Why Do I Need This?
The answer: you probably don’t.
The truth is that, because mathematics was deemed “important” early on in the history of public education, elementary and secondary students have to learn it. I can understand the argument for those in elementary school, since a lot of what they learn has to do with arithmetic and thinking about numbers in one’s mind. I think that’s a fairly useful skill that allows people to not need to pull out their calculators every time they have a small problem.
Once students get to secondary school, however, the actual usefulness of mathematics in every day life drops. I’m most definitely not saying that mathematics is useless and that nobody should learn it. On the contrary, I believe mathematics to be a wonderful subject to study. But we mustn’t kid ourselves that it often has limited use compared to something like learning a new language. Mathematics can help us answer serious questions and further our understanding of the universe, but the truth is that a good majority of people will most likely not be in that group of people who need to use mathematics.
The problem lies in how we frame mathematics. I constantly read and hear about people trying to make mathematics more relevant to a student’s every day lives. One only has to look at a textbook to see that this is true. Dan (link) even writes a weekly post where he shows his audience a picture of a textbook and asks us what we think the image is trying to convey. The idea he is trying to show is that we wrap up mathematics in a lot of “pseudocontext”, which isn’t really helpful to the student at all. This is exactly what I’m saying. Mathematics has its utility, but it isn’t necessarily useful for everything in our lives.
In my mind, core requirements in a curriculum should be there for a reason, which is that they are important to the education that a student is getting. Mathematics is one of those components that is required in every single year of secondary school education, so I think it is fair to assume that a class that takes up this volume of space should be absolutely critical to students. Unfortunately, this isn’t the case for mathematics. Those who enter the sciences will use mathematics, but there isn’t that much more application elsewhere (that I can think of). Therefore, is it really necessary to give students this much mathematics education when most of them will simply forget it all from lack of use?
I don’t have the full alternative solution to this problem, but I do think we could make a few changes. As mathematics is valuable in any career that involves the sciences (and here I mean both social science and natural science, so I am including the economic industry as well), we need to stress this importance at the beginning of secondary school. Since I know that people change their minds all the time with respect to their future (since they are young, after all), I think they should have mathematics classes until maximum secondary three. After that, we already have a fork in mathematics education as students must decide if they are going to take the “regular” mathematics course or what students like to call “high” mathematics, which is basically a more scientific and general mathematics course (a quick example is the students in the regular mathematics course learn about parabolas, but only fixed at the origin, while students in the other mathematics class have parabolas going everywhere). The same sort of decision is also made in secondary five.
Instead, of this, I can envision two options. On the one hand, those who want or need to take a mathematics course in order to go into higher education can take the scientific flavour of the course (which I don’t think is that much different than the regular one), and every one else can take a different course. Not a different mathematics course. A different subject entirely. As much as I love mathematics, there’s no hiding the fact that it has terrible reputation because so many students are taking it when they both hate mathematics and have no use for it. Should we really keep on giving them these classes when they aren’t helpful? Personally, I think the answer is “no”.
I don’t think the problem with mathematics is that it’s too abstract or that there aren’t enough relevant examples to pique the interest of students (though that may sometimes be the case to a small degree). I think the problem is that we expect too much of mathematics. It doesn’t need to be a subject that fits for everybody, just like fine arts or music or theatre aren’t for everyone. By removing the need for students to always take mathematics in order for them to graduate, we should set out the path for them for three out of their five years of education, let them decide, and teach those who want the mathematics education. That way, there will be a definite answer to why students need mathematics. It will be for their future career choice, as they themselves have chosen.
Finally, you might be thinking something along the lines of, “Art and physical education are mandatory for graduation, so why shouldn’t mathematics be as well?”
For physical education, I think the answer is simple enough. Even if students aren’t necessarily “getting” any skills from that class, it does provide one thing: a path for students to stay active and see that there are an enormous number of activities that one can do to stay healthy. To me, that’s reason enough to keep it as a mandatory requirement.
For art classes (I’m talking about all art, such as music and theatre), they are indeed mandatory to graduate, but it isn’t a requirement until a student’s fourth or fifth year. In this same way, I think it’s great to have students do some mathematics, but only enough to give them an idea of if they want to pursue it more. My point is that I don’t want it to be forced upon them for years and years.
Of course, I want to end by pointing out once again that I don’t know how reasonable this is, nor of all the factors at play. However, I can see that what we have now can be improved, so I’m hoping we can step up and do something about it.
Within One’s Expertise
Because I am a tutor for secondary students, most people assume that I know everything that needs to be known about those classes. However, that could not be further from the truth. In fact, I frequently encounter problems that the students I help have that I cannot answer. It’s not even that I don’t know the answer. Sometimes, it’s just so far back in my mind that I don’t remember what the exact steps are.
When this happens, I don’t beat around the bush and pretend I know the answer. I tell them I have no clue, and that we’ll work it out. The reason I do this is twofold. First, it’s probably true, since a lot of the secondary mathematics is straightforward after doing more advanced mathematics. But the second reason is that I want to show them that I am not a robot that knows all the answers to their questions. I’m a student just like them (even if I become a teacher as well), and I’m constantly learning. I want them to know that I am just like they are, expect a few years further down the line. That’s it. I’m not a mathematical prodigy or someone who can do all the calculations in their head. I’m just there to give them a hand when they need it.
This is why I’m not too scared of straying from my expertise. Yes, I may not have much experience in all the classes in secondary school (in fact, the majority of students I tutor this year are taking a class I’ve never taken), but there’s no harm in learning. Plus, learning this way also helps me become a better teacher in the future.
I’m definitely not advocating for you to proclaim expertise in every area of your life. I’m just saying that if you are willing to put in the hard work to learn, it’s entirely possible to become more knowledgeable at anything. Therefore, don’t restrict yourself to your personal expertise where you are comfortable. Experiment, and try to learn new things.
I find it a bit of a mild tragedy that we as students don’t get to feel the joy of discovery while learning new scientific concepts. Classrooms talk about DNA, magnetism, electricity, gravitation, chemistry, and evolution as if they are mundane things. Ideas are introduced, but rarely is there any sort of “discovery” by the students. Instead, the information is clearly meted out in logical sections with almost no flair.
Imagine if you’re reading a book that is typeset in all capitals with very little spacing between lines. No matter how amazing the book is, it will be difficult for a person to get engaged. This is why books aren’t made like that. They are designed carefully so that you don’t experience any drawbacks from the book itself, allowing you to enjoy the words within.
In this same way, I feel like many classes are poorly designed to stimulate student enjoyment. I’m not saying that science and mathematics classes need to be the highlight of every day for each student, but I’m saying there’s probably a slightly more exciting way to deliver information instead of spoon-feeding it to them in class lectures every single day.
I’ve written about this before, but students can be given bits of context to situate themselves in whatever they are learning. Answers to questions that are a bit more deeper, such as “Why did people care about this when it was first discovered?” instead of “Why am I learning this?”, can be helpful to giving students an appreciation of science and how it was developed throughout history. As I look back at my lessons on history, the only time I’ve learnt about scientific history is during brief snippets from my various science teachers, and those were rare. I think if those moments were shared more, I could gain a much deeper appreciation for why a scientific idea is important.
Additionally, most science classes follow the same tried-and-true formula: lecture with notes, homework, test, final exam. Each day is a new lecture, and the students are expected to take notes and absorb the information as it is handed to them. In this scheme, it’s easy to push productivity. A teacher simply needs to write more on the board in a given lecture, while minimizing questions. The problem with this, however, is that the class is reduced to a mere transcription of information. The students aren’t necessarily learning, they’re just absorbing. In my mind, learning happens when one is engaged, and there’s no better engagement than discovering something. After all, that’s the dream of basically every science student. Why shouldn’t we harness this to good use in classrooms?
I envision it working something like this. Students don’t get the full results. They get bits and pieces, and the teacher has to get them to think. What do these two bits of information suggest? What if we look at them in this way? By gently prodding the thought process of students in one direction or another, we can get them to think and deduce in place of just absorbing new information.
As I get older and continue my studies, I fear that I’ve lost many opportunities when I was younger to discover surprising truths about science. I don’t want to be a sponge that soaks up new information. I want to be the detective that takes in new information, thinks about it, and uses hunches to get new results. I believe this can be stimulated by having students discover more things in the classroom. Yes, it isn’t going to help the productivity of the class. But it will give them the tools to think, which is a big part of what science is all about.
On the Boundary
When I first started learning about physics, I thought it was amazing how we had these equations and patterns that emerge in nature to the point that we could actually predict what would happen if we threw an object or slid it on a certain surface. The classes were interesting to me because they allowed us to describe things we actually saw.
At this point, I wasn’t familiar with the notion of classical physics, or that there was anything other about physics that I was learning. I didn’t know there were inherent limitations into what we were learning. I imagined that physics was as intuitively simple as the principles I was learning.
However, the trouble came when taking those intuitive principles and extending them in my imagination. For example, imagine moving a book across your desk. When you push on it with your hand on one side, the whole book starts to slide across the desk. Have you ever wondered why this is?
It might seem like a stupid question, but hang in there with me for a moment. It seems obvious that the whole book will move despite only touch a small portion of it, but let’s extend this experiment. Imagine having a long broomstick that is one kilometre long. When you push it, what happens?
Intuition will give us the same answer: the broomstick moves as a complete “unit”. However, if we extend this already-crazy scenario even further, what happens when the broomstick is 300,000,000 m long?
If we were to continue with our intuition, we would be able to push one end of the stick and move the whole thing. But wait a moment. If pushing the broomstick means effectively moving some sort of “signal” throughout the atoms of the broomstick from one end to the other, you’ve just made a signal move faster than the speed of light! Since this goes against Einstein’s special relativity, something in our scenario is wrong.
As it turns out, the rod would not actually move together in one moment. Instead, it would take time for the signal to be passed from one atom to the other along the entire broomstick. After all, pushing the broomstick only means moving the atoms on one side of the broomstick, and so the atoms have to “communicate” to the adjacent ones that they too should move, all the way through the broomstick. It is much like what you can see if you film somebody letting go of a slinky with a high-speed camera. The top of the slinky falls first, followed by the next slowest ring and so on.
Personally, I find this extremely cool. It’s unexpected: we’re taught that an object in free-fall will, you know, fall. But here the situation seems to defy the usual assumptions (though a closer inspection would show that indeed, we could describe the slinky with tension as it falls).
What I wish I had more of in my classes of classical physics were boundary conditions: situations that showed the limitations of the wonderful equations we learnt. It’s great and all to know the ideal cases, but I think it would have been equally instructive to show where they don’t work. I feel like we often gloss over what doesn’t work in favour of what does, but when learning, the cases where something doesn’t work can be of great help.
When learning an idea, investigate where it’s domain of applicability exists, and what happens when one leaves it.