Discovery learning (or inquiry-based learning, or Moore method, or many other related variants) de-emphasizes lecture and reading in favor of allowing students to develop on their own as much of the material as possible. Euclidean geometry is an excellent playground for this because you can start with a few comprehensible common notions and postulates and run with them.
One step that is sometimes missing from the discovery learning process is the computer. Really understanding a result requires not just proof but probing deeply into the assumptions and finding examples that illustrate what is happening. In geometry in particular, there is software available to help students find the examples that lead to understanding, proofs, and new conjectures. Gerard A. Venema’s Exploring Advanced Euclidean Geometry with GeoGebra is a discovery learning text that embraces this approach.
GeoGebra is a software package that allows users to directly manipulate geometric objects and explore their properties and relationships to one another. It is open-source (i.e., free) and may be installed on various computing devices or plugged into a web browser. Its functionality is comparable to the commercial product Geometer’s Sketchpad.
Exploring Advanced Euclidean Geometry with GeoGebra is written for an inquiry-based approach, with lots of exercises and just enough narrative and historical commentary to hold it all together. It is not the sort of book you read without some paper and probably a computer in front of you. What makes the book special is the inclusion of GeoGebra exercises (clearly identified with a *) to encourage experimentation. Exercises may ask students to construct a visualization of a theorem, verify results, and build examples and conjectures. Eventually the student gets to proving a theorem, but not before playing with the statement quite a bit.
The book focuses on “advanced” planar Euclidean geometry, which the author defines to mean anything developed after Euclid’s Elements. This makes it an excellent candidate text for a second course in Euclidean geometry using inquiry-based methods that minimize lecture and maximize student discovery. There is also much value to be mined as a supplement to other Euclidean geometry texts. The author suggests a structure in which this text is used as something of a lab manual rather than a primary text.
Even if it does not fit for course adoption, this book is worth any geometry teacher’s attention as way to reconnect with the learning experience they want for their students.
As to content and prerequisites, the reader is expected to be familiar with elementary Euclidean geometry and generally be comfortable (or prepared to become comfortable) with definitions and proof. Some key Euclidean results are quickly reviewed in the first chapter. Topics include various notions of triangle “center,” the nine-point circle, the theorems of Ceva and Menelaus, geometric inversion, and (cheating the title just a bit) hyperbolic geometry. It has a lot of content for a slim volume (130 pages), but that’s the point — inquiry-based texts are necessarily short.
There are only a couple of short chapters about GeoGebra itself, which was a good move. Print guides to software do not age well and such material is better consumed online anyway. Venema keeps these discussions about building the tools relevant to the book’s exercises and less about “click here to do that.” This helps make the book adaptable to other software.
Advanced Euclidean geometry is an uncommon course offering these days. This book would work as sourcebook for directed study and as such is worth consideration for many academic libraries. However, practitioners looking for a comprehensive content reference or a traditional theorem-proof-exercise presentation should look elsewhere.
Inquiry-minded instructors should absolutely give this some attention. Working through these exercises feels very much like the process of doing mathematics, which is about the most one can ask of a book like this. The engaged student will learn much about learning as well as geometry.
Bill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa.