in the form of training to recognize patterns outside of where you would naturally expect to see them. This can be done through contest math training. Usually, during contests, you are not given a clue on what theorems or tricks you should use to arrive at the solution. It teaches you to pursue your guesses and try again when your previous attempt was unsuccessful. The element of discovery is thus intrinsic to this format, though it might render some students frustrated. Hence, I believe this process should also stay entirely out of the grading system at the beginning. I do not think that there should be policing when wobbling around does not produce any results, but there should be an incentive to do so.

However, as I mentioned previously, while this type of training exposes you to exploratory thinking, it remains limited in scale. You're solving problems intentionally designed to be manageable within a given timeframe, which means they can't involve too many layers of theorems and lemmas to compose together. One solution is to introduce problems that students can't immediately solve with their current knowledge. This might spark curiosity and motivate some students to explore material beyond the classroom, though it could also cause frustration and anger, which seems the more likely outcome. I propose introducing one such problem and having students submit their progress weekly or biweekly, showing how they've applied the tools learned so far. Finding a problem whose solution aligns perfectly with the order in which tools are introduced won't always be easy, but it's possible. By semester's end, students would have accumulated enough tools to write a complete solution to the problem. This final solution should be graded rigorously, but the intermediate submissions should be graded lightly — enough to encourage participation without penalizing wrong attempts.

Of course, the approach described above is not a perfect one, and it has a lot of flaws in it, but it is a step towards mentoring students to understand how mathematics is actually made. Isn’t mathematics all about trying new things and looking where they lead you?