SLO 3: Secondary Geometry Understanding. Understand the ideas underlying the typical secondary geometry curriculum well enough to explain them to their own students and use them to inform their own teaching.

While there are students who do not plan to teach geometry in Geometry for Teachers (GeT) courses at most institutions, 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 of 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. Thus, the GeT course should focus on helping students understand essential mathematical practices and develop problem-solving skills that can be translated to any geometry topic. 

The Standards for Mathematical Practices are the most 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 to incorporate them in their GeT course designs in a way that fits their teaching agenda. Many different professional organizations (e.g., AMTE and NCTM) contributed to these standards and suggested what faculty should be doing to prepare better secondary mathematics teachers. GeT instructors should also be aware of the national and state curriculum standards (Table 1) and introduce them to GeT students so that they can start to become familiar with the standards that they will teach. The majority of 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 required for students who will become secondary mathematics teachers, GeT instructors must understand the needs of this group of students when in their course. It is not enough for these students to know the content, these students must gain specialized 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 by others. This type of knowledge is necessary for future teachers.

Another aspect that helps with secondary geometry understanding is technology. Many teacher education standards and curriculum standards have addressed the use of technology in some way. For example, one of the grade 8 CCSSM geometry standards specifically mentions geometry software. Therefore, the GeT course needs to utilize technology to help understand and explore concepts in geometry. More information about the use of technology will be addressed in the Technology SLO.

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.