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.
I write and say this to myself over and over again, but it’s always worth repeating: we cannot obsess over work that is slightly imperfect. While it is nice to think of alternate scenarios or instances where we should have improved, the reality is that learning is the process of transforming those things that I’ve done wrong to things I’ll do right the next time. It’s never fun in the moment, no, but in the long run you’ll be able to solidify the lesson.
The reason I revisit this so often is that I am frequently reminded of things I have done wrong as a student. It’s the natural way of things. If I knew all the material, I would be given credit for the class and wouldn’t have to go. Therefore, I’ll make many mistakes and incorrect assumptions as a student. The goal of each class is to slowly eradicate them.
That becomes the question, then. How can one know that they have learned what needs to be learned in a particular class? There are different ways to go about answering this question. Tests are the obvious one, of course. As a physics undergraduate, I take tests in all my classes where the goal is usually some kind of computation or manipulation of variables and data. Therefore, I’m used to the stresses of tests and the knowledge that my mark on a test becomes more or less a quick measure of how I’ve understood a particular segment of the class. And despite most of my teachers acknowledging that this is the case (and that having a bad test can reflect a myriad of issues not related to how well one understands the material), we still have the traditional tests. They aren’t going to go away for the time being, so that is why students will undoubtedly stress about their mistakes on these tests. (I can personally vouch for this. I’ll frequently spend a whole day after a test is completed thinking about where I might have made mistakes.)
The second way to answer the question is much more subtle, but I believe it is much more indicative of learning. Unfortunately, it also is the one that is difficult to measure. The way it works is that I’ll catch myself in another class using the material I used in a previous class. In my mind, this is the essence of learning. The prime example for me is my progression through mathematics. When I first entered CÉGEP, I had virtually no idea what limits, continuity, and derivatives were. My background knowledge concerning calculus was essentially zero, and so learning the subject was a whole different situation than what I was used to. When I first took my tests concerning how to identify limits and what an approximation for the tangent line to a curve was, my knowledge was mild at best. I still did well (particularly well, according to some people) on tests, but the foundation wasn’t built. I could do the assignments, but I don’t know if I would call myself a master at the material. In essence, I was learning, but I hadn’t learned the whole subject.
Now that I am removed from that time by two years, I have a whole different perspective on the knowledge I gained in those first calculus classes. Instead of being somewhat sure of the concepts, I feel like I can explain them with minimal help. Indeed, I may even be in the position to tutor some students who are at that stage in their lives, and it’s exciting to see how far I’ve come.
Additionally, the main way I see that I have truly learned the concepts from years ago is in how I apply them every day. I like to stop and marvel at times how I’ve gone from being tested on derivatives and limits to simply applying these ideas in more advanced problems. I remember first learning about the power rule for derivatives and how it seemed almost magical at the time, but now it is a routine occurrence. Derivatives and calculus has become a daily fixture in my life. Even the more complicated ideas like the power rule are now something I am expected to whip out of my toolkit at a moment’s notice. The fact that I can do this is a testament to the fact that I have learned what there was to learn in those first calculus classes.
Evidently, you’re probably thinking: that’s great, but there’s no way you can actually test this in a way that is practical. And unfortunately, you’re probably right. I don’t know how we can move away from traditional tests, but what I remind myself whenever I begin getting hard on myself for making stupid mistakes on a test of assignment is that the tests don’t matter to my long-term development. As long as I get through the course and work hard at understanding, I will learn the content. Having a better mark than someone else doesn’t make me better than them, and it certainly won’t matter in the end.
Being hard on yourself is something I see too many students (including myself) do, and I want us to acknowledge the fact that it is almost never as bad as we make it out to be. Sometimes, getting just good enough on a test is alright, and the world won’t end.
Being Smart Doesn’t Have to be About Mathematics or Science
Pretty much anywhere I go in the region that I live, people will react to me saying that I’m going to be a scientist with this: “Of course, you have to be smart if you’re a scientist.”
This is the refrain that I hear all too often amongst my friends. While I’m getting my physics degree, they’re busy becoming nurses, learning the trade of construction, or looking to work in agriculture. They all have different lives, yet they all seem to agree that being a scientist requires you to be smart in a way that the others don’t.
As you may have imagined, I wholeheartedly disagree with this.
We assume a hierarchy of information, as if some information is more important than others. Or rather, understanding the information is only available to those who are intelligent. While this can be reasonably accepted, actually classifying said knowledge is more than a little difficult. It’s frankly a ridiculous proposition, and creates this implicit agreement where being a scientist means you’re smarter than the rest of the population. This isn’t true, and diminishes the way we think of other professions, because, if being a scientist or doctor requires you to be amazingly smart, it implies the reverse for other careers.
But this is simply not true. If it were so, then being smart would automatically mean you’d be a scientist or thereabouts. Why can’t we have incredibly intelligent people in other domains of life? If I were to go on a farm, I’d know little about what is actually happening there, yet many of my friends could explain this with ease. Learning about science and mathematics hasn’t helped me in that regard.
It’s the same story for many other walks of life. I don’t know much about law, history, or even literature. I doubt people would call me stupid because of this since I’m educated in science, but these other areas of life can be just as vital to humanity (and even more) than a lot of science. And I can guarantee you that there are plenty of smart people in those domains. Being smart does not necessarily equal being good at mathematics or science.
What I hope for is a larger acceptance of being smart. There are incredibly smart people who don’t need to be involved in mathematics or science, and that’s perfectly fine. If all the intelligent people group up into one domain, what’s going to happen to the others?
As many people have said before me, the idea of “multiple intelligences” seems to be correct. People can be incredibly smart in different aspects of their lives, which ultimately translates to being good at different things.
Let’s push back on classifying someone that does science as smart, as if anything else means one is not.
As a runner, I’m expected to give myself goals. It’s the thing that runners do. We set goals, train hard to get enough fitness to achieve them (usually in the form of some sort of race time), and then we evaluate and set new ones. This cycle is familiar to anyone who is a runner.
However, there are some people who are exceptions, and I happen to be one of them. I’m someone who hasn’t set any “real” goals in running for over a year now. An injury was certainly part of the problem, but there was (and still is) something else: paralysis.
When you’re working in a field, you usually want to look at others to know what they are up to. This helps inspire us to keep chugging along and grinding away at our work. Unfortunately, I’ve found that looking at others’ goals paralyzes me. An easy example is the past Olympics. During this time, I watched as some of the fastest in the world gutted it out on the track and On the roads. This got me all ready to once again train for some faster races. However, my goal at the time was not to do that. I was thinking about other races.
Fast forward to now, and I’m torn between what I want to do. I think I really want to see what I’m made of in a long trail race, but then I’ll watch or hear about another runner who has made a breakthrough in their training for a short and fast race, and suddenly my goals go out the window as their kind of racing seems more appealing. Instead of sticking with my goal, it’s as if I get lured away by the next shiny thing. I say this with no small amount of irony as I type away while facing my wall with race bibs that aren’t the kind of race I want to run next season.
There’s always the null solution: ignore everyone else. But that’s no fun, so I’m in a spot where I want to lock down on a goal but not feel like I’m a slave to it. At the same time, I’m trying to convince myself that I don’t need to do what everyone else is doing. We all live our own lives, and it’s okay if I don’t follow the path everyone else does.
I can extend this to other areas of life in general. I’ve found that it’s alluring to chase the new thing that others are doing, but it’s much more difficult to buckle down and do the work that I need to do. I’m getting better at it – my writing here is a testament to that – but I regularly need a reminder. Don’t let yourself be derailed by the goals of others. Pick something, and work towards it. And most important of all: remember that you aren’t tied to this goal for the rest of your life.