A group of Geometry for Teachers (GeT) instructors was formed in Fall 2019 to work toward the goal of identifying a list of essential student learning outcomes (SLOs) for inclusion in GeT courses—where *essential *means the identification of content knowledge that all prospective secondary geometry teachers should have the opportunity to learn. Below are the narratives of the SLOs that the group coalesced around as of June 2022.

We envision these SLOs as a starting point for further discussions. To prepare you to enter the conversation, please start by reading the introduction to the SLOs. Then join in by registering for the forum.

## SLO 1: Derive and explain geometric arguments and proofs.

Proof is a cornerstone of mathematics, and a GeT course should enhance a student’s ability to read and write proofs of theorems, apply them, and explain them to others. Geometry offers one of the best opportunities for students to take ownership of proof in both the undergraduate and secondary curriculum. Because geometric proofs come with natural visualizations, they provide a rich environment for students to think deeply about the arguments that are being made and to make sense of each statement. Students should understand that proof is the means by which we demonstrate deductively whether a statement is true or false, and that geometric arguments may take many forms. They should understand that in some cases one type of proof may provide a more accessible or understandable argument than another.

## SLO 2: Evaluate geometric arguments and approaches to solving problems.

Students should have opportunities not only to write proofs, but also to evaluate proofs and other types of reasoning. Geometry courses provide a natural setting for students to reflect on their own reasoning, share their reasoning with one another, and critique the reasoning of their peers. If students only ever see correct arguments given by teachers and textbooks, they may not carefully evaluate the arguments because they assume they will always be correct. The ability to evaluate other people’s arguments is an important real-world skill that is related to, but separate from, the skill of proof writing. Critiquing reasoning is a competency that needs to be practiced in order to improve it and is an essential skill for future geometry teachers.

## SLO 3: Understand the ideas underlying current secondary geometry content standards and use them to inform their own teaching.

Future secondary geometry teachers must deeply understand specialized content that is aligned to national and state secondary standards, know the best practices for teaching the content, and be able to reflect on their teaching. Due to limited time and instructors’ varied preferences in content selection, it is not practical to suggest a list of geometry topics to be covered in a GeT course. Thus, the GeT course should focus on helping students understand essential ideas emphasized in secondary geometry standards and use them to inform their future teaching. GeT instructors should be able to incorporate teacher preparation standards into their course designs in a way that fits their teaching agenda and introduce the national and state curriculum standards to future teachers. In addition, a GeT course should foster the construction of pedagogical content knowledge by sharing teaching techniques and by engaging students in conversations about teaching geometry content.

## SLO 4: Understand the relationships between axioms, theorems, and different geometric models in which they hold.

Geometry courses are one of the few places where students have opportunities to engage explicitly with axiomatic systems. Mathematical theories can be developed from a small set of axioms with theorems proven from those axioms. However, GeT students may have limited prior experience working with axiomatic systems. Students in GeT courses should gradually develop the ability to:

(a) recognize and communicate the distinction between axioms, definitions, and theorems, and describe how mathematical theories arise from them,

(b) construct logical arguments within the constraints of an axiomatic system, and(c) understand the roles of geometric models, such as the plane, the sphere, the hyperbolic plane, *etc*., in identifying which theorems can or cannot be proven from a given set of assumptions.

## SLO 5: Understand the role of definitions in mathematical discourse.

In a geometry class, definitions can be a fruitful area for students to explore. Students can propose their own definitions for geometric objects and relationships, engage in class discussions about mathematical definitions versus vague descriptions, and compare and contrast definitions that refer to different properties. Determining whether and when definitions have equivalent meanings prepares prospective teachers for the varieties of geometric definitions they may encounter in teaching secondary geometry. Prospective geometry teachers should understand the role of precision in definitions for geometric terms and relationships, including understanding that some geometric terms and relationships must remain undefined. They should also understand that there are a variety of acceptable choices for some geometric definitions and that these choices can influence the structure of proofs. For example, proving that two lines are parallel because they do not intersect can be very different from proving that they are parallel because they are everywhere equidistant.

## SLO 6: Effectively use technologies to explore geometry and develop understanding of geometric relationships.

Geometry courses need to utilize technology to help develop geometric reasoning and deepen students’ understanding of geometric concepts and relationships. Two types of technologies that are beneficial for teaching geometry are dynamic geometry environments (DGEs) and digital proof tools (DPTs). DGEs support student understanding by allowing students to explore properties of geometric figures dynamically, which provides advantages over using paper and pencil. These explorations help students generate their own conjectures, test their conjectures, and provide justification and understanding for theorems. DGEs have been an important tool for teaching geometry since the 1990s. DPTs are an emerging technology that provide students with interactive figures to manipulate and opportunities to practice writing proofs with immediate feedback.

## SLO 7: Demonstrate knowledge of Euclidean geometry, including the history and basics of Euclid’s *Elements* and its influence on math as a discipline*.*

The study of many mathematical subjects can be illuminated by looking at their histories; this is especially true of geometry. Euclidean geometry is named after Euclid, the Greek mathematician who lived in Alexandria around 300 BCE. Euclid systematized the knowledge of geometry and included it in thirteen books called *The Elements*. In *The Elements*, Euclid set out a sequence of Definitions, Postulates (axioms for geometry), Common Notions (axioms common to all mathematical subjects), and Propositions (theorems derived logically from the preceding materials). For most of the 2400 years since it was written, it was considered to be an essential text and the gold standard of mathematical rigor. Students need to know this history to place modern ideas about proof into context and to understand mathematics as a human endeavor.

## SLO 8: Be able to carry out basic Euclidean constructions and justify their correctness.

Traditional geometric constructions are those done exclusively with a compass and straightedge. However, the term can be used more generally to include other tools and manipulatives such as paper-folding (e.g., using origami or patty paper), dynamic geometry environments, or transparent mirrors (e.g., MIRAs). Constructions remain an essential part of Euclidean geometry and therefore of a GeT course. Constructions support the development of mathematical thinking in several essential ways: they provide a natural opportunity for making mathematical arguments, encourage the use of precise mathematical language in communication, impart a sense of where assumptions in building mathematical systems come from, open discussions of the historical development of geometry—especially the work of Euclid, and give future teachers experience with the curriculum they will be expected to teach.

## SLO 9: Compare Euclidean geometry to other geometries such as hyperbolic or spherical geometry.

Just as visiting another country can offer one a richer perspective on their own culture, the study of non-Euclidean geometries can help students to develop a deeper understanding of Euclidean geometry. The term “non-Euclidean geometry” is interpreted broadly here, referring to any geometry different from Euclidean, including spherical, hyperbolic, incidence, and taxicab geometries, among others. The choice of which non-Euclidean geometries to consider might depend on the demands of a particular GeT course, but all non-Euclidean geometries offer rich opportunities to explore and visualize novel and engaging worlds. Learning the properties of non-Euclidean geometries puts preservice teachers in the position of their students who may be learning Euclidean geometry for the first time. Moreover, non-Euclidean geometries challenge student assumptions about what is “true” or “obvious” in Euclidean geometry (e.g. the shape of a parabola or the angle sum in a triangle). In this way, exploring non-Euclidean geometries naturally encourages questions of “Why?” and “How?” and supports the development of mathematical thinking, especially the need for justification.

## SLO 10: Use transformations to explore definitions and theorems about congruence, similarity, and symmetry.

Two main types of transformations arise in GeT courses: isometries (also known as congruence transformations) and similarity transformations. Isometries include reflections, rotations, translations, and glide reflections; a similarity transformation is the composition of an isometry and a dilation. A GeT course can contain a dedicated unit on transformations, or transformational concepts can be integrated throughout the course (even to the extent of a purely axiomatic treatment). In order to enhance and facilitate prospective teacher learning, familiar function notation can be incorporated in an introduction to transformations. This may lead to a better understanding of a sequence of transformations as a composition. While some GeT courses may begin with an informal approach to the understanding of reflections and rotations and extend the concepts of symmetry to study geometric shapes using transformations, others may instead begin solely with sequences of reflections, which can be used to generate all other isometries of the plane.

## Course Process Objectives

All GeT courses should give PSTs many chances to experience and develop proficiency with the mathematical process skills of problem solving, oral and written communication of mathematical ideas, and productive collaboration. They should also have opportunities to engage with the progression of geometric exploration followed by conjecturing and construction of arguments.