Teaching Big Ideas in Mathematics: Equivalence
Our mathematics relies on the idea that the difference between 1 and 2 is the same as the difference between 99 and 100. This idea feels intuitive to most of us, but is actually counter-intuitive to the way we evolved. You see, from an exponential standpoint the difference between 1 and 2 is huge, the latter is double the number of fruit trees or saber-tooth tigers. That's an important difference to an evolving species. Less important is whether or not you have found a grove with 99 fruit trees vs one with 100.
The evolutionary move towards arithmetic sequences (1, 2, 3, 4...) as opposed to geometric (1, 2, 4, 8, 16..) enables the math and science we have today. To get there we have had to trick our minds into a new way of understanding quantity. Radiolab has a great episode on this where the cite the work of Stanislas Dehaene and Susan Carey to explain "the trick" of our number system:
"This is a trick...the point is, once you have that trick you build on that. And that opens the whole world of mathematics to you and you can build buildings and launch rockets into space."
In other words, very big ideas lie behind even very simple mathematical relationships.
One problem with math curriculum developed under pressure to cram in as many topics as possible, is that they often fail to adequately explore these big ideas. Instead, jumping right to the trick.
This is a mistake. Randall Charles has a great article outlining some of the big ideas covered in K-12 math which should become a greater focus of our course planning.
I want to take some time this spring to outline a few Big Ideas that my students and I encounter in our work together and explain performance tasks that students complete to show understanding of them.
This is the first entry in this series which I'll come back to a few times over the course of the next few months.
Last week my students created Alexander Calder-style hanging mobiles while grappling with this Big Idea from Charles: "EQUIVALENCE: Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value."
Instead of just applying algorithms to solve a page of equations for their own sake, students studied torque and used the understanding they developed to create something beautiful which shows an equivalent relationship. (Note: Students also did work with equivalence properties — ie divide both sides of 2x = 34 to isolate the variable. SolveMe Mobiles provided an opportunity for them to practice this kind of thinking while remaining in the mobile context.)
Here's a few quotes from them:
"My mobile is in equilibrium because all the numbers balance each out depending on the distance certain objects are from the fulcrum"
"the goal is to have the torques of each arm balanced by varying mass and distance, resulting in zero net torque and no rotation"
How do you foster understanding of equivalence as something big and important with your students? What other big mathematical ideas are your students working to understand?
Photos by author of student art work "Peace Doves" and "My Room"