2nd Year Mathematics (Europe), my Uni results are declining. How do I improve higher level mathematics problem solving abilities?

Question

Context: I am a final year BSc. Math student (Europe).

I got some decent results in my first year, top of the class in Calculus I and II (with simple proofs), good results on basic Linear Algebra (determinant, systems, etc). But the second year has been brutal, I barely got through my Topology, Probability, Linear Algebra (Abstract), Abstract Algebra (Groups, Fields), Complex Analysis, Numerical Analysis(theoretical), Diff Geometry courses. Noted, the level of abstraction has considerably increased and no longer do the exams require simple, plug and chug formulas calculations.

What are some remedies to such a decline in performance? Higher level mathematics requires a lot of Theorem, Lemma, Corollary memorisation. I am trying memorisation techniques, writing statements on post it notes, etc.

Is there some applied mathematics specialisation that is suited for a student that excelled in Calculus, even some Real Analysis, has good spacial ability/memory but is mediocre(the grades happen to show this is not an understatement) at all the rest?

Grades is a motivation factor for me, it makes me feel fulfilled and relaxed. I am feeling quite frustrated and ashamed of myself (although I worked really hard), even scared higher level mathematics might not be for me( I was looking forward to enrolling in a MSc. in applied mathematics)?

Answer 1

You have come up against a couple of issues in mathematics study and perhaps both of them are causing you trouble.

The first issue is that many people have sort of natural "boundary", though not a hard one. But before you hit the boundary everything seems easy and fairly obvious to you and after that it gets hard. Hard isn't impossible but you have to "think different" to continue your progress. It is a lot of work. I learned fairly early that I had to work to learn anything and managed to do it. One way was to solve many more problems/exercises in a course than were assigned. A couple of times I just went into sort of trance mode solving a lot of similar problems. This led to insight into the behavior of real functions (Analysis, if you like). It also gave insight into relationships that I ultimately exploited in my doctoral dissertation. My sister was much better/smarter than I was and hit her boundary much later. Sadly, she gave up in the quest rather than learn to fight through it.

And yes, the level of abstraction increases dramatically in every mathematical field. Expect that. Work from the concrete (exercises) to the abstract (insight).

The second issue you've come across is that insight into math isn't universal. You can have deep insight into some math fields and very little in others. Partly this is due to the first issue, above, since it is a lot of work to excel in several math fields. Most don't do that. I also had some pretty good insights into topology and might have been able to do a dissertation there, but almost none in abstract algebra, even though it is heavily axiomatic like topology. Rings still elude me almost entirely.

So, if you want to excel in math, work hard in a chosen specialty. Don't try to learn everything at the same level lest that level be low. Work on a lot of exercises. Get guidance on your solutions. Try to abstract out the deeper meanings of how and why things fit together. Keep notes to record your progress. Keep notes as to where you need to make progress.

Sadly, though, Analysis is a bit past its prime. But, pick a field that you like and are comfortable in. There is some room still for good people in most fields, even some that have cooled over the years.

But, to really reach the top in a narrow math field, you need to reach the point where you can imagine what might be true based on what is known already. I don't think I'd achieved that level yet at the point of completion of my doctorate. It came a bit later, I think, though I had to move to another field.

---

Note that I don't believe nor mean to imply that there are people who can't do math. Short of a brain injury all of us can. You just need to learn how to work effectively (and hard) at it. Good teaching helps, of course. And the same is true in CS and perhaps some other technical fields.

Answer 2

What are some remedies to such a decline in performance?

Firstly, a mathematical characterisation of your problem. A decline in grades as you advance in the degree program does not usually mean a decline in your mathematical skill/knowledge --- it means that the rate at which your mathematical skill/knowledge is improving is less than the rate at which the material is becoming more difficult. In this sense, your "decline in performance" is not absolute; it is relative to a standard that is moving higher as you go further through your degree. Imagine two continuous time-series functions, one representing your skill/knowledge and the other representing the median skill/knowledge of someone pursuing higher mathematics as a degree then a profession. Both functions are going upward over time, but yours just has a smaller positive slope. (Meanwhile, for the vast majority of people in the world not doing mathematics education or professional work, their time-series is virtually flat.)

So before you do anything else, step back and breathe a sigh of relief and realise that you are actually already making substantial forward progress. You are progressing through your degree, and evidently not failing it, and you are now in your final year of undergraduate studies. You are probably substantially better at mathematics now then you were in first-year, notwithstanding your lower grades. You have nothing to feel ashamed or scared about --- you are training yourself successfully in mathematics, but your rate of improvement is just a little slower than the median expectation in this type of program.

---

Given the above premise, one potential remedy here might be that you just need some additional time to "catch up" to the relative level you were at before. For example, if you were to do an additional one or two years of study (e.g., in a Masters program) then you would probably bridge the gap from where you are to where you want to be. Even assuming you were a mediocre student in a Masters program, you might exit it at a level that would be relatively high compared to a student completing their undergraduate degree.

In terms of learning methods, it sounds like you need to make the transition from "plug and chug" to first-principles knowledge and the ability to derive mathematical results. I would suggest you concentrate your efforts on learning the underlying logic and derivation of results where you know "the formula", to give you a deeper knowledge of the material. By practicing first-principles derivation of results, you will give yourself a good intuitive understanding of mathematical results, and be able to prove results more easily. (When I was an undergraduate, I was the opposite; I was forever forgetting the formulas, but I had a good understanding of how they were derived. Consequently, I could always re-derive them in exams when I forget. During my undergraduate degree, the number of times I forgot the signs on the quotient rule, or forgot the exact form of the infinite geometric sum formula, etc., was quite embarassing, but it was also a blessing because it forced me to understand these results from first-principles instead of just memorising a formula.)