Teaching GeT (led by Nat Miller)

Unlike more standardized classes such as calculus, future instructors of the collegiate geometry course for preservice teachers are assigned to the course with vastly different geometry experiences, and many of them have never taken a post-secondary geometry course at all. The Teaching GeT (Geometry for Teachers) working group was formed to conceive and draft written products that could fill in such gaps, focused on answering the core question: What would be useful for a new GeT instructor to consider prior to teaching the GeT course?

During the Fall 2019 semester, we put together a long list of potential topics that we could address. So far, we have mainly focused on one in particular: Trying to come up with a list of Essential Student Learning Outcomes (SLOs) that most people would agree should be addressed in any GeT course. This is closely related to the work that the Geometry Knowledge for Teaching workgroup did last year, looking at SLOs for some of the different types of GeT courses identified by Grover and Connor (2000). Our hope is to get feedback about which SLOs are seen as truly essential from the perspective of the wider community of people teaching GeT courses, so you may get a short survey from us sometime in the near future!

Going forward, we hope to identify the skills and knowledge necessary to teach our Essential SLOs that potential teachers may have not yet obtained and to think about ways to help them acquire these. One approach we may take is trying to write guides to some particular content or pedagogical areas that we think are of particular concern. We would welcome participation from anyone interested in helping with this project, even those who haven’t been previously involved!

Transformations (led by Julia St. Goar)

In the fall we articulated our main goals for the Transformations group: (1) to create a system of axioms for transformation proofs; (2) to get ideas for improving existing courses for future teachers involving transformations; (3) to grapple with what teachers really need to know about transformations in the Common Core and how we can motivate their study of transformations; (4) to make activities for the purposes of teaching transformation to undergraduate geometry students; (5) to understand axioms in a transformation context so that teachers can compare these axioms to other contexts; (6) to get help creating transformation content for a course for the first time; and (7) to deepen our understanding of axiomatic structures in a transformation context.

So far, we have shared a list of content goals each of us has in our own courses, including key concepts and definitions as well as theorems proved. This resulted in the beginning of a content concept map for transformation courses. We discussed classroom activities, teaching strategies, and technological tools each of us has been using to further student understanding of transformations, especially for the topics that our students have found tricky. We also shared information on other aspects of our courses, such as student audience composition, prerequisites, textbooks used, teaching goals, and even classroom culture, in order to discover the similarities and differences.

We have debated our goals for any axiomatic system we may create. One question was whether the axiomatic system should be adjusted for the undergraduate student audience (for example, a statement that is really a theorem may be called an axiom in an undergraduate classroom context). Axioms could be written to create a smoother classroom experience, to avoid having to put students through proving too many very basic theorems, and to allow students to get to more interesting theorems more quickly. Properties that high school students have learned about transformations in the past, for example, could potentially be considered axioms in a classroom setting. It became the view of the group that creating a student-friendly axiomatic system would be the most helpful for our purposes.

This spring we have begun by sharing various existing axiomatic systems to compare and contrast. It is likely that our discussion will focus on: the appropriateness of the axiom statements themselves in a classroom context, particularly for student understanding; the connection of these axioms to the Common Core; and the theorems that these axioms will allow students and teachers to prove throughout the course. This discussion on axioms will likely reinforce our ongoing discussion of ways to help undergraduate student understanding and to develop classroom activities focused on transformations. The group would welcome any new voices that could help us in these ongoing efforts.