a/k/a Worshipping The Real World
Here are three e-mails I received from three different people over the last three months. Spot the common theme.
My co-teacher and I were puzzling over what kind of problem would create an intellectual need for systems. Do you have anything you could send, by chance?
We are planning to launch a unit on systems of equations in early January (after December break) and wanted to try out your approach to create an intellectual "need."
Showing two straight lines on a piece of graph paper and finding points of intersection has very little significance to most people. I'm looking for a real-world problem that has an answer that is not self-evident, but which requires a little thinking and finding the intersections and is infinitely more productive and satisfying and will stay with them for the rest of their lives. That is what I am looking for.
I receive these questions on Twitter also. I find them almost impossible to answer because I don't know what your class worships.
Here's what I'm talking about:
You start class by asking your students to write down two numbers that add to ten. They do. Most likely a bunch of positive integers result.
Then you ask them to write down two numbers that subtract and get ten. They do.
Then you ask them to write down two numbers that do both at the exact. same. time. "Is that even possible?" you ask.
Many of them think that's totally impossible. You can't take the same two numbers and get the same output with two operations that are natural enemies of each other. They'd maybe never phrase it that way but the whole setup seems totally screwy and counterintuitive.
Then someone finds the pair and it seems obvious in hindsight to most students. We've been puzzled and now unpuzzled. Then you ask, "Is that the only pair that works?" knowing full well it is, and the class is puzzled again.
You define systems of equations as "finding numbers that make statements true" and you spend the next week on statements that have only one solution, that have infinite solutions, and the disagreeable sort that don't have any solutions.
Students learn to identify the kind of scenario they're looking at and how to find its solutions quickly (if any exist) using strong new tools you offer them over the unit.
The same lesson plays out but this time, after we've determined the pair of numbers that solve the system, a student pipes up and asks, "When will we ever use this in the real world."
Worshipping the Real World
David Foster Wallace wrote about worship – the secular kind, the kind that applies to everybody, not just the devout, the kind that applies especially to us teachers in here:
If you worship money and things — if they are where you tap real meaning in life — then you will never have enough. Never feel you have enough.
If your students worship grades, they won't complete assignments without knowing how many points it's worth. If they worship stickers and candy, they won't work without the promise of those prizes.
If you say a prayer to the "real world" every time you sit down to plan your math lessons, you and your students will never have enough real world, never feel you have enough connection to jobs and solar panels and trains leaving Chicago and things made of stuff.
If you instead say a prayer to the electric sensation of being puzzled and the catharsis that comes from being unpuzzled, you will never get enough of being puzzled and unpuzzled.
The first prayer limits me. The first prayer means my students will only be interested in something like The Slow Forty – a real world application of systems. The second prayer means my students will be interested in The Slow Forty (because it's puzzling) but also the puzzling moments that arise when we throw numbers, symbols, and shapes against each other in interesting ways.
The second prayer expands me. Interested people grow more interested. Silvia writes, "Interest is self-propelling. It motivates people to learn thereby giving them the knowledge needed to be interested" (2008, p. 59). Once you give your students the experience of becoming puzzled and unpuzzled by numbers, shapes, and variables, they're more likely to be puzzled by numbers, shapes, and variables later. That's fortunate! Because some territories in mathematics are populated exclusively by numbers, shapes, and variables, in which cases your first prayer will be in vain.
That's why I can't tell you what to teach on Monday. Your classroom culture will beat any curriculum I can recommend. I need to know what you and your students worship first.
Silvia, PJ. (2008). Interest — the curious emotion. Current Directions in Psychological Science, 17(1), 57–60. doi:10.1111/j.1467-8721.2008.00548.x