Mathemagical Moments

XKCD is the best!

One of the best things about my job: watching 3 of my students in an intense discussion of a problem on the homework. One of my often less-motivated students holding her own arguing with the two other guys about how to solve the problem.

I realized today that the LaTeX my students are producing in their reports is so much cleaner and nicer than what I’ve seen in previous semesters. I gave them LaTeXercises at the beginning. I took a list of my pet peeves from student reports and marched this crop of students through correcting them one by one for the first homework assignment. I instructed the TAs to be draconian about grading; you either reproduced the document perfectly or points came off. That did not endear me to the students; I got complaints on the early course feedback, “I missed one space and a whole point was taken off!” Sigh. Indeed, I am so mean.

Complaining aside, I think it was worth it. For me. Maybe not for them. I’ve been preaching the gospel about composing in LaTeX or a text editor rather than composing in a word processor; this should also help. None of that would matter if no one was listening to me. Clearly quite a few someones are. I have to remember to share this with them.

At the beginning of the semester, I felt bad for not doing much math, and now we are in lots-and-lots of math mode. We just has the most mathematically brutal assignment of the semester; some cleverness, some common sense, and a whole lot of algebra.

Project 2 is about population models; we do curve fitting to three reasonably well-known population models. Exponential growth, the logistic function, and the Gompertz function. All are non-linear. We will use linear least squares to get an initial estimate for the function parameters; then we use a nonlinear least squares optimizer to improve our parameter estimates.

We go through a basic calculus lesson about linear least squares, in which we calculate the squared error, see that it has a minimum, and then take derivatives and set them equal to zero to solve. We get two gnarly equations in two unknowns. Then I walk them through the same problem formulated via linear algebra, where you have an overdetermined system with full rank. Then we discuss how to use this tactic for doing exponential functions. Later on, with one initial guesstimate, we use this for logistic and Gompertz.

We also cover the differential equation formulation of these three models, and how you get the Per unit Population Growth Rate (also called Per Capita Growth Rate) abbreviated PPGR. This is how the population grows per individual in the population per unit time. Exponential growth has a constant PPGR. If you look at the US Census numbers from 1790-1840, you will find the PPGR for the USA was about 0.3 in that time, meaning for every one person in the initial population adds 0.3 persons over the course of a decade. In more recent decades this number is much lower!

If the population in a logistic model is close to zero, the PPGR is constant and it looks like exponential growth. But in a logistic model, we take into account finite resources and space, and it has the PPGR go to zero as the population approaches the limiting population.

The Gompertz model, like the logistic model, takes into account finite resources and space, and its PPGR goes to zero as the population approaches the limiting population. What’s weird about Gompertz is that as the population goes to zero, the PPGR goes to infinity. This model hypothesizes that if there are abundant and unlimited resources, a woman can decrease her genstation period in order to increase her number of births without bound. Clearly unreasonable. Yet the Gompertz model does a good job of fitting population data!

Each student picks his or her own dataset. Any US state or city is open (everyone does the US Population), and any foreign country, state, city, province is open.

When I first put the project together, I was dutifully paying homage to the necessity of teaching curve fitting as a mathematical modeling topic. I thought this was one of the most boring projects in the universe, but we all have to suffer sometimes. I’m surprised at how much rich learning there is in this project. In order for a student to succeed with writing the results and discussing the models and the data, s/he has to know something about the history of the population s/he is working with and be able to connect that history up with what s/he sees in the data and curve fits.

The first semester I taught the course and assigned this project, one of my students had been on a mission trip to Micronesia. He wanted to work with the population of Micronesia. Fine with me. Of course, Micronesia wasn’t really in contact with Western civilization until about the turn of the 20th century. Then things got disrupted by WWII. So there wasn’t a lot of data, and what there was wasn’t very good. My student was off to the library to see if he could dig out more and better. He didn’t get much. We did get some decent curve fits in the end. He learned a lot and I encouraged him to talk about this for his final project presentation. He punctuated the mathematics with photographs he took while he was there, discussing the models, the data, and the history, all at once.

Dumb project indeed. That was a definite win.