You are probably aware that . Sometimes when I’m tutoring I wind up teaching this to young students. Here is the story I use:
You already know that for a very simple reason.
Forget the reason for a moment, and just focus on the rule. When you multiply exponents with the same base, you can add the powers.
Evidently, is a number such that if you multiply it by itself, you get three. But that is exactly the meaning of the square root! Hence .
This is a very simple idea, but when I try it on students, it usually fails.
After going through the story, I ask what is. I’m hoping to hear “four”, but that’s not what happens. Sometimes they say it’s eight. Sometimes they say they don’t know. But the most common response is to go through the whole thing again. The student writes down
They stare it at for a while. Then they look up at me and say, “Is that right?” We discuss it a bit further to clarify. Circuitously, we stumble upon . After that we do a few more half-powers and they get it right. Then I ask what is. The student will write down
“It doesn’t work for that one,” they say. “You just get a 2/3 power, and we can’t do that.” So we talk about it some more, until after some time the student can go between roots and exponents.
Then I ask what is, but they struggle with this, too. Once that’s down we try for , but that is also impenetrable (I usually hear that it’s -6). When I suggest trying to figure it out based on the rule of exponent addition, the student feels frustrated and defeated.
It’s curious that I have such difficulty teaching this idea. It is not too complicated or too difficult, even for a young child. It is far simpler than long division and far less abstract than “set the unknown variable equal to x”. The problem is not the sophistication of the idea, but a more fundamental error in communication. When I give my little presentation, the students simply have no idea what I’m doing.
An analogy: I’m teaching someone how to lift weights (this is very hypothetical). I take a dumbbell and I start doing some bicep curls. It’s only a 5-lb dumbbell, and the motion is very simple, so I figure the guy I’m teaching will get it for sure. I hand him the weight and say, “You try.”
When I hand over the weight and the student starts yanking it up and down. He purposely mimics the way I grunt in exertion and copies my facial expressions. He remembers how I looked over my shoulder to talk to him while I demonstrated the exercise, so he looks over his shoulder when trying it out. The weight ultimately does go up and down, but only with a great deal of extraneous commotion. I straighten him out with some effort, but when we move over to the bench press we’ll repeat the whole confused process.
The problem is that before we began, my student didn’t know what weight-lifting is. He didn’t know the point is to make your muscles stronger, or the counter-intuitive idea that to make your muscles stronger, you first have to tire them out by working them hard.
Similarly, my math students watch me do this strange algebraic exercise with exponents not knowing that the goal is to discover new things. They think, instead, that I was simply teaching a new procedure, as in, “This is how you solve problems where the exponent is one half.”
This is not really a big problem. Students can learn new things; that’s what being a student is about. The problem is that students’ ineptitude at this task frustrates me. At times, when watching a student struggle with a problem, I’ve felt ironic wonder at the student’s remarkable creativity – how do they find so many unexpected ways to get everything totally wrong? I wind up concluding that the student is “stupid”, and the student leaves the lesson with only the impression that they have somehow failed at a task they never even understood.
I make these grievous errors in judgment because I assume that since I’ve seen the student handle far more complicated tasks, they should master this one right away. That is not so. ‘Simple’ isn’t ‘easy’. Computing a determinant of a 4×4 matrix isn’t simple, but my students can blaze through it. Showing that the determinant will be zero by noticing that the last row is equal to first row is very simple, but I’ve never had a student use that method.
The things we’re good at are not what’s simplest, but what’s most familiar. The converse also holds: things that are unfamiliar are difficult, even if they’re simple. I personally find it much easier to solve geometry problems using coordinates, algebra, and calculus than using Euclidean geometry, even when the Euclidean approach may be just a few lines of sketching and finding a similar triangle.
When I first noticed that students were having a hard time with problems because they required unfamiliar thinking, and not because they were too hard or because the students were bad, I tried to remedy the situation with speeches. I would talk about how interesting it is to figure out where a formula comes from. I would say over and over that no, I don’t have all the formulas memorized, because as long as I know most of it, I can figure the rest out. I would prove my point by waiting until they embarked on a difficult calculation, and then solving it quickly in my head using some trick or other, supposedly demonstrating how useful it is to be able to approach a problem many different ways. Then I would describe how it’s done. “You’ll like this thing I’m about to show you,” I would say. “It’ll make your life easier.”
This backfired. It mostly led the students to believe that I either gained some ineffable voodoo skills in college or that I am in possession of an extraordinary native intellect that they could never hope to emulate.
I still don’t know quite how to handle the “simple isn’t easy problem”. I have become far more patient when trying to push students’ boundaries, and far less ambitious. I regret the many times I compromised a student’s chance at learning and my own at equanimity by failing to recognize “simple isn’t easy” in practice. I continue to search for simpler and simpler teaching stories, but I don’t spend enough time searching for ways to make the unfamiliar territory easier to navigate. I don’t know how complicated a task that is – to figure out how to build a stepladder to a new level cognition – but I know it isn’t yet easy.