The Derivative Project:

Using Appropriate Technology to Teach the Concept of Derivative Via a Graphical Approach

Lisa Denise Murphy

I developed an instructional unit that uses a motion sensor and related software to introduce the concept of a derivative. I tested this unit in a first-semester calculus class, with very good results. I am now engaged in expanding this project in two ways: I want to teach a wider variety of calculus concepts, and I am testing a version of this project that can be used by students without access to a motion sensor.

The first version of this project uses a motion sensor and related software, available from Vernier Software. As the students walk back and forth in front of the sensor, they can watch the computer create, in real time, a graph of their distance from the sensor as a function of time. This helps the students understand how the graph represents motion. In particular, it helps them get past the popular "road map" (or "graph-as-picture") misconception, in which the graph is interpreted as a map, with the vertical axis representing north/south motion and the horizontal axis representing east/west motion. After working with the motion sensor for a while, all of the students in my study appeared to have a very strong understanding of the horizontal axis as representing time.

Students work through a packet of questions and problems, using the motion sensors to test their hypotheses. The goal is to develop an understanding of how motion is represented by the graph. Students should be able to identify from the graph whether the person is moving toward or away from the sensor, moving quickly or slowly, speeding up or slowing down, or stopping or changing direction. When the students have made a firm mental connection between the speed and direction of the motion and the slope of the graph, they are led to quantify this in various ways and then move on to a more formal definition of the derivative.

A draft of the lesson packet is available, although I have had a little trouble getting it on the web. Some of the grid lines in the graphs did not show up. I'll try to get a better version up soon.

The success of the first version prompted me to try to expand the applicability of the project. (Many thanks to my advisor, Kenneth J. Travers, for a good idea here.) I have written Java applet which can replace the motion sensor, so that students with access to Macintosh or PC computers can use the instructional unit without any extra equipment. The applet allows the students to create a graph by using a mouse to drag a stick figure back and forth across the top of the screen, instead of walking back and forth in front of the motion sensor. The applet creates graphs similar to those created by the motion sensor software. It remains to be seen whether students will get the same understanding from graphs generated by moving a mouse as they get from graphs generated by moving themselves, but I am optimistic. I have used the Java applet with a lesson packet very similar to the packet I used with the motion sensors, with apparently similar results.

You can try the Java program if you can get it to work--be sure to read the hints about how to get around its quirks. So far, I have been able to get it to work only on newer PC browsers and on Internet Explorer for the Mac. It hates old browsers and Netscape on Macs. Apparently the highly-touted platform independence of Java is not quite a reality.

As part of a pilot study for my dissertation project, I used the lesson packet to teach two calculus classes at a community college last fall. One section used motion sensors, and the other used the Java applet. I have video and audio tapes of the students at work, and pre- and post-tests for most students. The result of this study are reported in the paper I wrote for my oral preliminary exam. (Yes, I know it is called "oral," but you still have to write a paper.) I'll try to get the prelim paper up soon.

The pilot study had a few limitations, notably small sample size, a diverse population, and the need to use intact classes. Community college students are very diverse in their ages, occupations, life situations, academic backgrounds, and prior performance in mathematics. One class met during the day and the other in the evening, so of course the two classes attracted different types of students, making any comparison of their performance questionable. There were twelve students in each of the two classes, but only a total of 17 students participated in all phases of the study.

The main study gets around these problems by using volunteers from first-semester calculus classes at a major university. Nearly all of the university students are recent high school graduates, around 18 to 20 years of age, and full-time students. Some similarity in academic preparation is enforced by the university admission requirements. This results in a population that is much more homogeneous than the community college population. By taking volunteers and meeting with them outside of class, I avoid the problems inherent in using intact classes. In September of 1999, 30 students completed all phases of participation in the study. I will be repeating it in February of 2000 with a new group. Progress in that study will be reported on my dissertation page.

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This page last revised January 19, 2000.