We’ve all got black boxes in our lives.
A black box is a little mystery that you take for granted. It’s something you use without thinking, without skepticism, without once opening the lid to peek at the workings inside. For all you know, it might be powered by wind, water, cold fusion, hamster wheels—or even some fantastic combination thereof (i.e., nuclear hamster windstorms).
I’ve got a lot of black boxes. Too many. Cars, computers, non-stick skillets—every day is a decathlon of technologies and tasks I don’t actually understand. That’s why I love math, which (ideally) has no black boxes. In math, you only use tools once you’ve studied every gear and lever so closely that you could assemble them in your sleep. Math is the realm where all boxes are transparent.
This is, in fact, the exact opposite of how most students experience math. Too often, math is an inscrutable recipe book, describing foods you’ve never sampled (and probably wouldn’t care to). You must follow each recipe to the letter, because you have no clue how the ingredients taste, or what will happen if you combine them in new ways. And when you finish cooking, you throw the meal in the garbage disposal, and begin again.
For many students, math is just a big pile of black boxes. It’s like the worst Christmas ever.
Here’s an example. When I taught Geometry, the 9th graders arrived cheerily able to recite “the distance formula”:
It’s for finding the distance between two points in the xy-plane. For example, say your points are as follows:
Then, if you’re one of my 9th-graders (now college freshmen—my babies are so old!), you plug and chug:
The process is like kissing a robot: cold, mechanical, and not particularly romantic. You’re left wondering what’s going on inside. There is, of course, a better way to learn this. You begin by connecting the two points:
To find the distance between those points means finding the length of that line segment. But how do we find that length? Well, we furnish that line with some legs:
Look at that—a right triangle! And how long are those legs? Well, the vertical leg stretches from 2 to 6, so it has length 4.
And the horizontal leg stretches from 1 to 4, so it has length 3.
What’s next? Those who know the jungle will tell you that where you find a right triangle, the Pythagorean Theorem often lurks, too.
This is more satisfying if we’ve already proved the Pythagorean Theorem, but even if we must accept the Pythagorean Theorem as a black box of its own, that’s okay. It’s still worth breaking the distance formula’s engine into components, even if we don’t fully understand each part. Either way, solve the equation that Pythagoras furnishes, and look what we have:
This more natural method agrees with the formula, and doesn’t lead us through a dense thicket of mysterious notation.
Do enough of these problems, and you start to see patterns. Consider the distance from (1,1) to (2,5):
We don’t actually need to draw the triangle. We just need to know the length of the horizontal leg and the length of the vertical leg. Then we can apply the Pythagorean Theorem. We can even do it with generic points, using variables that could stand in for any coordinates—call them (x1, y1) and (x2, y2).
We consider the horizontal leg:
Then the vertical leg:
Then we apply the Pythagorean Theorem:
Yup—the original formula.
We’ve now built the distance formula box for ourselves. It’s transparent. Should we ever find we’ve “forgotten” the formula, we can quickly redevelop it, because we know exactly what’s going on inside. It’s powered by right triangles and Pythagoras’ theorem.
When it comes to distance, this might not change the way we compute, but it’ll change the way we think. This approach hints at how we might find distances in 3D, or distances along a curve. We’ve bought a better grasp on the formula with a single down payment. With any luck, future computations will flow more swiftly and error-free than if we’d adopted the formula arbitrarily, without rhyme or reason.
Best of all, we’re learning how boxes—all boxes—are built. We’re growing ready to build little boxes of our own, which is the loftiest goal of the mathematician.
As for the other black boxes in my life—cell phones, Bank of America, all-in-one shampoo/conditioner—I should probably investigate them, too. After I generalize the distance formula to n dimensions…
Thanks for reading! (And for not asking why the pink alien’s shirtsleeves are so long.) You might also check out Stupid Graphs!, Following Recipes, and The Quadratic Formula Must Die! (or, Long Live the Quadratic Formula!).