Applying mathematical topics and skills to the real world is an often-discussed tactic to engage students who are otherwise dismissive of, or frustrated with, mathematics as a whole. By connecting the math a student does with real life situations—particularly skills they will “need” in the future, but also more abstracted situations that pertain to real life—some people expect students to gain an appreciation for what mathematics can do, and feel inclined to study it so they can actively participate in these various applications.
It’s not worth debating whether this general idea has any evidence behind it, since for some students it is definitely effective.1It’s pretty likely that any reasonable educational approach is optimal for a handful of students. However, this philosophy about math education can be enacted in many ways that span all the way from lazy and misguided to promising or even proven. I have been thinking about these implementations, and want to explore what I have noticed so far. To be clear, I’m focusing on younger education. Calculus and above is so easily applied (or removed from application) as to not be worth discussing. However, what to do about multiplication in a fourth grade class is a different matter. Let’s compare two extremes.
Approach 1: “Word Problems”
A misguided and widely disdained approach is shoehorning “word problems”—that is, problems that happen to be written in complete sentences—that contain some “application”—a few nouns or verbs a student might know about—into the curriculum, with an expectation that this is sufficient to show the utility of mathematics. An example:
John is playing baseball. He has five baseballs in his bag, which is two more than Sarah has. How many baseballs does Sarah have?
Here, we’re trying to catch the young athlete hook, line, and sinker. Surely the mere mention of a sport is sufficient to get them engaged with the problem?
The issue with this problem is not that it’s bad inherently. If a student is expected to show their work, and if this problem is understood to solely act as a way to practice going from words to mathematical operations, this problem is perfectly fine. Students need to develop those abilities.
However, as a method of engaging a student, it’s rather poor. Sure, it has baseball. A few kids might connect with that concept. But the problem itself is rather silly, because it contains information that nobody would have: Why would John know he has two more baseballs than Sarah? Unless they were both brought up using these types of problems, and like to quiz each other, this is an unlikely interaction. Problems like these begin to feel condescending to most students, because they can see the ridiculousness of them and intuit the laziness of the writers. Much more can be done.
Approach 2: Projects or Frameworks
Rather than dictating a set of problems that are loosely “connected” to an application or engaging idea, a more ideal approach involves designing a full project or framework of problems where the various steps involve using the relevant mathematical skills. While some of the first approach still has a tendency to sneak into these projects, they have a directionality to them that encourages students to stick with it. The goal of the project, as seen by the student, is not to learn math, or even accomplish a mathematical goal, but rather to produce something, perhaps gain more knowledge about a field of interest, and by virtue of going through this project a student is required to engage with math.
These projects are difficult to develop, and that ceretainly contributes to their minimal presence in most schools. Of course, organizations like Desmos are continually working to build out better activities, and thus make applicable and engaging math more accessible to students and teachers alike. Another issue with this approach, at least at a younger level, is that it requires students to reckon more with the real world and try to learn mathematics in a broader context. For students with poorer reading skills, this can be troublesome. For curriculum developers, it’s not always obvious what your average student of a particualr age will be familiar with, so gauging overall difficulty and appropriateness remains a challenge.
What To Do?
As I said, the drive to make mathematics engaging or applicable on a level outside the realm of art, let’s say, in a way that would make both G.H. Hardy and Paul Lockhart both rather frustrated, is unlikely to remove itself from the school system. So, while that fight is ongoing, there is still a question of how to improve some of the rougher “intuition” and approaches about how to engage students in other ways.
Focusing on communication is a good start. If students are not put in a position where they must explain their reasoning and provide feedback on other students’ reasoning, then we are failing them. The best way to get students to engage with a topic is by getting them to talk about it. Once students are talking to each other, and to the instructor, you have something to work with. It doesn’t matter very much what gets them talking.
In the end, mathematics education is full of chicken-and-egg problems. Is mathematics enough on its own, or does it need application to be engaging? We don’t know unless we fully commit to a good implementation of one or the other. Once an implementation is chosen, it’s now a question of whether teachers are being appropriately trained, and are fully sticking to the implementation or falling back on their own experiences; there’s the additional question of whether an implementation is properly budgeted for, and if sufficient time is taken to determine the results. A student doing one thing in 4th grade, then suddenly doing another the following year, may get thrown off. We can only really tell how something went if we compare a full set of students across multiple years.
It’s a tricky problem, but one that is worth working towards solving.
- 1It’s pretty likely that any reasonable educational approach is optimal for a handful of students.