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.
@JoelRosenfeld Жыл бұрын
There are more layers to this too. For a class that is going to be central to your career or future classes, you'll want to spend a lot more time studying than with a one off elective. I always advise students to never take more than two difficult classes at once. You'll need to time and space to learn things, and if you organize your college career well, you can certainly manage it. I didn't, but I know people that did, and they always impressed me.
@abcxyz9723 Жыл бұрын
@@JoelRosenfeld First one is a good point. Towards the second: only taking two difficult classes at once is impossible (in Europe at least, no idea about US). All of my classes (so 4-6) are similarly mathematical and demanding.
@JoelRosenfeld Жыл бұрын
@@abcxyz9723 Yeah, we do have different systems. Bachelors degrees in the US are typically 4 year affairs rather than the 3 of European universities. Just less time overall. I definitely sympathize.
@abcxyz9723 Жыл бұрын
@@JoelRosenfeld Ah, yes. This too I guess. Particularly talking about a 2 years MSc though😄
@JoelRosenfeld Жыл бұрын
@@abcxyz9723 lol Even that is a bit different between our academic cultures. In the US, you usually don't do a separate master's degree, if you goal is a PhD. It's all lumped together into a PhD program that last 5-8 years typically.
@surrealistidealist Жыл бұрын
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. 😅
@JoelRosenfeld Жыл бұрын
Yeah, I wish more authors would use the hints vs answer model. I've seen it in some textbooks, but not many. Just one stop before you really have to go and read the solution could help a lot of students.
@SplittingField Жыл бұрын
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.
@JoelRosenfeld Жыл бұрын
If you have a huge bank of problems to pull from, like in a calculus textbook, then I agree. The solution to the previous problem could be chalked up to another "Example" in the text. For higher level courses, the problem sets are a lot more sparse, and there might only be one or two exercises to really get the idea of a method across. Of course, if you have to get things done on a deadline, you have to shorten the length of time you put to a problem before giving up and asking for help. It's just important to not entirely bypass the process.
@asn65001 Жыл бұрын
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.
@ambassador_in_training4 ай бұрын
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!!!
@JoelRosenfeld4 ай бұрын
I'm glad you enjoyed the video! Good luck on your journey. And there is nothing lowly about being a nurse. There is a lot of honor in making that sacrifice to keep people healthy and save lives.
@JohnDoe-kh3hy Жыл бұрын
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)
@duncanw9901 Жыл бұрын
The first clause is a logical contradiction 😉. You're not trying to become an algorithm for generating "the" answers. You're trying to become an algorithm to generate _the right questions to ask,_ and then produce, not "the" answers to those questions, but new methods, tools, and ways of proof to tame this terra nova.
@JohnDoe-kh3hy Жыл бұрын
@@duncanw9901 That is your approacho but not mine, even if you are trying to support yourself in "logic". For me, it is not a cotradiction. It is a method of reassurance on my abilities.
@CalBruin Жыл бұрын
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.
@douglasstrother6584 Жыл бұрын
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.
@JoelRosenfeld Жыл бұрын
Sure, that was the final straw that led to his understanding, and I do say if you spend a good amount of time, then ask for a hint. It's just important to give a serious attempt at the problem before you speak to someone else about it, which is what Halmos did. When I was an undergrad, I would often meet with friends to discuss problems. But I usually solved most if not all of the problems before we met. One of my friends would do the same, and we compared answers. The one that came without doing the work ahead of time would barely scrape by the exams.
@AxiomTutor Жыл бұрын
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.
@addemfrench Жыл бұрын
@s v definitely (highly contingent on your professor, if you even have one, though).
@perspicacity89 Жыл бұрын
This video helped me so much, thank you.
@JoelRosenfeld Жыл бұрын
You’re welcome! I’m glad it helped!
@Mahmood42978 Жыл бұрын
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.
@justinma172810 ай бұрын
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
@JoelRosenfeld10 ай бұрын
I get it. We all get it. It is very frustrating, but then again, that is the point. Math is very frustrating as a topic, and the real tools you need to develop are those that help you work despite that frustration. Sometimes, a result will occur to you within an hour, and other times, it’ll take a whole week. We’ve all been in the same position that you are now. And in the real world, there will be many problems where the solution is not obvious by any means. This is what your training is all about. Learning how to abstract concepts, and then trying your best to make a statement using those abstractions. I’ve used it in mathematics, engineering, and many other places throughout my career. I understand how you are feeling. But no matter how many different people have tried to revamp math education, this is still the best way to learn it.
@jks2342 ай бұрын
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.
@JoelRosenfeld2 ай бұрын
I really like this way of explaining it. I’ll have to use the stress test terminology next time I talk about this
@vicktorioalhakim3666 Жыл бұрын
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.
@carljones9640 Жыл бұрын
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.
@JoelRosenfeld Жыл бұрын
Yes, those are all good points. I know of this approach, but haven’t taken the time to look at it myself. Are infinitesimals actually presented today in the same way Newton did? More fundamentally, real numbers weren’t really well understood at the time either.
@carljones9640 Жыл бұрын
@@JoelRosenfeld Leibniz's differentials and Newton's instantaneous rates are updated to modern terminology and notation, of course, but the core properties remain. Proofs of the chain rule, for instance, are done almost like what you would see in an engineering course, where the differentials are simply canceled across division. That is almost exactly how Newton proved it.
@JoelRosenfeld Жыл бұрын
@@carljones9640 cool! I am planning to dig into the history in a lot more detail for some future videos. I’ll keep what you said in mind. Thank you!
@CalBruin Жыл бұрын
Non-Standard Analysis addresses this.
@duncanw9901 Жыл бұрын
Non-standard analysis is a bit of a red herring; while nice, it feels a bit like mere model-theoretic syntax sugar for ordinary limits. The truly based infinitesimal analysis is smooth---there's a wonderful monograph by John Bell, who lays out an infinitesimal theory that falls out of synthetic differential geometry, that's _exactly_ what our 16th century friends were trying to get at. It turns out the problem wasn't with Newton but with Aristotle: the theories are necessarily constructive.
@speedbird7587 Жыл бұрын
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.
@coreyevans5734 Жыл бұрын
Intermediate analysis is my current hurdle. And yeah, you hit the nail here.
@AJ-et3vf Жыл бұрын
Awesome video sir. Thank you
@JoelRosenfeld Жыл бұрын
You’re welcome! I’m glad you liked it!
@samhickel9616 Жыл бұрын
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.
@jennifertate43978 ай бұрын
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!!!
@numericalcode Жыл бұрын
Patience is the key
@spacetimemalleable771810 ай бұрын
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.
@sanjursan4 ай бұрын
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.
@JoelRosenfeld4 ай бұрын
I don't have personal experience with Sterberg, but I was able to find a table of contents online. It looks like it's a pretty comprehensive introduction to real analysis, and is much larger than Baby Rudin. I don't think you'd go wrong with either Rudin or this text. There is a lot in Sternberg, which in my experience is typically reserved for a first course in measure theory or functional analysis, such as Baire's Theorem. If you choose to go with this book, I think you'd be well prepared to move forward. If you use a different book, just expect some of the concepts covered in Sternberg to just come up in later classes.
@johnmccall40356 ай бұрын
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?
@MrSmackdonkey Жыл бұрын
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.
@Brea9023 ай бұрын
I figured this out in grade 7 and quit school forever. I taught myself.
@weinihao3632 Жыл бұрын
Ah, Halmos, not Helmholtz. That makes sense.
@JoelRosenfeld Жыл бұрын
Halmos is the man! I'm a big fan of his work.
@theplasmacollider6431 Жыл бұрын
Is it wrong to have the answers without the steps? It is nice to know if you are doing things right.
@JoelRosenfeld Жыл бұрын
It can be misleading. You may have a matching answer for completely wrong reasons, or the answer itself might not be the correct one. There is a meta skill of going back and verifying that you did each step of the process correctly. The answer is rarely the goal, but the process is what is important. Practice learning how to check yourself without an answer guide. If it were a class like calculus, make sure you do the problems without corresponding answers in the back of the book as well. That way you can train that skill.
@hunterklein9892 Жыл бұрын
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?
@aikidograndmaster1781 Жыл бұрын
Very True , which is the reason to stay away from books like Rudin...
@jeffrey8770 Жыл бұрын
@@aikidograndmaster1781 What's wrong with rudin?
@aikidograndmaster1781 Жыл бұрын
@@jeffrey8770 It is a lack of solved examples ..look at the indian math analysis books to see how absurd American texts are...
@kristjanbirnirivansson528 Жыл бұрын
I wish someone had told me this 20 years ago.
@k.butler8740 Жыл бұрын
That list at the end about things to check, like a relaxed hypothesis, should be fleshed out and made into a poster!
@vincentzevecke4578Ай бұрын
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 😊
@JoelRosenfeldАй бұрын
@@vincentzevecke4578 the biggest thing is to be compassionate with yourself. Understand that things aren’t going to click as easily as they did before your injury. Just like with any injury, ease yourself into it. Small victories are still victories. Try simple problems at first and try building up to bigger ones. Give yourself a break if you start getting headaches. Put down the work for 10 or 15 minutes and see if that helps. TBI’s can start to get better over time. But for the most part, you are going to need to work with it. Speak with your psychiatrist or neurologist to see if they can give you any professional advice. I am just a mathematician after all.
@vincentzevecke4578Ай бұрын
@@JoelRosenfeld I work with neurologist. My Trauma brain injury happened when I was child
@vincentzevecke4578Ай бұрын
I totally love my mathematics. I was misdiagnosis for 46 years
@vincentzevecke4578Ай бұрын
I will read this over and over. You do not how much I appreciate this.
@migfed Жыл бұрын
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.
@cozuu9190 Жыл бұрын
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
@JoelRosenfeld Жыл бұрын
Yeah looks like a bad cut. I say the full sentence right a minute or so earlier in the video. Can’t remember why I duplicated that. Thanks for catching it. I might actually be able to cut that out without reuploading the video.
@gentlemandude14 ай бұрын
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.
@JoelRosenfeld4 ай бұрын
If there isn’t time to throw themselves at the problem to really figure it out, then the student should take fewer classes. The problems given in undergrad are honestly not that difficult, and should honestly be solvable within the time constraints of a class. If they can’t be done on time, then either the student has become over reliant on solutions manuals or external help, and haven’t developed the real tools they need, or they are taking on too much at once.
@gentlemandude13 ай бұрын
@@JoelRosenfeld Wow! That was a very prick-ish response. Remember "difficult" is relative. What is straightforward for a Mathematics PhD might be very challenging for a novice. Try to have some empathy.
@JoelRosenfeld3 ай бұрын
@@gentlemandude1 I am not being a prick, I’m being raw and honest. When you exit college there is no guides and no direction to help you solve your problems. You need to develop skills in project management, time management, and problem solving. If you turn to a solution manual every time you hit a snag, you aren’t going to develop the skills you need. I also was a student once, and this is exactly how I went about it. I locked myself in the library and worked on problems for hours and hours until I got them. The more I did that, the easier it got. Because I practiced my problem solving skills. Then in later classes it took much less time to solve new problems because I had established strategies and learned how to parse the mathematics from the books I was reading.
@JoelRosenfeld3 ай бұрын
@@gentlemandude1 also, I do have a lot of sympathy. When I assign homework, I specifically pick things that are well within the reach of my students. But they often don’t even try to do them without an answer guide.
@gentlemandude13 ай бұрын
@@JoelRosenfeld I'm happy that you were able to devote hours and hours to study as an undergrad. As I mentioned before, that sort of situation is a luxury. It's good that you took advantage of that position. I'm not sure how long ago that was when you completed your undergrad, but the kinds of people doing undergraduate degrees has changed a lot over the past 25 years. There are more and more people studying who have children, part-time jobs, full-time jobs and other responsibilities beyond their studies. Are you saying that these people don't deserve to study mathematics at a high level because they are not in a position to devote every free moment of their lives to problems sets? I'm sure that you can see that your words can be interpreted as being a bit elitist. I want to be clear: I do not think that you are an elitist. I think that you want everyone to engage with mathematics and become strong mathematicians. I commend that. My problem is not with the hard work, struggle and determination involved in studying mathematics because, as someone who has studied in several different fields at the undergraduate and graduate levels, these requirements hold for any academic discipline.
@PhillipRhodes Жыл бұрын
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.
@danishsamir88075 ай бұрын
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
@JoelRosenfeld5 ай бұрын
How effective that method is really depends on the level you are at. If you are doing Calculus 1 or 2 problems, there really isn’t much harm, because there are literally hundreds of problems you can practice on. If you are getting to upper division mathematics, such as analysis, then looking at the answer can give away the lesson you are supposed to get from it. This is a big deal for people who want a career at mathematics, since the whole job is about figuring out proofs to conjectures no one knows the answer to.
@danishsamir88075 ай бұрын
@@JoelRosenfeld how would you approach a guy like me, who also wants to do this at a professional level in the future bc of my quest to become like Leonardo Da Vinci a polymath in the modern era, but also is stuck at school for now in 12th grade we do calculus but not so much bc here we study little bits and pieces of different disciplines within mathematics that compound and are tested on
@solaris413 Жыл бұрын
halmos interview link is not working please provide the link again
@JoelRosenfeld Жыл бұрын
Looks like the original creator made that video private. The link is valid, but the video has been hidden. Can’t control that on my side. I’ll see if I can find the same interview elsewhere, but there are no guarantees.
@anuraggurudev77616 ай бұрын
Good evening sir
@JoelRosenfeld6 ай бұрын
Good afternoon!
@safeyyaalyahia7198 Жыл бұрын
I wish I can give 100000 likes
@zoedesvl4131 Жыл бұрын
On my blog I have a link "how to study maths really quick" and it is a rickroll.
@JoelRosenfeld Жыл бұрын
Lol that’s hilarious. In the past I have put links on my course website labeled “Exam 3 Answers” before Exam 3 actually happened. It was also a Rickroll.
@smizmar8 Жыл бұрын
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 :)
@JoelRosenfeld Жыл бұрын
The process of verifying a solution is to look at each line and figure out what theorem or proposition makes it valid. Checking the proof or solution step by step. If everything checks out, then it’s likely the answer is correct. Alternatively, post your solution on a site such as r/learnmath, and people will come out to help you.
@smizmar8 Жыл бұрын
@@JoelRosenfeld Thanks so much for your reply. I need to learn more theorems then, but as far as finding out if I'm wrong, reddit is definitely the place haha. But genuinely, thank you, its an excellent suggestion :}