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When I am reading a mathematical textbook, I tend to skip most of the exercises. Generally I don't like exercises, particularly artificial ones. Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..

Sometimes I try to prove a theorem before reading the proof. Sometimes I try to find a different proof. Sometimes I try to find an example or a counter-example. Sometimes I try to generalize a theorem. Sometimes I come up with a question and I try to answer it.

I think those are good "exercises" for me.

EDIT What I think is a very good "excercise" is as follows:

(1) Try to prove a theorem before reading the proof.

(2) If you have no idea to prove it, take a look a bit at the proof.

(3) Continue to try to prove it.

(4) When you are stuck, take a look a bit at the proof.

(5) Repeat (3) and (4) until you come up with a proof.

EDIT Another method I recommend rather than doing "homework type" exercises: Try to write a "textbook" on the subject. You don't have to write a real one. I tried to do this on Galois theory. Actually I posted "lecture notes" on Galois theory on an internet mathematics forum. I believe my knowledge and skill on the subject greatly increased.

For example, I found this while I was writing "lecture notes" on Galois theory. I could also prove that any profinite group is a Galois group. This fact was mentioned in Neukirch's algebraic number theory. I found later that Bourbaki had this problem as an exercise. I don't understand its hint, though. Later I found someone wrote a paper on this problem. I made other small "discoveries" during the course. I was planning to write a "lecture note" on Grothendieck's Galois theory. This is an attractive plan, but has not yet been started.

EDIT If you want to have exercises, why not produce them yourself? When you are learning a subject, you naturally come up with questions. Some of these can be good exercises. At least you have the motivation not given by others. It is not homework. For example, I came up with the following question when I was learning algebraic geometry. I found that this was a good problem.

Let $k$ be a field. Let $A$ be a finitely generated commutative algebra over $k$. Let $\mathbb{P}^n = Proj(k[X_0, ... X_n])$. Determine $Hom_k(Spec(A), \mathbb{P}^n)$.

As I wrote, trying to find examples or counter-examples can be good exercises, too. For example, this is a good exercise in the theory of division algebras.

EDIT Let me show you another example of self-exercises. I encountered the following problem when I was writing a "lecture note" on Galois theory.

Let $K$ be a field. Let $K_{sep}$ be a separable algebraic closure of $K$. Let $G$ be the Galois group of $K_{sep}/K$.

Let $A$ be a finite dimensional algebra over $K$. If $A$ is isomorphic to a product of fields each of which is separable over $K$, $A$ is called a finite etale algebra. Let $FinEt(K)$ be the category of finite etale algebra over $K$.

Let $X$ be a finite set. Suppose $G$ acts on $X$ continuously. $X$ is called a finite $G$-set. Let $FinSets(G)$ be the category of finite $G$-sets.

Then $FinEt(K)$ is anti-equivalent to $FinSets(G)$.

This is a zero-dimensional version of the main theorem of Grothendieck's Galois theory. You can find the proof elsewhere, but I recommend you to prove it yourself. It's not difficult and it's a good exercise of Galois theory. Hint: Reduce it to the the case that $A$ is a finite separable extension of $K$ and X is a finite transitive $G$-set.

EDIT If you think this is too broad a question, you are free to add suitable conditions. This is a soft question.

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doesn't seem like a very useful question –  anthonyquas Jun 28 '12 at 2:43
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The fact that this can vary with the book (including many books, e.g. Lang's Algebraic Number Theory, that have no exercises) makes it seem too broad. –  Zev Chonoles Jun 28 '12 at 2:57
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A few professors have told me on several occasions that it is often not important to understand the proof of a result, but just that it is true. That being said, I agree that this is a broad question, and depends on personal preference. But, I think most people do a combination of what you seem to be doing, but also solve some exercises. One book that comes to mind for which solving exercises seems to be essential to grasp the subject better is Atiyah-Macdonalds's Commutative Algebra. I don't think I would have learnt a lot just by reading the text and skipping exercises. –  Rankeya Jun 28 '12 at 3:05
    
@anthonyquas See my EDIT. –  Makoto Kato Jun 28 '12 at 3:07
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Deciding on just understanding the theory (and constructing the proofs of theorems) but not actually using the theory to attack any problems reminds me of the following apocryphal tale. A (non-athletically-inclined) faculty member read tennis great Bjorn Borg's comment that tennis was a very simple game; all that needed to be done was to hit the ball over the net one more time than the other guy. He felt he understood the theory perfectly and so promptly entered the tennis tournament at the faculty club. He could not understand why he lost in the first round without winning even one point! –  Dilip Sarwate Jun 28 '12 at 14:16

7 Answers 7

Of course this is entirely subjective and depends on your intelligence and memory. I'd suspect most all people on this site are high in both areas, at least in logic/mathematics.

Understanding theorems and working through them like you explained is a very good way to understand material, especially if you can keep it in mind when needed. Exercises are usually repetitious, but also can give you some context that you can/should apply the theorems in.

So, no going through exercises isn't a requirement, but going through some is a good idea to both confirm you understand the material and to help commit that material to memory. That said I wouldn't suggest doing the first few exercises (in most textbooks they're the easiest) rather pick a few in the middle or some that the answer or how to solve them isn't obvious to you. Usually the ones toward the end of a section are either tough, long-winded or both. Doing some of those might be worth it, but some might just be long and strewn out and ultimately not worth the time.

This is just my experience with Math Textbooks, but I've only gotten to the undergraduate level, so how true this is at higher levels I don't know.

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I'm with the OP on this, I skip exercises too.

Here's the logic: A true understanding of maths is about being creative in its applications and not just the material initself.

Exercises, by definition, stifle creativity by presenting a sandbox in which to think.

It's a bit like Rocky 3, do you learn your craft by working out in a gym or by lifting logs down in the forest?

...and another thing, I reckon following examples in books leads to degenerate mathematics. They perpetuate particular styles of thinking about problems.

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But without exercises, how can you tell that you really understood the definitions and are able to use the definitions and the theorems given in the book? –  Asaf Karagila Jul 4 '12 at 16:17
    
Indeed! Both understanding and the value of the mathematics learnt should flow from three things: 1) Internal consistency i.e does it make sense 2) Independence from the body of mathematics already understood 3) Consistency with future mathematics acquired. This approach provides both a valid measure of understanding with the fewest constraints on creativity. –  benny rimmer Jul 6 '12 at 9:46
    
@AsafKaragila I think that one can create one's own exercises by asking as many questions as possible. In this way, one practices not only asking questions (which is crucial in research mathematics) but also answering one's own questions. I think doing challenging exercises, if they exist in the textbook, is worthwhile. However, I agree with "@bennyrimmer" in the sense that understanding mathematics on the line by line scale (which people learn by doing exercises) is far less important than understanding mathematics on the larger scale of research (although both are important). –  Amitesh Datta Jul 25 '12 at 8:31
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@Amitesh: Before one do research, one should learn how to do mathematics. Your comment is the root of all crankery, and the lot of people suggesting that they, pure amateurs without any real knowledge or experience can find holes in Godel's theorem; or inconsistencies in set theory. Solving exercises helps you be certain that you understood the theorems and definitions thoroughly. Trust me, I know. I skipped the exercises many times, only to find out many days later that I lacked an essential understanding that would have come from solving the exercises. Doing original research comes later. –  Asaf Karagila Jul 25 '12 at 10:16
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@Amitesh: What I wrote is coming from an extremely big-picture oriented mind. I hate the details and for me the big picture is the important thing. However mathematics is about proving, not conjecturing. It is rarely the case that my initial guess is successful, and when it is then it always the case that there is some trick holding me back. I still think that one should first solve some exercises I never wrote that one should solve all of them. However if one skips them entirely, they will fail at some point. I learned this the hard way. –  Asaf Karagila Jul 26 '12 at 5:41

And why are exercises so important if we want to do some research ? I have always seen research as developping new tools, new theories and new approaches to prove theorems...

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If your goal is to become a research mathematician, then doing exercises is important. Of course, there will be the rare person who can skip exercises with no detriment to their development, but (and I speak from the experience of roughly twenty years of involvement in training for research mathematics) such people are genuinely rare.

The other kinds of exercises that you describe are also good, and you should do them too!

The point of doing set exercises is to practice using particular techniques, so that you can recognize how and when to use them when you are confronted with technical obstacles in your research.

In my own field, two books whose exercises I routinely recommend to my students are Hartshorne's Algebraic geometry text and Silverman's Elliptic curves text. The exercises at the end of Cassels and Frolich are also good.

Atiyah and MacDonald also is known for its exercises.

One possible approach (not recommended for everyone, though) is to postpone doing exercises if you find them too difficult (or too time-consuming, but this is usually equivalent to too difficult), but to return to them later when you feel that you understand the subject better. However, if upon return, you still can't fairly easily solve standard exercises on a topic you think you know well, you probably don't know the topic as well as you think you do.


If your goal is not to become a research mathematician, then understanding probably has a different meaning and purpose, and your question will then possibly have a different answer, which I am not the right person to give.

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Dear Matt I think many exercises of the Hartshorne's book could be in the main text. Actually many of them are in EGA with full proofs. –  Makoto Kato Jun 30 '12 at 0:21
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@MakotoKato: Dear Makoto, Yes, that's right. But having them as exercises to be solved rather than results to be studied gives them a different role in one's training. (I always approach these sorts of questions from a "training" view-point, whether I'm thinking of my own training or that of my students.) Best wishes, –  Matt E Jun 30 '12 at 2:12

I think that the most important point in mathematics is to think about the subject for long periods of time. If you think about mathematics, then you will often develop intuition which is very important. Of course, if you think about something for long periods of time, then your memory of the material is better as well.

Ultimately, the point is that people generally learn more by doing (compare active learning to passive learning). Of course, there are exceptions to every rule and you are the person who best understands your own strengths and weaknesses. The important point is to identify your weaknesses and work hard on them through a combination of active thinking and problem solving.

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Depends on the textbook, I suppose. Some textbooks introduce a lot of material in the exercises that isn't developed in the main text.

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absolutely you can't truly say you understand something until you work through it

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