How long will you go without a solution.

[nascentmind]

Hi,

I am doing some self study in maths and was wondering how long would somebody go about thinking for a solution to a problem? This includes both simple and complex problems. I find that the subject matter gets clearer as the days go by. Sometimes a solution takes a week or more to come up. Its kind of frustrating because of the amount of time it takes. I am just normal with more of curiosity in maths and mainly its applications to what I am doing. I would like to know whether its normal and want to hear member experiences.

Thanks in advance.

 

[apt403]

It's normal not to be able to come up with a solution immediately, I've poured over questions for days, and I've found, after a certain point, it's best just to let your mind take a break, give the subconscious some time to mull it over. For me at least, it seems like if I can't figure out an answer, I just take a while and focus on something else, and when I go back to the problem, there's a much higher rate of those 'Eureka!' moments where everything just clicks.

Last night, I spent about 3 hours looking at the same problem, gave up, slept on it, and when I got up this morning, I looked at it again and found the solution in under 5 minutes.

And if you're really having trouble, just post it up here! You've got the combined force and knowledge of the Internet at your back, man!

 

[nascentmind]

Yes it happens to me very often when I take my mind off the problem and work on something else. So I developed a strategy where I work on 3-4 different problems completely unrelated to each other but math application is in involved in most of these. Sometimes I get more ideas and sometimes I get hints from these other problems. The problem is that people start pointing saying I am not focusing on one thing at a time or there is not much progress on anyone thing. How do you get around this?

 

[apt403]

If it works for you, just tell them you're multitasking!

 

[nascentmind]

hehe. Thanks! :)

 

[rasmhop]

Its kind of frustrating because of the amount of time it takes. I am just normal with more of curiosity in maths and mainly its applications to what I am doing. I would like to know whether its normal and want to hear member experiences.

How many problems do you feel this way about? If it's only an occasional problem then it's perfectly normal. If it's every other problem, then you may not have a good enough grasp on the subject matter.

And if you're really having trouble, just post it up here! You've got the combined force and knowledge of the Internet at your back, man!

I would suggest that he does not do this. Not being able to solve a problem and thinking about it for days (or weeks) is healthy. In my experience the problems you learn the most from are the ones you're having a hard time with. After having giving it some serious thought and effort you can of course ask for suggestions, but sometimes it's good to get stuck on problems even if it can be frustrating. At some point you will reach a level were you can't just ask an older student/professor/forum member who took the course earlier because you're doing real-life work or original research.

I realize I may have inadvertently been discouraging you from discussing problems and that is not my goal at all. You should definitely discuss them with other people, just not every problem and not always before you've spend 5min on it yourself. I have seen one too many cases of students totally incapable of thinking a thought for themselves because they can get the answer handed to them (I think study groups can be dangerous in this respect). While a traditional educational system judges you by your ability to get answers correct realize that the goal is to learn, not to answer as many problems as possible. Sometimes working 4-hours on what should have been a 30-min problem is what makes the concepts finally click.

Sometimes I get more ideas and sometimes I get hints from these other problems. The problem is that people start pointing saying I am not focusing on one thing at a time or there is not much progress on anyone thing. How do you get around this?

A couple of points:
1) Who are these people? Are they parents, fellow students or friends trying to help you? Are they teachers trying to guide your study habits? No matter what remember that it's ultimately up to you. I don't remember who said it, but a famous quote goes along the lines of: "I have never let my education get in the way of my learning". This is the right approach to have. If you really feel you're doing things in a way that maximizes learning, then I don't see why you should let their opinions get in the way. Just tell them that this is the most efficient method for you.

2) Maybe you should consider what they're saying. Working on 3-4 problems at once can get inefficient. When you say at the same time what exactly does that mean? You should probably spend at least 10min continuously on one before moving on, but really I don't see an issue with going from one problem to the next. Also if you're able to draw parallels and make connections, then chances are good that you're understanding the material.

I used to do math competitions where you got 4.5 hours for 3 problems. My approach to this was usually to use the first 45min by working on each problem for 15min. This would allow me to get a feel for all the problems. What approaches definitely didn't work. What approaches likely were to be used. Keeping all this in my head greatly increased my efficiency as I found that while working on one problem I might get ideas for the others. However I also learned that too frequent task switching was a bad idea. If you got stuck on a problem it was all too easy to switch problems. This is fine as long as you have approachable problems to attempt, but once you're only working on hard problems you need to commit to thinking for a while on one problem or you're never going to get anywhere.

These are of course my personal experiences and yours may differ, but remember that too frequent task switching is inefficient. However there is nothing wrong with having a couple of problems in the back of your hard. Some mathematician (I'm not sure who, but it may have been Erdos) said that you should always keep a couple of problems you're working on in the back of your head. Then whenever you hear about a new technique you can try it.

EDIT:
And to actually answer your question I just used 9 days working on an algebra problem I had thought of myself. I then shared it with a grad student doing research in algebraic topology who said that it was trivial because of some incredibly easy argument (basically boiling down to some corollaries of the Sylow theorems on the factorizations of certain subgroups as a product of Sylow p-groups). I was a bit ashamed because I felt I should have been able to tackle it, but at the same time now I feel it helped both my problem solving skills and my understanding of that area of algebra.

I spend 3-14days working on a problem 1-2 times a month. After that I either put it aside to revisit it later or try getting some feedback from others.

 

[nascentmind]

How many problems do you feel this way about?

Some problems but then I re read the material to see if I have not understood something right. Usually after that I can solve.

At some point you will reach a level were you can't just ask an older student/professor/forum member who took the course earlier because you're doing real-life work or original research.

This is what is happening sometimes. My day job has nothing to do with this work and its very tough to talk about it with my colleagues as they don't have any idea about it. Also if the problem's I come across are from journals the other party should have read them or have access to it as most of them are paid.

Who are these people?

Mostly colleagues because I try to get them interested in what I am doing. Sadly they say I am working on too many things and lacking focus as this won't help me with my future or anything.

I generally keep the problems in my head during my day job and read about some materials during my free time and yes a lot of times I might be spending time on thinking on problems which are trivial.

I am an average student but a very curious one. It takes time for me to assimilate ideas and have to work harder than most people but finding the truth as to how they do it and the researching makes it worthwhile for me.

 

[doive]

It really depends on the complexity of the problem, only recently Grigoriy Perelman solved the Poincaré conjecture which was proposed way back in 1904. So it can take a very long time indeed!

In my experience, what separates the good from the bad mathematicians at university undergrad level and above, is the length of time they devote to focussing on a single problem. Someone who can focus on many aspects of a complex problem over weeks or even months is much more likely to be successful, if purely because they explore more potential solutions. If you have the concentration span to consider a problem for a week or more at a time then power to you!