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Newsletter Volume 2 |
Fall 1997 |
Insights into the Integration of Mathematics and Physics Concepts
Ted CorleyDepartment of Mathematics
Glendale Community College
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Newsletter Volume 2 |
Fall 1997 |
How do you teach a particular math concept within a science context? How do you embed good mathematics instruction into science explorations? The ACEPT Physics-Mathematics Workshop during the summer of 1997 arrived at some great answers to these questions. The meshing of a number of critical elements-great leadership and organization, clear goals and expectations, dedicated participants with broad backgrounds, quality technology, and financial and moral support-resulted in a workshop that produced usable, quality curriculum materials that successfully tie mathematics and the sciences.
As a mathematics instructor, the mathematical "function" focus set by the workshop organizers made sense. Dealing with the physics focus of "motion" and "forces" was someone else's responsibility. How wrong I was. To my surprise, I had to become a learner again-to learn or re-learn many physics concepts. And in this re-education process I discovered the danger of teaching, knowing, or doing mathematics in isolation. That danger emerged, very obviously, during the development of my group's Module-The Coffee Filter Drop.
Amanda Beasley, REU student, investigates how coffee filters fall and how she can maximize their speeds.
Our module was designed to help students explore physical forces and the distance versus time function of a falling object. Having "solved" many college algebra and calculus problems that were based on dropping or throwing a ball, I immediately began with the assumption that all the math textbooks encourage students to make; namely, that air resistance is insignificant and the distance versus time function is always parabolic. This assumption is made in math courses before second semester calculus because the math concept being studied is the parabola. Discounting air resistance ensures that the physical phenomenon of dropping an object models the parabolic concept. It was, therefore, necessary for me to be mentally "shaken." I had to quit operating within a contrived environment that would support an abstract concept and move to a real environment-that of falling objects on Earth, and the air resistance that affects them.
My group started by talking about mass and terminal velocity as the focus for our module. But terminal velocity would never exist within my contrived parabolic model of the motion. The reality of air resistance came into focus for me when I saw the graph of distance versus time created by the Calculator Based Laboratory (CBL) and TI-83, and knew I was not looking at a parabola. In an attempt to respond to this physical reality, I immediately jumped to a different algebra concept-piecewise functions. The graph of distance vs. time for the freely falling coffee filter certainly appeared to start out parabolic, but then became linear. Because the physics people in my group were looking for any good function concept to tie into their science exploration, they readily accepted my new assumption, namely, that a piecewise function could be fit to the data, without noticing its limitations. Not immediately anyway.
Will 3 coffee filters nested together fall faster than one coffee filter?
It was when we started using the graphing calculator to fit curves to the different "pieces" of the graph that problems arose with my piecewise assumption. Fitting a line to the tail of the graph was what the physics people wanted to do. The slope of that line translates as the terminal velocity-a value that they wanted students to draw from the data. The physics folks started to dig in their heels, however, when I fit a parabola to the first part of the graph. They argued, "Air resistance begins as soon as the coffee filter is dropped so the distance vs. time function is never really a parabola."
After working through the physics formulas and solving the related differential equations, I became convinced that neither "piece" of the graph could be fit, exactly, with any piece of a piecewise function. At that point I realized something else that was rather interesting. Physics teachers use approximations when it suits them--but they also keep the purity of the science when it suits them. That's exactly what we do in mathematics! Sometimes assumptions and estimates are just fine, but at other times we insist on purity. Deciding when to be pure and when to "fudge" was causing difficulty in communication and integration between the two disciplines. Interesting discoveries such as these led the participating teachers and faculty to some very creative and flexible thinking.
Ted Corley, Glendale Community College, tests his calculations. Will the egg get crushed or barely tapped after the string is cut?
The final product of my group's effort-a module that successfully integrates the two disciplines in a study of falling objects-is one that I will most definitely use. That module, along with the eleven others developed during the ACEPT Physics-Math Workshop, offer practical and thought-provoking connections between the sciences and mathematics.
Although I am not scheduled to teach a college algebra course this fall, I do have my department chair's permission to present some of the workshop materials to the rest of the math faculty at Glendale Community College who will be teaching college algebra and science courses. My department chair has also identified a teacher and a section of college algebra where I will team-teach some lessons using the ACEPT modules. When I do teach college algebra again the following semester in spring 1998, I am looking forward to using the modules created in this very successful workshop.
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