Upload a Math Problem for People to Help
parent of 1 of our students wrote today about his daughter'southward occasional frustration with the difficulty of some of the problems in our courses. She does fantastic work in our courses, and was easily amongst the very top students in the form she took with me, and notwithstanding she still occasionally hits problems that she tin't solve.
Moreover, she has access to an first-class math teacher in her school who sometimes can't assistance her get past these issues, either. (This is no slight to him—I have students bring me problems I tin't solve, too!) Her question: "Why does information technology have to be so hard?"
The Case for Doing Hard Things
We inquire hard questions because so many of the problems worth solving in life are hard. If they were easy, someone else would have solved them earlier yous got to them. This is why college classes at tiptop-tier universities take tests on which well-nigh no one clears seventy%, much less gets a perfect score. They're training future researchers, and the whole point of research is to find and respond questions that have never been solved. You tin can't learn how to do that without fighting with problems you lot can't solve. If you are consistently getting every problem in a course correct, y'all shouldn't be too happy — it ways you aren't learning efficiently enough. You need to discover a harder class.
The trouble with not beingness challenged sufficiently goes well beyond not learning math (or whatever) as quickly as you can. I call back a lot of what we do at AoPS is preparing students for challenges well exterior mathematics. The same sort of strategies that go into solving very difficult math problems can be used to tackle a great many problems. I believe we're teaching students how to recollect, how to approach hard bug, and that math happens to be the all-time way to practice so for many people.
The first pace in dealing with hard problems is to accept and sympathise their importance. Don't duck them. They will teach yous a lot more than a worksheet full of piece of cake problems. Vivid "Aha!" moments about always bound from minds cultivated past long periods of frustration. But without that frustration, those brilliant ideas never ascend.
Strategies for Difficult Math Bug — and Beyond
Here are a few strategies for dealing with hard bug, and the frustration that comes with them:
Do something. Yes, the trouble is difficult. Yeah, you have no idea what to practice to solve information technology. At some point you have to stop staring and showtime trying stuff. Most of it won't work. Have that a lot of your effort will appear to take been wasted. Simply there'due south a chance that i of your stabs will hitting something, and fifty-fifty if it doesn't, the endeavour may prepare your mind for the winning idea when the time comes.
We started developing an elementary school curriculum months and months earlier we had the thought that became Beast University. Our lead curriculum developer wrote 100–200 pages of content, dreaming up lots of different styles and approaches we might use. Not a 1 of those pages will be in the final work, but they spurred a great many ideas for content we will use. Perhaps more importantly, information technology prepared us so that when nosotros finally hit upon the Fauna Academy idea, we were confident enough to pursue it.
Simplify the problem. Effort smaller numbers and special cases. Remove restrictions. Or add restrictions. Set your sights a piffling lower, and then raise them once yous tackle the simpler trouble.
Reflect on successes. Yous've solved lots of problems. Some of them were even difficult problems! How did yous do it? Outset with problems that are similar to the one yous confront, but also call up nearly others that have nothing to exercise with your electric current problem. Recall about the strategies you used to solve those problems, and you might merely stumble on the solution.
A few months ago, I was playing around with some Projection Euler problems, and I came upon a problem that (eventually) boiled down to generating integer solutions to c² = a² + b² + ab in an efficient manner. Number theory is not my strength, but my path to the solution was to recall get-go the method for generating Pythagorean triples. And so, I thought most how to generate that method, and the path to the solution became clear. (I'm guessing some of our more mathematically advanced readers take then internalized the solution procedure for this type of Diophantine equation that you don't have to travel with Pythagoras to get there!)
Focus on what you lot oasis't used nonetheless. Many problems (peculiarly geometry issues) take a lot of moving parts. Await dorsum at the problem, and the discoveries you have made so far and ask yourself: "What haven't I used yet in any effective way?" The answer to that question is often the key to your side by side step.
Work backwards. This is especially useful when trying to find proofs. Instead of starting from what y'all know and working towards what yous desire, offset from what y'all want, and ask yourself what you lot need to get there.
Ask for assist. This is hard for many outstanding students. Y'all're so used to getting everything right, to being the 1 everyone else asks, that information technology's hard to admit you demand assist. When I first got to the Mathematical Olympiad Program (MOP) my sophomore year, I was in way over my head. I understood very lilliputian of anything that happened in class. I asked for help from the professor one time — it was very hard to get up the courage to do so. I didn't understand annihilation he told me during the fifteen minutes he worked privately with me. I just couldn't acknowledge it and enquire for more help, so I stopped request. I could have learned much, much more had I only been more willing to acknowledge to people that I just didn't understand. (This is part of why our classes at present have a feature that allows students to enquire questions anonymously.) Get over information technology. You will become stuck. You volition need help. And if you ask for information technology, you'll get much farther than if yous don't.
Start early. This doesn't help much with timed tests, but with the longer-range assignments that are parts of college and of life, it's essential. Don't expect until the final minute — hard bug are hard plenty without having to deal with time pressure. Moreover, complex ideas accept a long time to understand fully. The people you know who seem wicked smart, and who seem to come with ideas much faster than you possibly could, are frequently people who take merely thought about the issues for much longer than you have. I used this strategy throughout college to great success — in the starting time few weeks of each semester, I worked far ahead in all of my classes. Therefore, by the end of the semester, I had been thinking about the primal ideas for a lot longer than most of my classmates, making the exams and such at the finish of the form a lot easier.
Have a break. Go abroad from the problem for a bit. When you come up back to information technology, you may observe that you lot haven't entirely gotten away from the problem at all — the background processes of your encephalon take connected plugging away, and y'all'll discover yourself a lot closer to the solution. Of course, it's a lot easier to take a break if you first early.
Start over. Put all your earlier piece of work aside, get a fresh canvas of newspaper, and try to start from scratch. Your other work volition still be at that place if you want to depict from it later, and information technology may have prepared you to accept advantage of insights you brand in your second go-round.
Give upward. You won't solve them all. At some signal, it's time to cutting your losses and move on. This is peculiarly true when you're in grooming, and trying to learn new things. A single hard trouble is usually going to teach y'all more in the start hour or two than it will in the next six, and there are a lot more issues to acquire from. And then, ready yourself a time limit, and if you're yet hopelessly stuck at the terminate of it, then read the solutions and move on.
Be introspective. If you do give up and read the solution, then read it actively, not passively. Every bit you read it, call up about what clues in the problem could have led yous to this solution. Think near what yous did wrong in your investigation. If there are math facts in the solution that y'all don't understand, then go investigate. I was completely befuddled the first fourth dimension I saw a bunch of stuff virtually "mod"southward in an olympiad solution — we didn't take the net then, then I couldn't easily observe out how straightforward modular arithmetic is! You have the internet now, so you have no excuse. If y'all did solve the problem, don't just pat yourself on the back. Remember about the key steps you made, and what the signs were to endeavor them. Retrieve about the blind alleys you explored en route to the solution, and how y'all could have avoided them. Those lessons will serve you well later.
Come back. If you gave up and looked at the solutions, so come dorsum and try the problem once again a few weeks later. If you don't have whatever solutions to look at, proceed the problem alive. Store it away on paper or in your mind.
Richard Feynman one time wrote that he would proceed four or five problems active in the dorsum of his heed. Whenever he heard a new strategy or technique, he would quickly run through his problems and see if he could use it to solve one of his problems. He credits this practise for some of the anecdotes that gave Feynman such a reputation for being a genius. It's farther evidence that being a genius is an awful lot near effort, preparation, and being comfortable with challenges.
Source: https://artofproblemsolving.com/blog/articles/dealing-with-hard-problems
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