Jumps in Abstraction
If I ask an adult to tell me what 3-5 is, there’s a good chance that they would tell me the answer is 2 without much thought. This kind of arithmetic is simple to us, since we’ve had to do it over and over again through elementary and secondary education. Even if we haven’t used mathematics in a long time, these questions are straightforward.
But it wasn’t always this easy. Remember, we aren’t born with this innate sense of negative numbers. In fact, we wait until the end of elementary school before seeing negative numbers. Before, this, if you ask a young student what 3-5 is, chances are that they will answer “2” (because they figured you meant it the other way), or if you explain that you are indeed talking about 3-5, they will tell you it isn’t possible.
You and I can both imagine what comes before the number zero. But to the young student, it’s not that they don’t have the imagination. They don’t even know that this extra richness is there. This is an important insight, because it signals to us that we get comfortable within our usual mathematical spaces. Consequently, we can become blind to the generalizations that are possible, just like the young student who can’t imagine that there is even such a thing as a negative number.
Going up a few levels in education, a lot of secondary school concerns itself with geometry. Students learn about perimeter, then area, and then volume. But the kinds of topics that are covered during these explorations are limited in scope. Students learn about regular solids like cubes, prisms, pyramids, cylinders, cones, and spheres. These are nice, because they allow teachers to combine them to form more complex solids. The task for the student then becomes figuring out how to separate a large, composite object into a bunch of smaller objects and add their corresponding volumes.
This is a great exercise for a student. However, by focusing on these core solids and only dealing with different combinations of them, the students don’t get to see the richness beyond those solids. The world isn’t only made up of those solids! We have plenty of other interesting forms that we can find in nature, from a sprawling tree to a curvy egg. These aren’t the simple solids that students are used to. Furthermore, what about the amazing objects that mathematicians have come up with, such as Torricelli’s Trumpet? I can just imagine the interest that would be generated when showing students how this particular object has an infinite surface area, yet somehow has a finite volume! Of course, one would have to work within the constraints of limited calculus knowledge, but I’m certain that this could work.
Sometimes, we have to get out of the thicket of working through particular problems, and figuring out where we are on our mathematical journey. By doing this, students get better at understanding the context that surrounds what they are learning, rather than simply keeping their heads down and working on problem after problem. That strategy may be “productive”, but it will ultimately hamper students’ awareness about the wider mathematical world.
I’m not advocating here for a radical change in how one teaches (that’s a different story). What I’m arguing for here is to give students a broader idea of what the results they see mean. It can be as simple as sowing the seeds for deeper connections for students to think about. The goal should be to make sure the students know that there is always more to uncover if they so choose. I absolutely don’t want students thinking that they have learned all that exists in mathematics by the time they are done secondary school.
It’s also good to note that this isn’t only important within secondary and elementary schools. This is something that should be done in all levels. Throughout this past semester, I’ve been pushed to consider many more mathematical spaces in my abstract algebra class. While many spaces take cues from spaces like the integers, the rational numbers, or the real numbers, there are many other spaces which have similar attributes, and can be studied beside our usual settings. The essence of algebra is preserved, even though we don’t know which explicit space we are talking about. This is both neat and difficult to wrap one’s mind around. It means that you have to step away from the comfort of the familiar spaces and explore new ones. It means opening up your perspective to a vast new world of possibilities.
I want students to get a sense of that while studying, to show them that there is always more to learn, if they are so inclined. Let’s do our best to get students out of their comfort zones, and be surprised and delighted every once in a while.