Our first approach in preparing a new lesson is often to ask, “Where does this skill apply in the world of work or in the world outside the classroom?” There may well be a great answer for some skills, but this strategy generalizes very poorly to lots of mathematics. So instead, I try first to ask myself, “Why did we invent this skill? How does this skill resolve the limits of older skills? If this skill is aspirin, then what is the headache and how do I create it?”
Two examples from my recent past.
Combining Like Terms
Why did we invent the skill of combining like terms in an expression? Why not leave the terms uncombined? Maybe the terms are fine! Why disturb the terms?
One reason to combine like terms is that it’s easier to perform operations on the terms when they’re combined. So let’s put students in a place to experience that use:
Evaluate for x = -5:
3x + 5 + 2x2 – 7 + 8x – 5x2 – 11x + 4 – 5x + 3x2 + 4 + 3x – 6 + 2x + x2
Put it on an opener. The expression simplifies to x2, giving students an enormous incentive to learn to combine like terms before evaluating.
[I’m grateful to Annie Forest for bringing the example to mind. She also adds a context, if that’s what you’re into.]
When students first learn to graph points, the parentheses are the first convention they throw out the window. And it’s hard to blame them. If I told you to graph the point 2, 5, would you need the parentheses to know the point I’m talking about? No.
So why did mathematicians invent parentheses? What purpose do they serve, assuming that purpose isn’t “tormenting middle school students thousands of years in the future.”
It turns out that, while it’s very easy to graph a single point with or without parentheses, graphing lots of points becomes very difficult without the parentheses. So let’s put students in a place to experience that need:
Graph the coordinates:
-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3
You can’t even easily tell if there are an even number of numbers!
[My thanks to various workshop participants for helping me understand this.]
The need for combining like terms is Harel’s need for computation and the need for parentheses is Harel’s need for communication. I can’t recommend his paper enough in which he outlines five needs for all of mathematics.
My point isn’t that we should avoid real-world or job-world applications of mathematics. My point is that for some mathematics that is actually impossible. But that doesn’t mean the mathematics was invented arbitrarily or for no reason or for malicious reasons. There was a need.
Math sometimes feels purposeless to students, a bunch of rules invented by people who wanted to make children miserable thousands of years in the future. We can put students in a place to experience those purposes instead.
We explored these ideas in a summer series.