![]() ![]() This can be viewed as a learning experience, of course, but I have found that it requires some real degree of mathematical sophistication on the part of students to see why these “obvious” results even need proof in the first place. ![]() It is quite time-consuming, for one thing, and, more seriously, much of that time is spent on proofs of results that the students think of as fairly obvious to begin with. On the other hand, a formal development like this, if done in a rigorous and intellectually honest way, has some pedagogical drawbacks. Such an approach seems very valuable and also provides good practice in writing proofs. On the one hand, I am very sympathetic to the idea that students, particularly future high school students, should see a rigorous development of Euclidean geometry. ![]() One of the issues on which I was most conflicted was the question of how much of a formal axiomatic development of Euclidean geometry should be done. Other decisions, though, were much more difficult. Some of these decisions were easy: I knew from the outset, for example, that I wanted to do Euclidean geometry the first semester (including topics that might be considered “advanced”, such as the nine-point circle and the theorems of Ceva and Menelaus) and then, in the second semester, talk in more detail about foundational questions and introduce the students to non-Euclidean geometry. Because I had not taught this sequence before, and because the syllabus was fairly flexible, I had a number of decisions to make before teaching the class. ![]() For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.This book arrived during the last week of classes at Iowa State University, just as I was finishing up a two-semester senior-level geometry sequence, attended mostly by mathematics majors, many (but not all) of whom were planning to go on to teach secondary school mathematics. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. The investigation of the parallel postulate leads to the various non-Euclidean geometries. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The theory of area is analyzed by cutting figures into triangles. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. To shore up the foundations we use Hilbert's axioms. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The course begins in Chapter 1 with a critical examination of Euclid's Elements. I assume only high-school geometry and some abstract algebra. This book has grown out of that teaching experience. In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. ![]()
0 Comments
Leave a Reply. |