We are all members of the Euclidean Archetype workgroup. As we summarized in our report, a GeT course organized around the Euclidean archetype will focus on the axiomatic development of fundamental principles of geometry. Informed by the spirit and organization of Euclid’s Elements, this course emphasizes mathematical precision, rigorous proof, and clear communication. We have all taught geometry with varying amounts of experience. We agree on many goals that a course should have but each of us prefers a different balancing of the ingredients. What follows is our discussion of the essential elements and the plusses and minuses of teaching a class using the Euclidean Archetype.
THE ESSENTIAL COMPONENTS
SS: The Euclidean archetype centers on axiom systems, and any GeT course following this framework should emphasize that structure: precise language, identification of agreed-upon undefined terms and axioms, and the development of the theorems of geometry from those foundations. A worthy highlight of this course is the independence of the parallel postulate. This requires some work, including a careful development of the concepts of models and independence and an exploration of alternative axiom systems for Euclidean geometry (including Euclid’s axioms and some other modern system).
SC: I agree. I would add that the structure naturally leads to an emphasis on proof writing. I find it useful to spend some time in a simpler axiom system such as an incidence geometry to enable students to practice writing proofs with fewer subtleties and issues.
NM: I think there are two key components here, that don’t necessarily have to be combined, but often are. This is sometimes referred to as the Euclidean Axiomatic archetype, and the two components are Euclidean Geometry and an axiomatic approach. You could have a course focused purely on Euclidean Geometry; you could have a purely axiomatic geometry course; and putting them together, you could have an axiomatic geometry trying to get at the main ideas of Euclidean geometry. There are certainly courses that mostly do one of these without the other. For example, some books focus on explorations of Euclidean geometry using dynamic geometry software without an axiomatic approach. On the other hand, some completely axiomatic courses don’t get very far into Euclidean geometry because it takes so long to prove elementary facts about incidence geometry and betweenness proceeding carefully from elementary axioms. Probably to be considered part of the Euclidean Axiomatic archetype, you need to explore some of both. Getting to both probably requires that we broaden both pieces, though. As SS notes, we will want to talk about models and independence, which will require us to work, at least a bit, with some non-Euclidean geometries; and to get to the interesting parts of Euclidean geometry, we will probably have to move away from the idea of proving absolutely everything from a purely axiomatic standpoint.
OTHER TOPICS TO INCLUDE
SS: Proving the independence of the parallel postulate opens up the world of non-Euclidean geometry, and exploring the seemingly strange world of hyperbolic geometry is a natural branching off point for this archetype. It provides students with an alternative axiom system to consider and by developing its major theorems students gain a stronger understanding of the more familiar Euclidean world. The archetype also provides an opportunity to study Euclidean straightedge and compass constructions. Careful development of these tools provides significant payoff if the instructor chooses to investigate models of hyperbolic geometry in some detail. Dynamic geometry software can be a powerful tool in this investigation.
SC: Compass and straightedge constructions are foundational in Euclidean geometry. Students can use these to make conjectures, prove theorems, and develop geometric intuition. Students can also consider models where various axioms fail to hold, such as geometry on the sphere, or on the Cartesian plane using the taxicab metric to measure distance.
NM: I agree with all of these ideas. Spherical geometry is also natural to look at in the context of parallel lines– with spherical, Euclidean, and hyperbolic geometry, we have cases with no, one, and more than one line(s) through a point parallel to a given point. I think spherical geometry is more accessible to students since they already know what a sphere is. There is also a sense in which spherical, Euclidean, and hyperbolic geometry are the building blocks for all 2 dimensional geometries. There is dynamic geometry software for each of these, and I also like to have students work with physical models.
ADVANTAGES
SS: I love the structure of this archetype. Building geometry from a set of axioms and undefined terms allows students to see a logical development of the subject. Even a short exploration of the Elements gives students an appreciation for the monumental achievement of Euclid while helping them recognize the need for precise language and rigorous proof. In addition to focusing on strengthening students’ logical reasoning abilities, the archetype also offers natural opportunities to build in a historical examination of geometers, from Thales to Saccheri to Bolyai to Riemann to Hilbert, as well as many others in between. I believe that a strong foundation in the axiomatic structure of geometry is an essential component of the preparation of future teachers of the subject.
SC: Euclidean Geometry has long been a model of deductive reasoning and teaching students to write proofs. Teaching it also presents a great opportunity to incorporate the humanities (art, history, western civ.) into the math curriculum. Most exciting part of teaching it for me was following the long and technical journey through Neutral geometry not allowing students to assume familiar results such as 180 degrees in a triangle. When, finally we bring in the Euclidean Parallel postulate, the parallel projection theorem, similar triangles, the Pythagorean Theorem, and trigonometry immediately enrich the study. Finally, it is natural to discuss practical applications.
NM: Geometry has long been a place in the mathematics curriculum where logic is discussed in a mathematical setting. I don’t think there is a better setting than a geometry class to get students thinking about the roles of axioms, definitions, and theorems, and to start thinking about metamathematical ideas about when statements are unprovable in a given system.
DRAWBACKS
SS: With its emphasis on an axiomatic development of geometry, this archetype does not as naturally lend itself to applications or pedagogical conversations as some other archetypes might. Moreover, Euclid’s axioms have little to say about geometric transformations, an important component of the Common Core State Standards for Mathematics. However, these topics could be included with careful planning by the instructor.
SC: Preservice teachers need additional perspectives, extended time with transformational geometry, and opportunities to do the kind of exploration emphasized in the common core. It is possible but much more challenging to include these features in a Euclidean course.
NM: One big drawback of a purely axiomatic approach is that there isn’t an axiomatization of Euclidean geometry that is fully complete and rigorous that is at the appropriate level for most undergraduates. If we use something like Hilbert’s axiomatization, we end up spending a lot of time giving fairly technical proofs of trivial results. Actually, Euclid’s treatment is still one that is at about the right level for most students, but it does make some unstated assumptions. The other piece that this approach usually leaves out is the opportunity for students to explore and make conjectures before trying to prove them, which is another giant piece of doing mathematics that geometry courses are especially well suited for. That’s why I tend to structure my courses around the experiencing geometry archetype, but for all the reasons we have discussed, I almost always include a section of the course structured around the axiomatic Euclidean archetype. One way to do this is to spend several weeks in the middle of the course having students prove basic theorems of neutral geometry from a simple four axiom system.
Steve Cohen is Associate Professor of Mathematics at Roosevelt University.
Nat Miller is Professor of Mathematical Sciences at the University of Northern Colorado.
Steve Szydlik is Professor of Mathematics at the University of Wisconsin Oshkosh.
