Here again I effected a compromise: I began the course by taking a careful look at the beginning of the Elements and pointed out why it did not meet current mathematical standards of rigor. (For example, we did not require formal proof of the fact that in triangle ABC, the angle bisector of angle A intersected BC between B and C for that matter, we didn’t even formalize the notion of what “between B and C” means.) Using this “naïve” approach, we went on to prove results about Euclidean geometry that went beyond what is currently done in high school, including, for example, the various triangle centers, the nine-point circle, the theorems of Menelaus and Ceva, and straightedge and compass constructions from a fairly serious viewpoint, including issues such as the classical impossible constructions.įor the second semester, I followed (but not terribly closely) the book from which I learned this material 40 years ago and which was, I was pleased to see, still in print: Basic Concepts of Geometry by Prenowitz and Jordan. At the beginning of the first semester (the text for which was Isaacs’ Geometry for College Students) I talked a little bit about Euclid’s Elements, but then told the students that we would not make a fetish about a rigorous axiomatic development but would instead allow reasonable reliance on diagrams for what most people would accept as “obvious” conclusions. 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. 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.
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