Understanding Concepts in Problem Solving: A Guide for Engineers

[cs23]

My engineering professor has always stressed that once you understand a concept you don't have to solve a lot of problems. He has also said that you don't need to memorize anything if you understand.

My Question is, how do you understand a concept?

 

[kote]

My engineering professor has always stressed that once you understand a concept you don't have to solve a lot of problems. He has also said that you don't need to memorize anything if you understand.

My Question is, how do you understand a concept?

You memorize the equations and do a lot of problems .

Trying to understand why things work the way they do, possibly by working through derivations, etc, can also help. That's why textbooks are set up the way they are. I also found that working through problems with others helped. Being forced to teach the concepts that I understood best solidified them, and having others teach the parts they understood offered fresh perspectives that I wouldn't have come up with doing the problems by myself.

 

[Dembadon]

I find talking to the professor (office hours) to be the most effective for me. Also, as kote mentioned, discussions with other people can help. Your textbook can be a good resource as well.

 

[Klockan3]

You have to think. Really, everything you do where you aren't getting mentally stuck is worthless in terms of understanding. When you solve problems, don't look through the book on solution strategies, instead read the relevant chapter, think about everything that is writte, then try to solve a problem without looking back at the chapter. Don't give up if you can't do it, that is the hard part, instead think about everything you know about the subject and try to use it to solve the problem. Do this for a long time or until you solve the problem, if you really can't solve it then go and review the material again. But never solve problems with the relevant pages up, that is just the lazy and bad way of doing things. The only way to understand things is to think, most don't think that much when they solve practice problems but instead tries to remember different things which is almost useless except that it reinforces those memories which don't have anything at all to do with understanding.

 

[clope023]

You have to think. Really, everything you do where you aren't getting mentally stuck is worthless in terms of understanding. When you solve problems, don't look through the book on solution strategies, instead read the relevant chapter, think about everything that is writte, then try to solve a problem without looking back at the chapter. Don't give up if you can't do it, that is the hard part, instead think about everything you know about the subject and try to use it to solve the problem. Do this for a long time or until you solve the problem, if you really can't solve it then go and review the material again. But never solve problems with the relevant pages up, that is just the lazy and bad way of doing things. The only way to understand things is to think, most don't think that much when they solve practice problems but instead tries to remember different things which is almost useless except that it reinforces those memories which don't have anything at all to do with understanding.

There's more than one way to learn.

I think you're making a lot of false assumptions.

I'm not saying your way doesn't have merrit but if I stuck to that exclusively I know I wouldn't get anything done.

 

[jeebs]

I think just about everything hard to grasp can be related to some sort of analogy that we are familiar with in every day life. In fact now that I think about it, that's probably what loads of physics is, translating something confusing into a familiar analogy that works.

 

[Klockan3]

There's more than one way to learn.

I think you're making a lot of false assumptions.

I'm not saying your way doesn't have merrit but if I stuck to that exclusively I know I wouldn't get anything done.

Well, it isn't about how many problems you solve or so, it is mostly about how much you think about the concepts that matters. Since most don't have a clue on how to think about things I try to give a way to provoke the right way to think, when you sit with a problem and look in the book for ways to solve it you aren't thinking about any concepts instead you are thinking on the easiest way to solve this specific problem.

The classical way of doing things builds confidence but not knowledge, which is why students flock to it. It sucks but it feels good when you do it, kinda like doing drugs.

Edit: Better link:
http://www.purdue.edu/newsroom/research/2011/110120KarpickeScience.html

 

[clope023]

Well, it isn't about how many problems you solve or so, it is mostly about how much you think about the concepts that matters. Since most don't have a clue on how to think about things I try to give a way to provoke the right way to think, when you sit with a problem and look in the book for ways to solve it you aren't thinking about any concepts instead you are thinking on the easiest way to solve this specific problem.

The classical way of doing things builds confidence but not knowledge, which is why students flock to it. It sucks but it feels good when you do it, kinda like doing drugs.

Edit: Better link:
http://www.purdue.edu/newsroom/research/2011/110120KarpickeScience.html

Solving the problems are mostly mathematical tricks, but in physics nature leads the equations; I find solving lots of solved problems opens up the concepts for me and after enough practice I can apply the concept without looking at a reference. Obviously I read the material before attempting a problem of any kind, that should be a given, but I prefer to learn by doing; I know I'm not a special case here either. I might go so far as to say yours is perhaps the more effective method but some people literally can't learn to solve problems without actually solving problems.

 

[gb7nash]

Everyone learns differently, but what's worked for me is doing examples. If I feel I'm not really understanding the examples, I meticulously look through my notes from class and ask my colleagues. If they're not sure, I read through the relevant chapted. If that doesn't work, I ask the professor.

 

[cs23]

When you solve problems, don't look through the book on solution strategies, instead read the relevant chapter, think about everything that is writte, then try to solve a problem without looking back at the chapter. .

When you say solution strategies, are you referring to the worked examples in the book. So when working on a problem, instead of studying the solution of a similar example, I should read the text(theory).

 

[jhae2.718]

You'll learn more by working through a problem without looking at a solution manual. You have to think about the concepts and try to find a way to apply them to your specific problem. I've had physics professors outline the following strategy:

• read the chapter
• work on the problems with the book closed
• after 30-40 minutes of being stuck, formulate questions that you think will help you solve the problem
• look the answers to those up in the book
• try it again, and if you can't get it, ask a friend/TA/prof for help.

Solving problems like this helps you build a certain intuition that will allow you to solve problems quickly and easily. Note that real life problems don't come with a solution guide.

 

[Klockan3]

I might go so far as to say yours is perhaps the more effective method but some people literally can't learn to solve problems without actually solving problems.

I am not so sure about that, I think that it is more a mental barrier than anything else. People think that they can't learn without doing tons of problems so they don't really give anything else a real try.

When you say solution strategies, are you referring to the worked examples in the book. So when working on a problem, instead of studying the solution of a similar example, I should read the text(theory).

If you are currently relying on solved examples then that is the first thing you should stop with. As I said the goal is to force yourself to think about the things, the goal is not to try to find which aspects you can copy from the examples. It requires more work and is more tiresome, yes, but it is tiresome since you are conceptualizing the ideas. In a way it is kinda like working out, after a good session your mind should be exhausted or you aren't doing it properly so don't be afraid when things are feeling tough. When it starts getting tough don't back down and take a coffee break, instead go at it even harder since this is when **** starts to happen in your head. Don't think "Ugh, this is annoying, I better stop!", instead think "Awesome, I can feel my mind working!".

And of course you can solve many problems as well if you are afraid of just doing a few, just try to rely as little as possible on material not given in the problem, especially the extremely refined material found in examples. Instead try to solve the problem using as general concepts as possible, if you can't do it with what you got on top of your head try to find some in the book and solve using those.

An example of a general concept is "Acceleration is equal to force divided by mass, aka F=ma", while a refined would be "A car with mass 1 ton is being pushed by a force of 1000 Newton, find acceleration. SOLUTION: F=ma gives a=1m/ss".

 

[chiro]

My engineering professor has always stressed that once you understand a concept you don't have to solve a lot of problems. He has also said that you don't need to memorize anything if you understand.

My Question is, how do you understand a concept?

From my experience, the best thing to do is to first understand any initial assumption, and once that has been done understand all of the steps that lead to a particular result.

In science, many assumptions have not really been understood (thats something for the scientists and scientific philosophers to ponder), but they never-the-less come from results verified by experiment, or the by-product of an experiment.

Also don't feel too bad about feeling confused when facing concepts from modern science. Things like quantum mechanics are hard or impossible to be understood even by many experienced scientists.

If you understand the assumptions, chances are you will understand the final results. If the assumptions are troubling you, ask your professor to clear any misconceptions you may have about any aspect of the initial assumptions.

 

[Ryker]

I am not so sure about that, I think that it is more a mental barrier than anything else. People think that they can't learn without doing tons of problems so they don't really give anything else a real try.

I fully agree with this, I think a lot of it has to do with the fact that people see doing lots of problems as working hard, and since everyone is saying you have to work hard and that's how it should be done, they just go with it, because it feels better than learning theory and thinking about it.

 

[clope023]

I am not so sure about that, I think that it is more a mental barrier than anything else. People think that they can't learn without doing tons of problems so they don't really give anything else a real try.

If you are currently relying on solved examples then that is the first thing you should stop with. As I said the goal is to force yourself to think about the things, the goal is not to try to find which aspects you can copy from the examples. It requires more work and is more tiresome, yes, but it is tiresome since you are conceptualizing the ideas. In a way it is kinda like working out, after a good session your mind should be exhausted or you aren't doing it properly so don't be afraid when things are feeling tough. When it starts getting tough don't back down and take a coffee break, instead go at it even harder since this is when **** starts to happen in your head. Don't think "Ugh, this is annoying, I better stop!", instead think "Awesome, I can feel my mind working!".

And of course you can solve many problems as well if you are afraid of just doing a few, just try to rely as little as possible on material not given in the problem, especially the extremely refined material found in examples. Instead try to solve the problem using as general concepts as possible, if you can't do it with what you got on top of your head try to find some in the book and solve using those.

An example of a general concept is "Acceleration is equal to force divided by mass, aka F=ma", while a refined would be "A car with mass 1 ton is being pushed by a force of 1000 Newton, find acceleration. SOLUTION: F=ma gives a=1m/ss".

The two methods are after the same thing, you're committing the excluded middle fallacy.

 

[Feldoh]

A good way to do figure out if you know a concept is to get a problem that covers a particular concept. Go through and solve the problem and every step along the way ask yourself: "why does this step work? why am I doing this?" if you can answer that then you probably know the concept.

Also try teaching the concept to a classmate. If you can teach it, you know it. A famous physicist by the name of Feynman once remarked that if you can't explain a concept to college freshman in intro physics then you don't understand it.

 

[Klockan3]

The two methods are after the same thing, you're committing the excluded middle fallacy.

They are not the same thing, I urge him to work with as fundamental concepts as he can muster instead of trying to jigsaw together a fitting solution from the set of examples in the book. The easiest way to achieve that is to do serious tries to solve problems without looking up things, even though solving problems this way would take a lot longer you would only need to do a handful of problems in each course since each problem would basically force you to derive the current chapter.

 

[A. Neumaier]

Also try teaching the concept to a classmate. If you can teach it, you know it. A famous physicist by the name of Feynman once remarked that if you can't explain a concept to college freshman in intro physics then you don't understand it.

Yes. many here stressed ''thinking'' without telling what sort of thinking is required.
If you look at worked examples (or at your own solution after you found it by trial and error), you need to ask yourself: _Why_ did this work here? On which property of the problem did this argument depend? How could I recognize when to apply the same trick/argument/transformation/substitution/etc. in a different situation? and similar questions.

Your understanding is the better the better you can give a running commentary on every step of a worked solution (or a textbook presentation of the matter), in the form: At first <abstract action> results in <progress in the written solution>. Then <abstract action> leads to <formula in the written solution>, now the key step is to do <abstract action>, reducing the problem to <statement in the written solution>, etc., in a way that everything is clear to you (and to someone listening, if there is one).

Learning a concept means learning how it connects to other concepts, techniques, tricks, examples that you know already. Science is a very dense web of relations, and concepts are nothing else than named things expressing such relations. The better connected your web of concepts is (and the fewer false connections it has) the better is your understanding, and the clearer your perception of what you study.

These connection grow mainly by practicing, and by reading the same things from as many perspectives as feasible. Therefore instructors, textbooks, additional readings, discussions with classmates or here on PF, worked exercises, doing your own exercises, and explaining things to others complement each other.

 

[clope023]

They are not the same thing, I urge him to work with as fundamental concepts as he can muster instead of trying to jigsaw together a fitting solution from the set of examples in the book. The easiest way to achieve that is to do serious tries to solve problems without looking up things, even though solving problems this way would take a lot longer you would only need to do a handful of problems in each course since each problem would basically force you to derive the current chapter.

They are not the same thing but they are AFTER the same thing, which is conceptual understanding, you're still committing the excluded middle fallacy.

 

[mathwonk]

you try to understand. you ask yourself "why?" a lot. you make understanding your goal. you keep at it as long as you feel you do not yet get it.

 

[Klockan3]

They are not the same thing but they are AFTER the same thing, which is conceptual understanding, you're still committing the excluded middle fallacy.

It doesn't matter what you are after when you do it, what matters is the results. Doing problems without thinking properly do not help you understand much.

 

[elfboy]

My Question is, how do you understand a concept?

by studying & practicing it

 

[clope023]

It doesn't matter what you are after when you do it, what matters is the results. Doing problems without thinking properly do not help you understand much.

Excluded middle yet again, what happens when you do problems while thinking properly? You seem to think it has to be one way or the high way and that is not the case.

 

[Klockan3]

Excluded middle yet again, what happens when you do problems while thinking properly? You seem to think it has to be one way or the high way and that is not the case.

No, I don't think that there is only one way, I however know that most do it the wrong way when they do it like they usually do. As I said before I try to give ways to force them into thinking in the correct ways, if they already knew how to do it then they wouldn't have a problem, would they? Understanding how to understand concepts is extremely nontrivial, most thinks that they do understand that as well but as studies have shown over and over again it isn't true. People in general have no clue on what makes them understand things or when they understand things.

What you can do is to try different things till it starts working satisfyingly well. The problem here is that since people don't know when they understand things or not even this method is flawed from the students point of view...

 

[cs23]

They are not the same thing, I urge him to work with as fundamental concepts as he can muster instead of trying to jigsaw together a fitting solution from the set of examples in the book. The easiest way to achieve that is to do serious tries to solve problems without looking up things, even though solving problems this way would take a lot longer you would only need to do a handful of problems in each course since each problem would basically force you to derive the current chapter.

so when solving a problem, i must not refer back to the relevant pages that would help me. Instead, i should think of the concept and try my best to apply it.

 

[Klockan3]

so when solving a problem, i must not refer back to the relevant pages that would help me. Instead, i should think of the concept and try my best to apply it.

Yeah, at least give it a try. You might find that it is easier than you think to come up with the correct solution by yourself.