One of the unfortunate side effects of having a curriculum and set schedule in mathematics is that one never gets to think about concepts for too long. Instead, the goal of a class is to simply throw a bunch of ideas to students and let them “ponder” the ideas on their own time. This is seriously backwards, and it’s at the heart of what is wrong with a lot of mathematics education today.
Think back to your time as a student in mathematics. When was the last time you were presented with a problem, worked on it, and then had to stop and mull over the idea for a few days? If you’re education has been anything like my own, the answer is nearly never. That’s simply not how school works. You never hear the teacher give you a problem that you can think about and come to a solution on. What happens is that a teacher will explain the problem, and then jump right into the solution, with no explanation in between. And that’s if you’re lucky. Sometimes, an answer will be given to you without there even being a question!
I’m willing to bet that if you’ve ever been asked a mathematics puzzle, you usually think about it for a few seconds, decide that you may as well just listen to the explanation, and jump right ahead. It’s what I do, because I know I will understand the explanation, and I don’t want to pause the video or stop reading because thinking about the problem will be difficult.
However, I’ve realized that this is a way to make me think that I know a lot about mathematics, but it’s a sort of pseudo-knowledge. After all, why do we engage in mathematics if not to try and solve problems and puzzles? Once the answer is given, there’s no more fun to it anymore. It’s just an algorithm or step-by-step process to be applied.
That is why I’m in favour of learning how to struggle in mathematics more. And I don’t mean the kind of struggling that occurs because you don’t know what formula to use. I see this so often when I tutor students, and I’m starting to harbour the belief that it’s because schools are just presenting them with a vast amount of information without any use for it. They don’t know what to do in problems because they don’t even understand what the problems are about. The reason: they have no link in their minds between the material they learn in class and the problems presented to them. And since it doesn’t match up, they are at a loss for what to do.
In place of just giving them a bunch of theory on different concepts that they will have to remember how to apply an algorithm, we should have them struggling on the problems that led to the various equations and concepts that are learned. By doing this, it gives the student an idea what it is they are learning about. These aren’t just abstract symbols on a piece of paper. They are ideas about objects, and not giving students this information and not allowing them to struggle on these sorts of problems before giving them the answer as class notes is doing more harm than is worth the productivity boost.