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.

Lead By Example

In my mathematics classes in CÉGEP, a lot of the content became more and more abstract and theoretical as I learnt more and more about calculus. As such, it became easy to lose perspective of what I was actually doing, since a lot of it was simply symbols. Fortunately, my teacher understood this and gave us plenty of examples to learn from.

I’ve been thinking about this topic in the context of first learning content. Here’s a question: should you start by giving students definitions, or examples?

Arguably, the former gives students a more complete view of the mathematics. After all, a definition is created precisely because people wanted to know when a certain concept applied or not. Therefore, a definition should be useful for students to figure out precisely what does and does not apply in the situation.

While that may be true, it’s probably not the way to introduce a new concept to students. I doubt I’m an outlier in saying that I always enjoyed seeing a graph or a picture of what we were doing in my mathematics class versus getting a rigid and complete definition. It’s not that the latter was bad, but that it didn’t give us any sort of intuition as to what we were doing.

For example, when I began to learn about derivatives and integrals in my calculus classes, my teacher didn’t immediately jump to showing the general definition of a derivative or integral. Instead, constant examples were provided, giving us a feeling for what these two mathematical concepts were. It wasn’t general by any stretch of the imagination. Instead, it was specific, and it gave us a foothold in the concept. Then, my teacher was able to step back and give us the general definitions.

In many disciplines, the opposite approach is what generally works. First, you focus on the general concepts, and then you drill down into the specifics for your goal. However, it’s easier to do the reverse in mathematics, because seeing an example (preferably, an easy one), allows a student to look at the example and think, I get the idea, how can we generalize it?

This is so crucial in the beginning of learning a mathematical concept, because definitions can be daunting. Often, they look like a bunch of symbols that just don’t make sense, whereas one can concretely get the idea of concepts like the directional derivative or the gradient vector by looking at examples or drawing graphs. Sure, they can also be understood by definitions, but it’s much more difficult to process. In general, we excel at absorbing information in a visual way, so it makes total sense that we would look at examples before generalizing.

The key idea here is that mathematics is about zooming out. We want to generalize information, formulas, and theorems as much as possible. It’s much more satisfying to have one equation that covers a whole gamut of possibilities instead of having one specific equation for each possibility. However, grasping these general equations aren’t immediately easy, since definitions are abstract. Therefore, the use of examples before broadening the picture allows students to understand what is happening for a specific concept, giving them a better idea of what the different components of a definition or theorem do.

If you just throw definitions and theorems at students, they will have a much more difficult time to grasp a concept than if you draw a simple graph or diagram.