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

SLO 3 Summary

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 3 Narrative

While the Geometry for Teachers (GeT) course at most institutions contains students who do not plan to teach, it is required for those who will become secondary math teachers. To be a good secondary geometry teacher, one must understand the content, know the best practices for teaching the content, and be able to reflect on one’s teaching.

Although high school geometry is described as “devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates)” (NGA & CCSSO, 2010, para. 2), for many reasons, students in the U.S. often enter a GeT course with varying levels of knowledge in Euclidean geometry. As GeT instructors, it is our job to fill in the students’ knowledge gaps so they are prepared to teach secondary geometry. However, due to limited time in a GeT course (usually one semester) and GeT instructors’ varied preferences in content selection, it is not practical to suggest a list of geometry topics to be covered in a GeT course.² Thus, the GeT course should focus on helping students understand essential mathematical practices and develop problem-solving skills that can be applied to the variety of geometry topics they may find themselves teaching.

The Standards for Mathematical Practices are an important piece of the Common Core State Standards for Mathematics (CCSSM), which “describe varieties of expertise that mathematics educators at all levels should seek to develop in their students” (NGA & CCSSO, 2010, para 1). Even though some states have moved away from using CCSSM and have developed their own state standards, their new state standards typically include these eight practices or something similar to them. These practices form the foundation for good mathematics teaching. They are

1. Make sense of problems and persevere in solving them. 

2. Reason abstractly and quantitatively. 

3. Construct viable arguments and critique the reasoning of others. 

4. Model with mathematics. 

5. Use appropriate tools strategically. 

6. Attend to precision. 

7. Look for and make use of structure. 

8. Look for and express regularity in repeated reasoning. (NGA & CCSSO, 2010) 

Furthermore, these practices provide the structure for mathematical problem solving, and any GeT student can benefit from becoming a better problem solver. We also want pre-service secondary geometry teachers to be able to model these practices in their future classrooms so GeT instructors should model these in our own classrooms.

All GeT instructors need to be aware of teacher preparation standards that have been created to help prepare secondary geometry teachers (Table 1) and incorporate them into their GeT course designs in a way that fits their teaching agenda. Many different professional organizations (e.g., AMTE and NCTM) have contributed to these standards and suggested what faculty should be doing to prepare better secondary mathematics teachers. GeT instructors should also be aware of their state and national standards (Table 1) and introduce them to GeT students so that they can start to become familiar with the standards that they will teach. Many states in the U.S. have adopted the CCSSM for their K-12 schools (see this map), and if your state does not use CCSSM, it is best to Google “State K-12 Mathematics Standards.”

Table 1: Resources for Standards

Because GeT courses are taken by preservice mathematics teachers, GeT instructors must understand the needs of this group of students in their course. It is not enough for preservice teachers to know the content; these students must also gain specialized pedagogical knowledge to teach effectively. Shulman (1986) describes this as pedagogical content knowledge; it includes, in part, an understanding of what makes learning some topics easy or difficult. To have this type of understanding, students must have opportunities to reflect upon and compare/contrast analogies, illustrations, and examples. Ball, Thames, and Phelps (2008) describe pedagogical content knowledge as a bridge between content knowledge and the practice of teaching. GeT instructors should foster the construction of this knowledge by sharing teaching techniques and through conversations about teaching geometry content. For example, by taking the time to discuss multiple approaches to solving problems or by examining different frameworks for writing proofs, the GeT instructor is providing students the opportunity to reflect on misconceptions and ways that make the content more understandable to others. This type of knowledge is necessary for future teachers.

² [Clarification, March 2023]  To clarify some confusion that has occurred,, we don’t mean to focus this SLO only on process standards.  Rather, we acknowledge that relevant standards will vary over time and from place to place. We suggest that teachers of GeT courses should be aware of all of the current content and process standards where they are and should consider how to incorporate them into their courses.

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.

National Governors Association Center for Best Practices [NGA], & Council of Chief State School Officers [CCSSO]. (2010). Common Core State Standards for Mathematics. Available at http://www.corestandards.org/Math/

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.