I remember well the day I first met Ms. Trump. I had been hired to teach mathematics at Pulaski County High School in Southwestern Virginia. My arrival in August 2003 coincided with the retirement of the mathematics department chair, and Diana Trump was his replacement. She explained that with their seniority system, I was assigned to teach three courses of geometry. To my surprise, she explained that she was the only teacher in the department who enjoyed teaching geometry; most everyone preferred teaching algebra. Although I had not completed student teaching in geometry, I felt prepared to teach the course. I had taken three college courses focused on Geometry for Teachers: an independent study of Euclid’s Elements as a sophomore as well as senior-level College Geometry and Capstone courses.
Although I had earned my teaching certification at a university only 30 minutes away, teaching that first year was a culture shock. I endeavored to find ways to connect geometry to my students’ lives, as many were not intending to attend college. I recall one activity from the Glencoe text we were using involved the identification of symmetries in automobile company logos. I was surprised that so many of the students in my classes knew the corresponding brands. It may have been due to the popularity of automobile racing in the community or that the largest employer in the county was a Volvo truck plant. I explained that some of the brands were actually owned by the same company (e.g., Honda and Acura). In addition to asking them to identify the symmetries in the decals, I tasked them to consider patterns in the symmetries of decals owned by the same company and why the company might or might not want a consumer to form associations between brands. My recollection is that no other lessons I taught that year had as engaged and active student participation.
Many things have changed in the past 19 years. I had introduced symmetries in my high school geometry class toward the end of the course, which was how it was organized in the Glencoe text. The role of transformations and the development of transformational reasoning has since become central to curricular standards, as evidenced by the Common Core State Standards (CCSS-M). The CCSS-M defines two figures to be congruent if there exists an isometry mapping one figure onto the other (rather than congruence of an image and its preimage being a property of an isometry). My high school students had categorized decals by the type of symmetry: point (half-turn or 180-degree rotation), line (reflectional), rotational (including point), or “none”. Categorizing symmetries this way is mostly excluded from the geometry Standards. An exception is a third-grade standard, CCSS.MATH.CONTENT.4.G.A.3: “Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.”
Another change is that I am now teaching College Geometry in Portland, Oregon. Entering my GeT course, most of my students (many of whom are older than traditional college age) have learned about symmetries as properties of figures. This may be because transformational reasoning has only recently become more central in geometry. A learning goal I havefor my students is expanding their understanding of a plane symmetry as a type of isometry. Based on my previous experiences teaching College Geometry, I had thought of three stages in GeT students’ understanding of a symmetry as a transformation, marked by changes in the domain of consideration. In the first stage, a symmetry is a transformation that acts on a proper subset of a figure. A student with this conceptualization might explain that a reflection that maps the left-half of a figure onto the right-half of the figure is a symmetry. With the second conceptualization, the student considers an entire figure as the domain. The student would consider that the same single reflection also maps the right half of the figure onto the left half. For the third conceptualization, the student considers that as an isometry, a symmetry is a bijective mapping of the entire plane (and thus they would not just consider the figure of interest as the domain and codomain).
My experiences learning and teaching about Adinkra symbols have broadened my views of the ways GeT students may reason about symmetry. I was first introduced to Adinkra symbols by a colleague at Portland State University, Dr. Joanna Bartlo, who had forwarded me the Culturally Situated Design Tools (CSDT) website (https://csdt.org) as a potential resource for simultaneously integrating computing (via Scratch) and mathematics from non-Eurocentric cultures into a Geometry for Middle School Teachers course I was teaching. I did not have time to incorporate the coding activity but instead asked my students to use the CSDT website to explore Adinkra symbols and their meanings, as a pre-assignment for class discussions of symmetry. Recalling my positive experiences teaching about automobile logos with my high school students, I believed learning about Adinkra would provide opportunities for students in my GeT class to form experiences connecting geometry to culture they could build on in their teaching.
The first two times I incorporated this assignment into my GeT courses, during remote teaching, I tasked students to record videos introducing to the class an Adinkra symbol they felt connected to, explain the symbol’s meaning, identify its symmetries, and share why they chose to highlight it. I recall that subsequent class discussions often focused on statements of appreciation, as well as the absence of symmetry. Students corrected those who mistakenly attributed reflectional or rotational symmetries to Adinkra that were “not quite” correct.
It was not until the following term that I learned that this notion of “not quite symmetrical” was discussed on a different section of the CSDT website. In particular, I learned about the concept of mutuality, and I suggested it to the Transformations Working Group as a potential topic for us to explore how our students reasoned about symmetries, isometries, and their definitions. Determining the appropriate level of rigor and the appropriate properties and axioms for transformations in GeT courses have been part of ongoing discussions with the Transformation Working Group since its formation. We decided to embark on a lesson study to provide an opportunity for us to focus on our students’ reasoning.
We ultimately decided to focus on an activity exploring Adinkra and mutuality because it provided our students (and us) with an opportunity to expand our knowledge about connections to mathematics from non-Eurocentric cultures. Furthermore, because “mutuality” is not a standard symmetry (i.e., described by rotation, reflection, or translation, or a composition thereof), and because it does not (yet!) have a commonly accepted mathematical definition, we saw an opportunity for students to experience genuinely open mathematical inquiry. (Boyce et al., 2021, para. 7)
It is important for GeT instructors to support prospective teachers’ learning of geometry that is relevant to classroom teaching. Even though symmetries are not an explicit goal of the secondary curriculum, they provide opportunities to motivate transformational reasoning and extend students’ understandings of connections between geometry and culture. Introducing mutuality is powerful because it encourages GeT students to reflect on the characteristics of their geometric reasoning, on what they are acting upon mentally when they conceive of symmetry, and how they might think about the absence of symmetry. It fosters an anti-deficit perspective of differences in others’ reasoning and provides opportunities for prospective teachers to think about the role of cultural identity in teaching geometry. I hope readers of the newsletter check out the blog post (https://blogs.ams.org/matheducation/2021/05/06/best-laid-co-plans-for-a-lesson-on-creating-a-mathematical-definition/) and materials the Transformation Working Group has created and that we continue to build on our experiences learning and teaching about Adinkra and mutuality.
References
Boyce, S., Ion, M., Lai, Y., McLeod, K., Pyzdrowski, L., Sears, R., & St. Goar, J. (2021, May 6). Best-Laid Co-Plans for a Lesson on Creating a Mathematical Definition. AMS Blogs: On Teaching and Learning Mathematics. https://blogs.ams.org/matheducation/2021/05/06/best-laid-co-plans-for-a-lesson-on-creating-a-mathematical-definition/
