Incompatible Ideas

Imagine I were to tell you that a circle and a square were really the same thing. Not in some fancy word-finesse, but that they are literally the same thing.

Most people would deem such a proposition ridiculous, and even go so far as to show me that I am wrong. A circle and a square would be drawn in front of me, and you would look at me triumphantly.

In this simple scenario, there’s no way I can convince you that a circle and a square are the same shape. It’s not possible, therefore, you don’t see many people arguing about circles and squares being identical.

However, we do see people arguing and debating about what kind of fundamental world we live in, and I’m not referring necessarily to a bunch of physicists. I’m talking about people who cling on to ideas like souls and free will. The former is basically a relic in science now, but the latter is quickly losing favour as well.

The problem is that these sorts of ideas are rampant in modern society, yet people don’t see how these ideas are in conflict with our fundamental ideas about reality. If we accept that our fundamental ideas are correct, then we can’t have things like souls or free will. It can certainly feel like it, but they are fundamentally illusions. These situations are basically just like that of me saying a circle and a square are the same thing, simply more complicated. However, because the situation is more complex, it’s easy to not see the connections between ideas that make certain concepts not reconcilable with our fundamental ideas of the universe.

We could always be wrong, but overthrowing the existing paradigm requires one to have a new model that is able to explain everything we know about the universe while being consistent with everything else we know. This is the hurdle that needs to be overcome, which is why many “novel” ideas aren’t likely true in the universe.

Time Remaining

The room is silent except for thirty or so sounds of scratching at various paces. Some scratches are going at a furious pace, while some are more relaxed. Still, there’s a certain tension in the air that can be felt from the lack of noise.

“Okay, five minutes left everyone,” the teacher says, and the pace of the noise is picked up even more. It’s the homestretch, where the sounds get even louder and more frantic. One can nearly pick out those who are struggling to finish by the speed of the sounds of the scratches near them, betraying how far back they are.


As you can probably gather, this is the typical experience of being in a room while writing a test (at least, for me). As a science students, I’ve had many chances to write tests, so I’m quite used to the whole procedure. There’s a pent-up energy at the beginning while the students discuss the potential questions on the test, and then everyone files in as if they’re being sent to death row. The test is then written at a furious pace, and then there is always the reminder with a few minutes left in the test where students start getting worried about not finishing the test.

It happens during nearly every test. There always seems to be someone who runs out of time. Often, this happens to multiple people, and it is not a fun feeling at all. As someone who routinely does well on tests and nearly always answers every single question, I am quite frustrated when I cannot get to a question on a test because of time. I feel cheated, because I get marks taken off as if I got a question wrong, when really I did not get a chance to really look at it.

This has got me thinking about the way we make tests and if there are ways to gauge student learning better. To do this, I want to question one of the things we take for granted during tests: question density.

Simply put, I’d define “question density” as the number of questions in a test per unit of time. Basically, it’s the measure of how much time one gets per question on a test.

Then, I’d attempt to calculate the number of minutes one should be taking per question on the test. Obviously, there are difficulties in getting an actual number out of this, since different levels of ability will be capable of doing more or less questions per unit of time.

Nonetheless, then one can take the ratio between these two numbers to find out how much margin a test has. If the number approaches one, the test does not have a lot of margin. As the number tends towards zero, more and more margin is available for the student.

Why is this important? What I’ve found for many of my science and mathematics courses is that there isn’t a lot of margin. This means that the teacher’s perception of how long a test will take mirrors the amount of time allotted for the test.

In general, I’ve found that this leads to worse test scores. I hypothesize that it’s due to the inability to check one’s work, which creates a rise in “stupid” mistakes. I then find it ironic when a teacher goes over a test and comments on how many people made such a silly mistake. In my mind, the answer is clear: they were in a rush, and so didn’t fully reflect on what they were answering.

I’m sure you’ve written a test that was too long, and the teacher basically admitted it afterwards. Fortunately, my teacher bumped everyone’s grades up to make up for the long test, but I would have much rather having a shorter test. While this negatively affected those who didn’t even get to certain questions (as I outlined my frustration for above), it was also surely a cause for silly mistakes in my exam. If the test was shorter, I perhaps wouldn’t have made those mistakes.

This is why margin is so important on tests. It gives students an opportunity to take a deep breath and calm down during the stressful time which is exam writing. By bringing the ratio down, students get more time to think about their answers, which I’m sure we can all agree is a good thing. You don’t need to write questions just to “fill” the test. Instead, the important questions need to be emphasized. If there really is that much content, I’d be of the mind of writing two parts of a test on separate days. This would allow teachers to keep the number of questions they want while still increasing the amount of margin for a test.

Rarely do I see an instance where the margin for a test is low, but I have seen it before. When I was in my astrophysics class in CÉGEP, the final had an amazing amount of margin. I was a very good student in that class, but I finished the test in about half the time, which was one and a half hours out of the three hour exam.

What I loved about the test was that it still asked many questions and had a lot of topics. It didn’t have a million things to do though like other exams where I would have to regularly check the clock. Here, I was able to sit, relax, and really answer the questions to the best of my ability. In my mind, that is what we want to see out of a student.

Margin during tests needs to be factored into the process of test creation because it has such a profound effect on how a student feels during an exam. From personal experience, a test with little margin is an extremely nerve-wracking experience where I am basically on autopilot. This is good for answering concepts in general, but this approach usually misses the finer details and is prone to making silly mistakes. Therefore, I am always happy to see a test with margin.

How much margin is enough? Personally, I love to look over my exam again when I am done in order to check for any small exams. This would mean the margin ratio would be about 0.75, but I think that’s a bit unrealistic. Still, it would be nice if a person in a fifty-five minute test could have about ten minutes to look over their answers, meaning the test should be able to be completed in about forty-five minutes. It may seem like too much wasted time, but I firmly believe students would be able to perform to a level that reflects their actual ability more than the current way tests are usually made.

In the end, packing a test with as many questions as you can during the time of an exam is a good way to have students feel frustrated and make stupid mistakes. On the other hand, by giving a fair amount of margin to the students, they will be able to relax and focus on the work at hand instead of at the ticking clock.

My hope is that the five-minute warning will become one where students are only checking their answers and not where a majority are still furiously writing in order to finish.

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.

Good Enough

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.