### The Short Route

If you think about what students learn in mathematics at an early age, it isn’t too difficult to realize why many kids find it useless.

To begin, students learn about arithmetic and patterns, which is relatively useful. They also learn about money and time, which is practical. However, they then start to learn about algebra, which starts to make things more complicated. Suddenly, variables as well as constants are flying all over the place, and keeping track of them can be a pain.

Because of this, teachers will often give students easy examples in order to “show them the ropes”. Many times, the question will be in the form of a word problem, with the student having to write down two equations and then find the solution. These aren’t particularly difficult problems, but they often confuse students because of their wording. Worse, a student can sometimes solve the problem *without* resorting to algebra and solving it the “long” way.

When I go through such problems with students I tutor, they often look at me with an expression that asks, *why do I have to do this?* And frankly, I don’t have a good reason for that particular problem.

The issue I see is that students are only being introduced to problems that are trivial to solve, which means they don’t get to see the full power of mathematics. It’s like watching a world-class archer shoot from only ten metres away. Sure, their shooting will probably be impeccable, but you kind of expected that anyway. In order to *really* be impressed, the archer will have to shoot from their competition position. This will show just how good the archer is.

Likewise, making students solve questions that are relatively trivial means they will only see mathematics as a tool that *works*, but not one that is super powerful. If instead we gave difficult or tricky questions to students, they would end up seeing just how useful mathematics can be compared to mentally “finding” the answer.

For myself, calculus and the three-dimensional coordinate system are what really demonstrate the power of mathematics. The former lets us precisely analyze the behaviour of curves, while the latter lets us understand how different curves and vectors compare and contrast when placed in the same space. After using these two tools a lot, I can somewhat visualize them in my head, but it’s quite taxing on my mind and is much easier on a computer. Therefore, doing problems that require more robust tools than our minds to doing mathematics shows how useful mathematics is.

This is why I fear many students don’t “get” mathematics in secondary school. A lot of it seems to be arduous for no reason, sort of like trying to get a computer to do one “simple” thing that ends up taking hours to figure out what the write code is. The students don’t see the incredible utility of mathematics because the teachers don’t give problems that are complicated. Instead, they favour giving a bunch of problems to get the “muscle memory” of that type of problem working. This gets students good at solving a problem like this, but they will be in trouble once a more tricky problem comes along.

As educators, we should be worried about how young students perceive mathematics. It should be thought of as a really great tool to analyzing situations. Since we cannot necessarily give them advance material, at least *telling* them about what this “simple” formula or concept is used for can help them understand that mathematics isn’t just a fancy and long route to getting an answer. In fact, the idea is that it should be the most efficient for difficult problems.

To illustrate this last idea, I want to bring up an example that Dan Meyer recently posted to his site about billiard balls. I’ve written about his problem before, but essentially its goal is to show how mathematics is the best way to figure out where a billiard ball will go after it hits the bumper.

However, instead of just asking where it will go after it hits *one* bumper, he asks where it will go after hitting multiple bumpers. Suddenly, the problem goes from something that could easily be solved in one’s mind (and therefore make the mathematics inefficient and useless), to where doing the mathematics is basically the *only* way to accurately solve the problem. In this situation, Dan Meyer shows how simply extending the problem can prove the utility of mathematics.

If we want students to not lose interest in mathematics in secondary school, we cannot always give easy examples to them. This makes mathematics look like the long route to take, which no one will *actually* do in life when facing a problem. Instead, we need to switch mathematics to being the *short* route, which in turn will show how useful mathematics really is.

### Where to Start

It’s almost certain that you once had a test go badly. Whether that means you failed, or that you got a much lower score than you are used to depends on who you are and what your expectations are during a test. However, I would argue that the reason you had a bad test largely stems from the problem of not knowing where to start.

Indeed, this is problem that will often occur to me on tests. It’s not that I don’t know the general sort of thing I need to do in order to complete the problem. It’s that I’m unsure what my first steps should be.

The best method I’ve found to combat this is to try and find a general “rhythm” for solving problems. Typically, each class has different “types” of problems. In my calculus class, we had problems dealing with area and volume, we had constraint problems, and we had problems involving space curves. These are all different types of problems with their own normal routines. As such, a better way of studying than just memorizing formulas is to embed the pattern – or rhythm – of the problem into your mind. That means you *always* do your first step when you encounter this kind of problem. Then, the only thing you need to do is recognize what “kind” of problem you have, which is usually easier.

Once you are able to pick out the patterns of problems, it becomes a lot easier to know where to start and how to solve. Don’t reinvent the wheel. There’s a joke in physics about spherical cows (link), and how we try to tackle a bunch of our problems by reducing them into easier problems. This isn’t necessarily a bad way to tackle sets of problems. Try and figure out the common patterns between them, and use that to inform what your first step should be. This will help mitigate the “I didn’t even know where to start” problem in tests.

### 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

[latexpage]

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.