A Place To Be Wrong

When I was in secondary two, my mathematics teacher asked if my class if anyone wanted to participate in a mathematics competition outside of class. A few people volunteered, but I did not. I suppose at the time I might have been a bit self-conscious about participating in an activity that seemed like it wouldn’t be a popular thing to tell people you do. (Of course, I wouldn’t mind telling someone that I did that right now.)

However, my teacher was surprised that I did not volunteer, and she eventually asked me about it. I told her I just wasn’t that interested (which was true), but she insisted that I sign up. I told her I would think about it, and eventually, I did sign up.

I wasn’t quite sure what it was going to be like. Apparently, the competition was just a quiz between others in my grade who signed up. I did not know how well I would do, but I was confident that I could do well since I was one of the best mathematics students in my grade.

The day of the test came, and the rules were explained to us. The test was all multiple choice, and you’d get a certain number of points if you got a question right, and none if you got one wrong. However, there was a catch: to reduce simple guessing as a strategy, you would get some points if you left a question blank instead of choosing an answer (to a total of ten unanswered questions).

With the rules explained, I dug into the test. It was designed such that the first questions were fairly trivial, and the ones further along became more and more difficult. I answered the first questions with little problems, but I slowly began to have trouble getting the answer for the next questions. It was frustrating because I had almost no idea what to do on these questions. I was baffled. Not wanting to take a chance, I did what I would never do on a test and skipped the question.

I remember at the end of the competition thinking, “There’s no way that I could win. I skipped so many questions that I’m sure someone else did better than me.” I then talked to one of my friends and asked how the test went. He told me that it went great.

Confused, I asked, “How many questions did you skip?”

“Maybe two,” he answered.

“Two?” I said in disbelief. “Man, I must have skipped at least five questions.”

Miraculously, I did end up winning, and I continued winning throughout the rest of secondary school. Each time, I would wonder how in the world I won when I skipped so many questions. I was forced to conclude that either everyone else made a lot more mistakes than I or that I skipped more questions than they did and got more points, or some sort of combination of the two.

As I competed later on in secondary school, I started to actually enjoy these quizzes. While usual tests were stressful since I knew I always had to perform perfectly, I knew that I could “drop the ball” a little bit on these quizzes and it wouldn’t be the end of the world. This gave me a lot more satisfaction during the competitions.

What I take away from that environment is that it was fun to challenge oneself, but you didn’t feel like crap if you screwed up or couldn’t figure out what to do. The pressure of regular tests wasn’t there.

This is something that I’ve been trying to figure out how to incorporate in schools. The goal of course is for students to know how to tackle problems they come across in disciplines like mathematics, but often the atmosphere of an exam takes away from the experience. Done right, these challenges should be fun, and encourage the students to try new methods of solving a problem. Experimentation should be key, and in the end it can teach the students an important lesson about the concept they are learning.

I think tests and quizzes are a good thing, but I think it would be even better if we included instances where quizzes were given that don’t necessarily count for marks, yet can still challenge students. Of course, some sort of incentive could be given, but the essence of the idea is to bring back problem solving without necessarily having to worry about grades.

When I was doing that competition, I loved not having to worry about how this will affect my overall grade. I could just focus on doing the task at hand, and I think it helped me do well on those tests and actually win.

Instead of having a class be only about assignments and tests, perhaps some sort of in-class problem solving could help foster more experimentation and creativity in the way students approach mathematics.


Throughout all my years in school, competition is rampant among students. It’s not always against other students, but often times it is. The reality is that students are always trying to get amazing grades, and this can be made worse in systems like CÉGEP where the grades of your peers are taken into account to give you your “global” grade. This is a problem, because it encourages you to be good in school while simultaneously hoping that your peers don’t do as well. I won’t lie: I’ve thought and said this before, because it is the way that you can get your best grade. Unfortunately, it’s also a terrible way to encourage learning among students.

Additionally, I am always troubled by my disappointment when it comes to tests or assignments. Truth be told, if I don’t get a hundred on a test or assignment, I am usually disappointed. I can get good grades, but I still won’t be happy with them. And this is a shame, because it introduces stress into a situation that should not be filled with stress.

In my mind, I yearn for a day where I can just enjoy learning and practicing my craft. When I run, I don’t even try to make every day an amazing day. Yet, this is precisely what happens with tests and assignments. Everything needs to be done perfectly, and anything else is unsatisfactory.

I don’t know how I’m going to resolve this, but it is an issue I actively reflect on. Becoming a great scientist or mathematician or teacher or anything else does not require one to be amazing at every single moment in his or her life. Instead, it requires dedication to the craft, and a willingness to work through the difficult days. That’s what I want to do.

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.