#### Evaluate geometric arguments and approaches to solving problems.

**SLO 2 Summary**

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 2 Narrative**

Geometry courses provide a natural setting for students to reflect on their reasoning, share their reasoning, and critique the reasoning of their peers. If students only ever see correct arguments given by teachers and textbooks, they may not learn how to critically evaluate arguments. 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 and is an essential skill for future geometry teachers. Students should have opportunities to critique reasoning throughout a geometry course. This can take on many forms: critiquing their own or other students’ proofs, working together in groups to solve a problem, classroom discussions of problems or proofs, posing problems that lead to student disagreement, and learning about the geometric thought process in the Van Hiele Levels.

Providing opportunities for students to critique geometric reasoning is also important for understanding nuances in geometric definitions [see SLO 5] and in geometric notation. An essential opportunity for students to practice critiquing reasoning is when GeT students present their reasoning and proofs to the class, with the instructor modeling and moderating positive and negative feedback and depth of questioning. Some instructors have also found it valuable to introduce others’ arguments that could arise in high school geometry contexts, such as sample student proofs or video approximations of secondary teaching situations. Instructors can position GeT students in the role of a teacher, rather than a peer, and invite broad and diverse observations about “students’” thinking. These discussions allow GeT students to practice critiquing reasoning while simultaneously deepening their understanding of secondary geometry content [see SLO 3].

While some of this reasoning will be in the form of formal proofs, other reasoning will be much more informal. Class discussions and work solving problems in groups can be great opportunities for students to practice critiquing the work of others. If groups are solving non-routine problems together, there are bound to be many opportunities for students to discuss their reasoning and to listen to and evaluate the reasoning of their peers. Likewise, whole-class discussions are further opportunities for students to hear and evaluate the reasoning of their peers. In an inquiry-based geometry classroom, one of the goals of the instructor is likely to be to create an environment in which students feel supported in sharing their reasoning and their thoughts about other people’s reasoning in a constructive way.

If a classroom environment is achieved in which people feel safe to share their reasoning and their thoughts about other people’s positions, then genuine disagreements may arise in the class. When this happens, it can be one of the situations that is most conducive to deep learning. When students feel that they have an emotional stake in the outcome of a discussion, they start paying deep attention to arguments on both sides. Instructors can look for problems to pose that are likely to lead good students to disagreements. Non-Euclidean geometries [see SLO 9] are particularly fertile for leading to productive disagreements, as everything there will be new and unfamiliar, and it will take some time to reach an agreement as to how to proceed. For example, students can try to decide which of Euclid’s Postulates and/or Propositions [see SLO 7] are true in a new geometry or what familiar definitions [see SLO 6] give rise to in other geometries. Any situation where students are making conjectures and trying to evaluate if they are true will lead to opportunities for students to come up with competing ideas they will need to resolve.

Interestingly, GeT instructors anecdotally have reported that a growing number of college students are exhibiting gaps in their geometric understandings and that students in their college classrooms sometimes struggle with visualizing relationships among quadrilaterals and have difficulties characterizing them. The Van Hiele levels are levels of learners’ geometric thinking and understanding (Mason, 1998). The five sequential levels include Visualization, Analysis, Abstraction, Deduction, and Rigor. This model of geometric learning posits that students at all levels will move through these different levels each time they encounter a new geometric subject. Although we expect preservice teachers to reach a high geometric thinking level (level 4 or 5), students who enter a high school geometry class typically perform at the lower levels. Therefore, it is recommended that GeT instructors create opportunities for preservice teachers to critique reasoning at various thinking levels. While it is natural for group activities to provide opportunities for the analysis of reasoning, the use of individual assignments can also be useful. For example, the use of end-of-class self-reflection assignments can provide GeT instructors feedback regarding gaps in student understanding or provide evidence of creative thinking and insightful connections.

Some other ways that instructors have implemented this standard in their classrooms include:

- Grading mock proofs on a test;
- Having students create rubrics for an assignment and evaluate their own work;
- Looking at possible K-12 classroom activities and asking students to critique them and discuss them in reference to their own future teaching; and
- Doing a jigsaw, think/pair/share, or “speed dating” activities discussing proof.

Regardless of the form that critiquing takes on, it is an essential aspect of a GeT course as it helps students think critically, improve their reasoning skills, learn how to develop solid mathematical arguments, and become better mathematicians.

**References**

Mason, M. (1998). The Van Hiele levels of geometric understanding. In L. McDougal (Ed.) *The professional handbook for teachers: Geometry* (pp. 4-8). McDougal-Littell/Houghton-Mifflin.