By Lizza Vachon
There are layers to the world, I tell my eighth graders.
Here at school we get a chance to uncover some of those layers. We look at the world and peel back a layer of art, studying the quirk in Mona Lisa’s smile. We lift an edge of history, and hear the roar of a populace in revolution. In another corner, another curl, and poetry spills out. Each layer can be looked at independently, but there’s something unique that comes from looking at the whole. In biology, this is called emergence; it simply means that the whole is more than the sum of its parts.
Equations of numbers alone are number patterns, relationships between variables and constants. As much beauty as there might be in those numbers and patterns, it is a static, meaningless beauty – a beauty of order without significance. When we talk about equations in math – linear, exponential, quadratic – I think it’s important for students to know why, as human beings, we’re so interested in these particular patterns. Which is where my prospector’s hammer comes in.
For the first part of the year, my students have been debating the definition of an equation, trying to create them from word problems and given data, and graphing them. As an end of unit project, I asked them to put all of these skills together, and, given a physics equation, create a demonstration to show others what their graphs represented. Each group – Force = mass/area, Frequency = 1/time, Density = mass/volume, and Weight=mass (gravity) – got to work reading provided material and talking among themselves to understand their equation and think about where and how it might appear in the world.
A prospector’s hammer has two ends, one with a square head, and the other with a pointed tip. Can you guess which group used it?
Another group dropped a squishy dodge-ball (with permission of course) from the second story window to show their equation . . .
Yet another group brought a flute along . . .
And the last group used some materials from the lab to show what happens when density is kept constant . . .
There’s only so much telling and talking about truth to students; good teaching is when they experience it for themselves. Here are a few quotes from the experience:
“But if we keep mass the same and change weight, gravity changes! How can gravity change!?”
“Okay, so what is a Newton? I mean, if we want to keep force the same, how many Newtons should we use?”
“So what exactly is density? I mean, if something weighs more it’s bigger, right?” (Several examples offered) “Ooohhh, yeah, I get it!”
Student: “Ms. V! Our equation isn’t working! The graph is all weird and we can’t figure out what we did wrong!”
Ms. V [after some looking at and fiddling with numbers]: “Aha! I know what happened! Would mass ever be negative?”
Student: “Oh! Ha ha! No!”
“How can we show frequency though? Do you think it would be okay if we just show what happens [when frequency increases or decreases]? I mean, you can’t exactly see it directly.”
And finally, from the reflection I asked my students to do:
“I didn’t realize how useful equations are.”
“I definitely understand how this [equation] works and what happens when you change each part.”
“I learned that the world is more complex than I thought.”
There are layers to the world, I tell my eighth grade students, and each of them, with a wonderful curiosity and a set determination, use all their experiences to observe, understand, and create. They pull from art, history, science, language arts, and math. What emerges is a world alive with equations – leaves falling, stars spinning, balls on the playground arcing (hopefully away from windows), there to illuminate our understanding of the world.