It’s not exactly a secret that I don’t like how mathematics is done in secondary four and five. I feel like the mathematics course for those not pursuing a career in STEM isn’t exactly the best use of a student’s time, because the curriculum doesn’t give students the full picture.
I just want to talk about one particular concept that drives me nuts: the parabola function. In secondary four and five, I studied them in essentially their full glory, looking at how a parabola is constructed (using a directrix), as well as the full equation (given by $f(x) = ax^2 + bx + c$). In the “science” flavour of mathematics for secondary four and five, you look at all of that. It’s more or less all you need to know about the function.
However, when you’re in the regular flavour of mathematics in secondary four, you still look at the parabola function, but it has been castrated. I’m not kidding when I say students literally look at the function as only this: $f(x) = ax^2$. Said differently, while I studied any kind of parabola on the Cartesian plane, those in the other mathematics class look solely at the parabolas with their vertex at the origin.
The only question I can think of is, “What’s the point?” If they aren’t even going to look at parabolas in general and confine them to the origin, is there any real use to showing them? I mean, you literally cannot make a physical example of throwing an object because the parabola at the origin won’t describe that sort of motion! The classic use case of the parabola isn’t even applicable to these students.
I think it’s a shame that these kinds of topics are in the regular secondary mathematics. In my mind, they are just filler for the year, because there’s no further explanation of them when there is so much more they could look at (without too a lot of added difficulty, either). I just don’t get how concepts like these can be shown in the form that they are now.
Going Through The Motions
When you know how to do something, it can often be repetitive and tedious to continue practicing. After all, you know exactly what you need to do, so why should you do more of it? This is something I’ve frequently asked myself, particularly when I’m in the middle of doing strength work after a run. I know I have to do it, but it’s not exactly easy to go and actually commit that time every single day. Likewise, many of us know that we aren’t giving our eyes the proper rest before sleep (and we probably aren’t sleeping enough), yet we stop ourselves from going and doing the thing we know we should.
This isn’t a new idea, and it’s one that I’ve mentioned in various forms here before. If we want to get better at doing something, we need to do it. It’s a nice idea that merely thinking about the thing will result in our improvement, but it’s not true. That’s why I find it so important to practice doing a bunch of questions before a test, or why I bother putting so much effort in assignments. It’s not because I necessarily like doing it all the time. Rather, it’s because this effort directly relates to the grades I want to get.
It’s not always fun to go through the motions of an activity, but it’s the best way to improve. On the whole, hard work is rewarded with better grades, and so that’s what you should do if academic achievement is your goal.
In my personal life, I have several different kinds of hobbies and interests. Obviously, you can see from my site that I love to write, and so it probably isn’t surprising that I enjoy reading as well. Additionally, I enjoy sports as a whole, but specifically I have a knack (and a love) for running, basketball, hockey, badminton, squash, and many other ones as well. However, if I had to choose which sports I participate on a serious basis, it would have to be solely running. I like all the other ones mentioned, but running is the one where I put a huge portion of my time and energy in. Likewise, I spend a good amount of effort writing and reading, so I would consider those my three big areas of my life.
But that’s just my personal life. Of course, my education is also extremely important to me, and in terms of my professional life, I’m working on becoming a better teacher. So that’s a quick list of the things that are important to me in my life, and they are what I spend most of my resources on.
Now, that’s not to say that I don’t have other interests. I’m the sort of person who is curious about many different fields. My interests include things like technology, AI and computer science, design, photography, typography, music, and video games. I imagine that I am like many others in the fact that I know a lot of geeky and seemingly random information concerning fields and areas of life that people would not expect me to have any knowledge about. I know a bit about the characteristics of type, the call-outs of many maps in Halo, principles of design, and many things about computers. Yet, my “big areas” are physics, mathematics, running, reading, and writing, not exactly applicable to the other things I listed above. But the point is that I have only dabbled in these other interests. I haven’t dug deep into them like I have with my core interests, and it makes all the difference.
Last year, I tried to learn a bit more on the guitar. This didn’t go so well. I learned some chords, but nothing came of it. Ultimately, I shelved this interest because I didn’t want to dedicate as much time to it as I did with my running. I saw that the two were competing for attention, and that I cared more about running, so I gave up my musical ambition.
Likewise, you cannot expect to simply add a new hobby into your life without encountering issues. More likely, you’ll have to make a decision between something you already do. The simple reality is that we have a limited time to do what we want each day, so we have to choose what to spend it on. That means making a difficult decision between things you enjoy. If you split your time, you’ll also split your focus. You can go that route, but you will always be limiting yourself.
There is always a juggling act going on in your life. Some things get more attention than others, and that’s alright. The important part is that you choose which things you want to be a focus in your life. Spend the time on things that interest you, and remember that it won’t be locked in stone forever.
One of the most hated things in all of school is the speech. Students hate talking by themselves in front of a room full of peers. This usually has something to do with students not wanting to look ridiculous in front of their peers. Personally, I never had too much of a problem with presenting in front of a bunch of others, but I wasn’t necessarily a natural either. My strategy consisted of trying to memorize my presentation as much as possible, but at the same time, I never fully had my presentation down. Therefore, I’d be walking a thin tightrope to not trip up on my presentation while still trying to sound natural.
Most of the time, it worked, but I knew that I was more or less faking it the entire time. I was trying to make it seem like I knew everything, though I didn’t. (To be fair, I was still good at presentations, but I always noticed this internally.)
I can’t help but notice the similarities between those presentations and how students need to approach tests at school. Whether it be in physics, chemistry, biology, or mathematics, students rarely go into a test feeling completely comfortable. Therefore, they are banking on the fact that the teacher will not ask difficult questions on the test that are in their “weak spot”.
Despite being pretty good at mathematics and physics, I’m frequently in this position. It’s not that I don’t know my material. It’s that there’s such a large amount of concepts that need to be known that it can be difficult to keep track of it in one’s head without forgetting a few things. Honestly, the only thing way I’ve found to permanently store concepts in my mind is through continued use over several semesters’ worth of classes, which is obviously too long to be useful for any specific test. I frequently catch myself in a new class and doing something that seemed so difficult just a few semesters ago and finding that it is straightforward now.
The problem is that there’s a tension between time and becoming comfortable with the material. Two and a half years after my first calculus course, I’m pretty comfortable with taking derivatives of most functions. But now, taking a derivative is supposed to be like doing simple arithmetic. New challenges have arisen in my courses, so those are what cause me trouble.
There’s no easy solution to this. The simplest solution is to do a lot of practice problems. Unfortunately, that takes a lot of time and so you are limited in what you can do. Therefore, the other strategy I take is to try and cover all my bases by doing problems that are vastly different in what they pose. Then, when I find one that I am struggling with, I focus on doing a few of those kinds of problems, and then I move on. This way, I’m not staying too narrowly focused on one topic.
The name of the game in school is to arrive on the test with as little need to “fake it” as possible. By exposing oneself to all the different scenarios and questions that can pop up, you’re giving yourself the best chance to be prepared to do well on the test.