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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.