Gaining a Deep Understanding of Theorems

Question

As one might infer from my user name, I am currently pursuing an autodidactic path in pure mathematics while awaiting the real math classes next year at university. I recently bought Dummit and Foote's (yes, I know this is a more "advanced" text for a beginner but my mentor in mathematics told me that I was up for the challenge) text to learn abstract algebra and while I absolutely love reading it and trying to prove the theorems (before I read the proofs)/do the exercises, I am having a hard time judging my level of comprehension.

For example, I had some success proving the first isomorphism theorem before I read the proof but I became stuck on the second isomorphism theorem. Once I had reached an impasse, I would scan the proof just to try to pick up on the "trick"/direction of the proof, then close the book and try to go at it again without looking at the proof. I've been fairly successful going about it this way and I've made it a point to drill myself when ever I have a spare moment of boredom to get a sheet of paper out, write down a theorem and then prove it off the top of my head (I wouldn't say it is from memory though as I always think through the logic of the proof, just remembering key points along the way).

So anyways, with all of that said, is being able to reproduce the important results of a text from scratch a good metric of my understanding? I will say that every time I work through a proof, I feel like I notice something new about it or I gain some degree of clarity about why the proof is structured a certain way. However I know that when it comes to mathematics, it is especially easy to delude myself into thinking that I understand something when I really don't. I have tried to be ruthless with myself and force myself to prioritize deep learning as opposed to superficial breadth but I am struggling to know when exactly I have achieved a deep understanding and how to keep this deep understanding while moving through the text. Any insight/experiences into the matter would be of tremendous value.

Also it might be worth noting that I currently attend a non-traditional high school where I get a great deal of freedom to work on what I choose so time is not really a factor for me.

Answer 1

The book How to Think Like a Mathematician — A Companion to Undergraduate Mathematics might be helpful to you. The following is a checklist for "true understanding" in Chapter 34 of the book.

You understand a definition if you

• can state it precisely,
• can state it in your own words,
• can give concrete examples of it, including trivial and non-trivial examples,
• can give non-examples of the definition,
• can recognize it in different and unfamiliar situations,
• know theorems in which it can be used,
• know why it can be used in those theorems,
• know why this particular definition is made,
• know other similar definitions of the same word and know the differences between them.

The last item on the list occurs because different mathematicians use different definitions and this has important consequences for theorems. By adding in an extra hypothesis to a definition many theorems become easier to prove.

You understand a theorem if you

• can state it precisely and in your own words,
• can give concrete examples of its use, understand its proof,
• can apply it in new and unfamiliar situations,
• can give a counterexample to statements given by weakening hypotheses,
• know its inverse and converse,
• can see some consequences from it (corollaries for example),
• can encapsulate it in one sentence, e.g. this gives me a method to calculate distance,
• see where it fits in the theory, is it an end in itself or a theorem used on the path to a greater theorem,
• know whether it refers to a small or large class of objects.

You understand a proof if you

• can state it precisely and in your own words,
• know where the assumptions are used,
• know the structure, i.e. how it breaks apart and which techniques (direct, contradiction, etc.) are used,
• can see every step as simple rather than as a miracle,
• can use the ideas in the proofs in your own proofs of other statements,
• can fill in any gaps,
• know how rigorous the proof is,
• can summarize it, i.e. leave out the details but keep key points,
• know where problems occur when hypotheses are dropped.

Answer 2

I am a novice as well, so this should be taken with a grain of salt.

In my experience, there are some proofs whose proofs are fundamental-- not knowing them somehow suggests that you are missing something fundamental about why the statement is true, and how one would typically use the theorem. (A lot of the time, understanding the use can be more relevant than recalling the details of its proof.)

There are other proofs that are somewhat technical and I find that if I need to go back and look at some details to reprove it, that is okay.

The isomorphism theorems are of the former type. Understanding this proof is tantamount to recalling the first isomorphism theorem. Given normal subgroups, and $K \subseteq H \subseteq G$, define the map $xK \mapsto xH$. After verifying that this is a homomorphism, notice that the kernel consists of closets $yK$ where $y$ belongs to $H$. Apply the first isomorphism theorem.