When I do nearly any assignment for homework, I’ll make a rough copy of my work before writing the copy that I will hand in. I do it for a practical reason: while it may take extra time to write my work twice, the truth is that I often take a lot of detours on my first try tackling a problem. I go down dead-ends, make little mistakes here and there that need to be corrected, and generally do a lot of messy work. Once I get the correct answer, I can tidy all my work up in order to make my final copy as concise as possible.
This work well for handing in assignments, but unfortunately this strategy is often adopted too much while teaching students new concepts. Instead of giving students time to think about how to prove or derive a statement, teachers often give the instructions to the student at face value. Additionally, there’s no “rough” work given, making students sometimes wonder why a certain strategy is being used. I’ve heard many times the phrase, “Don’t worry, you’ll understand what I’m doing in just a second.” It’s a nice thought, but I think it creates an environment where students don’t have to think about what they need to do themselves. Instead, they just need to follow along as the teacher feeds them the steps and the answers. This might be easier, but I definitely don’t think it’s as good of a strategy in the long run as getting students to struggle and go down those wrong paths.
Along the same line, this tendency to always show the “final” work sends the message to students that this is the only way they should have seen the problem, and having any other ideas wasn’t going to work. However, that couldn’t be further from the truth! We need to send the message that rough work is essential in science, because no one is likely to get the answer right on the first try. You need to choose a direction and make some progress before you can decide if you’re going to reach your goal.
If we want students to be more engaged in the act of learning and taking notes while looking at a derivation, we can’t spoon-feed them the method to use. Sure, it will give them the way to do it in the future, but it’s not a good way to get them to learn the concept. What we need to be doing is encouraging them to try ideas, see if they work, and refine them. That’s the basis of intuition. You get it because you’ve seen many problems like this and you have a feel for what’s going to happen.