*By:*

*Steven Boyce, Portland State University*

*Michael Ion, University of Michigan*

*Yvonne Lai, University of Nebraska-Lincoln*

*Kevin McLeod, University of Wisconsin-Milwaukee*

*Laura Pyzdrowski, West Virginia University*

*Ruthmae Sears, University of South Florida*

*Julia St. Goar, Merrimack College*

*All authors contributed equally to the preparation of the document.*

How do students typically engage with new definitions in undergraduate mathematics classes? Are students provided with a definition, and then instructors help students make sense of it? Do students have opportunities to create their own definitions? Often when instructors choose to involve students in the process of creating a definition, the role of the instructor may be to encourage students to structure or word their definitions in a particular way, with the goal of leading students toward a definition found in a textbook. This can be a daunting task. After all, enacting this kind of lesson means anticipating what students may do or say, deciding when to let students keep talking and when to interject, and responding to unexpected contributions. Designing a lesson that is mathematically substantive but also provides opportunities for students to do a lot of the talking (including students providing feedback to other students) is really hard! Even with the most well-laid plans, surprises can still happen.

One way to take on this challenge, and have support as the unexpected arises, is to collaborate with other instructors. The authors of the post are all instructors of geometry courses for prospective high school teachers, who participate in a “GeT: a Pencil” community meeting every other week, and sometimes more often. These community meetings gather university geometry instructors from across the country to collaborate on issues related to the teaching of the geometry course primarily taken by preservice teachers. Among us are mathematics and education faculty, whose academic backgrounds range from mathematical physics to difference equations to hyperbolic geometry to student cognition to teacher education. We saw a pandemic-era opportunity to co-plan and co-teach a common lesson. On Zoom, we can be more than 3000 miles away and learn from each other in the same room. While practices involving the design of lessons (such as the Japanese “lesson study”) have been established for decades in some K-12 settings, it is still rather rare in undergraduate settings, though there are some exceptions.

In this post, we share our experience of developing a lesson that could be taught in any of our courses and how this lesson did not go according to plan. We intended the lesson to focus on creating a new definition. Although the class did not reach a consensus on a definition, the process opened many mathematical questions.

We first show the key example of the concept to be defined. Then we describe why we chose to use this example, how we built a lesson around it, and the unexpected outcomes. Finally, we discuss what we learned (and hope to continue to learn) about collaborative planning and teaching.

# Defining mutuality

Consider the following image:

**Figure 1. Boa Me Na Me Mmoa Wo Adinkra symbol
How would you describe the aesthetic appeal of this figure, mathematically?**

(source: https://csdt.org/culture/adinkra/geometry.html)

Boa Me Na Me Mmoa Wo is an *Adinkra*. Adinkra are symbols created by the Ashanti people of Ghana to represent concepts. Its name in English is “Help me and let me help you”.

Our activity to engage students in constructing definitions focuses on the mathematical properties of this Adinkra symbol that make it visually appealing. Often, as mathematicians, we think of “symmetry” as a way to describe aesthetic elegance. Yet the only standard “symmetry” here is a single reflection. Intuitively, it seems incomplete to describe the “symmetry” of this Adinkra as merely a single reflection. The ethnomathematics educator Ron Eglash suggests that Boa Me Na Me Mmoa Wo exhibits mutuality: “The upper triangle is missing a square, but has an extra circle. The lower triangle is missing a circle, but has an extra square. Each has what the other needs to complete [itself].” Our main task focused on how one might define mutuality. At this point, we encourage the reader to attempt to create a mathematical definition that describes salient aesthetics of the Boa Me Na Me Mmoa Wo symbol.

We had several reasons for co-designing a lesson around mutuality. At the onset, we wanted our students to learn from each other and talk about geometric concepts, definitions, and axiomatic systems in productive ways. We also wanted a lesson that would allow students to compare definitions, and do so in a way that could be connected to secondary geometry from a transformation perspective. We considered several task ideas related to secondary geometry standards for transformations, such as comparing definitions of glide reflections, or identifying symmetries of frieze patterns. 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.

We also note that we use Adinkra with the implicit permission of at least some creators of Adinkra. All resources on the site Culturally Situated Design Tools, from which we learned about Adinkra, are disseminated by Ron Eglash with the explicit permission of people he visited to learn about their designs, and knowing that students and teachers may take mathematical directions not necessarily directly aligned with a culture of origin.

# Lesson task: Sorting Adinkra

We decided to open the lesson with a sorting task, completed individually by each student:

**Figure 2. How might you sort these twelve Adinkra symbols?**

(source for symbols: https://www.adinkrabrand.com/blogs/posts/african-adinkra-symbols-and-meanings)

We designed Google Jamboards with the arrangement shown in Figure 2, featuring the same set of symbols on each, and prompting each student: *Group these symbols by their aesthetic; put each image in exactly one group; then identify names for each group. *

We next put students into teams and asked them to review the individual classifications and discuss: *What did the groupings have in common? How were they different?* These questions had two purposes. First, they could elicit discourse needed to create a definition; for instance, articulating properties, handling disagreements, and coming to consensus. Second, they allowed the instructor to hear students going through this process, to give support as needed, and to adapt later parts of the lesson as needed.

In choosing symbols to include, we included several symbols with rotational and reflectional symmetries, the Boa Me Na Me Mmoa Wo symbol (Figure 1), as well as symbols that we anticipated students might group with it. We hoped to plant a seed for students to see a need to define (and refer to) the standard mathematical symmetries with precision.

# Lesson task: Defining the aesthetic of the Bo Me Na Me Mmoa Wo symbol

Next, we planned for students to formulate a definition of mutuality based on Boa Me Na Me Mmoa Wo, first individually, then in small groups, and then as a whole class. We note that in the end, students formulated and revised individual definitions after reading Eglash’s description quoted above.

We asked the students:

**Figure 3. If you had to create another symbol with that same aesthetic, what would you produce? How would you define an aesthetic category for the Boa Me Na Me Mmoa Wo symbol? **

To prepare for this, we started by brainstorming some potential definitions and experienced for ourselves the uncertainty of what interpretations might arise. For instance, one of us defined mutuality as:

Consider a figure A. If there are two subsets of B, C of A, and two isometries f and g, such that A U f(B) U g(C) has more symmetries than the original figure A, then A has mutuality.

For example, in the Boa Me Na Me Moa Wo symbol, we might take B to be the bottom square, C to be the top circle, f to be an isometry that maps B to the “empty” square, and g to be an isometry that maps C to the “empty” circle. Then A U f(B) U g(C) has a line of symmetry. A second among us defined mutuality as requiring the requirement that the rigid motions used to relate part of the symbol to each other be an involution and yet a third among us defined mutuality as requiring at least one line of reflective symmetry. A fourth among us pointed out that developmentally, students may look at concrete visual features, such as whether a figure has a vertical or horizontal line of symmetry, or whether a figure includes spirals or polygons. We wondered whether students would describe mutuality as the existence of a sequence of actions or operations on the entire figure, or as something that two sub-figures exhibit, and whether they would reason about the black/white contrast as a presence/absence of points or as different colors.

We planned to close the lesson by debriefing the lesson with students, discussing how the process they engaged in might apply to defining activities in secondary geometry classrooms, and providing resources for them to explore Adinkra, including their origins, names and meanings.

# Our plans meet with surprises: Co-teaching the lesson

Dr. Boyce and Dr. Sears co-taught the first iteration of the lesson in a 150-minute class with six students who were enrolled in a masters degree program in mathematics education. Three of the other co-authors attended as observers. We were fortunate that this class had already developed a warm, welcoming, and supportive environment. This was due in large part to emphasizing “The Five R’s” each class session: rigor, relevance, collaborative responsibility, cultural responsiveness, and authentic relationships. “The Five R’s” represent a point of view that Dr. Sears has emphasized within her classes to support the development of rigorous mathematics and a sense of community in which everyone works collaboratively to co-construct mathematical meaning and develop a conceptual understanding.

# How did the students sort? How did the students define?

Here was our first surprise: None of the students sorted by symmetry! Instead, students had categories such as “ovals”, “stars”, “swirls”, “thick lines”, “4’s”, and “2’s”. Figure 4 shows some of the students’ groups.

**Figure 4. Some groups of symbols created by students**

And here came a benefit of co-teaching: making collaborative in-the-moment decisions. An observer noticed that the students, working in groups of three, were focusing on types of polygons required (i.e., that it should consist of triangles, circles, and squares) in their definitions and we were concerned that the discourse might be headed away from our goal of eliciting transformational reasoning.

Dr. Sears predicted that if Dr. Boyce asked the student who had used the phrases “4’s” and “2’s” to speak up earlier about their reasoning behind their categorization in the whole-class discussion, then this may invoke transformational reasoning for the rest of the students in the class. Dr. Sears observed students in one group come to immediate agreement on “4’s” and “2’s” as a characterization, perhaps because of the practical nature of the description: the number of apparent pieces in the symbol. The students in this group then talked about specific locations of bolder lines and thinner lines in the designs. Noticing the emphasis on concrete visuals, Dr. Sears suggested that Dr. Boyce could ask a question such as, “If I were to reconstruct the grouping, would you know how?” By “reconstruct the grouping”, Dr. Sears meant re-sorting the symbols in a way that would replicate a particular grouping of symbols. Her instinct was that these students would think in terms of geometric transformations to describe relationships between figures. These tactics worked: students then talked about symmetry with respect to x- and y-axes and envisioned lines of reflection on symbols.

We then split students into two groups and asked them to define mutuality. One group wrote the definition: “should consist of an oval with shapes on the end”. The second group defined it as a “shape with one line of symmetry, where the line must be vertical, and where the shape consists of squares, triangles and circles”. The two groups’ definitions focused on what types of shapes could be used to form this particular Adinkra. Although there were opportunities to discuss precision (e.g., what does it mean for shapes to be “on the end”?) and come to a consensus on these definitions, we anticipated the discussion would stray from concept of mutuality as described by Eglash. So, we abandoned our plan for students to compare and revise these definitions. Instead, we decided to set up the task of defining mutuality based on Eglash’s description of the concept, by introducing an unplanned prompt for students to consider individually: “What properties make a definition mathematical?”

Students posted their ideas on a Padlet and then asked each other clarifying questions. The class came to a consensus that one student’s description captured their thoughts: “A definition is mathematical when mathematical vocabulary can be used and to create enough specificity to be proven.” They acknowledged that determining what constitutes “mathematical vocabulary” remained unresolved.

This prompt improved the students’ next attempts, though it also showed us places where we might need to continue building students’ mathematical language and reasoning. When students next read Eglash’s description of mutuality and constructed revised definitions of mutuality, they showed more attention to precision. For instance, one student wrote, “Translating the absence (complementary piece) of both shapes to create symmetry in the whole figure”. A second student wrote, “Two or more shapes are mutual if certain areas of each shape can be translated onto another shape, making all of the shapes congruent.” This student continued, “This applies to shapes that are monochromatic, but what about shapes with multiple colors or something?” A third student wrote, “Moving pieces of the same size, shape, and color around to preserve the symmetry of the original figure.” In these revisions, we can see an attention to geometric transformations and more precise language that was absent in their first attempts.

Finally, a student inquired about the candidate definition in a way that mathematicians would: Wondering about the boundaries of examples and non-examples. This student wondered: “Do areas of the translated pieces have to be equal? Can an exchange between shapes be mutual if one shape gives 100 cm^{2} and receives 1 cm^{2}? Just because an exchange is equitable, does that make it mutual?” The student sketched the figures shown in Figure 5, asking: “Are these two shapes mutual?”

**Figure 5. Do these two figures exhibit mutuality?**

This student’s comments relate to an observation that Eglash made in a recent talk attended by the authors. Eglash mentioned that the point of “mutuality” is that the exchanged objects are unequal, so that exchanges create a system of mutual obligations.

In a whole class discussion following these revisions, Dr. Boyce noted that students all wanted to work with the idea of “complete”, and suggested that some students thought about it in terms of geometric transformations, and others thought of “complete” in terms of congruence.

Two students then publicly debated the meaning of “complete”:

“It’s just an intuition, what it means to be complete.”

“Well, what about ‘whole’? Can ‘whole’ be a mathematical word for ‘complete’?”

“Well a whole doughnut has a hole in it, you know?”

“That’s a different whole, that’s ‘hole’.”

“But is a whole doughnut ‘whole’, is it missing a part, or is it complete?”

“Because it’s a circle without a center piece.”

“But for me, if I were looking for a doughnut, I don’t want to fill that in, I’d be cool with that.”

At this point, there were 5 minutes left to the class, and Dr. Sears pointed out that one might need to begin by defining the “whole” as a way to determine “complete”. Dr. Boyce then closed the class with a brief overview of Adinkras.

As this lesson unfolded, our best-laid plans went awry, with no class consensus on a definition of “mutuality”. Yet the lesson also suggested that areas with no consensus could be openings to further mathematical discussion. For instance, the meaning of “complete” and its dependence on a given “whole”, with the example of a doughnut, could be referenced later as an example of how some mathematical definitions depend on an ambient space. The question of whether mutuality requires “congruence” alludes to the idea that congruence is not the only way to conceive of equivalence. Similarity is also a way to consider equivalence. And ultimately, one might envision a new equivalence relation on shapes based on mutuality.

# Debriefing and next steps

We met as a group the next morning to debrief after the lesson. What had we learned from the process of planning, facilitating, and reflecting on the outcome of the lesson? We had field notes and screenshots of student work that were collected by observers during the lesson. We were able to record both whole-class and small-group discussions for subsequent viewing, and we reviewed the video-recordings together and documented what we noticed.

Sometimes when teaching a collaboratively designed lesson, there can be pressure to “stick to the lesson plan”. But as our experience shows, instructors may have to make adjustments to the plan, even in the moment. Regardless of co-planning, instructors need to adapt lessons on the fly. When co-teaching, we have the opportunity to figure out modifications with a partner. When co-planning, we have additional opportunities to learn from previous modifications.

When making adjustments, instructors need to keep in mind their intended learning goals, and also whether different intentions might better suit a lesson. In the lesson’s first iteration, the students were able to compare definitions and also increase their precision. In the process of writing definitions with more precision, the students drew on the language of geometric transformations. We believe the improvement in their definitions was based on an in-the-moment modification: asking students to articulate the properties that make a definition mathematical.

One intention of this lesson was for students to come to consensus on a class definition. This did not happen, and perhaps it could not have in the time that we had. However, something arguably more important happened. Namely, the students began to see areas where a draft definition could be improved. The way that the students took up the mathematical ambiguity of “complete” has hallmarks of genuine mathematical inquiry. They drew on their mathematical knowledge to articulate properties of a new concept, they identified areas where equivalence could be interpreted in different ways, and they identified where their definition needed more precision. As we move forward to subsequent iterations, we will continue to reflect on how students take up precision, what might hamper precision, and what will support precision. Just as importantly, we will attend to where mathematical inquiry is happening and how to help students see the doors that they open.

# Parting Thoughts

It is a rare opportunity for multiple instructors to plan a lesson together, see the result together, and learn about teaching and learning together. Because of the need for online teaching, we were able to collaborate across six different states. We benefited from each others’ different experiences and expertise when planning the lesson. For instance, the Adinkra context was one that Dr. Boyce had previously used. The focus on definition came from Dr. McLeod, with support from others. The sorting task resembled a task that Dr. Lai had written in an entirely different context. The kinds of questions we planned to ask students, and the emphasis on collaboration through the “five R’s”, came from Dr. Sears’s experiences. Together we were able to envision a lesson idea that was more powerful than what we could have designed individually. In the coming months, as we see the lesson unfold in more of our classes, we will learn more about teaching and learning than we could have individually. Most importantly, perhaps especially in these times, we also found a teaching community in each other through this experience.

Acknowledgments. The reported work is supported by NSF DUE-1725837 and NSF DUE-1937512. All opinions are those of the writers and do not necessarily represent the views of the Foundation. We are grateful to Mark Saul and Carolyn Abbott for helpful comments.