Dynamic Graphing Helps Cadets Visualize Concepts
When Lt. Col. Greg Hartman wants cadets in his Calculus III class to really understand what it means to define a parabola as the curve of all points in which the distance to the focus equals the distance to a baseline called the directrix, he can just show them how it works.
That’s because VMI now has a site license to Mathematica, a powerful computer program that allows dynamic graphing of equations.
“Instead of just static pictures, you can grab a point and move it and see the results,” said Hartman, associate professor of mathematics. The site license has put Mathematica in the classroom and in the lab, as well as making it available to the entire faculty.
This definition of the parabola was the starting point for one of Hartman’s early experiments with the “manipulate” command in Mathematica. After-class speculation with Cmdr. Dan Joseph ’91, also an associate professor in the math department, about the effect a curved directrix would have on a parabola inspired a static graph that seemed to the two mathematicians not to make sense.
Further testing on software the department had been using to create static graphs would have required the creation individual graphs associated with a succession of equations that changed the parameters. With static graphs, said Hartman, “You have to do it over and over again.”
Mathematica changes all that.
“Instead of making 50 pictures, I can drag the picture around and see what happens,” he said. At first, they moved the focal point around, but then they changed their approach to move the directrix instead. “If we kept the focal point at origin [(0,0)], all the equations were simpler.”
That process of creating new equations – new parameters – was so easy, and the result so interesting, that they pursued the project until they created a parabola with a circle as a directrix and started moving that around.
“I thought, ‘That looks an awful lot like an ellipse,’” said Hartman. “I got really excited. Dan got excited too.” Parabolas, circles, and ellipses are “important” shapes because of their reflective properties – used in telescopes and flashlights, for example.
“We were creating important shapes in a new way,” he explained. An understanding of the focal points of the ellipse can be used to control the reflective qualities of the parabola and to focus light and sound. “We fell in love with Mathematica.”
Hartman said the program should be of interest to faculty in the science and engineering fields, as well as math. It is useful for demonstrating known concepts, especially ones that might be difficult for cadets encountering them for the first time to visualize.
“It’s a great way to manipulate things,” he said. A molecule can be explored, for instance, by turning its different parts on and off. The effect of seasons on distribution of daylight can be observed in action as the professor changes the date and the line between daylight and darkness shifts accordingly over a graph showing the whole planet or any part of it.
“In my calculus class, I’ve been using it to plot certain pictures. If I teach a new kind of function, a parametric function, for example, it takes them a while to understand what this concept does. … By thinking hard about how to plot it, I realize what that parametric function’s really telling me.”
Parametric functions, Hartman said, are a way of describing how points behave in space. Adding a third dimension to the mathematics the cadets have mastered in Calculus I and II can be a big leap. With Mathematica, Hartman and his students can create an instant graph and turn it to view all sides. And they can graph its parts separately and fit them into the whole to see if they’ve got the equation right.
The drawback to all this fun with math?
You have to learn the syntax. Every program speaks a slightly different language, and faculty and cadets must learn the language to create the graphs. Hartman, who gave a faculty tutorial on Mathematica in September, would say it’s worth the work.
“Often in the department we marvel about how much better we’ve understood something because we’ve created a graphic to show someone else,” said Hartman. Cadets gain this same understanding by creating the graphs for themselves in the lab.
And there’s a benefit, too, for cadets as they graduate and move into careers in mathematics.
“Ultimately what we would like to see people do is recognize what a computer can do for you and what it’s not good at so that when you get to a job and they give you whatever software they use, all you have to do is learn the syntax – you know the mathematics behind it. It’s just a matter of getting this particular program to do what I want it to do.”
For now, Hartman is committed to helping cadets use Mathematica to master their math and helping faculty learn the syntax and appreciate its value in the classroom.
“Now that we have the site license,” he concluded, “I think it’s going to grow.”
By Sherri Tombarge
IR November 2011