Department of Mathematics, LSU Baton Rouge
Math 4005: Geometry
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Euclid Proposition 35, Book I. | From an 1830 reference book. | Hartshorne on Euclid. | Moise on area. | Lebesgue on area.
1963 High School Text. | 1966 High School Text. | 1972 High School Text. | Discovering Geometry. (1997). | UCSMP. (1997). |
Glencoe Geometry. (1998). | NCTM Standards 2000. | Math Connections. (2000). | Connected Geometry. (2000). | Connected Mathematics. (2004).
We begin our study of geometry by investigating a very basic piece of knowledge: the formula for the area of a triangle.
This formula might appear in the elementary-school curriculum as early as 5th grade. It is very common in middle-school math, where students not only learn to use the formula, but also are often expected to understand the reasons behind it.
The Sacramento City Unified School District Standards for 5th grade--which I site only to demonstrate that someone, somewhere believes this topic belongs in the 5th grade--includes the suggestion that one should:
Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram). (See: http://www.scusd.edu/elem_curriculum/docs_pdf/math_5_standards.PDF.)
Other examples of the area formula appearing in late elementary school math include the following lessons: example 1, example 2. Again, I include these references only to illustrate what one may encounter, not to suggest that they are positive examples, a matter on which I make no judgment. Numerous examples of the topic in middle-school math appear in the readings.
My goal is to ask this class to look deeply into what we know about the triangle area formula, how it's connected with the other things we know, what it is that ties the various parts of what we know together into a whole, what makes our knowledge trustworthy, solid and reliable, how and why it becomes useful, what limits it and what special tools we have made or acquired that assist us in knowing. I have several reasons. First, simply looking deeply at whate we know is, I think, a worthy thing. Second, examining the structure of our own knowledge will be preparation for reading Euclid, whom we will begin studying shortly. Euclid and his contemporaries developed a particular way of organizing and presenting mathematical knowledge that all mathematicians since Euclid have stuck to. (I actually believe that we all have a natural inclination in the same direction as Euclid, and I hope to be able to convince you of this.)
The triangle area formula is not a stand-alone simple idea. If it's examined with great care, it is complex and subtle. It rests on other ideas, and it's part of several idea-stories that start at quite distant places. When we examine the way that this formula sits among the things we know, sorting through and ordering our ideas, we find some that basic and others that are built upon these basic ones. Interestingl;y, the ideas that seem simplest before careful analysis do not always retain the appearance of simplicity. The details sometimes sometimes fall into unexpected patterns and force us into what may even seem unnatural contortions. To escape from those contortions we need to pull some surprising and unexpected moves.
When we talk about understanding the area formula (as opposed to merely knowing it in the sense of being able to quote it), we envision a person who can move about in a web of related ideas with ease, and we expect that person to have the ability to talk explicitly about the connections in that web and to be able to guide other people through them. Our knowledge soemthing we tend and care for, much as we tend and care for other important things in our lives: our relations with other people, our homes and our possessions. We re-evaluate, re-arrange and improve our representations in a never-ending cycle. If we tend and care for our knowledge, then we learn about our knowledge, and what we learn has the character and structure that knowledge of knowledge must have.
Most of what I regard as the most important themes related to the triangle area formula are at least mentioned in the annotations to the reading list.