Proofs Can Be Useful
I find it’s funny how I always hated to see a proof when I was in school. In my eyes, it was always just so boring. I knew that the teacher was giving an equation that was correct, so I didn’t see the point in trying to drag through the derivation of the equation. To make matters more confusing, my teachers would show us the proof once, and then we usually didn’t have to worry about how it came about anymore. At that point, we could simply apply it.
However, I now see the vital importance in going through a proof. It shows the inner parts of mathematics, the way one can reason in order to get a wanted result.
The problem is that derivations are generally very, very boring. Depending on who is teaching you, proofs can range from “hand-wavy” to thorough. I’ve sat through proofs that seem to be only a bunch of syntax movement, which is both tedious to write out and is difficult to follow. Unfortunately, boredom usually increases with thoroughness. Therefore, it’s of utmost importance to work on the presentation of a proof if you want students to understand both the proof and why it is important.
The first suggestion I have is to ask a lot of questions during the derivation. A question starts by breaking up the proof into approachable challenges. As I’ve written before, a proof is an answer to a question (link). It makes sense to ask questions then because it will both set goals for the proof (helping the students understand what is trying to be accomplished) and gives motivation to the students for why this is important.
In my experience, giving a good amount of motivation to the students has resulted in better derivations. I believe this has to due with the student being able to follow what is happening in the derivation versus scrambling to get everything written down with no reflection on what is happening. (This also has to do with the speed at which the teacher lectures.) I’ve been on the receiving end of many proofs that I could barely follow because there wasn’t enough motivation for it and so I simply copied down the notes. This is definitely not a good way to appreciate proofs.
The second strategy I highly recommend is giving the students a different perspective on the proof while going through the various portions of it. Instead of simply interpreting the mathematics (which tends to be unfamiliar to students), give the students a graphical interpretation or even an analogy in order to solidify the derivation. Sure, the mathematics are the important, but the student needs to understand what they mean.
Now that I’ve learned a fair amount of mathematics (but nowhere near complete), I better understand how important knowing a proof is. It’s not that the proof will give you an upper hand when solving a problem with the formula, but it will help you have a better grasp of the concept in general. That’s reason enough to want students to understand proofs and not just have them see them.