Reading over + memorizing steps of problem- bad way to study?

[pjmarshall]

(before reading this, I'm currently enrolled in engineering, not math, which I switched from, so i don't deal with advanced proofs and such):

Okay, so for my lower division math courses (using elementary textbooks such as stewart or lay), my method of studying was:

-Read the book
-Do all the problems
-Read over every type of problem and try to understand and memorize their steps
-Look through a notebook and strain mind trying to drill steps (and method) into head

This worked for some time, but as of now there are obviously too many types of problems to apply various methods to, and I HATE this method of studying, though it worked quite well for my grades. Now, I know for math, I heard that a lot of it comes from having a good 'toolbox' to work from, and it seems the only way I came obtain such a toolbox is by drilling these damn methods into my skull. But it really takes the a lot of the fun out of mathematics... I'd rather just recognize basic steps of a problem and solve it. However, the thing is, after looking through the steps to a solution, I'm sometimes distraught when I see that I didn't even know that the trick used to solve it was 'legal' or possible! Obviously, somehow it's implied, but I can't see how someone can, after reading over the basics, come to such a conclusion. A lot of times, due to constrained class time, I just skip over the way these methods are obtained and simply apply them (though they're almost never used on tests, since the bulk of the tests just test the basics, not the hard problems).

I know that a lot of it is because I don't have the intuition that some have, but I want to know how someone, after knowing only basic concepts, is able to imply these things. I realized that I'm not amazing at math, but I still like designing/performing experiments, and I still like math in general. I want to try to get above Bs in my upper division courses, but it seems like I'm stuck in that range.

So is it feasible to go over EVERY SINGLE problem, because I have to be able to do all of them or else that means I don't understand the underlying concept? I don't think my brain can handle memorizing every single concept and trick and problem in my textbooks. My method of study has failed me now, and i want to know a better way to work with applied math.

 

[D H]

Yes, this is a very bad way to study. As you have discovered, it simply doesn't work once you get to advanced topics. Instead of learning to see the forest for the trees you have learned to memorize the number of spots on each of the red bugs that live on the south side of one particular tree in the forest.

You need to learn what motivates the steps that were taken rather than memorizing the details of each step. You must have a phenomenal memory to have been able to make it through your lower level classes with this approach. Learn to use that phenomenal memory to remember patterns rather than details. You have to relearn how to learn. You have taught yourself some bad learning habits. You need to unlearn them.

Learn to look for patterns. Instead of memorizing the mechanical details of each step, step back and ask why those steps were taken. Then learn to ask yourself why a sequence of steps was taken. Soon you will start seeing some common themes. You don't need to remember the exact details if you remember those common themes. A huge side benefit is that things will suddenly start making sense. Instead of seeing a bunch of apparently disparate steps taken to solve a problem you will start seeing the path from the start to the end and the reasons for taking that path.