I find it amusing that I could probably impress my family at the dinner table by using terminology from my mathematics classes. If I used words like logarithmic, multivariable calculus, hypersurfaces, tangent planes, and linear approximations, I could get them to think that what I’m doing is pretty advanced stuff that they wouldn’t even be able to wrap their heads around.
But that’s not the case. Many of these terms can be explained much more simply. Tangent planes have an easy geometric visualization that is easy to understand, and the linear approximation is essentially the tangent plane! Even those two terms sound complex but are really the same and not that difficult to understand.
Mathematics is good at defining concepts rigorously, but the terminology used in mathematics can make the discipline seem very confusing, even when it isn’t. In this sense, there’s a sort of “masking” going on. The terms can seem more difficult than the concepts themselves.
This is why it’s difficult to communicate mathematics with people outside of the discipline. A lot of the concepts can make sense, but the terminology is difficult if one does not have a solid background in mathematics. And since explaining a concept requires terminology, those outside of the discipline tend to find the whole thing just too complicated.
For some subjects within mathematics, it’s possible to sidestep this issue through the use of visuals. If the concept you’re trying to explain has any sort of geometric intuition, it’s usually enough to show them the geometric situation to gain an understanding of what is going on. Unfortunately, this isn’t possible with many subjects, such as linear algebra or anything involving more than three dimensions. At this point, the mathematics becomes the sole anchor point for a concept, and terminology is important.
Terminology in mathematics is usually precise, offering little wiggle room. However, the language that is used is often inaccessible to the casual observer outside of mathematics. This is a shame, since many mathematical ideas seem complicated from their names, yet are downright simple when explained. As such, there’s a potential to get more people interested in mathematics by breaking down these language barriers. Sure, it’s not necessarily rigorous, but the truth is that those outside of mathematics won’t really care about the more subtle points of a definition. Instead, simply introducing them to the subject is more beneficial in the long run.
Let’s try not to mask mathematics from the public with a veil of complex terminology. Instead, let’s try to move it away whenever someone asks a question concerning mathematics. Hopefully, we can then get many more people interested in mathematical concepts.
This Isn’t The Forefront
During primary, secondary, and CÉGEP, mathematics is pretty similar in its style. Essentially, the idea is to give students the tools and skills necessary to be able to solve various problems (both in pure mathematics and applications such as physics or chemistry). There is a large focus on formulas and strategies to solve problems.
However, there’s one key distinction between this education and the “real world” of working within these disciplines: the problems for the former have solutions, while the ones for the latter are waiting for solutions. This is important, because it means the skills and work ethic that got you to the discipline throughout school won’t necessarily work now.
It’s a reminder that we cannot stay complacent about our progress. Yes, it’s incredibly important that we learn and retain the knowledge from our education, but there is a large difference between struggling with a question and being able to peak at a few steps of the solution to help you get on the right track and banging your head against the wall because you’re breaking new ground and you find yourself stuck.
Remember, the forefront of a discipline is much different than the path to it.
If there is one thing in particular I dislike about many tests in science and mathematics, it’s how we link the ability to do well on tests with the ability to do science.
The biggest problem I have with this link is that the things we test are not the things we actually care about as scientists, mathematicians, or educators. A great example is the terminology question. This is a question that basically requires a student to know a definition without needing to understand what is important about the concept. Instead, they can “fake it” by knowing the correct definition. These questions don’t actually aid in understanding concepts, they just make sure you know what the proper terms are.
I saw a question like this the other day where I was tutoring a student. It asked, “Write down the above equation in functional form.” The equation was something like $ 3y -4x = 0 $, and I couldn’t immediately understand what the question was. I knew that it involved manipulating the equation, but I didn’t even know what functional form was. Thinking back on it now, I imagine it must be an equation in the form of $y=mx+b$, but I still thought it was strange at the time.
What is the actual value of the question? To me, it only seems like a way of saying, “Write the above equation for y in terms of x.” That makes more sense to me, and it’s only a few words more than “functional form”, yet it is more descriptive. Additionally, I fear it slots the learning of equations and algebra into “archetypes”, which isn’t productive.
Another issue is that science and mathematics (and perhaps other fields, but I am not as familiar with them so I do not know) have serious naming issues. If you were to imagine the body of knowledge science has uncovered as a book in which anyone who has something new to contribute writes their own page and inserts it into the book, you wouldn’t be far off. Science is oftentimes a jumble of confusing terms, with a lot of historical baggage.
For example, I remember learning in my astrophysics class about the different classification of stars. I learnt that we had a whole system of stars: O, B, A, F, G, K, M. At first, I wondered why on earth the classification wasn’t alphabetical, since it classified the brightness of stars. However, I soon found out that the reality was the the original classifications weren’t particularly good, so the names were kept and the order was switched around in order to have them increasing.
Therefore, what could have been an easy process to remember the classifications became a mnemonic: Oh Be A Fine Girl/Guy, Kiss Me. While funny, it isn’t exactly intuitive to learn this classification, and this is something we were indeed testing on. My question then becomes: does being able to remember classifications well on a test really solidify our understanding of which stars are brighter? I’d argue probably not.
This isn’t the only example, either. If you’ve ever been privy to a science or mathematics class, you’ll quickly learn that equations and phenomena aren’t necessarily described in ways that are entirely useful for a person hearing the name. Instead, the concepts will usually carry the name of the person who discovered, invented, or came up with the concept. This is a way for many scientists and mathematicians to stay “immortal” in time. After all, few physics students are ever going to forget who Newton is, just like few biology students will forget about Darwin, or how mathematics students will not forget about Leibniz, Newton, Pythagoras, or many of the French mathematicians. We don’t know these people because we are all fans of the history of science (though some of us are). Really, it’s due to constant barrage of hearing these names in concepts in classes that solidifies these people in our minds.
Unfortunately, this isn’t a great recipe to actually knowing what these concepts means. Saying “Newton’s Laws of Motion” only gives me one real piece of information: the laws will be about motion. But these could have easily been referred to as the three general laws of motion. Except you will almost never hear that. Instead, you’ll hear, “Newton’s Laws”, because it’s a succinct way to describe them. And there’s nothing wrong with describing concepts in concise terms. The problem to me occurs when students get tested on knowing these things, where the names have almost no use.
I’m reminded of some words I’ve heard from Richard Feynman, who described an encounter he had with a kid who criticized his father because he never really explained the names of birds to Richard when they went on walks. However, Feynman shoots back, saying that his father taught him about the birds instead of just what the names were, and that this was much more valuable and interesting. Indeed, this is the sort of thing I imagine. Not that we shouldn’t understand what things are called, but that it’s always more important to know the concept than just the name, and we should be tested accordingly. Personally, I know that it would knock off a lot of stress from remembering how concepts worked, because I wouldn’t have to link a name to a concept. Instead, I could just explain the concept. “Explain Newton’s Second Law” would be asked as “How does motion relate to forces?” The latter question is much more descriptive than simply giving the name of a scientist.
I hope that a shift comes in mathematics and science soon, bringing questions that dig deeper instead of looking at surface-level detail about names.
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.