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About solutions to problems

by P3X-018
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P3X-018
#1
Jan26-07, 12:05 PM
P: 144
In many science textbooks the solutions to the problems are provided in the book. Doesn't this have a negative effect on learning the subject? When you try to solve a problem, and then check the answers to see wether you have the correct, when you realize that the answer is incorrect you immidiatly try to figure how to get to given result, sometimes by for example manipulating equations without even understanding the solution... And when you get used to check the problems by the answers, later on you don't use enough time thinking about wether your answer is correct or not by really thinking about the peoblem, instead you can just compare with the answeres.
Isn't this a bad effect, because when you're finally to the exams there usually aren't anything you can compare your result to, and if you haven't learned to check through your solutions by considering what you could have done wrong, i.e. is the assumptions or the equations correct, then it's more likely that you won't see the existing errors.
Doesn't this mean that your ability to check through your own solution would be weakened, by too much comparision with answers?

I'm asking this because I think this is what I'm effected by, since when I complete problems I immidiatly check the answers, and sometimes the problem/solution isn't even understood, even though the solution is correct. And I've noticed that this has made my ability to solve problems weaker.
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cristo
#2
Jan26-07, 12:12 PM
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P: 8,307
I don't think that being given numerical solutions to problems is a bad idea; if you use them correctly. If you complete a problem, then look to the solution and see that you've got it right, then you know your method is correct. If, however, the solution differs to the one you obtained, then it should encourage you to check through your work, to look for mistakes. If you use the solutions in this way then, no it is not bad. However if, as you state above, you would try and manipulate equations in order to obtain the "correct" solution, then this is not good. Solutions to exercises can be wrong!
hopeless blonde
#3
Jan26-07, 12:19 PM
P: 2
In many ways i agree with you. when you have the availability to constantly compare your work against something you never learn to use your own initiative, and in many cases double doubt your own conclusions. I am similar where by even though i know i have come to the right conclusion i check and compare simply to settle my mind on the matter, whereas as you said in exams you dont have that. By the time im finished my maths module i had gone back 4 times simply to see if i can compare any of the theories or check again if i hadnt made some odd mistake. So i can agree that yes it has made my problem solving weaker especially with the temptation of checking the answers when you get stuck, and never ending up on learning anything. however, how would we ever know we were right or if we have ever done something wrong if we hadnt those answer to compare??

Moridin
#4
Jan26-07, 12:21 PM
P: 858
About solutions to problems

Yes, answers in the back of the textbook is bad for people without discipline and/or common sense. However, provided that one has discipline and/or common sense, given solutions is a great resource, preferably the entire solution and not just an answer.

The point of having exercises at all in textbooks is so that one can apply the knowledge on have learned to problems and ultimately increasing one's problem solving ability and technique.

Learn the information necessary to attempt to work with the problem successfully. Make a serious attempt at solving it, no matter if it doesn't come to you directly. If you think you have solved it correctly, look at the answers given in the book. If it matches, that is great, you have solved it and you move on to the next problem.

If it was wrong, then go back to the problem and re-check it, step by step. If it doesn't help and you spent enough time trying to understand what you did wrong and you come up with nothing, I'd say write down the solution from the section with answers and go over it, step by step, until you have achieved a satisfactory understanding. If you skip the understanding part, suit yourself in my opinion. Doing that is just irresponsible.

By writing the correct and complete solution from the answer section, you will have learned a satisfactory method to solve similar problems. Go back to the theory/information section and then work on the other problems. After a while doing other problems, you will probably have forgotten about the exact solution to the problem you didn't complete, so you can do it again. And succeed this time. Practice makes perfect. Don't be afraid to look at the solutions of problems after you have given your very best effort. In fact, you will most often save time by doing this on the problems you cannot solve after giving your very best effort, instead of just sitting in a void thinking about how bad you are in that area. Read the theory/information section.

You can both get a good problem solving skill, learn to manage time and learning as well as checking your own answer.

Oh and hopeless blonde, that initiative you speak of is called interest. You will take the initiative to learn if you have the interest in doing so.
complexPHILOSOPHY
#5
Jan26-07, 12:38 PM
P: 360
I am not sure what level of mathematics or science you guys are in but I wouldn't be able to learn very much if I wasn't able to check my answers and proofs with the ones in the book. It is not the resource itself that distorts the learning process, it is your application of the resource that does.

As Moridin stated, simple common-sense can direct a person to use the extremely powerful tools in the back of the book. You are given precise logical examples of difficult problems, design to help you understand how exactly the problem was solved. If you choose to simply accept the answer and try to reconstruct it without understanding why, then that is your failing and not the books.

Once you get to a certain level of mathematics and sciences, there no longer are answers or proofs in the back of the books (or if there are, there usually isn't very many or they aren't always comprehensive). I don't know how many times I have had to go searching for a proof or solution to a difficult problem because it is nowhere to be found in my book.

If I didn't go through and make sure that all of my answers and proofs were correct and understood why, I wouldn't get very far in maths.

I am actually completely baffled as to how so many kids in my classes can get through the class without ever really understanding it. They are plug and chug machines, something I fail miserably at (and the reason I did poorly in math in highschool).

Sorry for the rant, this post just kind of aggrivated me because people take the time to work out all of these problems for you, dedicate their time, effort and brilliance to writing a comprehensive book and you don't use it properly, then complain that it hinders you.
P3X-018
#6
Jan26-07, 01:00 PM
P: 144
True that it is my own responsibility to control this. I agree that if there is no answers then there is the downside that you would never know wether your answer is correct. I also do understand the problem and the solution, but when there is answers in back of the book you can't help being being curious on wether your solution is correct, and that you really have understood the theory. Usually when realizing that my solution is incorrect then that is not because I have misunderstood the theory, but rather forgot to take a certain consideration (or aspect of the problem) into account, which leads to error. If you usually realize that you forgot it by checking the solutions, then you'll probably also forgot certain considerations later on, because you'll get to check your solution with answers instead of analysing the problem.
But I agree, that this is my own problem, and that I should have some discipline regarding checking solutions.
Although you got to admit that you can't help being curious most of the time wether your solution is correct, and check the answers. When you do this too many times, then...
scorpa
#7
Jan26-07, 01:00 PM
P: 372
I hate it when books do not give the answers to the problems in the back. First you should make a serious attempt at solving the problem, then check to see if you did it right, and if you did it wrong you can take another look at try to determine where you went wrong for the next time. If you don't have the answer provided you may think you did the problem correctly but have actually gotten the answer wrong, and you won't change your ways because you are unaware you did anything wrong.
P3X-018
#8
Jan26-07, 01:02 PM
P: 144
Quote Quote by scorpa View Post
I hate it when books do not give the answers to the problems in the back. First you should make a serious attempt at solving the problem, then check to see if you did it right, and if you did it wrong you can take another look at try to determine where you went wrong for the next time. If you don't have the answer provided you may think you did the problem correctly but have actually gotten the answer wrong, and you won't change your ways because you are unaware you did anything wrong.
And that's the downside of not having answers. I agree.
complexPHILOSOPHY
#9
Jan26-07, 01:09 PM
P: 360
Quote Quote by P3X-018 View Post
Although you got to admit that you can't help being curious most of the time wether your solution is correct, and check the answers. When you do this too many times, then...
Nah, fer sure homie. I am always curious how it is worked out, I think that's what motivates us to actually do the problems in the first place! However, I know if I go see how it is done first, I might prematurely assume that I understand it because it's always easier to understand, if the proof is in front of your face.

For me, if I go read the proof without really trying to solve the problem, I will have a hard time reconstructing the proof on my own, not because I can't but because I have the picture of the proof in my head (granted it's not to complicated to visualize), so I have to come back later once I have forgotten the proof, although it tends to be familiar afterwards. Otherwise, I am essentially doing the same thing as copying it out of the book. This is why I have to make certain that I fully understand what I am doing and not just copying things down because I saw it done that way before.

Perhaps my own reasons for not reading the proof is subjective but I have a decent memory which supercedes my ability to think, so I can't distort my thinking process until I have exhausted all of my options.

Does anyone else have this problem, or do you guys just breeze through everything?
scorpa
#10
Jan26-07, 01:12 PM
P: 372
Why are you looking to see how it is done first? You shouldn't be looking at the answer until you have worked it out on your own. If you look at the answer before you even attempt the question you might as well not bother at all.
Kurdt
#11
Jan26-07, 01:16 PM
Emeritus
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P: 4,981
If you have no numerical solutions to check your answers with then you could be applying things completely incorrectly which is more detrimental. Numerical answers are essential so you can see you've used the methods correctly and also to show you if you haven't so you can correct them. Full worked solutions in a text book however would not be good.
P3X-018
#12
Jan26-07, 02:00 PM
P: 144
Indeed numerical answers isn't that much of a problem. But if you have to find some expression, and this expression is given in the answers, then if you look at the expression sometimes you can even see how the problem is solved.
And maybe this is what I have problem with, if I don't think of the method my self but just "read it of" from the answers, I might not learn to independently think about the method.
This'll be like, as complexPHILOSOPHY mentioned, "copying it out of the book", the method is this case that is.
It's like if you just read about a theorem (and it's proof) from a book, you might miss some information regarding that theorem. However, if you prove the theorem your self you might (and usually will) get a much better understanding of it, and be able to "picture" it (as complex said).
complexPHILOSOPHY
#13
Jan26-07, 02:08 PM
P: 360
Quote Quote by P3X-018 View Post
Indeed numerical answers isn't that much of a problem. But if you have to find some expression, and this expression is given in the answers, then if you look at the expression sometimes you can even see how the problem is solved.
And maybe this is what I have problem with, if I don't think of the method my self but just "read it of" from the answers, I might not learn to independently think about the method.
This'll be like, as complexPHILOSOPHY mentioned, "copying it out of the book", the method is this case that is.
It's like if you just read about a theorem (and it's proof) from a book, you might miss some information regarding that theorem. However, if you prove the theorem your self you might (and usually will) get a much better understanding of it, and be able to "picture" it (as complex said).
Now I am confident that I did misunderstand you, initially, in your first post so I apologize for that homie. I had assumed you were referring strictly to numeric solutions, in which you simply found them in the back of the book and then proceeded to solve the problem for that specific answer, instead of an unknown solution.

So, if you, like me, only suffer when you are exposed to the entire proof first, because your thought process is now tainted (you are no longer using your logic to solve the problem, you are modeling your proof after the one in your head), you are essentially just copying out of the book. This is why I have to remain extremely disciplined when it comes to reading through a math or physics text. Since I am pretty retarded, I have to read through my books using a flash card to cover up the proofs, otherwise, it hinders me.

If that is the case, then that is a different situation, I believe. It is just a matter of discipline and making sure you are doing everything you can to work the problems on your own. Once you start learning other peoples methods and stop trying to use your own, you are no longer a unique problem solver and more of a computation machine. Given the assumption that, on your own and given enough time, you can prove the problem on your own.

Obviously, if we are talking about really difficult problems, that require exposure to proofs first to gain more inuitive insight, then that is something different.
Moridin
#14
Jan26-07, 02:31 PM
P: 858
Quote Quote by P3X-018 View Post
True that it is my own responsibility to control this. I agree that if there is no answers then there is the downside that you would never know wether your answer is correct. I also do understand the problem and the solution, but when there is answers in back of the book you can't help being being curious on wether your solution is correct, and that you really have understood the theory. Usually when realizing that my solution is incorrect then that is not because I have misunderstood the theory, but rather forgot to take a certain consideration (or aspect of the problem) into account, which leads to error. If you usually realize that you forgot it by checking the solutions, then you'll probably also forgot certain considerations later on, because you'll get to check your solution with answers instead of analysing the problem.
But I agree, that this is my own problem, and that I should have some discipline regarding checking solutions.
Although you got to admit that you can't help being curious most of the time wether your solution is correct, and check the answers. When you do this too many times, then...
If there is time, one could always just erase everything one has written on that particular problem and start over, this time remembering the computational issue one had earlier.

complexPHILOSOPHY, I usually try to learn the initial conditions for deriving the proof and use that as a framework. Let me take an example - Although it is a pretty basic proof, it is pretty pointless to try to remember every step in proving that the derivative of tangent x with respect to x is one over cosine squared x. It is much easier to learn that tan x = sin x / cos x and then apply the quotient rule. Although that gets harder and harder as the level increases.
complexPHILOSOPHY
#15
Jan26-07, 02:42 PM
P: 360
Quote Quote by Moridin View Post
If there is time, one could always just erase everything one has written on that particular problem and start over, this time remembering the computational issue one had earlier.

complexPHILOSOPHY, I usually try to learn the initial conditions for deriving the proof and use that as a framework. Let me take an example - Although it is a pretty basic proof, it is pretty pointless to try to remember every step in proving that the derivative of tangent x with respect to x is one over cosine squared x. It is much easier to learn that tan x = sin x / cos x and then apply the quotient rule. Although that gets harder and harder as the level increases.
That is exactly what I try to do but sometimes for me, atleast, being exposed to a proof before I have developed a solid understanding of the conditions for deriving the proof, distorts my thought process and I pass over important but minute details and then have to come back and figure out what I missed a few problems later. This doesn't happen because I structure my reading since I know I have to read math a specific way or I won't understand it. I have to read slowly and summarize the things I am learning to compartmentalize my memory. I don't have a powerful photographic memory, so I have to organize information consciously in order to remember it. It takes me a great deal of effort but it's working so far. I read on here that people can just read math as fast as they can, memorizing definitions, axioms, concepts and theorems without ever having to do the problems. I wish I had that talent but for me, I have to work through every problem there is until I feel confident.

I have not worked through any higher-level, difficult mathematics yet, so I don't know if I can always hope to derive the proof.

At a certain point, does it start to become really difficult to derive everything without some memorization of proofs? I have no experience with that level of math, so I am curious.

Also, does just working through math, constantly doing proofs (whether they are different maths or not), just naturally give you a strong intuitive feel? Right now, I feel like I can't really solve problems that well or they take me a long time. Will it get better as you go? I was really only exposed to algebraic and geometric arithmetic, trigonometry and calculus, this year so I am really new to math and it scares me that I am having a harder time with it than any other subject.

I took algebra I three times in high school and never opened the book (although I went through it just fine earlier this year), so I feel sort of stupid now, although I am working through group theory without any problems right now. Sorry for such stupid questions, I am just a nervous student!
Werg22
#16
Jan26-07, 03:11 PM
P: 1,520
It depends on the way you learn. For me, the best has always been to come up with my own solution and then check the solution given. Often our solutions are identical, but when they're not it's an eye opener; it exposes you to new ideas, new ways of thinking.


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