Thinking about problems without solution?

[kakarotyjn]

There is a parlance from my teacher that we should think for hard problems as much as we can.Don't look for solutions,do problems without solutions and so you can be trained to face real problems.

What's your view of this opnion?Should we think for hours to solve a test problem?

 

[Kajahtava]

I'm not really sure if problems without a solution train one that much. A good start for me is to mentally verify at the beginning if a solution exists, if not, stop.

Also, people seem to be inclined to think that all problems have a solution, that's hardly true.

 

[zomgwtf]

There is a parlance from my teacher that we should think for hard problems as much as we can.Don't look for solutions,do problems without solutions and so you can be trained to face real problems.

What's your view of this opnion?Should we think for hours to solve a test problem?

I think that this is good advice, it sucks to take this approach when you have hours of other homework to do but I truly believe you learn the most from your mistakes the biggest problem is, how do you know if your wrong? I assume however at some point you will be given the correct answer and have guidance along the way (ie. you show the teacher what you did the next day and they help you out). If it's just sitting there on your own doing hours of math problems and you don't know if your doing it right or wrong and you'll never know your mistakes then I don't see much help in this method to you.

If you always rely on solutions you won't really learn anything, except for where to look for which solution.
When you advance further and further in math or physics there are plenty of cases where there simply is no set solution yet, and you have to take the time to work it all out on your own. People who enjoy doing math love these types of situations my friend who is currently taking math takes the most amount of pleasure when he works out proofs on his own... I don't really understand why he enjoys doing it but he does. :tongue:

I think this is a major problem in our modern society though. Students now have the internet which gives them free solutions to everything they ever needed to know. All they need to learn is where to look and how to verify... not very hard at all. The reason I say this is a problem is that it appears that they know a lot since they can do all the work properly and achieve great grades... but when something eventually comes up where there are no answers on the internet or solutions given you see that they don't know what they are doing (most of the time). So I think it's a great skill to build up as long as you build it up with guidance.

 

[Count Iblis]

There is a parlance from my teacher that we should think for hard problems as much as we can.Don't look for solutions,do problems without solutions and so you can be trained to face real problems.

What's your view of this opnion?Should we think for hours to solve a test problem?

I agree with Zomgwtf. Also, you should forget about seting deadlines like "hours". Sometimes you need to spend days if not weeks. There is a trend in the educational systems all over the world (prompted by cost cuting) to give students more and more simpler exercises that can be done in a small amount of time, so that students can more easily pass exams and finish their studies faster.

But good exercises are exercises that you may be working on for many days that will lead you to study certain theories. To understand those theories you may need to set yourself some other exercises, study other theories and set yourself exercises for those other theories etc. etc. When you finally solve the problem you may have learned a great deal about many, many different things.

When I was at university studying was a lot like that. Today it is more like a stupid high school system where students are given simple exercises that they can do by plugging one formula into another formula from the book without actually understanding one iota.

 

[zomgwtf]

But good exercises are exercises that you may be working on for many days that will lead you to study certain theories. To understand those theories you may need to set yourself some other exercises, study other theories and set yourself exercises for those other theories etc. etc. When you finally solve the problem you may have learned a great deal about many, many different things.

I was definitely glad at the end of my high school 'career' when I had realized this. My high school wasn't one of those 'normal' high schools that gave simple assignments to keep the averages up. I hated it while I was doing my course work but at the end of every assignment there was a sort of 'I accomplished it!' euphoria :tongue:. You tended to learn plenty more than the assignment was originally intended to show you as well. As a major bonus many universities recognized that my school may have had lower averages relative to others in the area but it was compensated normally.

Talking to other people my age who didn't attend my high school reinforces this. They just don't seem as 'intelligent' for the most part... many of them achieve 90 averages (which was extremely rare at my school) and know everything about solving the courses problems but when you talk to them you can just tell they are sort of 'ditzy'. That's why I love these forums, even though a majority of the people are much much older than myself they seem to respect intelligence and critical thinking.

 

[Mathnomalous]

I think it's an efficient approach as long as there's proper guidance as the previous posters expressed. I also think Kajahtava hit on an important point about people thinking all problems have solutions.

For a while early in high school, I always believed that solutions to math problems where supposed to be whole numbers; whenever I solved a problem with a fraction, I went back and checked over and over to see what I did "wrong." I also thought that all problems had solution so I ended up wasting time trying to find solutions to problems that didn't have them.

The point being, that training oneself to spot the problems without solutions faster saves one time to focus on the ones that do have them.

 

[kakarotyjn]

I'm not really sure if problems without a solution train one that much. A good start for me is to mentally verify at the beginning if a solution exists, if not, stop.

Also, people seem to be inclined to think that all problems have a solution, that's hardly true.

Oh,I'm sorry that I don't make it clear."Problems without solution" means they have solution but we can get,so wen can only think for it to solve the problems.

If we do problems we don't know whether they have a solution,that's so hard.

 

[erok81]

If you always rely on solutions you won't really learn anything, except for where to look for which solution.
When you advance further and further in math or physics there are plenty of cases where there simply is no set solution yet, and you have to take the time to work it all out on your own. People who enjoy doing math love these types of situations my friend who is currently taking math takes the most amount of pleasure when he works out proofs on his own... I don't really understand why he enjoys doing it but he does. :tongue:

I 100% agree. My solution manuals have ruined me in the past. I had to have my wife hide mine so I couldn't use it anymore. Whenever I'd do my homework I'd have that trusty manual by my side looking at every problem as I solve them.

The wake up call for me came when people in class would ask me questions and I'd have no idea how I solved them. Now that I don't use that stupid manual, I can explain these problems to other students with ease.

Of course I still like to have the answer in the back of the book so I can check my work.

 

[rootX]

I 100% agree. My solution manuals have ruined me in the past. I had to have my wife hide mine so I couldn't use it anymore. Whenever I'd do my homework I'd have that trusty manual by my side looking at every problem as I solve them.

The wake up call for me came when people in class would ask me questions and I'd have no idea how I solved them. Now that I don't use that stupid manual, I can explain these problems to other students with ease.

Of course I still like to have the answer in the back of the book so I can check my work.

I never figured out in 1-2 days period if it is better to solve 200+ questions while referring to the solution manual for hints/final answers .. or solving 50+ questions by working through each problem while referring to only the final answers or lastly solving about 100+ questions with nothing to refer to.

 

[kakarotyjn]

If it's just sitting there on your own doing hours of math problems and you don't know if your doing it right or wrong and you'll never know your mistakes then I don't see much help in this method to you.

Year,that's the point I'm anxious about.I know we should think for hard and good problems without solution.But if we only think it without anybody's guidence,how can we learn from it?How do we learn new method?How do we pass the exam?

I'm really depressed by my grades.I was suggested by a friend to learn mathematical analysis with the textbook written by Zorich.But our teacher don't use it,he use a less competitive textbook written by Chinese.

Maybe you know,there are lots of hard problem in Zorich's book and I think them all by myself without solutions and guidence,which took me much time.But I havn't work out a majority of them,and at the exam,I did really bad.I get low scores.I'm really heart-broken.

So I query myself and my behavior.Sigh