In this issue:

• Book Launch: Social Justice Mathematics for Secondary Students
• Rethinking Story Problems to Engage Students’ Mathematical Curiosity
• Feeling the Angle Sum of a Triangle
• The Art of Math Improv

This spring’s issue of the Math Circular is full of novel approaches that will challenge your thinking of what a Math Circle can and should be.

In Hortensia Soto’s “Feeling the Angle Sum of a Triangle,” she introduces the concept of embodied learning in mathematics through a simple yet powerful example. Jason Ermer’s “Rethinking Story Problems to Engage Students’ Mathematical Curiosity” questions the often negative sentiments around story problems, examining the rich history behind such problems and introducing the Arithmetiquities, a compelling classroom adventure merging mathematics and narrative. Pedro Morales-Almazán offers a fresh take on active teaching and learning by reminding us that both are improvised performances (“The Art of Math Improv”). With their newly released book (“Book Launch: Social Justice Mathematics for Secondary Students”), the Math Teachers’ Circle 4 Social Justice provides compelling, classroom-tested lesson plans for bringing relevant social justice issues into Math Circles as well as classrooms.

Taken together, the articles are a wonderful illustration of how Math Circles can serve as a forum for exploring how math connects to other fundamentally human experiences and pursuits.

Our free resources at the Math Circle Network continue to grow. We invite you to connect with other Math Circle leaders and members through our brand-new Forum. We’ve also made a number of recent additions to our activity database, along with developing curated lists of starter activities for student circles and teacher circles.

We hope you enjoy our magazine and look forward to hearing from you!

Sincerely,

The Math Circle Network (circles@aimath.org)

P.S. Interested in writing for us? Please check out our Submissions page.

Math Circles Beyond the Page (and Screen) was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

By Jason Ermer

In 2004, while proctoring the American Invitational Mathematics Examination (AIME), I encountered a problem that started me on a years-long adventure of mathematics and storytelling.

A unicorn is tethered by a 20-foot silver rope to the base of a magician’s cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is (a-(sqrt(b))/c feet, where a, b, and c are positive integers, and c is prime. Find a+b+c. (2004 AIME I, problem 14)

Since my youth I have been a fan of tales like The Hobbit, the Choose Your Own Adventure gamebooks, and fantasy roleplaying games like Dungeons & Dragons. Although the context of the unicorn and the magician’s tower is irrelevant to the mathematics of the AIME problem, my fantasy-loving inner teen was drawn so strongly to the context that I couldn’t rest until I had solved the problem. Something about the story motivated me to explore the mathematics it contained.

As I reflected on the feeling, I wondered whether the same might be possible for my students. Could I engage students’ mathematical curiosity and creativity through narrative?

Reconsidering the Potential of Story Problems

“Story problems” have a bad reputation with students for a variety of reasons. Story problems frequently require a student to extract the relevant information and decide what to do with it, which makes them more challenging than exercises that are purely computational. Contrary to my students’ opinions, I see the opportunity for mathematical modeling as the true virtue of the story problem, especially in a world where technology can assist us with mechanics and computation.

In other ways, though, I understand my students’ consternation. Story problems draw criticism because they so often embed mathematics in what educators have dubbed “pseudo-context.” A pseudo-contextual problem references real objects or situations, but simultaneously asks students to suspend their real-world problem solving skills about the situation. Consider, for example, the farmer who counts how many heads and how many feet are in their barnyard, rather than simply counting the number of pigs and chickens! But pseudo-context has been part of mathematical problem-posing since the Rhind Papyrus, and the prevalence of such “not entirely reasonable” scenarios serves to highlight the power of puzzles to engage the mind.

Furthermore, if we look beyond the two-sentence textbook story problem, we find many skilled authors who have demonstrated the value in merging mathematical content with narrative. An engaging story can draw the reader in and present mathematical ideas in a way that is more clear, compelling, and entertaining than a page of equations or diagrams.

The narrative context of Flatland: A Romance of Many Dimensions by Edwin A. Abbot personifies geometric figures as they explore multi-dimensional spaces. In The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, mathematical ideas arise as a boy named Robert meets the titular Number Devil in his dreams. The Number Devil guides Robert to unexpected locations where the pair discuss concepts like triangular numbers and infinite series. In both Flatland and The Number Devil, deep mathematical ideas are explored as the characters ask questions and have conversations, similar to what might occur in a classroom.

Lewis Carroll, known best for Alice’s Adventures in Wonderland, published many mathematical puzzles, including a serialized set of humorous short stories called A Tangled Tale, each of which includes a mathematical puzzle. Carroll’s goal was to get readers engaged in mathematical thinking by sneaking the puzzles into the story “like medicine so dexterously, but ineffectually, concealed in the jam of our early childhood.”

Introducing The Arithmetiquities

My own attempt to merge mathematics and narrative has followed in Lewis Carroll’s footsteps. The Arithmetiquities is a serialized fantasy adventure tale told through thirty-six mathematical story problems. The story is a Lord of the Rings spoof, featuring a quirky team of adventurers who fan out across the land in their quest to find the Arithmetiquities, a collection of ancient mathemagical artifacts, such as the Sieve of Eratosthenes and the Roots of Unity.

For example, in chapter 26 Egga, Floora, and Greeta — the dwarven shieldmaidens of Lumparland — have ventured into the snow-capped mountains around Yälä Peak in search of the famed Golden Rectangle.

The shieldmaidens, along with their mountaineering party, reach the Shrine of the Golden Rectangle only to find that Hämähäkki, matriarch of the ice spiders is already there, along with her minions!

Hämähäkki is surrounded by her royal guardian regiment, twenty fierce ice spiders with a long and valiant history of service to their matriarch. They have been through so many battles, in fact, that the members of this regiment have, on average, only seven legs (instead of the standard number of legs for an ice spider which, you may have guessed, is eight). One of them, the most grizzled guardian, has only 4 legs left!

Given this information, and knowing that every spider has a whole number of legs, what is the maximum number of spiders in the regiment who might still have all eight legs?

I have shared the Arithmetiquities with students and teachers, and together we have seen how an ongoing story can engage learners and draw them into mathematics:

Student and Teacher Reflections on The Arithmetiquities

It is a great way to share, be creative, and make math more fun. (Sixth grader)

It makes math seem less boring and the stories help me understand the problem instead of just some equation. (Sixth grader)

I’m always looking forward to the next chapter. I’m someone who likes to read, so the fact that a continuous adventure, with many heroic characters, was put into these math problems makes it even more fun! (Sixth grader)

I like how it's not just like typical equations, it makes me think about different things and stretch myself. I also like how you can follow the characters through the different chapters. (Seventh grader)

They have a fun little twist to them and more detail. It’s not like, “Sally had 3 apples,” but instead there are chapters that all come together and are actually fun for kids. (Seventh grader)

Having the problems be in a story form with characters and stories you can follow, is helpful for me and also helps me get more creative with how I solve things. (Seventh grader)

My students LOVE Dennis. I’m not sure why this was the character that grabbed my 7th grade class’s attention this year, but every Friday when we tackle a new chapter of The Arithmetiquities, they are eager to see if Dennis is involved in that week’s adventures. Dennis has even made it into other math problems because my students are so invested in him! (Seventh grade teacher)

Creating Narrative Settings in a Math Circle

Many chapters in The Arithmetiquities are suitable for discussion at a Mathematics Circle for students or teachers. At a recent session of MTC Austin, we discussed creating narrative settings to engage the particular interests of one’s students. We considered story problems that might take place in popular literary or video-game settings, or draw on popular trends from anime to superheroes. Participants were invited to flex their creative writing muscles and spice up some sample “boring” story problems. One participant chose this original:

I have a cylindrical tin of popcorn that is 18 inches tall and has a radius of 4 inches. I want to use the tin for something else and need to empty the popcorn into a box. The box is 8 inches long, 8 inches wide, and 14 inches tall. Will the popcorn fit in the box? Explain.

From that starting place emerged this epic quest with an ecological flair:

The maiden Zeelan carries the Master Flask, a flask of cylindrical shape that is 18 inches tall and has a radius of 4 inches. The Flask contains the last seeds of the great Normwood Tree. This Tree is the most important tree in the realm and there are only three left, which means the seeds are invaluable. She has been tasked with the job of retrieving the ashes of the most celebrated warrior maiden of the land and has been requested to use the famed Master Flask. She must store the most prized seeds in a safe place and has been offered a treasure chest by her trusted warrior, Zeldon. The chest measures 8 inches long, 8 inches wide, and 14 inches tall. Will all of the seeds be safe? Or will some have to be stored in a different container? The future of the Normwood Tree species depends on this answer!

The Arithmetiquities is available online for teachers to explore and use with students (www.arithmetiquities.org). Plus, every textbook contains a wealth of humdrum story problems ready for your own creative reinterpretation! Consider undertaking a mathematical adventure — either for yourself as a creative exercise, or to share with the students or teachers you work with!

About the Author

Equal parts storyteller, teacher, and mathematician, Jason Ermer (he/him) was once voted “most likely to lead children to their fictional doom.” Jason was one of the founders of the Mathematics Teachers’ Circle of Austin. He lives in Seattle, WA where he leads sessions of exploration and discovery with both young people and adults. Find Jason online at www.mythematics.org or on Twitter @mythematics.

Rethinking Story Problems to Engage Students’ Mathematical Curiosity was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

By Hortensia Soto

Learning and the Senses

Growing up, my mom used to say “usen sus cinco sentidos” (use your five senses) when she wanted us kids to use reason or to behave in a logical manner. This phrase gained new meaning for me when I discovered embodied learning, which is a philosophy that claims that learning occurs when the meaning of what is learned is grounded in body movement (action), perception, and sensation. As such, those of us who follow this philosophy of learning create learning environments where learners engage in seeing, hearing, and feeling — three of the five senses. Let’s see what this looks like.

Walking the Sides and Angles of a Triangle

In elementary or middle school, students may discover that the sum of the interior angles of a Euclidean triangle is 180 degrees by drawing a triangle, measuring each angle, and summing up their angles to obtain something that is approximately 180 degrees. The students might also draw a triangle, color the angles different colors, tear the angles, and line them up to illustrate that the angles create a straight line, which has a measure of 180 degrees as shown in Figure 1, above.

Both of these activities extend beyond hearing the teacher say or seeing the teacher write “the sum of the interior angles of a triangle is 180 degrees.” Another way to help students discover this fact is by having them walk the sides of the triangle while traversing the interior angles. This activity gets students to move, make mistakes, learn from them, and begin to feel the mathematics.

The activity begins by creating a large triangle on the floor using any materials that are handy, such as painter’s tape. A volunteer, whom we will refer to as the walker, then stands on a vertex of the triangle and walks on a side of the triangle. The walker then rotates by the amount of the first interior angle, then walks the second side, and so forth until they get back to the first vertex. The goal of this activity is to have the students recognize that they will end up facing the opposite direction from where they started. Thus, they will have rotated 180 degrees, as illustrated in Figure 2 with the red and black arrows at vertex A.

Distinguishing Between Interior and Exterior Angles

This may sound like an easy task, but it is not at all intuitive and in fact, it can seem unnatural for learners. Learners of all ages often end up facing the same direction they started in, indicating that they rotated a full 360 degrees instead of only 180 degrees. In this section, I describe what happened when I implemented the activity virtually as part of an AIM Math Circle. The short video clip below illustrates a participant traversing the exterior angles instead of the interior angles, which occurs quite frequently because rotating by the measure of the interior angle at vertex B, requires that one walk backwards along segment BC.

The fact that the walker ends up facing the same direction can result in cognitive dissonance for everyone in the room, not just the walker. This is an opportunity for a rich discussion about what happened. For example, at this Math Circle, one participant tried to explain by saying “so you turned and each time you turned an angle,” as she gestured a turn with her hand each time that she uttered the words “turned” and “angle” as shown in the video:

A bit befuddled, she then went on to explain that the sum of the angles is 180 degrees, so that she expected the walker to be facing the other direction.

A second participant then conveyed that the walker traversed the exterior angles instead of the interior angles as she gestured the exterior and interior angles:

This realization is a great learning opportunity because in traversing the exterior angles instead of the interior angles, students also discover that the sum of the exterior angles of a triangle is 360 degrees. At this Math Circle, the walker stated, “I actually can feel the math. I can tell I am doing that [walking the exterior angle].”

Another participant also illustrated that if one extends one’s arms as one walks and goes “to the tip of the vertex, then you rotate” — as she rotated her whole body with arms extended — “and you see you are sweeping the exterior angle so you can do this action,” as she re-gestured from left to right (see Figure 3).

After the walker realized that she had walked the exterior angles, she re-walked the triangle and the interior angles as the activity was designed and illustrated on Figure 2. Besides feeling the math, she also conveyed that she enjoyed walking the exterior angles because she felt like she was dancing. She remarked, “I felt like I was making a dance turn.”

While this activity is engaging, it tends to mostly engage the walker. Other students can also participate at their desk by drawing a large triangle on a piece of paper and pushing a small toy, such as a droid, along the sides and angle of their triangle as shown in Figure 5.

Notice that the droid in the last image is facing the opposite direction compared to the first image. Wanting the participant to also feel the rotation in her hand, I asked her to push the droid without taking her hand off the droid. This allowed her to see that her hand also ended up facing the opposite direction. She remarked that this made her “feel like a contortionist a little bit.” The short video clip illustrates this movement.

Assimilating New Movements

Certainly, a skier does not do a 180 degree turn in this way. Turns such as those are made up of a continuous rotation following the same path of an arc. This might be why the walker expressed feeling like a dancer when she traversed the exterior angles.

That rotating 180 degrees to traverse the interior angles of a triangle could make one feel awkward or like a contortionist sounds about right: This is a movement where one experiences rotating 180 degrees in a new way. Similarly, we might also say that learning itself is moving in a new way.

About the Author

Hortensia Soto is a Professor of Mathematics at Colorado State University. She is currently the Associate Secretary of the Mathematical Association of America (MAA) and the upcoming President-Elect of the MAA. Much of her teaching and research embraces embodied learning. Learn more about Embodied Mathematics Imagination and Cognition and follow her on Twitter @HortensiaSoto3.

Feeling the Angle Sum of a Triangle was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

By Pedro Morales-Almazán

“I hate math!”

That was my opening line for the October session of the Santa Cruz Math Teachers’ Circle. As a mathematician, I have heard this phrase multiple times. Students, friends, family members, and even random strangers, we all have heard this expression when they discover that we do — and love — mathematics. But, how do we counter those who have such an awful attitude towards math?

Simply put: we don’t.

This counterintuitive response is no surprise for improvisers. Improv, short for improvisational theater, has become popular mainly due to comedy shows, professional development programs, and improv courses. In improv, being spontaneous and unscripted is key. The challenge is, despite this uncertainty, to develop coherent and interesting stories. More than wit, improv requires training and practice. Just as any other skill, great improvisers learn and practice techniques to better navigate the unpredictable.

Teaching as Improvisation

As teachers, we are very familiar with the unpredictable. Even in highly planned lessons, we often find unforeseen questions or situations arising from our interactions with students. No matter how the , unexpected situations are bound to happen. There is a famous saying that goes: life is improvised. This is definitely true for teaching as well. An experienced teacher has learned how to handle these uncertainties, and even, to take advantage of them. We can also learn how to deal with this in a more structured way. We can systematically study how to handle uncertainty and how to use it to our advantage. This is one of the main values of improv for us as teachers.

I started in the world of improv about five years ago. I found it to be a very enjoyable and liberating experience. The playfulness and free-flow of the activities felt very different from the typical academic environment that I was used to. This made the experience very engaging. I started to analyze and to research a bit more about improv in academic settings. There is quite a lot of literature about the benefits of improv in soft-skill development. By no means do I consider myself an expert improviser, but I am passionate about the applications of improv to teaching and learning.

There are two main approaches to improv in the classroom. One is to incorporate it as an activity with students. Playing improv games with students increases engagement in the class and develops soft skills that sometimes can be overlooked in a mathematics curriculum. This is closely related to active learning. There is plenty of research done in this area, especially for mathematics. Another approach is related to what I call active teaching. It is possible to have a classroom with many active learning strategies conducted by a passive teacher. An active teacher is responsive and adapts to what is happening in the classroom. Even if a teacher does not use improv games or activities in the classroom, they can use the lessons promoted by it.

The Rules of Improv

Some of the most important lessons from improv are known as the Rules of Improv. These so-called rules are useful techniques that help navigate uncertainty. The most common ones are:

• Yes, and
• Active listening
• Being present
• Team work
• Failure

These rules develop important skills needed to interact with students and to promote mathematical thinking. We can say that these five rules focus on perception, communication, mindfulness, cooperation, and compassion.

© Aude Vanlathem / www.audevan.com

A Math Circle Approach to Improv

At our Math Teachers’ Circle, many of our participants had expressed interest in the connections between improv and mathematics, so we made our October 2021 session about “Math Improv.”

The session was held over Zoom and we had about 15 participants. We have a very diverse group of people that attend our Circle, including high school teachers, school administrators, retired mathematicians, and math enthusiasts. This diversity in backgrounds and skill sets converges in our passion and interest for mathematics.

The first rule that we approached for our session was “Yes, and.” We started with a simple game:

Game: Yes, and

Instructions: In a circle, players go in sequence saying a statement. (To facilitate “going around circle” on Zoom, participants can use grid view and rearrange their grids to follow the same order.) Players need to add to what was said previously by starting their phrase: “yes, and.”

Goal: The goal is to collectively construct a story. It is OK if the story is not realistic or defies logic. What is important is to account for what is being said before and not denying the previous players’ statements.

The main idea of this game is to acknowledge everyone’s reality. For example, the statement “I hate math” expresses a very common perception among non-mathematicians. We can certainly deny it and tell them that math is actually beautiful. However, implicitly we are saying that their perception is wrong. Denial does not move any conversation forward. Rather, it halts any progress.

By “yes-and”-ing an idea, we can move the conversation forward into something that incorporates the different perceptions present in a discussion. Beware: “yes-and”-ing an idea does not necessarily mean automatically agreeing with it. It means to recognize that the other person thinks, believes, or feels it. When the other person states “I hate math,” we can acknowledge that feeling while at the same time adding our own perspective to the story.

A typical situation inside the classroom that many of us often encounter is when a student says, “This problem is difficult.” Even well-intentioned teachers might respond with, “Let me show you how this is easy.” This is a subtle denial that implies that the student is wrong in thinking that the problem is difficult for them. A better approach is to “yes-and” the student’s statement — not to say that the problem is difficult, but rather to acknowledge that it is difficult for the student. In this way, the student and the teacher can team up, agreeing on their mental and emotional states.

Continuing with our session, we played another variation of the “Yes, and“ game:

Game: The Alphabet

Instructions: Similar to “Yes, and” but players don’t start with the phrase “Yes, and.” Instead, players cycle through the alphabet, beginning their statements with a word that starts with the next letter of the alphabet.

Goal: The goal is to pay attention to what the previous player said, not only in meaning, but also to actively listen to their first word.

Many times we listen only to reply. Active listening promotes listening to understand. In the classroom, teachers often answer their own questions rather than the students’. By being more intentional about the way we listen, we can improve our communication. This is key in the classroom setting, as it promotes engagement and trust with students.

It is particularly important to pay close attention to the wording that students use. As novices, students often don’t have a well-articulated mathematical vocabulary, which can prevent us from correctly interpreting the actual questions that students pose in class.

A common situation occurs when a student says, “This problem doesn’t make sense.” Often, students mean different things when using this phrase. This can be “I haven’t seen a problem like this before,” or “I don’t understand some of the terminology,” or “I don’t know how to start.” Active listening helps us to pinpoint the specific difficulty that a student is facing rather than assuming it.

We played a few more improv games in our MTC session focusing on the “Yes, and” and “Active listening” rules of improv. It was very interesting to see the interactions between people with different backgrounds and interests. Even in Zoom, we experienced a very engaging session.

It is important to remark that most of these activities and strategies are not math-specific in the sense that improv activities can be used in many different contexts. However, one aspect relevant to mathematics is that improv brings a human touch to it. Traditionally we experience math classes as very fact-oriented. Even in active settings, “yes-and”ing, listening, and embracing failure might not be present. Improv then becomes particularly relevant in math education, not necessarily from the content point of view, but from the communication and the metacognitive perspective. With improv, we can embrace both the student and the teacher as human beings, interacting together while doing mathematics. For this reason, I strongly believe that Improv brings many aspects that are particularly needed in the mathematics classroom.

Taking It Further

These activities are really just the tip of the iceberg regarding math improv. There are many other wonderful improv games that promote perception, communication, mindfulness, cooperation, and compassion — important skills to promote not only among students but also with teachers.

There are also many books and internet repositories focusing on improv. A very good resource is the Improv Encyclopedia. A list of additional recommended reading for learning more about improv is given at the end of this article.

Games generally make for engaging MTC activities and promote fruitful discussions. Improv provides a structured framework to engage with students and to handle unexpected situations, going beyond ice-breaker activities and games to give a more holistic approach to the teaching and learning experience.

Teaching is improvised, and it is much better when we have the tools to intentionally handle and enjoy the improvisation.

About the Author

Originally from Guatemala, Pedro Morales-Almazán is currently a Teaching Professor in the Mathematics Department at the University of California, Santa Cruz. He also directs the Santa Cruz Math Teachers’ Circle. He says, “I discovered mathematics when participating in math competitions in high school. Since then, I have been attracted to the puzzles and brain teasers that mathematics provides.” His teaching interests are related to inquiry-based learning, the use of technology for adaptive teaching, and improving student engagement. His research focuses on applications of Zeta Functions in Number Theory and Quantum Field Theory. Find him on Twitter @p3d40 and on Medium (in Spanish).

Recommended Reading on Improv
George, M. (2012). How Mathematics Teaching Can Be Like Improv Theater. MathAMATYC Educator, 3(2), 21–23.

Morales-Almazan, P. (2021). Improv practices in Mathematics active teaching. PRIMUS, 1–16.

Sawyer, K. (2019). The creative classroom: Innovative teaching for 21st-century learners. Teachers College Press.

Young, A. (2013). Improvisation in the mathematics classroom. PRIMUS, 23(5), 467–476.

The Art of Math Improv was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

By Kyle R. Evans and Megan Staples

Connecting Mathematics and Social Justice: Lessons and Resources for Secondary Math Teachers is an open-access book designed to be a resource guide of detailed lesson plans that bring together social justice themes and secondary mathematics content. The lessons and themes generally focus on issues in the state of Connecticut but are adaptable to any local context. Kyle Evans (Trinity College) and Megan Staples (University of Connecticut) are the co-founders of the Math Teachers’ Circle 4 Social Justice and serve as editors of the book.

Six chapters, each with lessons and resources, were written and designed by 14 secondary mathematics educators in the state who participated in the first Summer Stars cohort during the summer of 2021. As part of this two-week (virtual) summer experience, the cohort engaged in valuable conversations surrounding mathematics and social justice, heard from guest speakers on a range of topics, and learned about a framework to ground the educators as they developed their lessons for the book. A seventh chapter was contributed by master’s students engage in a related project.

Each of the lessons from the book is also being featured in one of our monthly MTC workshops during 2021–2022. During the workshops, participants are actively involved in problem solving and in critical conversations, from the points of view of both the learner and the teacher.

Please visit the MTC4SJ website for resources from previous workshops, information on upcoming workshops, to join the mailing list, and for Spotlight features on Connecticut educators.

About the Authors and the MTC4SJ

Kyle R. Evans is Visiting Assistant Professor of Mathematics at Trinity College in Hartford, Connecticut. Megan Staples is Associate Professor of Mathematics Education in the Neag School of Education at the University of Connecticut in Storrs, Connecticut.

The Mathematics Teachers’ Circle 4 Social Justice (MTC4SJ), led by Kyle Evans, Megan Staples, Scott Kapralos, and Georgina Rivera, is based in Connecticut and seeks to support teachers as mathematicians and social justice educators by promoting problem solving and critical conversations.

Book Launch: Social Justice Mathematics for Secondary Students was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

By Nick Grener, Fred Peck, and Matt Roscoe

By virtue of the fact that you are reading this, we assume that you hold a positive opinion of the phenomenon known as math circles. It is likely that you have experienced a math circle, either as a participant or perhaps as a facilitator. Furthermore, you probably enjoyed the experience and found it enriching, rewarding, or just plain fun.

Certainly, though, not every circle session generates the same experience for each participant and facilitator. For example, have you ever found yourself generating an interesting question about the topic at hand, but then feeling frustrated that you couldn’t pursue it because the facilitator is trying to direct the room in a different direction? Or experiencing stress while preparing to facilitate a circle where you did not know the audience well and were nervous about appearing sufficiently competent? Or even having to cancel a calendared math circle session for want of a willing facilitator?

In the years since we began to promote math circles in Montana for students, educators, and the general public, we have experienced all of these situations. Recently we have found a way to address some of these concerns that has fundamentally changed the nature of our gatherings. While this change is still playing out, we are excited to share here our new methodology, our observations on how we believe our Math Teachers’ Circle has improved as a result, and some resources that might help you implement it at your own math circle.

More specifically, over the years, our gatherings have evolved to adhere to a new organizing principle in which there is no facilitator or moderator, by which we mean that there is no “teacher,” “expert,” or “person who knows the answer and how to get it.” Instead, each month, a different member of our math circle community volunteers to bring a problem, question, or general setting that is mathematical in nature for the group to collectively consider. We call those who bring the problem “originators” of the session.

Importantly, for problems where an answer is pursued, the originator does not know the solution (or, at least, the full solution) to the problem. Oftentimes, the originator will steer away from the presentation of a specific question or set of questions altogether, and leave the question-generating aspect of the experience completely up to whomever shows up to the session. In any case, after the initial sharing of their seed idea, the originator steps back from the foreground and participates as an equal in the session with everybody else in the room.

The result looks something like this:

Perhaps the primary fear of running a math circle without a person at the “center” is that it will be too unfocused and will devolve into chaos, or — worse — disinterest. Our experience in Missoula, however, is that engagement is uniformly good, which we attribute to the fact that everyone in attendance is tacitly agreeing to be accountable for how the experience will play out. In other words, rather than one person organizing and structuring — and therefore shouldering all of the responsibility for — the gathering, the participants collectively take responsibility for engaging in sustained mathematical activity.

It also seems to us that the energy in the room is especially high in those cases where participants are invited to own the direction that the session takes by forming the questions that will be worked on. For example, a board game might be presented with the simple prompt of, “What math questions does this game make you think of?”, coupled with the freedom to actually explore the questions generated by the group.

Here are a few other advantages of unfacilitated math circles that we have experienced:

• They are stress-free to organize. It is far less imposing to originate a session than to facilitate a session, since the former merely requires sharing something of interest to you, whereas the latter imposes the expectations that you have thorough knowledge of the content of the session and keep the room directed during the meeting itself.
• They are more fun for the originator, since the originator gets to participate in the mathematical activity of the group, rather than having to orchestrate the activity. In fact, the circle is more exciting for all involved, since each discovery made by the group is an “aha” moment for every single person in the room.
• There is no clear ending to the session. In fact, our discussions sometimes continue well beyond the physical gathering as we continue to explore and make new discoveries and share them with one another via email. This, perhaps, is the single most important distinction from a traditional session where the facilitator feels pressure to direct the mathematical activity toward some predetermined end point, thereby discouraging the opportunity for further study.

So, how might you experiment with this method at your own circle? Here are some ways originators have recently begun meetings of our math circle that you might consider trying:

• Sharing a low-floor problem that (to the originator’s knowledge) has not yet been solved. A solid library of accessible unsolved problems from the history of mathematics can be found at MathPickle.
• Bringing an object or activity that can be mathematized. We have used games like Tenzi, Settlers of Catan, and Gobblet Gobblers to generate rich discussions. We have made and explored mathematical artwork including Celtic Knots and Spirographs. Finally, we have investigated interesting mathematical objects, such as 3D-printed Menger Sponges and sets of TANGLE toys.
• Providing participant handouts for one of the sessions in the Math Teachers’ Circle Network library. The session should look intriguing to the originator, but it should be something that they have not spent significant time working on themselves prior to participating in the circle. Often, the handouts themselves are quite structured. In this case, the originator might remove much of the structure while preserving the central question.
• Bringing a question that the originator is curious about. For example, in a recent gathering, a member of our group explained that they had, for a long time, been perplexed by the geometry of the formula for the tetrahedral numbers. The algebraic formula for the nth tetrahedral number suggests that 6 copies of the nth tetrahedron can be arranged to form an n × (n+1) × (n+2) rectangular prism. However, the originator could not figure out how to make such an arrangement, so they brought the question to our group. Originators have also brought problems that they saw on social media and found intriguing. James Tanton’s Twitter feed is a great source for these.

To be sure, the transition to a circle without a facilitator may take some adjustment, particularly if your sessions typically have a large number of folks attending. But if these ideas interest you, we encourage you to discuss them, give it a try, and see what happens. You might just find yourself renaming your “Math Circle” (the set of all points equidistant from a fixed center/leader) to a “Math…” — wait, what do you call a set of n points that are all equidistant from each other?

About the Authors

Nick Grener is a mathematics teacher at Hellgate High School in Missoula, Montana. Fred Peck and Matt Roscoe are mathematics professors at the University of Montana. All are leaders of the Montana Math Teachers’ Circle.

Circles with No Center was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

Involving Students in the Beauty and Power of Mathematics

By Molly A. Walker

The San Jose Math Circle (SJMC) in Northern California is one of the nation’s oldest math circles, with its beginnings in 1998. Prior to the pandemic, the math circle offered two parallel weekly meetings for middle (seventh and eighth grade) and high school (ninth through 12th grade) students. About 60 youth gathered in a classroom at San José State University on Friday evenings for 2 hours to tackle difficult, original and always fun mathematical problems. In September 2020, the circle moved to online instruction, running a single group of about 40 students in grades 7 through 11.

In addition to hosting weekly meetings, SJMC is also a co-sponsor of Bay Area Mathematical Adventures (BAMA), a series of public lectures featuring renowned mathematicians, which “circlers” are encouraged to attend. The circle is also part of the annual Bay Area Mathematical Olympiad, typically held the last Wednesday in February at Bay Area schools.

Origins and Leadership

How did SJMC get its start? In 1998, co-founders Tatiana Shubin and Tom Davis attended a talk by Zvezdelina Stankova, founder of the Berkeley Math Circle, who discussed math circles in Soviet bloc countries, and how she wanted to start similar math programs and a math contest in the San Francisco Bay Area.

While Tatiana is credited with doing most of the early work to get the math circle going, the backing and enthusiasm for the program offered by many individuals and two math institutes — the American Institute of Mathematics (AIM) in San Jose and the Mathematical Sciences Research Institute (MSRI) in Berkeley — were crucial. Other early SJMC co-founders and co-directors were Brian Conrey, director of AIM; David Hayes, San José State math professor, now retired; and Raghu Subramanian, founder and CEO of startup Nonadoo, then a recent PhD in electrical engineering. Longtime SJMC leaders also include Joshua Zucker, a veteran instructor with the Art of Problem Solving, along with Alon Amit, vice president at Intuit. Tom Freeman, retired computer software professional, served as an SJMC director for 10 years before moving to Arizona; he continues to help out with the math circle long-distance.

“The best part for me is working with the kids,” says SJMC co-founder Tom Davis. “There are always a few who are a lot smarter than I am, but I have the advantage of 50 more years of mathematics work, so we’re evenly matched. When I lead a session, I am totally engaged, and after an evening session, I am sometimes so energized that I have a hard time sleeping. I often attend sessions by other leaders and that gives me ideas for my own future sessions, and I learn some cool math.”

“Kids are naturally inquisitive and curious,” says SJMC co-founder Tatiana Shubin, who recently retired as a math professor at San José State University. “They are not satisfied with an approach often taken in their school math classes: ‘Here’s an algorithm, just memorize it and apply it to crunch numbers.’ True, algorithms are useful, but mathematics is so much more than that. At SJMC we want to involve students in the real beauty and power of mathematics through hands-on investigation and challenging yet deeply satisfying problems. We also believe that it is extremely important for students to meet their like-minded peers and to exchange ideas with them and with professional mathematicians.”

“Tatiana Shubin invites a variety of mathematicians to lead the sessions, each with a particular passion and teaching approach. I consider these differences and diversity of topics one of the best parts of the math circle,” says Aida Rivera, SJMC parent and administrator.

Tom Davis describes what it was like in the early days. “At first it was hard just getting students to attend. Nobody knew what a math circle was,” he recalls. “After just a couple of years, however, it became easier, with the siblings or classmates of attendees showing up. Sometimes local teachers would recommend the circles to their students.”

Joshua Zucker is equally enthusiastic about his math circles work. “Students get excited and discover new things — sometimes even new things for me! Last year at San Jose we explored what happens if you take all the ways of writing a list of numbers that have a certain total, say 7, and organize them based on how many 1s are used. For instance, with no 1s, you have 7, 5+2, 2+5, 4+3, 3+4, 3+2+2, 2+3+2, 2+2+3. If you use some 1s you get a lot more possibilities for a while, but not so many options with five or seven of them (and no possibilities with exactly six 1s!). We discovered a bunch of patterns that I had never seen before as we dug more deeply into this, and eventually found reasons why those patterns occurred.”

Joshua says the hardest part of any instruction is what to do with the students who seem disengaged. “You wonder if they are just confused, and with some help will get back to a problem. Or are they just not interested in the topic? Or maybe they don’t want to be there and are only sitting in the room because their parents are forcing them to come? How much should I slow down (and risk being boring for some of the students) to try to get these people back on track?

“Sometimes when they’re not very confident about what’s going on, it’s a challenge to get the students to take risks — to speak up or try to answer a question or offer an idea,” he adds. “On the other hand, one of the big successes sometimes is getting them to be more comfortable with the feeling of not knowing what’s going on. That’s a lot of what life as a mathematician is!”

Student Success Stories

Parent and math circle administrator Aida Rivera’s three children have all benefited from SJMC. “When Tatiana met my son Sebastián Urdaneta in 2015, she invited him to join SJMC,” says Aida.

Now a student at Foothill College in California, Sebastián joined SJMC in seventh grade and found the experience a revelation. “I encountered a new side of math I had not seen before. I was in awe at how the professors would introduce a seemingly simple topic and expand it into something deep,” says Sebastián. “I was able to explore many topics with people who really knew the material and were very happy to share their know-how with us.”

After going through SJMC’s middle school and high school programs, Sebastián decided he wanted to remain part of SJMC and now contributes to the math circle community as an assistant. “It is pretty neat to see the spark of curiosity that I had in me in the faces of the younger students!” he says.

Another involved parent who is also a math teacher is Iulia Rusu. Born in Eastern Europe, Iulia was familiar with the math circle model. When she learned about SJMC, she encouraged her daughter Inimai Subramanian to apply.

Last summer, Iulia and Inimai had the opportunity to attend the Alliance of Indigenous Math Circles (AIMC) Math Camp at Navajo Prep in Farmington, New Mexico, with Tatiana Shubin, who in addition to co-founding the SJMC is a co-founder and co-director of the AIMC. As a junior mentor, Inimai worked alongside university professors to help Native American students solve challenging math problems and also ran a table station with complex puzzles at the local Julia Robinson Mathematics Festival. She led one of the camp’s cultural activities by performing and teaching hand gestures and footwork from ancient Indian classical dance to students and staff. Every evening after dinner, she helped the students program sphero bots to autonomously move within the tracks they had taped on the gym floor.

“Being a mentor for the math camp taught me not only how I can help others, but also how to engage with different students in a considerate and tactful way,” says Inimai. “Coming from a wide variety of tribes, the students were quite rooted in their respective cultures and found it hard to be comfortable around an outsider. At first, it was difficult to connect with them. Once we started discussing mathematical concepts, their inhibitions quickly vanished; they became more open and eager to describe their ideas on how to solve the assigned problems. That is when I realized that mathematics is the universal language of all cultures, and that it has the power to bring sets of completely different people together.”

The experience of Timothy Ngotiaoco is also instructive. Now a graduate student at MIT studying the representation theory of Lie groups, he did math research during his gap year with SJSU mathematician Daniel Goldston, whom he met at the circle. Timothy joined SJMC as a sixth or seventh grader in 2009 and was involved through 2015. Because he had been homeschooled for three years, it was especially important to him to have found a place to learn math and peers to learn it with. He was the captain of the SJMC Math Wrangle team and served as a junior mentor at Baa Hózhó Math Camp in Arizona.

According to Timothy, “The math wrangle was amazing because we needed to present proofs and solutions to more theoretical problems. It’s a fantastic way to improve one’s mathematical ability, since you need to learn to prove things rigorously. My favorite part is the rebuttal process, where the opposing team can point out flaws in your proof and propose fixes, or perhaps even a better solution. Now that I’m a graduate student, I value being able to do things like this, since I often need to question the theorems I read or prove, and make sure the proofs I have are correct.

“I was certainly blessed to have a mother willing to drive me everywhere and bring me to so many math programs. I was also fortunate to live in the Bay Area, where there are so many math programs offered. I didn’t realize until I came to MIT how rare this is. Many math students I’ve met had never heard of math competitions or math circles. Those of us who did have these experiences are often better prepared for math subjects, simply because we have seen more math, and by that I don’t mean taking advanced courses like abstract algebra, multivariable calculus, linear algebra. I didn’t do much of that in high school.”

“Being part of a math community like SJMC is the sort of environment that many mathematically curious and math-loving kids are looking for,” observes Aida Rivera. “Yet they typically don’t find this within their schools. SJMC is an environment where what is being presented is new and challenging, and that’s completely okay. It’s where persevering brings you amazing ‘Aha!’ moments that will long be remembered and will serve as inspiration to tackle more uncharted territory.”

Favorite Math Circle Problems from SJMC Leaders

From Joshua Zucker: What are ways of organizing all the lists of positive integers with a total of 7, and what do you discover with each different way you can organize them?

From Tom Davis: Suppose you start with one pile of 10 coins. To make a move, you pick an existing pile that has more than one coin in it and divide it (not necessarily evenly) into two piles, so your first move might be to split the pile of 10 into piles of 6 and 4. Whatever those numbers are, your “score” is increased by the product of those numbers, so for the example above, your score would increase by 6x4=24. (If your first move had split the pile to make two piles of 5, you would have added 5x5=25 to your score instead of 24.) As you make moves, the piles get smaller and smaller, and finally you will have ten piles with 1 coin in each. What is the largest possible total score you can get? What is the smallest total score you can get?

About the Author

Molly A. Walker is a business communication consultant and writer living in Oakland, California. She is on twitter @mwalkercom.

San Jose Math Circle: Involving Students in the Beauty and Power of Mathematics was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

Boundary-Spanning Mathematical Play

By Teresa K. Dunleavy, Jennifer N. Lovett, Elizabeth K. Metts, and Shannon N. Reider

Overview: In this piece, we share the boundary-spanning ways we engage our work with the Math Teachers’ Circle of Middle Tennessee. We conceptualize mathematical play within and beyond its typical contexts — across a spectrum of higher education and PK-12 classroom professionals.

In our work as mathematics professionals for the Math Teachers’ Circle of Middle Tennessee (MTCMidTN)¹, we are deeply connected to engaging in mathematical play. As teachers and teacher educators, our hope for the MTCMidTN is to engage participants as mathematicians who both enjoy mathematical exploration and who learn from and with one another. We identify as boundary spanners: We are professors, teachers, and university students who engage in doing mathematics together, hoping to disrupt typical notions of what it means to engage in mathematics across and between higher education and PK-12 contexts.

The MTCMidTN was designed as a collaboration between in-service teachers who were graduates of the Vanderbilt University M.Ed. Programs in Secondary Education and M.Ed. in Teaching & Learning in Urban Settings, along with mathematics teacher education faculty from Vanderbilt University and Middle Tennessee State University. The first author is an assistant professor of math education and co-director of secondary education at Vanderbilt; the second author is an assistant professor of mathematics at Middle Tennessee State; the third taught high school mathematics for eight years and is now a doctoral student; the fourth taught middle school mathematics for eight years and was a numeracy coach during the 2020–2021 school year, when this article was written.

We center MTCMidTN around mathematical play, aiming to engage in complex mathematics for the sake of joy. We specifically use the term “mathematical play” to describe engaging in math exploration based on enjoyment and curiosity, as driven by doing mathematics. Mathematical play also implies that, against the typical norm, the purpose of engaging with a given math idea is intentionally not answer-oriented (see, e.g., Dunleavy, 2018). Instead, we are curious to hear from one another about creative ways to solve problems.

In this article, we examine our specific boundary-spanning learning context, in order to illustrate the potential ways that mathematics professionals can learn in and through practice, with and from one another (Hundley et al., 2018; Lambert, 2010). We also seek to illustrate how different kinds of boundary-spanning experiences can nurture the growth and development of mathematics professionals.

Across our work, we engage in mathematical play in a way that intentionally weaves mathematical pedagogy and practice, so as to support the ongoing development of who we each are as mathematicians and how we advance our own mathematical thinking, alongside the mathematical thinking of those we teach and with whom we interact. As mathematics teacher educators, we bring a situated view of learning (Lave & Wenger, 1991), such that professional learners develop through changing participation while engaging in our community. Due to the lack of joy and narrow success in mathematics nationally, we also center research on equitable teaching and learning (e.g., Aguirre et al., 2013; Cohen & Lotan, 2014; Dunleavy, 2018).

One key role for us as boundary spanners involves mediating the power structures of the participants, so that not all of the knowledge is perceived to be with the people who either have the most experience or the highest status. We work to establish a community where the developing expertise is distributed and valued across participants from different backgrounds and who bring different experiences. Fundamentally, our work involves unpacking who mathematics professionals are, where boundary-spanning mathematical play takes place, and what kinds of opportunities for mathematical play are bridged between individuals.

Design Principles for Boundary-Spanning Mathematical Play

Acknowledging the daily pressure and stresses that often prevent teachers’ enjoyment of mathematics in and of itself, we have conceived of the MTCMidTN as focused on three design principles that shape how we engage in mathematical play together:

• First, we are driven by the notion that mathematics teachers from different professional backgrounds have something to learn from one another.
• Second, MTCMidTN must exist outside university coursework and required professional development meetings.
• Third, mathematical play intentionally bridges coursework and traditional fieldwork.

In the following section, we unpack each of these design principles.

Design Principle #1: Who.

Boundary-spanning mathematical play includes professionals from higher education and PK-12.

One of the key features of a Math Teachers’ Circle, as defined by the national Math Teachers’ Circle Network (2019), is that we are made of professionals from different professional spheres. Powerful learning happens in the classroom in differentiated groups, when folks come from different backgrounds and have different kinds of previous experiences (Cohen & Lotan, 2014). This idea is applied to our intentional, random, heterogeneous grouping when we engage in work together. A heterogeneous group of professionals affords opportunities for individuals from different professional homes to learn from one another’s diverse experiences in the same way that students learn from one another, addressing status hierarchies that exist across the spheres of our professional work (Aguirre et al., 2013; Cohen & Lotan, 2014; Dunleavy, 2018). We have found that mathematical play across professional contexts is critical to our foundational design.

We have also witnessed how our diversity of expertise results in the most powerful kinds of opportunities for mathematical play. Across four years of working together, we have engaged over 175 individuals, spanning professional identities across university faculty, university students, school teachers, administrators, and district leaders.

One example of the power of boundary spanning across professionals happened during the first meeting of our second year. We passed out the paper-folding puzzle below as a warm-up:

The warm-up was a puzzle: Participants were invited to cut out the figure along the outer border, so that the squares remained connected, and to fold the paper such that the pictured eight letters ended up in alphabetical order on top of one another. During this time, participants asked questions about whether one could cut in between the letters in order to complete the task. When we replied no, that cuts in between letters were not allowed, two professors — a mathematics professor and a mathematics education professor — insisted that some cuts in between letters needed to be made in order to complete the task. Simultaneously, two middle school teachers and a teacher candidate solved the task without the requested “in-between cuts” that the professors insisted were necessary. After this MTCMidTN meeting, one of the mathematics professors reported walking around with the paper in his pocket for the next week, working to solve it every so often without being successful.

What we find particularly powerful about this example is that whereas mathematics professors might typically be perceived to be as or more capable to complete a given math task than middle or high school teachers, this task flipped that potentially predicted outcome. The fact that MTCMidTN was able to offer a task requiring mathematical thinking that featured middle school teachers’ success alongside mathematics professors’ struggles illustrates our potential for distributing mathematical knowledge across professionals. In this context, the power of who can do interesting and challenging mathematics has been expanded.

Design Principle #2: Where.

Boundary-spanning mathematical play exists outside of required coursework and professional development meetings.

MTCs bring together professionals across the spectrum of PK-12 and higher education mathematics contexts. We designed the MTCMidTN specifically to offer opportunities for participants to capitalize on math play that can happen outside of traditional higher education classrooms and professional development settings. Pragmatically, the varied professionals who participate in MTCMidTN work in contexts with different hours of operation. Philosophically, learning within a new structural space allows for new norms that characterize our respective boundary-spanning learning contexts and break down typical power dynamics, thus supporting learning through distributed expertise. We attribute a part of the ongoing success of the MTCMidTN to supporting the development of a positive identity for these communities.

By intentionally scheduling MTCs outside of required coursework and professional development, we place agency front and center. Part of the power of enacting agency is participation based on curiosity and choice rather than on professional responsibility, capitalizing on (and hoping for!) the potential for an individual’s positive association with and joy in mathematics. Reaching out to teachers across our local school districts and alumni networks, our per-event attendance has ranged from 16 to 45 people. Across the last 4 years we have had a typical attendance of around 25, about a third of whom attend regularly. Our participation, particularly because it includes people from across the spectrum, illustrates that professionals enjoy coming together to engage in math play.

Design Principle #3: What.

Boundary-spanning mathematical play intentionally bridges what happens in higher education math with what happens in school math.

Many times, math teachers do not have significant opportunities to discuss math outside of either teaching or engaging in a mandated course or professional development time (Campbell & Dunleavy, 2016). This lack of conversation outside of professional environments implies a lack of opportunities to build the connective tissue between what happens in higher education coursework and what happens in teaching and learning. Our boundary spanning exists in the way that we use our contexts to build bridges between what happens in universities and what happens in PK-12 classrooms. We promote this bridge both because we value the varying expertise that exists in these separate spaces, and because we know that bringing the expertise of these professionals together can only strengthen respect between and across these spaces.

During each meeting, we support participants to work in heterogeneous, randomly assigned groups on interesting mathematics problems. Groups engage in low-floor, high-ceiling problems that have an easy way to get started, offer multiple solution paths, and are often still yet unsolved by the time we wrap up. We act as boundary spanners for how professors, mathematics teachers, teacher candidates, and undergraduate and graduate students engage in math problems together.

Our hope is to enjoy mathematical problem solving while engaging in discussions about how to approach and how to think about mathematics in multiple ways. We aim to use mathematical play to bridge math knowledge with pedagogical content knowledge. Further, we assert that new contexts like MTCMidTN are needed, in order to bring varied professionals together in productive conversation. In other words, we use mathematical play to mediate the theory upon which university work is based with the practical knowledge of teaching, upon which classroom work is based.

The MTCMidTN addresses the development of mathematics professionals’ capacity to engage in mathematical play.

The work of the MTCMidTN invites participants to engage in mathematics in the ways that we would envision wanting our students and ourselves to engage in mathematical play. By simultaneously engaging faculty, teachers, teacher candidates, and undergraduate and graduate students in the same mathematics task, we intentionally work to engage in boundary-spanning work that redefines how, where, and with whom mathematical play can happen. Further, because our participants are randomly assigned to work with and meet new colleagues during each visit, there is great potential to bridge professional learning across mathematics professionals with different backgrounds. This heterogeneous, ever-evolving, and growing group affords us the opportunity to build a community that develops care, respect, and a desire to learn about mathematics from new perspectives that might not otherwise have been considered.

Growing our Capacity for Boundary-Spanning Learning Opportunities

Consistent with strong school-university partnerships, we utilize boundary spanning in order to engage “who, where, and what” mathematical play looks like.

Our commitment across university and school contexts pushes us to conceptualize opportunities for mathematical play that exist within and beyond the spaces where coursework and fieldwork typically happen. Specifically, we have sought to design our context so that we focus on supporting mathematical play such that it exists:

1. across the spectrum of mathematics professionals,
2. outside of traditional coursework and PD, developing capacity for content expertise alongside collaboration and innovative problem solving, and
3. to bridge university mathematics with school mathematics.

As we considered these design principles, we created the MTCMidTN such that educators have opportunities to learn alongside professionals across the educational spectrum and to increase their capacity for content expertise through collaborative and innovative problem solving, as well as to disrupt traditional status hierarchies for who is perceived as engaging in meaningful mathematics. As such, we intentionally work to weave pedagogy and practice in order to support a professional community committed to lifelong mathematical play. Boundary-spanning learning and teaching requires the simultaneous integration of both knowing and doing. We believe such learning contexts support and bind together the development of professional knowledge and skill, contributing to our success as a community.

Acknowledgments

The first author would like to thank Jeanne Peter and Amy Palmeri, who participated in early conversations about the concepts developed in this work.

About the Authors

Teresa K. Dunleavy is Assistant Professor of the Practice of Mathematics Education at Vanderbilt University. Jennifer N. Lovett is Assistant Professor of Mathematics at Middle Tennessee State University. Elizabeth K. Metts is a doctoral student at Vanderbilt University. At the time of writing this article, Shannon N. Reider was a numeracy coach at Metro Nashville Public Schools. All are co-founders of the Middle Tennessee Math Teachers’ Circle (MTCMidTN), located in Nashville, TN.

Footnote
[1] When we refer to the MTCMidTN participants as mathematics professionals, we mean that we are a cross-section of professionals that include university faculty from mathematics education and mathematics departments, in-service teachers, mathematics coaches, teacher leaders, and administration, and university students who identify either as future teachers or future teacher educators (including B.S., M.Ed., Ed.D., Ph.D., and postdoctoral researchers).

References
Aguirre, J., Mayfield-Ingram, K., & Martin, D. (2013). The impact of identity in k-8 mathematics: Rethinking equity-based practices. The National Council of Teachers of Mathematics.

Campbell, S. S. & Dunleavy, T. K. (2016). Connecting university coursework and practitioner knowledge through mediated field experiences. Teacher Education Quarterly, 43(3), 49–70.

Carter, S. (August 26, 2018). Eight Lettered Squares Puzzle. Retrieved from https://mathequalslove.blogspot.com/search?q=eight+lettered+squares.

Cohen, E. G. & Lotan, R. A. (2014). Designing groupwork: Strategies for the heterogeneous classroom (3rd ed.). New York, NY: Teachers College Press.

Dunleavy, T. K. (2018). High school Algebra students busting the myth about mathematical Smartness: Counterstories to the dominant narrative “Get It Quick and Get It Right.” Education Sciences, 8(2), 1–13. Retrieved from http://www.mdpi.com/2227-7102/8/2/58.

Hundley, M., Palmeri, A., Hostetler, A. L., Johnson, H. J., Dunleavy, T. K., & Self, E. A. (2018). Developmental trajectories, disciplinary practices, and sites of practice in novice teacher learning: A thing to be learned. In Polly, M. Putman, T. M. Petty, & A. J. Good (Eds.), Innovative Practices in Teacher Preparation and Graduate-Level Teacher Education Programs, (pp. 153–180). Hershey, PA: IGI Global.

Lambert, M. (2010). Learning teaching in, from, and for practice: What do we mean? Journal of Teacher Education, 61 (1–2), 21–34.

Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.

Math Teachers’ Circle Network. (2019). Retrieved from https://www.mathteacherscircle.org/.

Reider, S., Dunleavy, T. K., & Metts, E. (2019, November). Liar’s bingo with the Math Teachers’ Circle of Middle Tennessee. Presentation at the annual Southeast Regional meeting of the National Council of Teachers of Mathematics, Nashville, TN.

Building a Community of Professional Learners: Boundary-Spanning Mathematical Play was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

By Kyle R. Evans and Megan Staples

Math is often viewed by students as both important (for tips, taxes, and good test scores for college) and at the same time exceedingly useless (when will I use the quadratic formula in the real world??). Yet math has an incredibly useful role to play when it comes to issues of great import to our society and democracy. Beyond something written down in moral code, math is a key tool in determining whether something is or is not fair in our society.

Our Math Teachers’ Circle focuses on problem solving related to issues of social justice. While there are many components that can define “social justice,” part of the definition that we rely on is “ensuring the fair distribution of human and material resources in society” (Berry, Conway, Lawler, & Staley, 2020).

For a recent MTC session on voting systems, we reflected on the various ways fairness is operationalized in mathematics and how each can be seen in our society. Before reading further, pause and think about the question:

What makes something “fair” from a mathematical perspective?

Interestingly, just as students get more and more sophisticated in their thinking about issues of fairness, the math tools students learn that can identify and examine fairness become more and more sophisticated. We offer three examples that demonstrate the relationship between mathematics, fairness, and our society.

Elementary Grades

Fairness is “everyone gets the same amount.”

We see this readily throughout the elementary math curriculum, where students equally divide quantities of cookies, pizza, and marbles.

If we look at this version of fairness in the world around us, we think about people having equal representation in local elections (one person, one vote), or everyone having the same opportunity (at least on paper) to participate in an event.

Middle School

Fairness is “everyone gets an appropriate amount relative to their needs or circumstances.”

In other words, the more you need, the more you get. This version of fairness plays out in the grade school curriculum with proportionality. For example, students calculate how much a person makes for a given number of hours of work, and how much they should make if they work twice the number of hours.

Looking at this version of fairness in our world, we see that the number of House representatives for a state (in the U.S.) is proportional to the state’s population. (Note that this proportionality is not reflected in the Electoral College system.)

High School

Fairness is “outcomes that are true to a claim or standard.”

Students encounter this idea when they study statistics and engage questions like the accuracy of a company’s claim about how long their batteries last. In the context of our world, we look to statistics to help us ferret out faulty claims or systematic bias.

For example, a guiding principle for our system is that individuals should be tried by a jury of their peers. Lawyers in the court case Casteneda v. Partida[1] used the binomial distribution and z-scores to demonstrate that not only was Casteneda, a Mexican-American, not tried by a jury of his peers (a potential single-instance anomaly), but the system for jury selection in his Texas county effectively denied him that opportunity.

There are additional ways in which mathematics is used as a tool in our society to help us act fairly or examine issues of fairness. We use math to help us create measures and standards, and hope they approximate the world well, and fairly. We model potential outcomes as we try to figure out the impact of a new initiative and whether it disproportionately impacts different groups of people. We use Monte Carlo methods to create the set of possible outcomes, and their likelihood, in applications ranging from predicting election outcomes to quantifying partisan gerrymandering.

We leave our discussion here, with the hope that we have brought into view some ways that mathematics can help us examine fairness in our world, and have potentially inspired the desire to bring such contexts into classrooms!

About the Authors and the MTC4SJ

Kyle R. Evans is Visiting Assistant Professor of Mathematics at Trinity College in Hartford, Connecticut. Megan Staples is Associate Professor of Mathematics Education in the Neag School of Education at the University of Connecticut in Storrs, Connecticut.

The Mathematics Teachers’ Circle 4 Social Justice (MTC4SJ), led by Kyle Evans, Megan Staples, Scott Kapralos, and Georgina Rivera, is based in Connecticut and seeks to support teachers as mathematicians and social justice educators by promoting problem solving and critical conversations.

Footnote
[1] The authors thank Mark Myers for bringing this example to their attention.

Reference
Berry III, R. Q., Conway IV, B. M., Lawler, B. R., & Staley, J. W. (2020). High school mathematics lessons to explore, understand, and respond to social injustice. Corwin Press.

Math and Fairness was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.

In this issue:

• Circles with No Center
• Math and Fairness
• Building a Community of Professional Learners: Boundary-Spanning Mathematical Play
• San Jose Math Circle: Involving Students in the Beauty and Power of Mathematics

The Math Circle Network is proud to bring you the first issue of the Math Circular, a magazine for leaders and members of Math Circles and other math engagement programs.

Math Circles are communities focused on the enjoyment of mathematical problem solving. Meetings are lively, interactive, and often “funstrating”: challenging, but in a highly rewarding way!

Math Circles can take many forms, including after-school programs for students, professional learning communities for teachers and mathematicians, or groups of parents or families who want to become more involved with mathematics education. There are nearly 300 Math Circles around the U.S., including approximately 150 Math Teacher Circles and another 150 Math Student Circles. (Check out our map to see if there’s one near you, or contact us to learn how to start your own!)

At the Math Circle Network, we believe that Math Circles are powerful bridges among K-12 schools, higher education institutions, out-of-school programs, and families. We provide free resources — like the Math Circular! — for all Math Circles, with a special focus on supporting Math Circles that reach underrepresented or underserved students and their teachers. We hope you enjoy our magazine and look forward to hearing from you!

Sincerely,

The Math Circle Network (circles@aimath.org)

P.S. Interested in writing for us? Please check out our Submissions page.

Welcome to the Math Circular! was originally published in Math Circular on Medium, where people are continuing the conversation by highlighting and responding to this story.