In undergrad I never really cared about whether math had applications; it was beautiful and that was incentive enough. I would teach my students to value aesthetics of proofs and for them, that too would be enough. And then came Pre-Algebra in a 95% free and reduced lunch school in rural Northern Michigan. My students hated math. There was no beauty, only previous failure. All of a sudden, I had to think of some way to motivate my students to engage with the subject. For several years I floundered; Jimmy shovels at a rate of 8 ft every minute, if the driveway is... You know what? It doesn't really matter how long Jimmy is out there, because he's going to stay out there until he finishes shoveling the driveway. Trite doesn't work. Does math matter or not? If it's just some intellectual exercise I'm forcing on my students because politicians and textbook makers think its a good idea then we should change the good idea.
There's a long road to my infatuation with problem-based learning and using experiments in my classroom, but developing lessons for the Kuyers Institute was pivotal. Can teaching a math lesson be about more than just the math? Can we talk about social inequality using numbers? What role does math play in convincing people of the rightness or wrongness of an opinion in our society?
One of the few times I've been successful with such a lesson is in studying pay inequality between men and women in the U.S.A. I've done this lesson in both Algebra 1 and Algebra 2. I think the older students are better able to delve into the underlying factors (not surprising).
Then we look for trends in the tables. Almost always students say that men earn more than women and that as the years increase people make more. Then we see if the graphs can help us be more specific (page 2 of the doc). What do you see? Can you use math to describe that? Students suggest exponential curves. We try them and they don't work. Then we talk about cutting off parts of the data. If you cut the data at 1975-ish, it becomes approximately linear.
After laying the foundation for the lesson, it's group time. In groups of 3 or 4, students start to work through the questions in the lesson The lesson is divided up into 3 tasks to make for natural discussion breaks as a class for formalization of content. Here's a quick overview:
In Task 1 students find equations for both men and women incomes using lines of best fit and two points. Students must also explain what each aspect of the function stands for and whether it is appropriate to predict backwards using the functions.
Task 2 tackles linear regression, correlation coefficients and whether our equations have any merit when comparing with current data (I have 2010 incomes in the lesson).
Task 3 takes a brief foray into rational functions (cents on the dollar). I couldn't help bring up a conceptual problem by asking students to create a line of best fit for this function as well (it looks linear). However, this yields drastically different results than the findings from the previous tasks. Lovely discussion about why this doesn't work.
All in all, it's an interesting lesson that leads to great conversations about the structure of our society, concepts like 'justice' and how we use math to 'prove' our points.
I'd be interested in your thoughts and/or recommendations for improving it.