When I joined the Geometry for Teachers (GeT) group in the summer of 2018, one thing that was always clear to me was that the GeT course at various institutions is very different. During the 2019 – 2020 academic year the Teaching GeT group, led by Dr. Nat Miller, made a huge effort to create a common list of Student Learning Objectives (SLO) for the GeT course. The list needed to be concrete enough to help a faculty member teaching the course for the first time, yet flexible enough that it works for different institutions. 

Since I feel that proof should be the focus of the GeT course, I will elaborate on the first Proof SLO, “Derive and explain geometric arguments and proofs in written and oral form.” In the study of mathematics, true understanding is attained when one can read and write a proof of a theorem, explain it to another person, and apply the theorem. Geometry is an area of mathematics for which this skill may be more easily attained because figures are frequently used to illustrate theorems. While one should never rely on a picture when completing a geometry proof, having an image is often helpful when writing a geometry proof.

Why is it important that future teachers have a solid background in proof? While not every state has adopted or uses the Common Core State Standards for Mathematics (CCSSM), many have written new mathematics standards that are very similar to these standards. In the CCSSM, the first mention of proof is in the 8^th grade geometry standards. These standards focus on using rotations, translations, and reflections to demonstrate that two figures are congruent to one another, and students are expected to be able to explain a proof of the Pythagorean Theorem and its converse. In the CCSSM high school geometry standards, students are expected to “understand congruence in terms of rigid motion” and “prove geometric theorems,” (National Governors Association, 2010).

Can we expect secondary mathematics teachers to teach proof without undergoing it for themselves? Mathematics teachers need to experience geometry proofs from the student perspective so they can empathize when their own students struggle. They need to be able to understand different types of proofs, such as synthetic (from axioms), analytic (using coordinates), and proofs using transformations or symmetries. They should be able to communicate proofs in different ways (two-column, paragraph, or a sequence of transformations). It is also essential that they can choose the most accessible type of proof for the situation.

In an ideal world, students would enter the GeT course with a strong background in proof, but that is generally not the case. Some of the variation in their proof background depends on the level of the GeT course (200, 300, or 400), but much of it depends on their own high school geometry experience. Based on what the GeT students tell me, high school geometry likely has the most variability of all high school mathematics courses throughout our country. Some students never had to do proofs in their course, while others did a lot of proofs in various forms. This is one of the reasons that the GeT SLOs are so important—we cannot help improve the teaching of high school geometry without looking at our own teaching of geometry. Great teachers are reflective practitioners—what better way to produce strong mathematics teachers than to model this practice for our preservice teachers. 

Writing a synthetic proof requires the ability to put together a logical argument in a systematic manner. This skill leads to growth in critical thinking and reasoning. Daily we encounter situations where critical thinking is needed. Geometry is the best course to use proof to help students build problem solving skills, which is why I believe that proof should be the heart of the GeT course.

References

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC.