One of the great things about mathematics is that there tend to be multiple ways to solve a problem. This implies two things. First, there is no “correct” way to finding a solution. As long as the approach is logically sound and produces the answer, it is a good method to solve the problem. Second, multiple methods imply a fastest method. This method might take less work, be more obvious, or even be simpler. Of course, keep in mind that this is a subjective measure of “simple”. One method that might be simple or obvious to one student could seem bizarre to another. It’s a matter of perspective, yet the fact remains that a given student will find one approach to a problem easier than another.
If one approach is “easier” than another, it lends itself well to developing new mathematics. In particular, a lot of mathematics that is taught in school (in particular, secondary school) is there because it solves a historical problem. (Whether this historical problem had any practical use is a different question!) In other words, a lot of the tools of mathematics are taught to students because they solved a problem.
In post-secondary education, the standard example would be of integral calculus. How much area is under a given curve, and how does this relate to the derivative of that function? These kinds of questions are answered with the tools of calculus, and have applications all over science.
The problem, from the student’s point of view, is that this isn’t how the new tools tend to be presented. Instead, they are given the new tools without the prior motivation, which can make learning these new tools seem pointless. This is even more poignant when students are learning techniques to solve problems they have already learned how to solve. I can hear the question: Why do I need to learn about a new way to solve a problem I know how to solve?
This question isn’t answered as often as it should be. Remember, before a student gets into pure mathematics (where the emphasis is more on proofs and results), mathematics is computational in nature. This means the student is learning how to use new tools to solve problems. While “real-world” examples are given, they don’t tend to capture the heart of why these tools are useful.
Here’s an example where the motivation for the tools is good. Students in physics start by learning the kinematic equations of motion, which hold when the acceleration is constant (such as the case of gravity on the surface of the Earth). Once students learn calculus, they are able to tackle problems in which the acceleration continuously changes. Armed with these tools, they can now address complex problems that their old tools were not equipped to solve. The use of calculus is clear to a physics student. It’s not that it let’s you solve equations of motion. It’s that calculus is able to model the continuous change that is present in a lot of our world. That’s the point of calculus, in a nutshell. How can we quantify change, even when it isn’t constant? Physics students get to experience this sort of motivation.
On the other hand, a lot of students don’t get to see the motivation underlying their tools. Instead, they are introduced to a new rule or concept, and then are assigned practice in order to get better at applying the rule. There’s little mention as to why these rules are as they are, or why these tools are important.
I want to note here that I’m not advocating for motivating mathematics with “real-world” applications. I don’t care if the mathematics that one learns has any inherent use, because that’s not the point (at least, not the whole point) of mathematics. What I am interested in is the question, “What made a mathematician so inclined to invent this mathematical tool?” That’s an interesting question that isn’t dependent on an external application.
Teachers need to find these reasons and explain it to the students. It will achieve two important functions. First, it will be easier for students to answer conceptual questions, because they will know why this tool was needed. By knowing the “why”, students associate mathematical tools with a specific purpose. The second function is that students will be more on board with a new method. Have you ever tried teaching a new method of solving a problem and got a lot of resistance from students who prefer the older method? That’s not an accident. Either both methods are equally good (in which case the students have a point about not wanting to learn a new method), or else the new method is better. But, you aren’t teaching it in a way that convinces them to come on board with you. The latter part here is crucial. If the new method is truly better, you need to show that to the students. I always think back to addition and multiplication. If you have a bunch of objects you need to count, we all now know that it’s so much easier to arrange them into a rectangular array so that we can multiply them. We would much rather arrange 800 objects into 40 rows of 20 and perform (40)(20)=800 instead of counting each object individually. The “tool” of using arrays with multiplication just blows addition out of the water! You need to replicate this experience with your students for all topics.
If you feel like your students aren’t understanding why a new mathematical tool is so much better than an older one, I would argue that it wasn’t presented in a way that made this fact clear. Sure, in hindsight it might be clear, but by the time students have this benefit of hindsight, they will have long-completed the class. In order to get them to understand how a certain tool solves a problem the “best”, presenting the historical need for this tool is a great way to go. It’s not that teaching things in a historical manner is the best way to teach, but history does give us some guidance as to what problems can motivate a new tool. Use this as guidance when you’re planning your new lesson, and chances are the students will be both more on board with you, and will have a better conceptual idea of why they need to learn this new tool.