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I have recently started solving some problems from some math problem solving books, and I've noticed a difficulty. When I have to solve problems on the math homework/exam, it becomes a game of "find the way to apply the techniques taught in the section", because that is simply the best way by far to complete the homework and timed exams in a reasonable amount of time.

For example, if in class we are studying the Cauchy Riemann equations and I get a question about the differentiability of a complex function, I immediately try to incorporate the C.R. equations; it would be wildly impractical to try out all other ways, such as writing out the limit definition of differentiability, trying some complicated algebraic manipulations, thinking about the problem geometrically, etc . . .

When I move on to problems that are in a more general setting (outside of class), I find that this mindset is a little hard to shake off; I think only in the context of what was covered in the book, and I find it difficult to let my mind truly run free. If I can't solve a problem and read the solution, sometimes I have the gut reaction "that's not fair, there was nothing mentioned about X so far in the book, how was I supposed to know we were allowed to use that?" It's difficult to truly let my mind run free.

Has anybody else experienced this difficulty? What steps would you take to maintain an open mindset, while still solving classroom problems under time constraints?

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    $\begingroup$ It's why I practice solving problems on MSE. You practically never know how you should approach any particular problem, so you simply get good at figuring the first step out through practice, careful reading of the first problem, and reading other answers. $\endgroup$ – Simply Beautiful Art Mar 3 '17 at 15:05
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    $\begingroup$ This is very similar to solving chess problems versus playing a game of chess. It's easy (for some) to find checkmates in a checkmate problem book, but much harder to find them when playing a game and nobody has told you if there is a checkmate available or not. Something similar happens in computer programming (which I do for a living). In both cases (chess and programming), the solution that often works is seeing patterns in the problems similar to patterns you've had in your past experience. $\endgroup$ – Guy Schalnat Mar 3 '17 at 18:57
  • $\begingroup$ In American high schools there is usually a "math team" (like a football team) where you take tests throughout the year and the tests are sent into a central authority (a "league") who determines which schools "won" the tests and ranks the schools. It's great for building general math skills because the problems are not at all related to the chapters in the text books and most problems require creative thinking to solve them on time. $\endgroup$ – Todd Wilcox Mar 3 '17 at 20:55
  • $\begingroup$ In advanced college mathematics courses and graduate courses, the problems in the section do require the material in the section, but they also require some independent thought. Nobody shows you how to do the problems because the problems aren't formulaic, and seeing one problem done may not necessarily help you in solving another one. When I say "advanced," I mean more advanced than any math course someone who isn't majoring in math (or possibly physics) would take. It takes awhile to get there. $\endgroup$ – Matt Samuel Mar 3 '17 at 23:15
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It's why I practice solving problems on MSE. You practically never know how you should approach any particular problem, so you simply get good at figuring the first step out through practice, careful reading of the first problem, and reading other answers. – Simply Beautiful Art 7 hours ago

Indeed, you will find many quick-witted and possibly inspiring answers here on MSE. Try to answer questions yourself, and you will find that there are usually five scenarios:

  1. You've seen the particular problem or something close, and you're going to use past knowledge to solve it. After that, I recommend watching the question or checking it again in a few hours to see how other people solved the same problem.

  2. You haven't seen this particular problem, but you can still solve it. This is a good opportunity to test how well you can adapt from one problem to the next. Some simple problems that you haven't yet tackled before can really trip you up the first time around, and often when I post answers, it may actually have been my third or fourth attempt after failing multiple times. Such answers can take me hours to come up with, and are often weaved with different concepts, an ability you'll want to develop.

  3. You are in unknown territory and honestly have no idea, but can still understand the problem and can likely understand the solution, once someone comes up with it. These are great questions to favorite (hit the star below the voting area) and check back later. Pretty good way to, if not build technique for hard problems you can't seem to solve (yet!), then it is a good way to get more comfortable with material you don't fully understand at the moment (I often have moments where I'm like "Oh! Now it all makes sense!" and I'm actually referring to old things I've seen that I didn't fully understand before)

  4. These are the questions you avoid because you honestly can't understand anything, and it's just beyond your mathematical knowledge. No idea what to tell you about these questions. You can't learn everything I suppose :-/

  5. This is when you are (gosh darn!) late to the problem, and all the answers you could think of in 5 minutes are already taken. For these scenarios, I find it a great challenge to try the following:

    • See if you can answer it anyways! Being able to come up with more and more ways to answer a single problem is always a good thing, and it'll help you spot ways to answer questions by relating them to more things.

    • See if you can produce... a better answer. This is always very difficult, especially if you enter a question that already has an accepted answer, but, remember! It's not impossible to come from behind! I find that by posting significantly different answers, it has a few affects, 1) it adds not only to your experience (its seriously not easy), but 2) to the community, and 3) it, in the long run, will allow you to answer more questions... questions you may have glossed over due to already having answers. More questions to answer means more practice, right?

Of course, this is simply a good way to learn and practice techniques and problem solving, but it does not replace learning from a book or class, which are much better ways to learn new material.

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  • $\begingroup$ +1 Really great post! So I assume that in situations 2 and 3 you often times don't post an answer? In my experience, if I don't know the answer right away, by the time I figure it out there are already multiple answers, usually which include my method anyway. Though I suppose that if you are answering higher level questions, there are so many possible approaches that perhaps yours still has a chance to be original. $\endgroup$ – Ovi Mar 3 '17 at 23:56
  • $\begingroup$ No, in situations 2 and 3, I try to post an answer, which is a great learning experience, even if I never make it to the finish line. I suppose I'll add a 5th situation, which deals with your "but everyone else beat me to the answers I know" scenario ;) $\endgroup$ – Simply Beautiful Art Mar 3 '17 at 23:59
  • $\begingroup$ Does bulletpoint $2$ exclude bulletpoint $1$? That is, when you can't think of any new ideas to solve it, you just use an old idea but try to formulate the answer in a better/more explanatory/shorter way? $\endgroup$ – Ovi Mar 4 '17 at 0:48
  • $\begingroup$ Yes, of course. Practicing the similar answers to similar questions has actually led me to produce better versions of them. All in the process of evolution and natural selection I suppose ;) $\endgroup$ – Simply Beautiful Art Mar 4 '17 at 0:55
  • $\begingroup$ I imagine that you not only practice math from SE, but you also read your own books? How much time do you usually spend doing math each day? $\endgroup$ – Ovi Mar 4 '17 at 1:12
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This is a well-known problem. I don't know if it's an option for you but sometimes math departments offer a course which is called something along the lines of "problem solving" (in my home university it was called "Workshop in Mathematics"). In such a course, you are given each week a list of problems to work out on without any context, present your solutions and learn from the solution of other students. This isn't like regular homework where you need to solve all the problems - you need to try and solve some of the problems that interest you and hopefully succeed in solving at least one problem. Some problems might be easy, some will become easy once you'll identify which techniques should be used to attack the problems (so the difficult thing will be to put the problem in the appropriate context) and some will be hard no matter what you try. To toughen you up, sometimes an open problem might "slip in".

The point is that none of the problems are "routine application" of the concepts learned two days ago (for one, because you don't learn any systematic theory which builds up as weeks pass by). There is no time pressure (it is definitely possible to spend the whole week thinking on one problem) nor too much pressure to solve everything (you discover quickly that it is almost impossible). If the problems are chosen well and the atmosphere in the class is good, this kind of class really develops the skills that are often neglected in regular courses. You get to "play with the problem", identify various possible approaches, try and fail a lot, look for unexpected connections, etc. And naturally, those kind of skills are much more relevant later (whether you go into research or apply your knowledge to real world problems).

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    $\begingroup$ :-) sounds like the take home quizzes my physics teacher gives out. $\endgroup$ – Simply Beautiful Art Mar 3 '17 at 16:04
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Generally speaking, when you see a problem the first thing that comes into mind is comparing it to problems that you have already seen and hence you become tempted to use the tools you actually learnt from a book or your lecturer. Some people have the gift to think out of the box and generate new techniques and solutions that don't follow the traditional approaches. On the other hand some other people might have a good knowledge in different aspects such that they can connect the dots and think about new approaches. I think you shouldn't feel bad about yourself and at the same time you should try not to be constrained to the approaches you know. After some time you will have your own style that can impress other people. Everyone of us has his strengths and weaknesses but the door is open for everyone to learn and try to master new methods. Usually at school or university students are constrained to use some strict methods and I agree with you that this might restrict the mindset but this shouldn't be a hindrance from trying out new methods. Not necessarily you show off at exams or homework but in communities as open as MSE there is a window for creativity and to be inspired by some great minds we have here.

A piece of advice

Try to read from different resources about a certain problem. The university books are usually restricted to the traditional approaches and don't help your mind to think freely.

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I actually took the 'opposite' approach to what others are suggesting or what you might have been looking for. I focused on learning interesting mathematics for its own sake and solving problems in whatever areas I pleased, totally free from any constraints of techniques or prerequisites. The only rule I had for myself was that I would try to refrain from using anything I had not already proven or verified the proof, unless I felt that it was too difficult for me to do so at my current ability (such as the central limit theorem when I was learning statistics).

Then of course I had difficulty in exams that were very tight on time. I frequently re-derived the theorems I needed during the exam itself, and it did take up some time, but I still got by because my ability to tackle new problems reduced the time taken to find solutions to the harder problems! Of course, just before exams I did some practice questions just to get used to them temporarily, as it would be a bit foolish to intentionally disadvantage myself compared to other students who already were used to exam-style questions.

I feel that this approach is in the end the best way to learn mathematics. It is far easier to learn to restrict yourself than to learn to explore without guidance, so I think it is better to learn the latter and worry about the former only when it comes down to academic assessment.

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  • $\begingroup$ It may help to regularly attempt olympiad-style problems. While they are usually of very different nature from 'classroom maths', they will allow you to learn a myriad of distinct (and often cute) ways to use the same simple techniques to do crazy things. Take the extremal principle for example, with a few examples at artofproblemsolving.com/wiki/index.php/Extreme_principle. $\endgroup$ – user21820 Mar 3 '17 at 17:44

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