You are studying math WRONG 

Often time is a problem. Students have multiple courses and deadlines. Everyone seems to forget this. I wish I could ponder hours or days over a single problem, but it’s impossible time-wise. I got other stuff to do too.

I really love the idea of hints instead of solutions. I think it'd be really cool to have entire hint books! But I guess that would just be more of a study guide. 😅

Excellent. You're right. Reading theorems and thinking about the problem, trying to find out something, then failing , then going over and over again this algorithm until success.

I'm learning this late in the game, but since using this approach I've noticed a much stronger grasp of the topics. It's definitely much slower, though.

Awesome video sir. Thank you

Thank you so much for posting this encouraging video!!! Halmos' advice to put away the book and try to solve it yourself, even if it takes days and weeks is painful, yet precious! Your insight regarding building intuition while trying to go through the solution, perhaps a wrong one, nevertheless sensing the confidence that the solution is correct, is truly a gem!!! I'm definitely one of those people, who tends to quickly look at the solution 😅 I do want to understand the higher level maths & hopefully one day do it professionally. In the meantime, I'm doing the lowly job of a nurse!!! Thanks again for taking the time to post such great value!!!

Intermediate analysis is my current hurdle. And yeah, you hit the nail here.

This video helped me so much, thank you.

One challenge is the time-efficiency of a breadth vs. a depth approach. IF you had the time to work through hundreds and thousands of problems on some topic, then looking at solutions after a few attempts is not too bad. This is because you will encounter analogous problems time and time again and you will practice and often times rediscover the strategy you saw. Essentially, that previous encounter with the solution will be a "hint" next time you see a similar problem. And with the variations in how the strategy applies (or doesn't), you will deepen and reinforce your understanding. However, problems become complex, time-consuming and short in supply that you no longer have such a luxury. Hence, as a student who may only work through 20 or so meaningful problems a week per subject, getting the most out of each is very important. There is also the benefit of developing the meta skill of how to struggle through some challenge, making slow and steady progress.

So a bit of pushback. The major issue is students in most classes do not have the time to spend in struggling through a pset or "studying" because they are concerned, rightly so, for their grades. The prime objective is performance not comprehension. The stress and issue come from having a finite amount of time to "master the material". It is not (necessarily) the case the student is incapable of grasping some particular concepts, and given the building block nature of mathematics, not understanding -- i.e. "failure to master" earlier content produces a snowball effect wherein a student falls further and further behind in the class. To pick an example, in most Freshman Calculus classes, delta-epsilons or limits are covered in a week or two, at most, before moving onto the next topic. Whether using Stewart or baby-Rudin, worst still Spivak, it takes most a while for the concepts to marinate in the brain for them to understand fully a three quantifier statement. All this leaving aside the whole debate on the Discovery pedagogical approach. So what is my solution? I suggest studying ahead. If going to take that class on Trig or upper division Real Analysis in the Fall term, then find the textbook that will be assigned and other resources and start studying during the summer. Audit a class the term before taking the class for credit. Spend that early time going at own pace rather than the class pace. A month spent struggling on one's own over concepts is worry free is better than stressing during the crush time in having to "master the material" in time for the fast approaching upcoming exam.

This makes sense, as my professor has this exact same attitude has been telling me specifically that I need to do this more. However, as a student, it is extremely frustrating to have to come up with a dozen proofs a week, every week. I am probably not going on to grad school for mathematics, but it feels like the professor expects everyone to live and breathe the problem set all week. Unlike in other disciplines, I feel like in math classes, we are not taught the specific technique/strategies to tackle a specific type of proof and I don't understand why. Why can't we get a hint for every question saying what technique to apply or demostrate the technique during the lecture? Instead lectures consist of introducing new definitions and theorems and doing proofs for those. When my professor shows me the answer or releases the answer key after the deadline, I read through the proof and think to myself "in a million years, I never would have thought about doing it that way" because it used a super specific, super clever trick that we were never taught anywhere. So, I feel really frustrated when I sit there for hours on a proof question to later find out that I was not even close to solving it. After that, we move on to a new topic the following week and then the cycle repeats. Understanding that the advice in this video is valuable, I am still torn about what to think

Immediately searching for the solution is harmful. But you can also go to the other extreme and be harmful. Knowing when to judge that you're just not making as much progress struggling, as you would if you read some further information, is a hard line to judge. It's complicated by the fact that problems in texts can be written badly. A book that tells you a bunch of definitions about real numbers and then states "prove the fundamental theorem of arithmetic" would be a bad problem. Many textbooks contain problems that are not this bad, but still bad enough to not warrant spending a huge effort on. And I genuinely think lots of people make both mistakes, by looking too early or too late. I think a lot of people struggle for too long, get too frustrated, and even give up the entire project, when they would have been better off reading a solution and trying to work backward to figure out how they could have solved the problem -- and then take that lesson to future problems.

Paul Halmos' discussion with Warren Ambrose was the "Secret Sauce": talking with others about problems is indespensible. In many of my classes, we'd organize "Problem Set Parties" to outline/work-out problems.

I do not use solutions manual to cheat; I´m using them to check my answers out. I do reseach on the same problem to see other people approaches so I can come out with new/better ways to proceed or get the answer (improving my own algorithm)

This really drove home for me what problem sets and tests are intended to do. They are structured such that you can really "stress test" the foundations of your knowledge of this subject. If you have rock-solid fundamentals for this subject, then you will navigate the problems without issue. But ANYWHERE there are holes, you will get stuck. And like an expert inspector, you now know where to apply the healing cement. They are not really evaluators and "proof" of knowledge and quality. A 90% in a class does not mean "I am an expert in this subject". More accurately, it means "we tried to find gaps in this student's knowledge and succeeded about 10% of the time". Tests and problem sets are like a battery of quality assurance tests that show you where there is a weakness in your thinking.

I think there's a role for Solutions manuals but obviously they can be abused. A lot of times there are different ways to solve the problem, often more efficient ways. I think the role of solutions are to give guidelines and resolve misconceptions after attempting the problems. Of course the temptation to check the Solutions on first impulse when hitting a snag can undo that merit. However, if your goal is to learn the material, you can strategize a way to use the solutions as a check after you've exhausted your own personal resources. And in that, the solution can aid or enhance the learning experience because now you'll become cognizant of the pitfalls of your original attempts.

Absolutely true! However, if the textbook that supplies the basic definitions and stepping-stone theorems to solve the problems is not clear enough and/or well written, then students tend to get lost in what these definitions and theorems really mean, and how the discussed ideas can be used to solve a problem. That's why, having *good* examples of problems, along with their solutions in books is *essential* to build an understanding of how the theory is applied. Such examples can then be used as basis for constructing solutions to the exercise problems. Any book that doesn't supply these is a waste of time imho. A good example of a book that does this reaaally well is "Applied Linear Algebra and Matrix Analysis" by Carl D. Meyer. It's an excellent book. Another point I would like to make is this: it is one thing to understand concepts and theory, and another to learn techniques and methods to solve problems. In fact, I think that these two topics are nearly orthogonal to each other, and a lot of books (analysis books are the main culprits) tend to focus more on the first, and less on the other, which I think is wrong. Yes, technique can indeed be acquired by trial and error eventually, after banging your head on the wall sufficiently enough times while struggling on a problem, but what point is to learn technique from scratch by yourself, when it can be presented to you without exposing the solutions to the exercises. Also, I tend to do this when solving problems: if a certain problem becomes too much of a time-killer and starts to mentally bother me, I do look up the solution. However, I do force myself to come up with a new solution that is different than the supplied one, and often I do, since my understanding of the problem and theory is now much better. I think this is a good compromise :) Also, having math.stackexchange as a space to discuss problems and get/give help is amazing.

I find the understanding process is logarithmic. At first solutions (manuals) do help me, after enough worked examples the next step is taking the guard rails off and doing problems. Many books for upper level don't have much examples or solved problems making the barrier to entry hard. Example I am working to understand Hilbert's problem 90 for p-adic topology (and then of course Kummer theory). The proof seems almost magical and I would not be able to prove it without having seeing the proof first, additionally I need examples to gain understanding and context. I know about Dedekind's theorem on the linear independence of automorphisms but given that I would not know how to apply it to prove the cohomological version of problem 90 - for some they might be able to prove the theorem with just this however perhaps for many seeing the solution to the proof and examples would be the path to understanding.

Patience is the key

For the most part, I try to follow this philosophy. That being said, reading the solutions to certain problems can expose you to new techniques that you wouldn't have thought of before. I owe a great deal of my problem solving ability to studying the proofs and solutions of others, noting the strategies they use and then incorporating those strategies into my repertoire. But yes, ideally, you should make an honest attempt at every problem and get comfortable with not being able to solve things within the first 5 minutes.

Liked the video. Only thing I'd like to point out is that Leibniz and Newton have been vindicated by model theory - infinitesimals do exist and models of the reals containing them have been proven to exist. Some institutions have begun test-runs of teaching analysis using the infinitesimal approach, completely foregoing the epsilon-delta definitions because they are incredibly cumbersome, nonintuitive, and are, really, nothing more than a way to approximate the concept of an infinitesimal without explicitly saying one exists. So, it's not that the foundations of calculus were shaky - the understanding of models was shaky. The foundations of calculus as described by Leibniz and Newton are rock solid - as proven in the 1960's - and we know now they were simply thinking of a model of real numbers that others were not able to see.

Can you please offer an opinion on the two real Analysis books by Schlomo Sternberg. To me, his graduate level book, Real Variables, makes Rudin seem pretty easy. From the first page on metric space, Sternberg had me lost. I am using Kolmogorov for self study. I last had a class in math in 1966, Advance Calc, coincidentally.

I find that a lot of advanced math textbooks have poorly constructed problem sets. The best ones I’ve seen start with simple questions based off the theorems and definitions and then slowly extend into harder applications. Unfortunately these seem to be rare to me. Do you happen to have a list of your favorite textbooks and problem sets for topics in higher maths?

For self--study however, having at least every other answer is essential because, unless you have a private tutor--like person, you may not know if you're understanding the math and producing the right answers. In one of my proof-writing books, which is free online in PDF form, by the way, "Book Of Proof", by Richard Hammack, all of the odd-numbered answers are available. And since I'm in no hurry---self--study is the best---if I'm stuck, or after completing a problem, I peek at the answer little by little, like hints, so that if I don't know how to start, I can begin as the author did, with the first line of a proof, or only part of that first line, etc. I may get the rest of the proof correct after getting started the "right" way, before seeing the complete answer! That has worked well for me, and I can study, really study the answers and see why they are what they are, etc. All math books should have a good number of answers!!!

Is it wrong to have the answers without the steps? It is nice to know if you are doing things right.

That list at the end about things to check, like a relaxed hypothesis, should be fleshed out and made into a poster!

Yes! It's the journey and not the destination. Just proved a simple theorem I took days thinking about and yelling at myself. Had a brief moment of joy after proving it; then on to the next challenge/problem. The problem with this method of course is TIME unless you're a Von Neumann, Hilbert, or some other genius.

I could not agree more with this video! It is better to climb hill than get helicoptered to the top. Have you even really been there?

halmos interview link is not working please provide the link again

I think Halmos demands regarding studying math although are truly wise are not in real conditions, feasible. Of course if you don't detach from your solution manual you are learning nothing. But having solutions or even better a couple of hints to a hard exercise is something that one appreciate unless is keeping you from actually thinking hard.

Ah, Halmos, not Helmholtz. That makes sense.

I have a problem. I have real difficulties with patience, because of my trauma brain injury. Do you have any probably techniques. It would be deeply appreciate 😊

I figured this out in grade 7 and quit school forever. I taught myself.

I wish someone had told me this 20 years ago.

Good evening sir

Of course you're not going to learn much if you just copy the answers from the solutions manual (or stackexchange, or reddit, or wherever) without even trying the problem(s). But assuming you do at least *try* to solve the problem at hand, the reality is that sometimes you really are just stuck, and since time for us mortal individuals is not infinite, the best thing to do is - eventually - to go looking for answers or hints. What I disagree with here is that there's some fundamental "thing" that says "if you look at a hint or answers you guarantee you never internalize whatever it was you were supposed to be learning." That sounds like unsubstantiated folklore to me. Maybe you just need to do 5 similar'ish problems (with the solution), or 10, or 15, or 50, or 500, before you see the underlying pattern. But I see no reason to buy the idea that anything essential is automatically lost by looking at answers or hints. And if the alternative is to just stay stuck forever... well. You do the math.

2:28 Am I crazy or is this a really bad cut/edit where it begins with NEVER lol Surely you meant to be saying to internalize the lesson of the problems and not the other way around

But in Indian schools they place over emphasis on the destination rather than the journey. They care more about the solution rather than the journey otherwise we won't be able to solve hundreds and hundreds of problems in a short period of time and appear for the exams. Your insight is good but it isnt pragmatic for my situation. What's your view

I agree with the others who have mentioned this. Time is a luxury in math classes. Yes, the struggle is important to learning, no doubt. But the reality in most university and college course is that the work needs to be completed at a steady rate regardless of the student's ability to have an "epiphany". At some point, you have to hand in that proof or finish writing the exam or quiz. In this context, it is silly to bang one's head against the wall in order to achieve some sort of 'purity' of thought. I admit that this is a problem with formal education. Obviously, someone studying independently won't have these sorts of pressures. But for those learning math in a more institutional setting, spending hours on one question from a weekly problem set for one course is very foolish. If students were able submit assignments when they were completed rather than when they are due, they would never need to look up solutions and then reverse engineer their responses. Deadlines are the true reason why students look up solutions, not because they don't wish to learn the material at a deeper level. In the end, they are forced to choose to "get the job done" instead of learning the material thoroughly. It's a real shame and a waste.

On my blog I have a link "how to study maths really quick" and it is a rickroll.

I wander, as a non-university math student, I see a problem proposed online, I try to find the answer, the video online gives a different one, but I still think my answer is correct. With a university degree to back me, I might think, of course I'm right (to be fair, I might not), but in any case, what would be some steps you would suggest, that after doing the hard yards to figure it out, that you are indeed truly, incorrect. I guess you'd hope you'd be correct, but you'd just like to know :)

I wish I can give 100000 likes