I don’t like how applications of math are presented as a “special topic” once the theoretical has been understood. There are, admittedly, some aspects of concepts that are more thoroughly understood with background knowledge. I subscribe to an approach that bounces between applied and theoretical whenever possible.
This especially applies to differential equations. I tell my calculus students the story of my time at college when I took a course in differential equations. I spent a lot of time trying to understand how the processes of solving differential equations actually worked. It was the same way I studied multi-variable calculus the semester before, lectures for which I found fairly intuitive. The lectures for differential equations, on the other hand, were extremely technical and involved processes that were not clearly motivated by, well, anything from my professor.
This was also a time when my tolerance for pure mathematics was fairly low. As an engineering student, I needed to have an application nearby in order to push through the theoretical, otherwise my internal ‘what’s the point’ light would start flashing and I would tune out. There was very little of this in the lectures, but I pushed for understanding in my completion of problem sets and studying for the first exam.
My grade on the first exam was a 73. I was shocked. I also decided to give in to the suggestions of the sophomore students that had taken it before, who said just to memorize it all. I didn’t like it, but I did it, and my grade subsequently shot up. It was not until I took courses in system design, heat transfer, dynamics, and control systems, when I saw how differential equations actually worked and could work to understand much of the theory behind them.
Footnote: This entry is not in any way going to be an indictment of my university mathematics education (which was on the whole fantastic), a commentary on the perils of testing (which I do like discussing), or on how pure mathematics is not a rich course of study (which I further believe is NOT the case after teaching for several years and actually doing recreational mathematics on my own and with students). I don’t care to be boxed into any of those categories – the point is coming, I promise.
Last year was my first time teaching Calculus. Knowing how powerful differential equations are, I had prepared a full day where we spent looking at various differential equations and how they are used to model real phenomena from a bunch of different fields. What I found, however, was that my Calculus students reacted in much the same way as my other math students would when they sensed a day of word problems was ahead. There had to be a better way.
Here’s the plan:
Every day, I will show some physical situation with changes that can be modeled using a differential equation. No simulation allowed unless I can also show an actual apparatus that the students can visibly see, feel, hear, or otherwise sense changing over time, distance, etc. There’s something really powerful about generating data real-time – especially when it is related to something students can sense themselves.
My progress thus far:
Day 1: Newton’s Law of Cooling
I started class by asking a student to bring a mug of hot water from the water dispenser around the corner. When he returned, I tossed in a temperature sensor that I had connected to a National Instruments myDAQ board, and without much other commentary, started some review of antiderivatives. Close to the end, I stopped the LabVIEW program from running, and showed the resulting lovely graph of Temperature vs. time.
This resulted in all sorts of questions and discussions- when was the temperature changing the fastest with respect to time? What would happen to the derivative of temperature with respect to time as time went on? What is the physical meaning of this?
One of the students noted that this happened because the temperature of the cup was higher than the temperature of the room. This started a mini-discussion about situations where the temperature of the cup would rise. This all motivated the idea behind Newton’s law of cooling beautifully.
Day 2: Newton’s 2nd Law
The physics students weren’t impressed by this one. Part of the homework assignment from the previous day was to research and post information on the class wiki about a differential equation that (genuinely) held some meaning or interest for them. A couple of the students independently put on Newton’s 2nd, and I accepted it since they did it in slightly different ways. I then showed the students this apparatus (again, not a surprise to the physics students).
This time though, the focus was on the dynamics of an object on a spring. Giving the mass a nudge downward starting it oscillating nicely.
This led us to figure out what the forces acting on it were, namely gravity, the spring force, and possibly friction. This led to the differential equation form of Newton’s 2nd law. I did make available a Processing sketch I put together that contained the differential equation so they could see that this really was what governed the motion of the object.
We didn’t talk too much about the specifics of the program, as lines of computer code thrown at students tend to result in glassy eyes fairly quickly without proper preparation. We will look at programming again later on in the year though, so I’m not too upset that we didn’t talk about the details.
My hope is to include some lights attached to capacitors and resistors to show an RC circuit, a draining tank of water, deflection of a cantilevered beam, maybe even monitoring an oxygen sensor with a candle in a closed container. Part of me also wants to do a bar heated at one end, maybe a bit trickier since it is a partial differential equation, but I think it might also serve to get students thinking about how temperature might vary as functions of time and position. I don’t know what else, but I’m excited about the possibilities.
What are your favorite demonstrations of differential equations in action?