Some of my favorite questions to ask in class involve drawing up some sort of a mathematical model for my students and asking: does this make sense? Whether matching curves to the heating and cooling laws of my morning coffee, or studying how our shadows grow relative to our speed, it’s always a nice way to play with students’ quantitative intuition.
Recently, Dan Meyer of the blog dy/dan, wrote about this idea in The Difference Between Math and Modeling with Math in Five Seconds. He shows a video of a dog setting a world record for balloon popping (yeah, I know, I’m also simultaneously thrilled that’s a thing and deeply concerned about the fact that dogs love to eat deflated balloon bits) and then asks students to model the situation. He teases out the differences between linear and non-linear models, and the difference between just doing math and interpreting the realities of a situation.
I think that people underestimate the element of intuition and human touch in mathematics. Students ask questions like, how do I know which model to use? How do I know which integration technique to use? How do I know which convergence test to use? And I always feel like a bit of a jerk (slash, I also feel a bit like a wise old sage) when I say the only possible correct answer, namely “You use the one that works.”
But you have to develop some intuition for these things.
Models are interesting because people are typically using them to describe phenomenon that they already see happening. Trying to quantify the meaning of several different variables in the context of a larger system, and to forecast where that whole system might go if the variables change. Recently, I podcasted about another great example of mathematical modeling in several variables, sharing the recent work of Steven Strogatz and some folks at the MIT Sensible City Lab about the effectiveness of ridesharing.
Teaching calculus, the closest that we get to mathematical modeling is in studying the laws of natural growth, but this is already rich territory. If you really want to scare your class, have them watch this video from NPR abut population growth, tell them about the human carrying capacity of the earth. Then have them calculate how long we’ve got left.
For the less macabre among us, Meyer also points us towards the reindeer of St. Matthew’s Island.
Someone once said to me that computers will never put mathematicians out of business, because a computer can never match our human intuition. A computer can never look at a model with our human eyes and human sense of the world and say, does this make sense? I’m not sure if I believe that, what do you think? Let me know on Twitter @extremefriday.