Good news! “I” got an “A” in differential equations this semester! WAHOO!
The I is in quotes because it wasn’t actually me, it was my sister taking the class, turning in the homework assignments, and taking the tests. But I helped her all along the way. I taught her.
When we discovered her grade we spent some good time crowing about it. We talked about all the homework, and she told me about her tests. And this is the interesting part: she told me that she didn’t always know how to do all the problems going into the test, but I had taught her how to think about the problems so that even when she came across a problem she didn’t know she could think about it and figure out how to come up with the solution.
That was what made the difference in her grade. It took a considerable amount of time to teach her how to think about solving math problems, which is partly why we couldn’t always get through all the material, but knowing the steps in solving a problem is completely different from know how to figure out those steps and solve it from there, even create a new way to solve it. In fact, that’s how the great mathematicians do it: they just think about how to solve the problems, they don’t follow prescribed methods. The genius isn’t in the steps but in the process.
But this kind of teaching/learning could be applied to any field: if you can learn how to think about the problem to solve it you are infinitely better off because no two problems are exactly alike which means that one method may work on one problem but not the next despite the similarities. But, if you know how to examine a problem and discover solutions then you can usually find your answer.
This kind of learning and thinking is the hardest because it’s a right-brained activity that many analytical fields neglect. Yet math is more right-brained than people realize because it takes considerable creativity to do this kind of thinking. But it’s not all one brain or the other either. It’s the whole brain: the right brain creating and the left brain analyzing the creation and sending back corrections and the right brain creating new things to fix the errors and so on.
So the next time you get stuck on a math — or any field really — problem don’t think of it as an analytical straight-forward problem. Think of it as a creation to a solution and see what you come up with.