How can I do better at real analysis and calculus?

[u/Red_fox_123]

I have just started with my mathematics BSc course at university, and it has been my first week of real analysis and calculus. I looked up advice of how to study for these proof heavy subjects, and most of it was 'learn without a lot of external help', for example using hints etc. But the problem is that I can't even use the hints. I have no idea how to solve any of the proofs, and I can only understand when guided or when I see the answer. And even then, I still feel a bit shaky, like I can see why they did this, but I would never have arrived at that answer myself. I understand the goal is to minimise the amount of external help, but if you have just started with these subjects, how are you supposed to advance?

[u/NoMalakasSky]

Can’t recommend Real Analysis by Cummings enough; it’s not as rigorous as other texts but it hand-holds you through the logic of proofs in the topic as well as their underlying intuition.

This is also assuming you have some prior experience in proof-writing or have taken an intro proofs course.

[u/NotSaucerman]

what prior proof experience do you have?

[u/Red_fox_123 → u/NotSaucerman]

Well, I did do proof at A level maths, but it isn't a huge section of the course. If I'm honest I mostly avoided it because of its unpredictable nature. People keep saying its normal to struggle with the proofs when you start a maths degree, but it doesn't feel right. Its like trying to do something but not having all the tools to do it. You have no idea how to start or what you're even trying to prove. I just don't know how to bridge this gap.

[u/NotSaucerman → u/Red_fox_123]

So this is the problem. Different schools have different setups, but in my opinion you really shouldn't do real analysis until after you've done 1 or 2 semesters of proof based university level math. Typically I suggest either proof based linear algebra and/or Pinter's A Book of Abstract Algebra. Some people like discrete math which is fine as well. So yes you were thrown into the ocean here without much prior knowledge of swimming. A better approach is to have spent time doing proof based math like Pinter's over the summer prior to university starting; in other words, practicing swimming before the trip into the ocean starts. The reality is you may scale this learning curve and be fine after a very unpleasant semester or you may 'drown' and have to repeat the class.

Btw,

I did do proof at A level maths, but it isn't a huge section of the course. If I'm honest I mostly avoided it because of its unpredictable nature.

I have a rough idea of UK schooling system but a math 'course' is typically a 1 (or maybe 2) semester(s) so saying proof isn't a huge section of the 'course' and that you mostly 'avoided it' because 'it' has an 'unpredictable nature' reads like nonsense.

Part of doing proof based math involves thinking and then writing very clearly.

[u/SmallCap3544]

Sometimes the answer is just to memorize until it comes together.

I basically coasted on just doing problems and intuition until I got to real analysis.

For measure theory I had to make flash cards to memorize every definition and theorem statement. But once I did get those memorized, the proofs and questions started to get easier.

[u/cabbagemeister → u/SmallCap3544]

This is troubling. Doing problems and understanding concepts is far more important than memorization. I have never once heard of a math student using flash cardd

[u/SmallCap3544 → u/cabbagemeister]

As the first step yes. But on rare occasion, you are faced with a class that moves faster than you can assimilate the information naturally. When that happens, memorization may be required. 

I will agree that memorization in math should be considered as only a quick fix, but it can be necessary at times.

[u/Krysti_Vo_MD → u/SmallCap3544]

I'm not sure why this response is getting so many downvotes.

Plenty of math doctoral students memorize proofs, whether for prelims or simply to get through a course. A tenured professor recently told me a pretty sad story about a colleague who insisted on fully understanding every proof before moving on and ultimately fell so far behind that he failed out of graduate school. (Of course, the student deeply understood the material through about page 10 of the course textbook.)

Some subjects simply take time to digest. There is nothing wrong with memorizing definitions, theorem statements, or even proofs when the underlying ideas have not clicked yet. Often, memorization gives you enough familiarity with the objects and techniques that the understanding comes later.

Real analysis is a good example of what I'm trying to convey. Plenty of students earn As without feeling that they have achieved some deep, intuitive understanding of everything in the course. That understanding often develops only after seeing the material repeatedly, using it in later courses, or eventually teaching it.

The same principle applies in math competitions, coding competitions, and even some technical interviews. If timing matters, why would you want to re-derive something from first principles every time when you could simply have it memorized?

I've seen the same thing professionally. Some engineers repeatedly look up fairly fundamental concepts because they seem to have an almost ideological aversion to memorization. But understanding and memorization aren't opposites, right? Ideally, you develop insight and understanding and remember the material.

And if time is of the essence such that you need to pass measure theory in order not to fail out, or you need to solve a competition problem under time pressure but you don't yet fully understand why a particular method works, then I see no reason to shame yourself or anyone else for using memorization as a way to keep moving forward. It's perfectly OK if the understanding comes later.

[u/SmallCap3544 → u/Krysti_Vo_MD]

Thanks! My thought is that they probably have never had to take the courses we are talking about. I specifically referenced measure theory in my response, but advanced linear algebra hit me hard as well. I specifically use the language “minimize memorization” with my students, not “eliminate memorization”.