Why Is Abstract Algebra So Challenging to Visualize Compared to Calculus?

[epr2008]

I am a junior in college right now, and after finishing the Calculus sequence and having my first semester of Analysis, I am now taking Abstract Algebra. I did alright in Analysis but not as good as I had hoped to do. My biggest problems are that, unlike Calculus, which for the most part I could visualize and see the bigger picture, I simply cannot. Calculus was a breeze, I understood the broader scheme of things, and I understood why I was learning the derivatives and integrals, what it meant to do these things, and most importantly THE DIRECTION I WAS HEADED WITH ANY SPECIFIC PROBLEM. I was lucky. In high school I had an extremely good Calculus teacher. Unfortunately, in college, especially in understanding and proving abstract maths, I am totally lost. The reason is because not only do my university professors not really told me what it is I am trying to actually do. They use the definition-theorem-proof-theorem-proof-example approach which teaches nothing but memorization, and I cannot visualize it and therefore can't learn to understand it for myself. Like I said, I've finished an entire semester and I don't even really understand what constitutes a proof or the direction I should head in doing the proofs because I can't see what I'm working towards, nor do I know the questions to ask to give me a reference point. I am starting to lose faith in math and in myself pursuing a career in the subject. If anyone has any suggestions, advice, or any kind of help at all I would greatly appreciate it.

 

[Sprooth]

Oh no, at the thought of someone turning away from math, I had to say something!

Sorry this is your first experience with math in college. The further you advance in math during college, the more you get into "real math". During my second or third year, I realized the math I was learning hardly resembled what I did in high school, but I liked the new face of math all the more.

The first class that heralded my change in thinking was my Intro to Abstract Math course, which involved learning some basics like set theory, functions, cardinality, etc., and also learning how to structure proofs. We did a couple formal proofs in high school, but due to how they were presented I just considered them a waste of time. During my Intro to Abstract Math course, I started to realize to value of proofs and began to try to change my mindset to one of a mathematician.

It is difficult to give some advice that would help everyone in a situation like yours, since people's minds work so differently, but I'll explain what helped me.

Basically, I began considering math as a new universe, where I had to redefine everything I know is true, and learn the relation between all the new concepts. BUT it's extremely important to understand WHY things are the way they are. There are arguably no coincidences in math. Unlike any other subject, it is a subject born of truth and logic, and rigor is its godfather. Okay, that's my silly analogy, but you must convince yourself of the importance of mathematical rigor, which is essential to be able to appreciate the beauty (which you'll begin to notice more and more). I find it incredible to be able to form flawless arguments which can be defended from any angle, which is entirely different from what lawyers do. Lawyers create more or less solid (not flawless) arguments which only have to pass for a jury, but mathematicians must have an unyielding grasp on logic. Something that happens along the way is that you'll continue increasing your standards for yourself in terms of logic and conceptualization.

Some study-specific tips I would give are:
* Know your definitions and theorems well. These are your building blocks and tools. Try not just to do rote memorization, but understand the importance of each factor. If a definition or the hypothesis in a theorem has 3 factors, then try to understand what would be missing if you removed each one.
* Be aware of common counterexamples. These are often what demonstrate why a theorem is not true when one of the assumptions is removed, and they also help enlarge your mental bank of possibilities.
* As much as you have time, go through the proofs you're encountering, and make sure you understand each component of it. Just like the theorems, imagine what what happen if you removed a statement of a proof.
* Note that there is a very significant difference in understanding a proof entirely when you read it and being able to produce the proof on your own. Try sitting down and writing out certain proofs without any references. This is a good way to review for exams.
* Don't be afraid to ask your professor or friend for help. Ask them to explain something you don't understand. Sometimes all it takes is for someone to rephrase something, and sometimes people will have a very different way of thinking about something which could help you a lot.
* Do a lot of sitting back and pondering. The importance of this could vary between people, but if you don't understand something, don't settle for getting through it with a half-understanding. The human brain can't make large changes too quickly usually, so you'll need to give your brain some time to learn these new and sometimes mind-boggling concepts. For example, when I was first learning about infinite products in my topology course, I was struggling for a way to conceptualize infinite-dimensional spaces (can you blame me?), and so I had to labor over the work for a while, then sit back and try to wrap my mind around it for a while, then back to the work, and repeat. For the first couple days when the prof was talking about that stuff, I felt lost and began to feel hopeless, but a week later, I was surprised by the advancement in understanding I had accomplished.
Also, I remember hearing that one mathematician said he did his best work while riding his bike. A change of pace, like a short walk with some fresh air or something, can be really beneficial.

Well, that's all I can think of for now. Time to do some Dual n-back cognitive training before class!


Sprooth

 

[Sprooth]

Oh, one more thing. Yeah, abstract algebra is a tough beast. I took a couple abstract algebra courses in college, and they were pretty difficult. For me what made them hard was the lack of an inherent way to visualize the subject matter, unlike analysis or topology.

I am now taking my first graduate course in abstract algebra, and I'm still trying to figure out a way to excel in it. For some things it just takes repeated exposure and effortful study. You are capable of comprehending it though.

One professor of mine told me abstract algebra is "a different way of thinking". I wish he elaborated a little bit on that.

Does anyone else have any thoughts on abstract algebra and how the mindset is different?