The coverage of this book is what one would ordinarily expect to find in a high school curriculum without an AP program. Operations such as limits, differentiation and integration are mentioned when there is a suitable purpose and abstract algebraic structures such as groups and fields are also mentioned.
What differentiates this book is the added depth of coverage of what are the primary topics of the high school curriculum. For example, the last chapter, called “Axiomatics and Euclidean Geometry,” covers the geometry found in high school, but the approach is based on the solid foundation of the axioms. Most specifically there is a detailed examination of the parallel postulate in constructing the final aspects of Euclidean geometry and what is missing when it is absent.
Brief mentions of the historical development of the topics are also used to describe how a topic fits into the overall fabric of mathematics. This works very well at the high school level, because these students are approaching the point of rigor and they often like to know how all the material they are studying interrelates.
One of the best ways to describe this book is that it prepares the high school teacher for many of the likely “Why is this true?” questions that are often asked by the most advanced students. While the names of the chapters are routine, this is an example where it would be wrong to judge a book by the titles of its chapters.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
I. ALGEBRA AND ANALYSIS WITH CONNECTIONS TO GEOMETRY.
II. GEOMETRY WITH CONNECTIONS TO ALGEBRA AND ANALYSIS.