How to view undergraduate studies in maths?

Question

I am a undergraduate studying maths at the moment and I have been enrolled for 3 years almost. As I have not fully finished with the degree, I still have time and courses to take and there is this question/issue in my head that I don't always really feel comfortable with.

(This are all my perception) I think during undergraduate you don't become a mathematician but you lay the ground. At times I can get chance to see how is graduate life, how is doing research like and it sometimes look a bit like a fantasy to think that you can be productive in graduate years due to:

1. Tho learning maths can be easy or hard, what you learn in undergraduate may not make you ready to do research or even read papers and feel comfortable.
2. Even if you excel at undergraduate, solving unsolved problems is different from solving assignments and this if you think about it might make you feel like imposter that you are 'useless' actually.

I was wondering how to view undergraduate studies during your undergraduate years and benefit from it in a smarter way. One good advice was to go broad and get the big picture , you will have time to dive deep. And I just am looking for thoughts of people on the matter.

I am pretty sure this thoughts are common, so any link to similar issues or etc are welcome.

Answer 1

For the most part, what you study in undergraduate in most fields has been known for a long time. The same is true in maths. In fact, lots of things more than 100 years old aren't touched on at the undergraduate level.

Beyond that, maths are currently a vast field. It has been about a hundred years since it was possible for a single person to understand all of that (there are references to that, I'll let you look - see Henri Poincare, for example).

There are three things that you can learn as an undergraduate. The first, and simplest is what you mention in the question - a bit of skill in problem solving in a few of the major, basic, fields.

Beyond that, is gaining practice in "a mathematical way of thinking", or reducing complex problems to simpler parts that can be attacked with the tools available. Thinking in terms of patterns and relationships.

Still a bit further, and not always achieved as an undergraduate, is true insight into a few of the important mathematical concepts. What is the deeper meaning of, say the derivative, or of an axiomatic system. This includes something about the limitations of mathematics.

If you can achieve all of that, then the undergrad days provide a good foundation for moving forward and to go deeper into some, probably quite narrow, subfield.