When I first began my CÉGEP mathematics courses, one of the things I noticed that was new was how my teacher would give “motivation” to what we’re doing. Basically, the idea was to give us a reason for why we were doing something, instead of just throwing it out in the blue.
This was a welcome change from what I originally did in secondary school. There was some motivation for what we were doing, but most of the motivation was left up to us to find. We simply went from one subject to another. With my teacher in college however, there was nearly always some sort of motivation embedded into the lecture. Some of it might have went over our heads, but my teacher always made sure to give us some reason for what we did.
I’m still a student, but I’m also a tutor, and so I can see how both sides of the relationship see motivation. As a student, I like to know why I’m doing something or what the end goal is supposed to be. As a tutor, I know that giving students direction while learning new concepts is key to keeping their attention (and the inverse means a lack of interest). Therefore, I try to look for ways to motivate the line of reasoning I pursue with them in mathematics. It turns the learning experience from just myself giving information to the student to an experience where the student can feel like they are along for the ride.
And from personal experience, it’s a lot more fun to learn when you feel like you are an active observer with the teacher, instead of merely listening and taking notes.
A Bunch Of Formulas
At the beginning of my physics classes in CÉGEP, I would receive the formula sheet for my entire semester. That meant I had all the formulas on one page, allowing me to scope out what I would be doing in the semester.
At the outset, this is a bit overwhelming. Having all the formulas in front of me tends to make the course seem super complex. Plus, since I’m only beginning my journey through the world of mathematics and physics, I’m not accustomed to a lot of the new equations I see on these pages. Therefore, it can be difficult to understand what these equations even mean when the semester begins.
However, as I go through the material during the semester, it becomes easier to understand the formulas. It’s not magic or because I’m “smart”. Instead, working with the formulas becomes easier with practice and seeing how the formula comes about.
This sounds obvious, but I mention it because there are many people who don’t do science and think it’s some sort of magic that’s happening when we work with these crazy complicated formulas. And sure, to the uninitiated, they are complicated. But that’s because they are looking at the formulas for the first time, with no explanation or context. This is a surefire way to be confused. Indeed, I get confused if I approach new formulas in this way.
That’s why I try to play down situations in which others tell me that mathematics and physics is just too complicated for them. When they tell me this, I say something along the lines of, “Anyone can understand the concepts if they put their mind to it.”
Learn the why behind the equations first, and you’ll be much more satisfied moving forward because you will know what you’re doing.
If we were in the fourth century and I told you that the Earth was flat, a reasonable question you could ask me would be, “If you just continued moving in one direction, would you just eventually fall off?”
Now, that wasn’t what I said at the beginning. I simply said the Earth was flat. However, the logical implication of a flat Earth means that there is an edge to it (unless, of course, it is infinite). Apart from the infinity loophole, there is no other way out of this option. Either the Earth is infinitely flat or it has an edge.
Suppose you want to test my question because you are sure you are correct. You say, “Alright, we will continue moving in one direction, and we’ll eventually arrive at the edge, proving what I say is true.” From there, the we begin our journey. Assuming we can somehow move in exactly one direction without being obstructed, we would eventually return where you left on. Therefore, this would invalidate my hypothesis. The Earth could not be flat, or else we would have detected an edge.
So that means it’s flat, right?
Well, not so fast.
An astute observer could say that the Earth is a cylindrical shape, and you and I simply walked along the lateral face of the cylinder. However, barring this cylindrical Earth scenario, the Earth must be some sort of solid that one can walk around and arrive back at the same point once again – a sphere.
What I want to illustrate in this example is that one can get into some pretty interesting scenarios by taking statements to their logical end. It’s kind of like trying to reconstruct the pathway of a conversation, but in the forward direction. And almost always, the person who first made the statement will be surprised with the direction the conversation or train of thought has taken based off of one simple statement.
This has particularly relevant implications as our world gets more and more global. While issue like climate change, maintaining our environment for future generations, and spreading resources more equitably may seem far off and too abstract for many of us to think about, they are issues that will have profound consequences. Therefore, saying that you care about the environment but initiatives to preserve it will have to be put off for this year due to lack of funds means you are choosing to not allocate resources to an important issue. Perhaps it’s not the most pressing issue to you, but it certainly won’t improve (and has the possibility to be worse) through inaction. Therefore, the logical consequence of one person not doing something today to help the environment (multiplied hundreds of millions of times) results in an environment that does not improve.
More generally, this situation of saying something without thinking of its logical implications happens all the time in the media. Statements are made which may sound alright on their face, but are disastrous if one really thinks them through. It might sound like heresy to raise taxes because it affects you right now, but if the quality of your life (and others in the world) improves in the long term, is it worth accepting?
Climate change. Space and other life in the universe. Genome sequencing and editing. The allocation of resources. These are the difficult questions of our time, and quick responses might not be the answer. At the very least, we should require more thinking about them than we currently do. Our current answers tend to be quickly thought up, and averse to change because it’s not the status quo.
Instead of being the first or the loudest, how about we be the first to think through the logical implications of our actions and words?
In my multivariable calculus class, I learned about various types of curves that take different formats, from parametric to polar coordinates. Because the course was a sort of introduction to these notions, we weren’t given the “full” explanation on a bunch of these curves. Therefore, I want to touch on some interesting aspects of this category of curves (called limaçons) below.
First, I was only introduced to the curve by having the formula given to me. The formula was in polar coordinates, and is as follows:
As one can imagine, this formula makes little sense to a student when they haven’t learned much about these types of curves. However, what if they looked at this instead?
When I first saw this, I just thought, “Oh, that’s what it is.” The curve makes so much more sense when viewed in this type of frame. I am confused as to why this animation wasn’t shown to my class, because we saw a similar animation for the cycloid.
What’s cool about the cardioid is that it’s a curve that has a very nice formula to compute its arc length. The formula is $L=16a$.
In terms of the area of a cardioid, it’s formula is pretty nice, too: $A=6\pi a^2$, which means it’s six times the area of the circles above. I always find it neat to see comparisons of different shapes to see how they relate to one another.
More to come on these kinds of curves and ways to represent them.