In Where Mathematics Comes From, cognitive scientists George Lakoff and Rafael Nuñez assert that our understanding of abstract mathematical concepts relies upon our sensory-motor experiences:
“For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in the sensory-motor system. The mechanism by which the abstract is comprehended in terms of the concrete is called conceptual metaphor. Mathematical thought also makes use of conceptual metaphor, as when we conceptualize numbers as points on a line.” (p. 5)
If we understand numbers as points on a line, then we can conceptualize numeric variables as moveable points on a line, and functions as a particular kind of coordinated movement of two points on two lines. Just as the coordinated movement of two people often takes the form of a dance, so too we can view the coordinated movement of two points as a function dance, a dance in which the independent point’s location on its line determines the dependent point’s location on its line.
Try out the sine dance with the Web Sketchpad model below. When you press the Go! button, three seconds later independent point x will move at a constant speed to the right along its line. Your job is to drag the bunny up and down so that the red dependent point at the tip of the bunny’s arrow is in the right place to represent sin(x). Fortunately the blue point moves up and down to show you where you should be.
Try it out now:
As with a real dance, it takes some practice to get it right. Make it faster (and harder) by moving point challenge to the right (speeding up the dance) or move it to the left for an easier dance. You can also edit the parameters to change the the nature of the dance.
What did you find challenging about doing the sine dance? Did it give you a different sense of the relative rate of change—the way the dependent variable sped up and slowed down—than you might have expected? Doing such dances can provide novel kinetic insights as you realize how fast (or slow) y moves as x travels along its domain. You get to experience the function’s rate of change through your own bodily motion.
This sketch, and other “function dance” sketches, are still being developed into finished activities. I’d really appreciate your comments and suggestions, especially if you’re willing to try them out with your students. Please email me: stek at geometricfunctions dot com.
Daniel Scher and I will be presenting this and other thought-provoking activities at the NCTM Annual Meeting next week in New Orleans. If you’re interested, please come to session #638 (Two Birds, One Stone: Transformations, Functions, and the Common Core) on Saturday at 11 am.