leright said:
I don't think it is enough, personally, to simply do problem after problem. I think it is equally important to sit down and think deeply about the subjects being discussed in class and set aside working problems for a while. While you can get by and get an A in your classes by only working problems and memorizing procedures, you might not necessarily understand the ins and outs of everything. There are too many people out there that haven't got a clue what they are doing, yet are still able to ace the tests and work (SOME) or the problems...
I completely agree.
As a teacher, I see some students who redo all the assignment problems an dthe practice problems and still fail or get low marks on the tests. Because they did not try to understand what they were doing. It seems that they think that preparing for a test is like preparing for a competition in weight lifting. If you have lifted enough weights, you will get to the competition and lift the weight given to you.
The best way to learn is to do the problems, but before working them out (and *especially* before looking at the solutions), ask yourself: "is it clear to me how I should proceed? Do I see more than one way? Can I get started without looking at the solutions"?
Too many times people basically copy down the solutions and feel that they have mastered a subject or solve a problem without ever stopping to figure out why they are doing it in a certain way.
You will learn *a lot* if you tackle a problem and think about doing it one way and try this (without looking at the solution) and get stuck at some point or get a wrong answer. Then you go back and have to understand why this method is incorrect or why the result is wrong (at that point it is ok to look at the solutions). If you have used the right approach then it is just a matter of a math mistake possibly and you have to make sure you see your mistake. However, if the approach is wrong, you should see the teacher and understand what is incorrect with this approach (usually it makes you realize that you did not really understand the question, or that some formula applies only in some approximation or that you had a serious misunderstanding about some fundamental concept).
If for some problems you just don't know how to get started, look at the solutions but take a minute to really absorb why this is the way to do it.
After doing all that, if you go back over the problems and it is crystal clear why they must be done a certain way, then you have a much deeper understanding.
I can say that usually the students who come to me saying "but why couldn't we do this way instead" are the ones who end up getting the best grades. Because they *think* about how to solve the problems instead of just grinding through.
My two cents
Patrick