Parent of one of our students today wrote about the incidental frustration of his daughter about the difficulty of some problems on our courses.Courses, and was easy one of the very best students in the class that she took, and yet she still becomes the occasional problems that she cannot solve.
Moreover, she has access to an excellent math teacher at her school, who sometimes cannot help her to pass these problems (this is not easy for him - I have students to bring problems that I can't solve!) Her question: "Why would it be so difficult? "
The case to do hard things
We ask difficult questions because so many of the problems that are worth solving life are difficult.If they were lightweight, someone else would have solved them before you came to them.Nobody knew 70%, much less gets a perfect score.The train future researchers, and the whole research is to find and answer questions that have never been resolved..If you consistently get a problem in a correct class, don't be too happy - that means that you do not learn effectively.
The problem not to be sufficiently challenged goes much further than not learning math (or whatever).To solve very difficult math problems, you can use to tackle many problems.
The first step in dealing with difficult problems is to accept and understand their meaning ... It will not teach you much more than a spreadsheet full of slight problems."Aha!"Moments almost always come out of the mind cultivated by long periods of frustration.But without this frustration the brilliant ideas will never occur.
Strategies for difficult math problems - and beyond
Here are a few strategies for dealing with difficult problems and the frustration that goes with it:
Do something.Yi, the problem is difficult.Yen, you have no idea what to do to solve it.At some point you have to stop staring and start trying things..
We started developing a primary school management plan months and months before we had the idea that it becameBeast Academy.Important, it prepared us, so when we finally hit Beast Academy's idea, we were certainly pursuing it.
Simplified has problem.
Think about successes.You have solved many problems.Strategies you used to solve these problems and you can simply come across the solution.
A few months ago I played with some project Euler problems and I got a problem that (eventually) came down to generate entire solutions forC²= A²+ B²+ ABIn an effective way.The theory is not my strength, but my path to the solution was to remember the method to generate Pythagoras Triple.Some of our more mathematical advanced readers have internalized the solution process for this type of Diofantine comparison that you don't have to travel with Pythagoras to get there!)
Focus on what you haven't used yet.The key to your next step.
Work backwards.This is especially useful when you try to discover evidence.In the place to start with what you know and work on what you want, start with what you want and ask yourself what you need to get there.
Ask for help.This is difficult for many excellent students.Years I was on my way over my head.I understood very little of something that happened in the classroom.I worked with me in the 15 minutes.I just couldn't admit it and ask for more help, so I stopped asking.Do not understand.further than if you don't.
Started early.This does not help much with timed tests, but with the longer reach that is part of college and life, it is important.The people you know who seem bad and who seem to think of ideas much faster than you could be, often people who have just thought of the problems for much longer than you used.The first few weeks of each semester I worked far ahead in all my classes.At the end of the semester I had thought of the most important ideas for much longer than most of my classmates, who made the exams and the like at the end of the course much easier.
Take a break..Of course it is much easier to take a break when you start early.
Start over.In your second round.
Give up.You don't solve them all.At some point it is time to reduce your losses and continue.The first hour or two than it will be in the next six, and there are many more problems to learn from.So set yourself a time limit and if you are still hopelessly determined, read the solution and continue.
Be introspective. If you give up and read the solution, read it active, not passive.You must look mathematical facts in the solution that you do not understand.I was completely confused the first time I saw many things about "against" in an Olympiad solution - we had no internet when I couldn't have it easy to see how simple modular arithmetic is!To try them.
Coming backIs
Richard Feynman once wrote that he would keep four or five problems in the rear..
