What and how to study for a pure maths education at university?

[Garvett]

TL;DR Summary: The university pure maths courses feel rough and uninsightful, yet exclude (temporally) the possibility of studying good texts. What and how does one study in order to attain such understanding as to be profound, but allowing, by structure, still to follow the curriculum?

Greetings! I know that is not exactly a maths question, but it is something very pertinent with me in regard to the process of studying maths. I have got into a university, which is considered one of the best institutions in my country where you can a get a pure maths education from. However, I am still, for some reason, very sceptical of following lecture notes. Let me clarify what I mean:

Before I applied for university, I did maths in a way that was the most convenient for me, and which, I believe, was very effective in its ways of actually making me learn stuff, and to understand concepts profoundly, and to apply them confidently. That was by taking a single textbook, preferably with problems, and go page after page, theorem after theorem, until all had been done. There is in a textbook a certain authority, especially if it is a revered treatise. It often contains brilliant explanations, and extremely insightful problems. After you have read a section of it, and solved the problems, you feel like you really understand what you have learned and can apply the concepts confidently.

Now I encounter a strangest situation. I know that lectures are written with a great time constraint in mind; for example, that particular subjects must be covered by the end of the semester. Many lecture notes (they are distributed by the professors) are hand-written, often rather scant, and they feel more like a compendium of different topics, although they may possess a good structure. The standard course textbooks seldom fit my needs: for instance, our Analysis course uses a textbook that includes no explanations at all, and condenses the whole course into 300 pages of immense font size and utter rigour (which, although still good to pursue for its own sake, is still something that has to be backed up with good examples and explanations), which I think ill-suited for the needs of someone who dreams of a scientific career and wishes to advance quickly, and with a profundity and ampleness of understanding.

For some courses there are no textbooks at all, or for some the textbooks do not follow the lectures well.There is this immense disparity between lectures and textbooks, and the lack of a system drives me mad, as I feel like I am learning nothing at all, and I don’t want there in my head to be a hodge-podge of different topics covered superficially. This problem is further intensified by the fact that there is little time for a really good textbook to be picked up and intently studied, — the only proper way a textbook should be studied, — and often these good textbooks do not follow the structure or the rigour of the course. Also there is a need to take notes in order to understand a topic better, and if you don’t know where to take proper notes from, and wherefrom to take problems to solve, there is very little point in making notes at all — there is no structure!

I ask for advice as to how this problem can be countered, and how I can study quickly and profoundly the mathematical topics that are necessary for the course, as well as additional things, enhancing my understanding of the subject, giving it a certain completeness?

 

[gleem]

What year are you in at your university? Do you have this issue in all of your math courses?

 

[fresh_42]

Any advice for you depends on so many unknown factors that it is almost impossible to give one.

Let's make an example, say group theory.

• Lecture Notes
  
  Lecture notes are not necessarily handwritings. If you search for English ones, i.e. in the biggest stack of available lecture notes per language, by e.g. "Group Theory +pdf" or "Abstract Algebra +pdf" then you will find many of them - typed. As they are lecture notes, you can also find exams and their solutions by a slightly different search key. I plundered those in English and mostly German to make it more difficult to find the solutions for our challenge threads. You can find many examples of group theory questions in the solution manuals
  https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/
  This is, however, not a substitute for learning group theory, just a glimpse into its wide range of examples. Hence, there are lecture notes written with LaTeX in abundance if you search for English texts. Even many German universities have English texts on their servers.
• van der Waerden, Algebra I & II / Serge Lang, Algebra
  
  van der Waerden's two paperbacks were those I have used. They are written a bit old-fashioned but I liked his style. They give you a good basis, but they are not focused on group theory exclusively. If you like a more modern style, then Lang's GTM book could be a good alternative. Lang was one author of Bourbaki which already describes the style that can be expected.
• Alexander Gennadjewitsch Kurosch, Group Theory 1 & 2
  
  I have both of his books about exclusively group theory. They are wonderful, but written in a very old-fashioned way with plenty of text and a minimum of formulas. One has to like this style, it takes time to read it, and I had to find out too much time. Guess I'm more the Bourbaki student.
• Wikipedia
  
  This is probably the fastest way to learn facts. However, it is hard to construct a variety of keywords that actually serve the goal of learning something. You have to find out the gross of keywords you will need first and bring them into a reasonable order. I would prefer lecture notes.
• Insight articles
  
  We have many insight articles that shed light on contexts and the relationship between concepts that Wikipedia cannot provide. They really deliver insights instead of factual knowledge. Of course, these do not cover a classical treatise of e.g. group theory, but they are worth a look if you search for truths behind the formulas.

See, there are quite a few possibilities and a lot are a matter of taste, and what you consider as insightful. If you ask us here, then you get different answers from different users. My motto was always: I cannot know any book, but I have learned whom to ask!

 

[FactChecker]

For some courses there are no textbooks at all, or for some the textbooks do not follow the lectures well.There is this immense disparity between lectures and textbooks, and the lack of a system drives me mad, as I feel like I am learning nothing at all

This sounds strange to me. A good textbook should give you the fundamental information that should help you to understand the lectures. Maybe the textbooks or lectures are not good. Maybe the material is being presented in a different order. In any case, studying the textbook before you take the class might allow you to benefit from the lectures. That takes a lot of effort and requires that you are able to self-study from a textbook.

I found that the only thing that I retained, kept notes on, and really learned from, were good textbooks. I could read and reread them until I understood. I am not a fast learner. The class lectures were more to supplement my understanding. But that is me. I guess that many people learn better from lectures.

 

[gleem]

Oppositely, I used my lecture notes as my main source of learning. Since lectures may not cover every detail I followed along with the requisite texts supplementing the subject material as needed.

Only two courses did not have required texts grad courses in QM and Stat Mech. although they had recommended texts. I think I had well organized professors in UG and GS. Perhaps I was lucky.

Edit; Never was given a set of lecture notes. I think I was not fashionable at that time.

 

[Garvett]

What year are you in at your university? Do you have this issue in all of your math courses?

I am a freshman undergraduate, and I would say that in some form this issue translates to all the maths subjects out there. Especially Analysis, on our textbook on which I have spoken extensively as an example. I must add that one of the most feared scenarios of all for me is this: After I have committed myself to studying the way the university expects me to, to end up in a different mathematical community, and then to be asked to solve some kind of problem or provide a demonstration, and then I would get slapped on the face with something like „Why are you doing it this way?! There is this very simple and elegant and potent method, you can find it in that good textbook, I can’t believe you’ve never seen it!”

 

[fresh_42]

A widely underestimated method of studying something is teaching something!

If you prepare a teaching lesson then you have to be prepared for many, often unconventional questions. The only way to be prepared for such arbitrary questions is to have understood the subject in detail.

 

[Garvett]

A widely underestimated method of studying something is teaching something!

If you prepare a teaching lesson then you have to be prepared for many, often unconventional questions. The only way to be prepared to such arbitrary questions, is to have understood the subject in detail.

I agree completely, and that is what I should generally strive for, but this advice has two hidden aspects to it: (1) It assumes you have enough time to prepare such a lesson; (2) You still have to get the ample information required from somewhere, and that is one of the main points I try to raise in the post.

 

[gleem]

You still have to get the ample information required from somewhere, and that is one of the main points I try to raise in the post.

My interpretation of this statement is that you can't find or are not provided with a text that suits you.

 

[Garvett]

My interpretation of this statement is that you can't find or are not provided with a text that suits you.

Partly, that is the problem here, but if it were simply an issue of finding a good textbook and not much else, I wouldn’t have posted the thread in the first place) I need to find textbooks that match several other constraints: (1) they follow the lecture courses well enough, i.e. they run parallel with them; (2) they have many difficult and insightful problems; (3) they expatiate on the lectures greatly, and have a certain completeness and consistency of presentation to them; (4) they are not too voluminous, as it would render studying the texts an impossible task given the tempo of the curriculum.

 

[gmax137]

Have you talked to the professors about this? When I was in school, the profs had piles of textbooks and they were generous in loaning them out or giving them away. I think the book publishers provided samples, trying to get their books selected.

 

[fresh_42]

Partly, that is the problem here, but if it were simply an issue of finding a good textbook and not much else, I wouldn’t have posted the thread in the first place) I need to find textbooks that match several other constraints: (1) they follow the lecture course well enough, i.e. it runs parallel with it; (2) they have many difficult and insightful problems; (3) they expatiate on the lectures greatly, and have a certain completeness and consistency of presentation to them; (4) they are not too voluminous, as it would render studying the texts an impossible task given the tempo of the curriculum.

You cannot avoid the most important factor: think by yourself! Teaching is one way to enforce this. You can even do it with a virtual audience: pretend as if.

Lecture notes are at least a cheap alternative and you can switch from one to another one if it doesn't fit your preferred style. Even good books can be very different (see y post #3) and there is no guarantee they suit you.

 

[symbolipoint]

Edit; Never was given a set of lecture notes. I think I was not fashionable at that time.

The way I understand "lecture notes", is that these are documents which the student produces during lecture time, in the classroom, according to what he believes or feels needs to be recorded or written for his review later.

 

[fresh_42]

I use the term when I refer to the notes professors provide on their university servers for their students. They are the professors' manuscripts. Well, not manus anymore, LaTeX nowadays.

Here is an example of what I mean:
https://www.math.uni-bielefeld.de/~grigor/odelec2008.pdf
https://de.wikipedia.org/wiki/Alexander_Assaturowitsch_Grigorjan

 

[martinbn]

I am a freshman undergraduate, and I would say that in some form this issue translates to all the maths subjects out there. Especially Analysis, on our textbook on which I have spoken extensively as an example. I must add that one of the most feared scenarios of all for me is this: After I have committed myself to studying the way the university expects me to, to end up in a different mathematical community, and then to be asked to solve some kind of problem or provide a demonstration, and then I would get slapped on the face with something like „Why are you doing it this way?! There is this very simple and elegant and potent method, you can find it in that good textbook, I can’t believe you’ve never seen it!”

This is never going to happen.

 

[gleem]

(1) they follow the lecture course well enough, i.e. it runs parallel with it;

It's hard to imagine a lecture on any subject that does not in general follow any arbitrary text on that subject. The difference may be in emphasis, order or writing style.

they expatiate on the lectures greatly, and have a certain completeness and consistency of presentation to them;

You may never find this as every person is unique.

(4) they are not too voluminous, as it would render studying the texts an impossible task given the tempo of the curriculum.

Again each lecturer is unique so you are asking for something that probably does not exist.As for a course on Analysis ask us our opinion for a well-written text and correlate topics in the lecture to those in the text.

 

[Garvett]

This is never going to happen.

Why so? I know that I may be overthinking stuff, that is my mental specialty.

 

[Garvett]

The way I understand "lecture notes", is that these are documents which the student produces during lecture time, in the classroom, according to what he believes or feels needs to be recorded or written for his review later.

Essentially I meant the notes that professors hand out; as they are called in the reply below yours, those are ”professors’ manuscripts”.

 

[gleem]

The way I understand "lecture notes", is that these are documents which the student produces during lecture time, in the classroom, according to what he believes or feels needs to be recorded or written for his review

Garvett said "Many lecture notes (they are distributed by the professors) are hand-written, "

 

[Garvett]

Garvett said "Many lecture notes (they are distributed by the professors) are hand-written, "

Ah, yes, I should clarify that as well. Essentially many professors just write out their lectures by hand, then scan them, and allow to be distributed, so there is something in it quite unusual.

 

[gleem]

Essentially many professors just write out their lectures by hand, then scan them, and allow to be distributed, so there is something in it quite unusual.

Lecture notes can be terse being only an outline and points that the professor wishes to stress. The professor's lecture may be highly augmented with details from memory so one has to follow the lecture with the notes in front of him and fill in details and all the pearls of wisdom that may come to mind but not in the distributed notes. The lecture notes are not as complete as a book that may be written based on them.

 

[mathwonk]

I am sympathetic to your plight. It is typical first year adjustment to university level courses. There is never enough time, and all courses tend to move too fast. I recommend finding some more agreeable books to supplement your lecture notes, and trying to find time to consult them. A good practice, if feasible, is to go to the library immediately after each lecture and go over your own lecture notes, while the lecture is fresh in your mind, to try to understand it better, and marking points that need further research.

Another good practice is finding a classmate at about the same level as yourself and going over the material together, teaching it to each other.

Your description of the analysis textbook reminds me of the terrible book by Rudin, which many professional analysts love to inflict on students: rigorous but totally without motivation or explanation. Instead try anything by Berberian or Apostol. For rigorous calculus, Spivak, Apostol, Courant, and Fleming are quite good.

I might add that taking analysis as a freshman its already asking for pain, as this is usually a 3rd or 4th year undergrad course, at least in the US.

If you tell us more about which books you are finding unhelpful, we will give more suggestions.

 

[symbolipoint]

I recommend finding some more agreeable books to supplement your lecture notes, and trying to find time to consult them. A good practice, if feasible, is to go to the library immediately after each lecture and go over your own lecture notes, while the lecture is fresh in your mind, to try to understand it better, and marking points that need further research.

Excellent! That was something I practiced in my struggle for Intermediate Algebra in high school; and it helped, without any doubt. My grade was, just a C; but the thought comes up, if I did not put in that effort, I would have well earned less than a C. Later, when did the same course in c.c., I put in similar effort but earned A. No matter enrolling in course once, or twice, give MAXIMUM EFFORT!