Over the Hump

It’s interesting how I often hear other students say something along the lines of, “I can’t wait until we are done this course, because I can then forget about X.” I’ll also admit that I’ve been prone to saying these sorts of things as well. However, it’s interesting to me because it shows us how we place our knowledge into silos. It reminds me of something my mathematics professor said after one of our tests, “Yes, I put a physics question on the test. You’re all grown up now, so we don’t have to keep our meat and potatoes separate.”

Too often we see our classes as representing knowledge that is valuable in only that class. This starts early, with class names such as “Science”, “Mathematics”, and “Languages”. Of course, the truth is that these are only artificial constructions, and this soon becomes obvious for science students when they see the amount of mathematics that are used in their field. These subjects aren’t separate, yet we treat certain concepts we have trouble with as something we can “get past”, when the reality is that they will always be there, following us. You don’t graduate away from statistics when you’re in science, nor do you get away with not understanding calculus well if you’re in mathematics (generally).

This becomes quite evident as one moves past secondary school, but I wish there was more of an effort to show this interconnectedness of all subjects much sooner. It is definitely useful to show these distinctions between subjects, but just because it is useful does not mean these distinctions actually exist.

This is why I try to commit myself to learning things properly, not just learning them for the test and then never thinking about them again. There are some rare courses in which you can do this, but they are few and far between. Much better, in my opinion, to do the hard work of understanding something you’re struggling with. The best part is that it generally won’t be for waste, since

Understanding Graphs

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When I look at most types of functions in two-dimensions, I can more-or-less visualize what is happening. This is a byproduct of working with these functions for over seven years. At one point, you start to get a feel for how a parabola will look, or what placing a factor in the denominator of an ellipse will stretch or squeeze the shape. However, when you first start learning these functions, they aren’t immediately obvious.

Unfortunately, many students have problems with this (which is natural), but the bad thing is that many teachers don’t address this issue. I see so much potential for students to better understand how functions work and how changing the parameters affect the shape of the graph. With the tools we have today, I think every teacher should be using them in order to help solidify their students’ understanding, particularly at the beginning.

The tool I’m going to talk about here is Desmos (Link), but of course if you know another tool that works, then go for it. However, there are two real key features of Desmos that I would say put it above the rest in terms of graphing. For one, it’s buttery smooth. It doesn’t take five minutes for a function to be plotted. The function is updated as you write, which is fantastic. The real feature that I love though is the use of sliders. This is precisely what can give students a feel for a function.

Let’s suppose we have a standard form quadratic function:

For most graphing software, you need to specify what all the constants are. However, you can simply write them as variables in Desmos, and it will create sliders where you can dynamically change the constants and see the consequences on the graph. Additionally, you can specify the range of values for each slider, and by what step size it will increase/decrease every time. Once you have those settings as you wish, you can either manually move the slider or have it “play” through all the options.

When I first used the power of this specific feature, I was studying polar curves in calculus III. I used the sliders because I was interested in seeing how polar roses or cardioids would change as the constant parameters were altered. Desmos became a great help for gaining that sort of intuition which helps even when I don’t have an instance of Desmos running.

The point is not to create a dependence on graphing software for the students. Instead, it’s about giving them a foundation in which to base their intuition. Intuition is not a magical ability to understand mathematics. It comes from visualizing and deeply understanding the underlying principles that are at work. In that sense, I want students to be capable of getting that visual aspect of functions and graphs that is so often missing. I want them to understand what changing $a$ in the above equation does to the function.

I am now trying to use this great tool with all my students I tutor. I think it is quite obvious that using graphing software to dynamically show how a function’s graph changes when parameters are altered is much more interesting and easier to remember than just being told that increasing that $a$ symbol on the page will make the parabola narrower.

Understanding Terminology

I wrote about how getting traction in the beginning of our studies in school can really affect our trajectory. I want to take this a step further today and talk about a specific example that alienates many students: terminology.

This issue is most persistent in mathematics, where definitions and theorems abound. It’s as if everything has a special name and meaning for it. To make matters even more confusing, the experts have decided to make shortcuts and give all these terms symbols to describe them. Now, the student has to learn – not one – but two sets of vocabularies to understand what is being discussed. And if I’m being honest, that’s only in the good-case scenario. More often, we find concepts that have multiple terms for them, and I’ve seen students quickly disengage from it all.

It’s great to have descriptors for all mathematical ideas, but we are often teaching them to students just for the sake of teaching them. Just as I’ve never been in a language class that painstakingly goes over words that have a bunch of synonyms, this isn’t particularly useful in class. Worse, getting tested on this sort of thing is nonsense, since one should be able to get by with only one name for an object. Sure, it can make for smoother conversations while exchanging mathematical ideas, but that knowledge and retention comes from familiarity, from actual usage. I know this because a lot of useless mathematical “facts” have long since departed from my memory since we never use them anymore. I remember doing step functions in secondary school (where the function was essentially a series of horizontal lines of the same size and would “jump” to a new new function value after a certain interval). I remember the notion of the step function, but it has been about four years since then and I don’t remember looking at this kind of function once. Maybe it’s because I’m in physics and mathematics and not in some different program, but the time spent on that function wasn’t useful at all. In the same sense, students aren’t learning the terminology because there is such little use for it.

The solution I propose is simple. If we really want to keep all these terms in mathematics education, we need to use the terms while discussing with students. Furthermore, we need them to practice using the terms. It’s not enough to just give a bunch of definitions and expect students to understand what they mean. Make them familiar with the terms, not just something that was copied once into a notebook and then forgotten about except for on a test.

Traction

I reflect on this often, but I keep coming back to the same conclusion: mathematics, or physics, or any other science is not as difficult as people make it out to be. When non-science people roll their eyes as I tell them that the ideas I’m working with aren’t that difficult, I’m not just trying to be modest and say I’m not smart. That’s not the point. Instead, what I want to convey to them is the idea that mathematics and physics is like any other field. By working hard to understand what you are doing, you can become great at these subjects. It doesn’t take some innate ability to be good at physics or mathematics. It just requires patience and determination.

This is why it breaks my heart when I see those who struggle with mathematics or physics and act as if they will never be able to “fully” get it. To make matters worse, the subject becomes nearly an enemy to them, something that they’re doing the most they can to finish but then completely discard.

I think the main issue is one of traction. Too often I see young students being divided up into those who are “good” and those who aren’t. It’s frustrating to me because I know that these challenges will not last forever. But what happens is that they do end up persisting, simply because the student doesn’t “get” those first concepts as quickly. I’ve begun to wonder: how many people who proclaim to be bad at mathematics simply haven’t given themselves (nor have others) enough of a chance to really understand the subject?

If we want more people to be excited about mathematics and science, we cannot keep up this illusion of someone being “better” than the other. This may be true in the long run of someone’s career, but it isn’t a useful way of describing students when they are young and most prone to being shaped for life. If we start sending the message that learning mathematics and science is about gaining traction, than I think we can get more people into these fields. Once you’re off and running, the subjects aren’t quite as intimidating as they may have seemed.