Math and Problem solving

[srinath.r]

"I agree with those people who just dont care for imo, since if someone just solves problems all day, he will lose his imagination and just become a problemsolvingmachine without any thinking further, about new things.
The other thing is the time, who cares if one can solve 999 problems in x hours, when he becomes a mathematician there will be no time limit.
So what the imo organizers should do, is not a contest where you have to solve problems in a given time, but like find something new discover a new proof for a theorem etc."

What all you have to say on this ? Learning and investigate deep and advanced math or solve lot of problems and clear all the olympiads and examinations . ????

[kurt.math]

Moderator says: there is never a good reason to block-quote the entire post immediately preceding your own.

I don't agree with your assertion that you would lose you imagination if you keep doing IMO problems. Most of the IMO problems require the true work of a mathematician, you begin to play around with the problem, then you make some hypotheses, you test them and look at the problems in new ways and then try other things until you either brute force an answer or you have a mini-eureka moment and solve the problem, this is true with most national and international mathematical Olympiads.

One of the most important qualities a mathematician can have is the ability to problem solve, intuition and dedication. Although its true that mathematicians have no time limit but its fairly important to solve a problem in a short amount of time, a mathematician that can solve a certain problem in 2 days that another mathematician can solve in 5 months will do better and be more notable. Take Euler, in 1 year he wrote 54 papers, which is about a paper a weak (and the majoring of his work was significant, not derivative restatements). Or, more contemporary, Terrence tao in a year publishes between 26 and 60 papers. Its their ability in problem solving that allows them to see this faster than other mathematicians.

Also, consider the fact that since the IMO has been running, of the 40 mathematicians that have recieved the fields medal, 8 of them have done the IMO. And in 2006, of the 4 recipients, 2 had been to the IMO! So its clear that the IMO is a key feature that gives a way for gifted high school students the possibility to use their imagination and solve difficult problems.

But i am intrigued by you idea of being given a "research" problem and finding a new proof or discovering something new from it. I dont think this would work. Consider how the IMO problems are strictly non calculus, this is because most high schools around the world do not go past elementary calculus, and as such cannot pose problems that require knowledge of a topic in a higher range such as lie algebra or algebraic topology. If they did construct such a competition that required knowledge of undergraduate mathematics for high school students it would only appeal to the geniuses which havent already started their degree. And so there would not be that many people who fit the requirements.

Although i am intrigued about an undergraduate competition where you are given a difficult research problem and have to finish first. That would be interesting.

[Lazarus]

[srinath.r]

@kurt math
Actually those words were not mine ,I just copy pasted it from somewhere . Anyway thats immaterial
Even I disagree that doing problems and problems will lose your imagination skill . But I insist that there is no point in doing problem after problem after problem after problem. Usually people (there are exceptions) just solve problems and that according to them that becomes math for them . See it is like this
,"ok today take this X country sheet in the Yth year " after few hours "thank god I solved this sheet entirely and where is the Y+1th year's sheet "
A person liking math ,will not do like this . See it just becomes routine and monotonous and as I said he is one typical problem solving machine . When the mind is tuned in that way , they think only when some book or a problem demands ,say for example writing down powers of 2 and getting some nice conjecture may be interesting and also investigating the fibonacci numbers and pascal triangles ,but people who are typical problem solving machine will just think about them only when the religious problem solving book they use gives a problem demanding "Prove that nth fibonacci number is ............" or . That too he will leave the problem once he finishes solving it and go to the next problem without even extending the dynamics and boundary of the problem . If these are all there ,how can he become a research mathematician where self and automatic investigation is important .
I am not telling that problem solving is a disadvantage for a mathematician . I am telling problem solving is not a pre-requisite ....problem solving helps but without it still one can make a very good mathematician .

@Lezarus,sorry but I am not able to understand what you mean ?

[matt276eagles]

I think you're suggesting that the sole purpose of math competitions is to prepare people to do research math. I don't think this could be much farther from the truth. I'm a freshman in college who spent a lot of time during high school doing olympiad math, and I don't think I'd be well-suited at all to doing research math. But still, I enjoyed solving olympiad math problems and taking the olympiads themselves more than I can describe, and I value my experience with math competitions more than anything else I did academically in high school. Olympiad math taught me how to solve problems. It kind of taught me how to do math, but more importantly it taught me how to solve problems. I'm definitely not planning to go into research math, and I'm not even planning to major in math, but the math contest experience was still incredibly valuable for me.

Maybe you'd still argue that the purpose of math competitions SHOULD be to prepare people to do research math. But if this were true, then the target audience of math competitions would be much, much smaller than it is now, and other people like me wouldn't benefit from them. The people who really want to prepare for research math can just do math research. Problem solved.

[JBL]

This seems relevant.

[srinath.r]

@matt eagles
If that 'you' in your post refers 'me' ,I am sorry but I never meant olympiad mathematics trains people for research math . This statement is absolutely wrong as far as I am concerned . But since you are in college now ,are you saying that having a very good flair in olympiad type problems ,you have an exceptional scope in research mathematics .?

@JBL ,thank you very much for giving the link ,very apt in fact

[fedja]

[srinath.r]

@fedja ,You have got a point . It is mainly to do problems and be quite familar solving olympiad problems (having some tricks under the sleeves) ,so how does it matter when (age) you try problems . You said that learning deep theories at age between 9-15 cant train you well or solving problems and problems only will train you well at that age . This may be wrong ,what if we go about a theory centered problem solving approach, there is no point in absolutely learning just theory ,what if we think deeply about those things ,frame problems and try solving them .This is also a kind of "problem solving" right ,where you use the theory to frame problems and try solving them .This is also something like a research . Why cant young people just pick some stuffs from air and start working on them . The main point is to keep yourself touch with problem solving right ? So how does it matter when and where we solve problems and it doesnt matter from which 'olympiad book' we solve a problem ,we are still solving problems .Framing theories and problems will develop a more creative thinking isnt it ?Because most people are mad after olympiad problems and they solve lots and lots of problems and they think about some beautiful things only when a problem or statement or a book demands it,this hinders creativity and research thinking isnt it ? I think both learning theories and some good familiarity with problem solving even at a young age is important and essential for a research mathematician .
What do you say on this?

[Boy Soprano II]

[yenlee]

Math contest jingoists claim that contests are invaluable because they improve problem solving skills. This is tautologically true, but no one can know how well these math contest skills transfer to other problems, and consequently we cannot know the actual value of this skill set relative to the value of doing some other educational activity. Opinions on this issue break down something like this: Adults who enjoyed math contests and succeeded in them are more likely to think that these skills are valuable, while successful adults who did not have such experiences are more likely to see these skills as unimportant. (It's only human nature. The same goes for other questions such as, "Does learning music make you better at math, or vice versa," or, "Do video game skills help cognitive development, or are they totally useless?" Anyone who thinks highly of themselves will approve of most aspects of the way they were brought up.)

[fedja]

[srinath.r]

@fedja thank you very much for the detailed post.
See,by me telling "learning theories " ,I do not mean that spend your time entirely just by book reading and knowing some 1000 facts . You said that learning theories and thinking about it deeply pre-conditions your thinking. But I have one question to ask ,for example imagine a situation like this ,you are interested in learning some abstract algebra and you pick some book and start reading it .You learn what is a group ,next what I am telling is once you are introduced to this why not think deeply about it "Oh,this is a group ,so what relation and what significance this has " . How does it limit your thoughts .What you say is correct if I just get the book -see definitons ,see lemmas,see proofs and if I am not putting any own effort apart from understanding it . By seeing some definitions and getting some insights how does it limit my thinking if I am thinking about it deeply ?By you telling me to do more problems when I am young without learning theories as doing just the latter will limits my thoughts ,I can invent group theory? It is practically not possible for everyone who keeps on trying problems after problems to reinvent the entire mathematics . At one point of time you will be introduced or learn some theory and it can be at any time.Concept learning and problem solving are related when the concept is within the scope of a particular problem ,so I am telling once you sit on a problem and you manage to solve it and if it is interesting why not go deep about it -explore somethings and look for resources which have already dealt with that concept and improve yourself? This definitely contributes for becoming a research mathematician right ? I am mainly telling there is no point in solving 1000 problems only one after an other.
Please forgive if I have misunderstood what you mean to say.
Unrelatedly ,you had known so much compared to me when you were in 10th ( I am in 10th now ),maybe your education system is very good that way .In our country education system is horrible and they teach fundamental theorem of arithmetic in school only in 10th .

[Boy Soprano II]

[mavropnevma]

To bring in my tuppence of wisdom - how about the sheer satisfaction of getting a problem right? Nowadays everybody is so very much result-oriented and questions everything on the basis of the value-added and return-on-investment, that we forget the essence of just being there and able-bodied and able-minded to enjoy an abstract challenge. I direct you to (re)read Herman Hesse's "Magister Ludi or The Glass Bead Game". It is also true that I am somewhat of an eccentric, so I do not want to seem prone on proselytizing ...

[srinath.r]