Experimenting in Geometry
It’s Not a Spectator Sport
by Laurel Cooley
“Let no one enter who does not know geometry” —Plato
Plato had an enormous respect for mathematics, particularly geometry. He believed that mathematics prepares one to deal rigorously with abstract concepts and to develop the capacity to relate these concepts with each other in a systematic way. I would argue that geometry, like all mathematics, is best understood through reason. However, I also believe that geometry is best learned through experience.
We begin our mathematics education at a very early age as we touch and move items, compare and classify them, and bend or reshape them. Based on these early experiences, our perceptions form logico-mathematical structures that help us understand the physical world. Thus, each experience we have helps to broaden our mathematical understanding. As we have more experiences, we begin to reconstruct information and classify more specifically, increasing our conceptual understanding. As we understand more, we become less dependent on the tactile, external experiences we needed when we were young. Our experiences actually become more internal as we reflect on what we are seeing or reading.
The ability adults develop to apply previously constructed knowledge and concepts to solve new problems is very useful in all fields of mathematics. Geometry, however, is a subject that is both particularly conceptual and physical, in so far as the shapes on which it is based have properties that may be measured, adjusted, compared and discovered in a very concrete way. I would argue that geometry needs to be learned through examination and discovery in order for one to develop a true appreciation for all of its subtleties. These concrete experiences are a prerequisite to building the conceptual structures necessary for the reasoning of higher-order geometry.
It has always been a rather simple task for a teacher to set up learning activities in such a way that a student could explore elementary geometric relationships. With a compass, ruler and protractor, many relationships can be documented. However, as one moves to higher orders of geometry, such as trigonometry, it becomes more difficult to experiment in a real way. The sine and cosine functions, for example, can be sketched (and should be), but it is difficult to demonstrate these functions through concrete experiments. Fortunately, with the advent of several types of user-friendly, inexpensive or even free software specifically designed for geometry, the ability to explore and discover higher-level concepts in geometry has become much easier. I am going to focus on the sine and cosine functions, since they are the bases of the other four trigonometric functions.
Many College Now mathematics courses are centered on the review of the high school curriculum. I see this extra time with students as an opportunity to provide activities that engage them in ways for which there is usually not sufficient time in the regular classes. The Math A and B exams are high pressure and do not allow teachers much flexibility during the regular school day. The review classes could be a place where students and teachers engage in group activities, experiment, explore, and take time to reflect on their work.
An example of a simple topic, sine of an angle, is included in the problem presented here: How to Buy a Ladder [PDF]. A College Now instructor might use this problem in a cooperative learning setting. Students would work in groups of two or three to solve this problem and then present their work. This real-life problem is followed by an activity that allows students to explore more abstractly the sine and cosine of an angle on the unit circle. Such activity-based learning is very important at this stage of mathematical learning; it is very motivational and shows the students that mathematics can have some very clear applications.
The Internet is also a great resource for activities that aid in experimentation in geometry. There are many sites in which one may find “applets.” Applets are dynamic models written in Java that can be manipulated to demonstrate mathematical relationships in a very coherent way. One of the biggest difficulties students have with sine, cosine, and the other trigonometric functions is understanding the relationship between their graphs and the unit circle.
A three-hour [HS Algebra 1 Regents] review class could spend part of the time in a computer lab with a structured lesson that uses these applets to explore the trigonometric functions. The applet dynamically demonstrates this relationship in a clear way.
In addition to several software packages that are also excellent for exploring geometric relationships, such as Cabri and The Geometer’s Sketchpad, one can find shareware that can be used for exploring geometric relationships. Below is a list with a short description of each shareware.
Free Software (listed alphabetically)
MathPad Graphing Scientific Calculator
MathPad is a kind of graphing scientific calculator. One may input the text and then edit the whole calculations. It runs on a Macintosh and is distributed for free. It was developed by Mark Widholm at the University of New Hampshire.
MuPAD Computer Algebra System
MuPAD is a computer algebra system (CAS) , which means that you may perform calculations either numerically or symbolically. CAS’s are very good for graphing functions and helping to visualize the mathematics. It was developed at the University of Paderborn in Germany and is distributed for free.
This is a programmable modeling environment for exploring the behaviors of decentralized systems, such as bird flocks, traffic jams, and ant colonies. It runs on a Macintosh, and is designed especially for use by students.
There are, of course, so many different ways to help students build those mental structures needed to be able to reflect on and conceptualize geometry. These are a few ideas and, perhaps if you have more, you could pass them on to me.
Other related applet sites include the following:
- an applet that actively demonstrates the relationship between the graph of the sine function and the unit circle;
- an applet that demonstrates the relationship between the input value of the sine function in degrees and the corresponding y-value as it pertains to the y-value of the unit circle;
- an applet that demonstrates the connection between the graph of the cosine function with the unit circle in degrees or radians;
- an applet that demonstrates transformations of graphs of the sine and cosine functions; and, finally,
- a site with a plethora of applets and nice examples for various topics in geometry.
Laurel Cooley earned her doctoral degree in Mathematics Education at New York University (NYU) in 1995. At NYU, Dr. Cooley was awarded a Graduate Teaching Fellowship with which she developed and taught a liberal arts mathematics course emphasizing real-life applications. She had previously completed her masters degree and secondary mathematics licensure at New Paltz State University of New York in 1990 where she was awarded the New York State Empire Challenger Fellowship. She has served in the Department of Mathematics and Computer Studies at York College since 1994.