Joseph A. Montagna
The objectives of this unit are to expose students of 7th and 8th grades to the beginnings of geometry and one man who is credited with a large part of its early development, Pythagoras, and to teach students the basic principles of geometry. The life of Pythagoras is an interesting story, for it gives us a clearer idea of what life could have been like in the ancient world, and what education was like, as well. The history behind the mathematics should motivate students to learn geometry, but it is also important for them to see that each discipline does not stand isolated.
Geometry is one of mankind’s tools that has become indispensable. Our world today would be vastly altered had geometry not been developed. Modern society depends on the techniques and methods of geometry to build, to navigate, to design, and to calculate the vast distances of outer space. All around us we can observe evidence of geometry put to practical use; the new skyscraper reaching for the heavens, the jet airliner or ship that arrives precisely at its destination, the tunnels dug through mountains from both sides that meet precisely where they should, or the bridges that span our largest rivers. Geometry is utilized by engineers, builders, astronomers and even do-it-yourselfers.
There is also a great deal of evidence of geometry in nature. Ice crystals that are always hexagonal, the symmetry of living things, the orderly movement of the planets and other heavenly bodies, a snail’s spiral shell, or the economical use of simple shapes are each examples of nature’s geometry. All around us we can observe some geometrical principles, which is probably what started the ancients on their way to developing this field of study. Nature appears to have a plan, and man seeks to unravel its mysteries.
I think that this unit can be interesting to students. The narrative portion is by no means complete. Much is left to the teacher to fill in on his/her own. The lessons that are included may be altered in any way the teacher deems appropriate. It would have been impossible for me to offer all of the ways that the material in this unit could be taught. This unit represents one way to teach it. The creative teacher will use this unit as a resource and devise other, perhaps better, ways to teach the material. The important thing is to give students more than aspects of it.
The history of mathematics is inspiring. The subject is typically human, that is, it is a body of knowledge that was fathered by man’s curiosity. Nature was one obvious topic that gave the ancients their mental workouts. Look at nature. Observe the perfection in the arrangement of the petals of a daisy, or the seeds that develop in a sunflower. Observe the web of a spider, with its delicate pattern that is perfectly symmetrical. Within the enormous field of the study of natural occurrences man made some mathematical discoveries which, in turn, became stepping stones to further discoveries. Mathematics of this period was also wrapped up in ancient mystical societies whose teachings remained secret except to a small number of their initiates. Numbers had special qualities to those who believed in their magical powers. Even in modern times certain numbers conjure up thoughts of demons, devils, omens and good fortune. Numbers are also referred to in mythology. Aeschylus in
The Prometheus Bound
(line 459), has Prometheus describe, “ . . . number . . . which is the most ingenious of all devices,” as one of his gifts to mankind, alongside fire and writing. Whether it was religion, superstition, the search for nature’s secrets, or the mere practicality of a number system which served as the catalyst, man did create the field of mathematics and will continue to develop it.
The Greeks built upon a solid foundation that was laid by the Egyptians, Babylonians and other ancient civilizations. The Egyptians were the premier architects and builders of ancient civilizations. The pyramids and the buildings they created were tremendous in size, complex in their design and built to last forever. Three thousand years, or so, is hardly forever, yet their existence for so long does make the point. The Egyptians also laid the foundations of surveying and measuring, a skill made necessary by the annual flooding of the plain alongside the Nile River. Every spring the Nile floodwaters would erase all semblance of the markings laid out to distinguish boundaries. They devised a system of measurement that surpassed any other. Indeed, the term geometry is derived from the Greek “geometria”, measurement of the earth. It is fair to presume that the Egyptians inspired the coining of this term. Babylonian and Nordic civilizations of ancient times did much in the way of astronomical observation which, in turn, led to the development of dividing time into periods and the prediction of astronomical occurrences. All of this is to point out that the Greeks did not invent mathematics. What they did do is begin to formalize and compile much of the valuable work in the field that preceded them.
Why is it that the Greeks were able to do what no other civilization before them was able to do? I believe that an answer can be found if one takes into account the nature of the Greek culture within the context of the world as it then existed. The Greeks were relentless in their search for “truth”, be it in art, politics, or philosophy. Mathematics fit quite nicely into their culture. It was neat. Mathematics was perfection. It made sense. In addition to this, the world of the ancient Greeks existed literally at the edge of knowledge. Situated relatively close to Egypt and Mesopotamia, the Greeks stood on the shoulders of these two giants of the ancient world. The Greeks learned from the Egyptians and the Babylonians and made new discoveries of their own.
It is important for the teacher to be aware of this fact, that mathematics did not begin with the Greeks; that they formalized and refined what developed in earlier periods. How much of this history up to the teacher. However, I would recommend that, at the very least, a capsulized history be presented to aid the students in their understanding of the subject. A few titles for this purpose are recommended in the bibliography.