Last year, the Teaching GeT (Geometry for Teachers) Group started to think about the question: what materials could we produce that would be most helpful for a new teacher of a GeT course? We quickly ran into an issue: unlike many other mathematical subjects, there isn’t a standard curriculum for a GeT course. Many different kinds of courses are taught, even among the members of the group. We therefore started to wonder how much agreement there was about the goals for such a course. Are all the different kinds of courses attempting to meet a common set of goals in different ways, or do the different courses have fundamentally different goals? This question has important implications for how we view what we are doing in a GeT course, how we provide professional development for new instructors of GeT courses, and how we determine if a GeT course is successful. As a group, we tried to come up with a list of shared goals for the GeT course in the form of SLOs (Student Learning Objectives). What we discovered was that, at least among the members of our group, there was a strong consensus about the major goals for a GeT course.  

In order to arrive at our set of essential SLOs, we used a winnowing strategy. First, we asked members of the group to write down what each person thought were essential SLOs. I took everyone’s lists and combined them into a master list that contained all of everyone’s ideas. We looked at the master list as a group and winnowed it down to the ideas that everyone agreed were essential. I used these to create a draft list of SLOs. We put each SLO up on an online discussion board where different members of the group could make individual comments on each SLO. Finally, we went through all the comments as a group and refined the SLOs.

The SLOs that we developed dealt with 10 broad categories:

1. Proofs Derive and explain geometric arguments and proofs in written and oral form. 
2. Proof Verification Decide whether or not geometric arguments given by others are correct.
3. Secondary Geometry Understanding Understand the ideas underlying the typical secondary geometry curriculum well enough to explain them to their own students and use them to inform their own teaching.
4. Axioms, Theorems, and Models Understand and explain the relationship between axioms, theorems, and geometric models in which they hold (such as the plane, the sphere, the hyperbolic plane, etc.).
5. Definitions Understand the role of definitions in mathematical discourse. 
6. Technologies Effectively use technologies such as dynamic geometry software to explore geometry.
7. Euclid’s Elements Demonstrate knowledge of Euclidean Geometry, including the history and basics of Euclid’s Elements, and its influence on math as a discipline.
8. Straightedge and Compass Constructions Be able to perform basic Euclidean straightedge and compass constructions and be able to provide justification for why the procedure is correct.
9. non-Euclidean Geometries Compare Euclidean geometry to other geometries such as hyperbolic or spherical geometry.
10.  NCTM Standards Apply the following NCTM Geometry Standards: (a) analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; (b) apply transformations and use symmetry to analyze mathematical situations; and (c) use visualization, spatial reasoning, and geometric modeling to solve problems.

We included a statement that in addition to teaching these content standards, all Geometry courses for future teachers should give students many chances to experience and develop their abilities with the mathematical process skills of problem solving, reasoning and proof writing, oral and written communication of mathematical ideas, and productive collaboration within groups. They should also get a chance to engage with the progression of exploration followed by making conjectures, followed by trying to prove their conjectures.

Our hope is that we will be able to validate these goals with a wider cross-section of the community of people who teach GeT courses by verifying that most stakeholders also consider these to be the major goals of any GeT course. They can then serve as the basis for professional development, assessment, and evaluation efforts.