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⊇American Institute of Mathematics · Winter/Spring 2017 · MTCircular

Freeing the Clones Exploring Paradox

Spot It! Mathematical Structure in a Children’s Game

Connect Circle Dispatches and Special Opportunities

A Circle in the Delta A Math Science Partnership

Winter/Spring 2017

MTCircular 2017 Winter Spring.indd 1 12/21/16 2:48 PM

MTCircular · Winter/Spring 2017 · American Institute of Mathematics

Identities and Connections

TABLE OF CONTENTS

Dear Math Teachers’ Circle Network,

This issue of the MTCircular highlights several fac-ets of MTCs, including one that is incredibly strik-ing: MTCs have the power to affect the very identi-ties of those who participate. In “A Teacher and a Mathematician,” Katie A. Hendrickson traces this transformation in teachers experiencing MTCs for the first time, drawing on results from her recently completed dissertation. (Congratulations, Katie!) In “Circle on the Delta,” Liza Cope describes how MTCs can both complement and extend the impact of other professional development opportunities. For those of you looking for great spring MTC session ideas, write-ups by Kimberly Muller (“Freeing the Clones”) and Tom Clark (“Spot It!”) will provide plenty of food for thought and structure for exploration.

I’m excited to report that our national network is growing rapidly and currently includes 125 MTCs in 39 states. Perhaps even more exciting is the

development of connections between nearby MTCs through regional networks. For example, the CSU Dominguez Hills MTC hosted four other MTC lead-ership teams from Southern California at a one-day meeting in October, where they planned exchanges and brainstormed about pooling their resources for future meetings and workshops. The Montana Math Teachers’ Circle Network (http://hs.umt.edu/math/mtmathteacherscircle/) continues to pursue the initiative to bring MTC participation within reach of every math teacher in the state.

To help encourage the development of additional regional networks of MTCs, the national MTC Net-work will offer some support for other leadership teams to meet face-to-face this spring. This support is made possible through a grant from the NSA as well as through generous Giving Tuesday donations. Please contact me ([email protected]) if you are interested in finding out more.

Happy problem solving!

Brianna Donaldson, Director of Special Projects

A NOTE FROM AIM

The Math Teachers’ Circle Network is a project of the American Institute of Mathematics

600 E. Brokaw Road, San Jose, CA 95112 Phone: 408-350-2088

Email: [email protected]: http://www.mathteacherscircle.org/

http://www.facebook.com/mtcnetwork @mathteachcircle

02 MTCircular · Winter/Spring 2017 · American Institute of Mathematics


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Identities and Connections

TABLE OF CONTENTS

A Note from AIMIdentities and Connections

A Teacher and a MathematicianThe Identity Element

A Circle in the DeltaExtending a Math-Science Partnership

Spot It!Mathematical Structure in a Children’s Game

Freeing the ClonesExploring Paradox

News and Views

Dispatches from the CirclesLocal Updates from Across the Country

OpportunitiesSpecial Events, Recognitions, Careers

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A Math Teachers’ Circle immersion workshop is an intensive experience focused on

mathematics and creating community. Recommended as the first “big event” for a

new Math Teachers’ Circle, an immersion workshop has enormous potential to affect

teachers’ beliefs about themselves, their mathematics abilities, and the nature of

mathematics—or their mathematics identity. The MTC immersion experience puts

teachers in the sometimes uncertain and uncomfortable role of mathematics learners.

The experience of working through this discomfort can have a profound effect on

teachers’ identities, which underlies their mathematics teaching actions.

by Katie A. Hendrickson


⊇American Institute of Mathematics · Winter/Spring 2017 · MTCircular 05

The Importance of Identity

Anderson (2007) defines identity as “the way we define ourselves and how others define us” (p. 8). Mathematics identity, then, can be thought of as how people see themselves as mathematically capable (Martin, 2000). It may include elements of productive disposition (NRC, 2001), beliefs about mathematics, and self-efficacy or self-concept beliefs. In the case of MTC participation, it is also important to consider the unique role of collaboration and community-building. Normative identity, or the set of desirable practices for members of a group, interacts with personal identity, the way that individuals identify with, comply with, or resist the normative identity (Cobb, Gresalfi, & Hodge, 2009). Each MTC site creates a normative identity, and participants’ personal identities reflect their engagement with or acceptance of those practices.

Why should we, MTC leaders, care about identity? Teachers hold multiple interacting identities. Their mathematics identities are closely tied to their math-ematics teaching identities; that is, their mathematics experiences affect how they teach, and vice versa (Hendrickson, 2016). Teachers’ identities are strongly influenced by their past experiences doing and learn-ing mathematics. Math Teachers’ Circle can provide a positive, productive, challenging experience in doing mathematics.

MTC Impact on Teacher Identity

In summer 2016, I studied teacher identity with the help of MTC leaders and participants across the country. Participants answered surveys, completed written reflections, and participated in interviews, as I attempted to explore and understand shifts in the teachers’ identities after participating in a weeklong immersion for the first time.

Many of the teachers in this study began the week feeling nervous about their abilities compared to oth-ers in the MTC. They felt frustrated or uncertain when they first encountered open-ended problems—but by focusing on the problem-solving process rather than on the solution, exploring multiple solution paths, and using their strengths to approach problems, the teach-ers began to experience success. Improved confidence around mathematical problem solving stimulated perseverance and persistence with future problems.

As one respondent put it, “When I was able to actually solve some of the problems, I was like, ‘Oh, hey. This isn’t so bad.’ It helped boost confidence in regards to the ability to be a problem solver.”

Collaboration was essential to the process, not only because it contributed to successful problem solving, but also because the discussions helped the teachers understand that their peers had similar anxieties about problem solving. “By the end of the week, my sense of math security had greatly improved,” said one teacher.

The Role of MTC Leaders

The following recommendations for MTC leaders, based on my research, can help teachers to develop positive and productive mathematics identity:

“It helped me to see myself as a mathematician.”

Identity is an important outcome of the MTC. To ultimately affect teachers’ classroom practice, it is important to consider the effects of the MTC experi-ence on teachers’ identities. Identity is complex and deeply held, and influences beliefs and actions in the classroom. Math Teachers’ Circle can influence how teachers see themselves as doers and teachers of math-ematics. MTC leaders should consider the evolution of identity through such experiences, and carefully work to create a community of practice (Cobb et al., 2009). The goal of such a community is for teachers to see themselves as members of the group and to develop their personal identity in relation to the normative identity of the group.

“As the week progressed, I realized that every-

one felt that same way at one time or another.”

The immersion experience is key. A weeklong MTC workshop immerses teachers in problem solving and allows them to experience both success and failure. Teachers who are nervous or anxious during the first few days begin to feel more confident taking problem-solving risks later in the week. Professional develop-ment that takes place over a single session or a single day does not allow enough time for teachers to fully engage in the activities and begin to internalize the teaching methods. It takes time to develop the MTC community and establish the normative identity of the group (Cobb et al., 2009).

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MTCircular · Winter/Spring 2017 · American Institute of Mathematics06

“It’s easy to fall back into old ways.”

Provide follow-up sessions. The teachers in this study had already begun to experience significant shifts in their mathematics identities by the end of the immer-sion workshop. They intended to use MTC problems and pedagogies in their classrooms, but anticipated challenges when the normative identity of their schools differed from the normative identity of the MTC. Regular follow-up MTC sessions can give teachers opportunities to continue to rethink their ideas of effective problems and pedagogy, and to ask questions about implementing these ideas in their classes.

“I’m trying to become more student led. And I

need help.”

Consider having pedagogical conversations. Teachers’ mathematics identities and teaching identi-ties are intertwined and it is important for professional development to consider both. When leading an MTC, it is easy to focus on the mathematics content to the detriment of problem-solving pedagogy. To assist teachers in using new pedagogical strategies in their classrooms, sessions should focus explicitly on transferring ideas from the MTC to the classroom. The teachers in this study learned from both content-focused activities and pedagogy-focused activities. However, the teachers who were the most confident and specific about changes they planned to make in their classroom were most likely to have attended MTCs with sessions devoted to making this transition.

“Having to do the math puts you in the role of

student.”

Engage teachers as learners. In MTC, many teachers experience mathematics learning differently than they ever have in the past. The mathematics content and the pedagogy engages them as adults and allows them to experience perplexity, frustration, and success. Authentic engagement as a learner is important for teacher identity development (Cohen, 2010). Because their mathematics identities and teaching identities tended to be closely intertwined, the MTC can help

Scenes from 2016 summer immersion workshops across the country. From top: AIM MTC in San Jose, Calif., Crooked River MTC in Cleveland, Ohio, Smoky Mountain MTC in Cullowhee, N.C., and SouthEast Ohio MTC in Athens, Ohio.



teachers to build connections between mathematics problem solving and mathematics teaching.

“I took comfort in knowing that my peers all felt

the same way.”

Teamwork makes the dream work! Collaborative groupings in the MTC are important for successful problem-solving experiences, developing perseverance, and modeling a successful pedagogical technique. Collaboration can also assist in the development of positive mathematics identities. The teachers in this study who worked with their peers began to realize that their fears and anxieties were not unique to them, and that getting stuck is a natural part of the problem-solving process. They also saw different ways of approaching the same problem, gave assistance, received assistance, and began to understand that everyone’s contributions—including their own—were valuable. Taking It Back to the Classroom

Teachers’ reflections on their MTC experiences indicated the desire to help their students to develop positive mathematics identities through similar experiences. One teacher said:

“I hope to give them maybe more challenging problems at a higher level than I maybe would have attempted previously, and I think they can do it with collaboration. I think that they would probably have the same experience I did, be apprehensive at first. If they stick with it and endure, and are successful, I think they can build a lot of confidence that way.”

The development of identity, and mathematics identity in particular, can be a critical outcome for teachers participating in MTCs. An intensive MTC immersion workshop is an opportunity for teachers to develop a productive and positive mathematics identity.

Katie A. Hendrickson works on state policy and advocacy for Code.org. She recently earned her doctorate in Mathematics Education from Ohio University, where she co-founded the SouthEast Ohio MTC.

ReferencesAnderson, R. (2007). Being a mathematics learner: Four faces of identity. Mathematics Educator, 17(1), 7–14.

Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in math-ematics classrooms. Journal for Research in Mathematics Education, 40, 40–68. doi: 10.2307/40539320

Cohen, J. L. (2010). Getting recognised: Teachers negotiating professional identities as learners through talk. Teaching and Teacher Education, 26, 473–481. doi: 10.1016/j.tate.2009.06.005

Hendrickson, K. A. (2016). (Doctoral dis-sertation). Math Teachers’ Circles: The effects of a professional development community on mathematics teachers’ identities. Retrieved from Ebscohost. (EACCD63BC675FBD1)

Martin, D. B. (2000). Mathematics success and failure among African-American youth: The roles of sociohistorical context, community forces, school influence, and individual agency. Mahwah, NJ: Erlbaum.

National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press. Retrieved from http://www.nap.edu/catalog.php?record_id=9822

This research was supported by the Ohio University Patton College of Education Graduate Study and Educational Research Fund.



In 2013, the College of Education and Health Science at Delta State University (Cleveland, Miss.) was awarded a three-year Math Science Partnership (MSP) grant to provide professional development for

math teachers in grades 6-8. This program included a two-week intensive summer institute, three Saturday work sessions, a monthly two-hour online discussion opportunity, and three classroom coaching visits. Participating teachers earned four hours of graduate credit and 10 continuing education units.

In 2013, the first year of the grant, the mathematical focus was number, operations, and algebraic thinking. In 2014, the focus was geometry and measurement; in 2015, statistics and probability. In addition to the changing mathematical content, the grant also focused on the Mississippi College & Career Readiness Standards and best practices for implementing and assessing these standards.

The summer institute was offered annually to two cohorts, each lasting two weeks. Instructors included Delta State University math professors along with current and retired middle school and high school teachers who shared successful strategies and techniques from their classrooms. Representatives

from local industries—including real estate, aviation, photography, agriculture, and factory production—also dropped by to give professional learning presentations.

The overall goal of the project was to increase math scores among students in grades 6-8 and to better prepare them for high school mathematics instruction, college readiness and career development. By 2015, 34 teachers from 24 schools across 11 Mississippi counties had participated in the program.

In 2014, during the second year of the MSP grant, three Delta State University professors – Liza Cope, David Hebert, Clifton Wingard – and two teach-ers from the MSP program – Whitney Janous and Kathleen Lott – began plans for creating a Mississippi Delta Math Teacher’s Circle.

A Math Teachers’ Circle would further enrich teach-ers’ experience of mathematics. Teachers would be able to experience open-ended problem solving firsthand in lively discussion sessions led by mathematicians. In so doing, we hoped teachers would feel empowered to bring more problem solving into the classroom—a key component in student learning and engagement in mathematics.

In summer 2014, our team traveled to Washington D.C. to attend a weeklong intensive training hosted by the American Institute of Mathematics. The next year, we launched the Mississippi Delta Math Teacher’s Circle with a two-day summer immersion program in July 2015 and six Saturday afternoon Circle meetings throughout the 2015-2016 school. The programs served roughly 25 teachers, all former participants of the Delta MSP program.

The Math Teachers’ Circle and the Delta MSP program complemented each other and shared similar goals. Both aimed to develop a culture of professional learning, to deepen teachers’ math content knowl-edge, and to increase use and understanding of the Mississippi teaching standards.

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by Liza Cope

A Circle in the DeltaExtending a Math Science Partnership

The Mississippi Delta MTC leadership team, from left to right: Liza Cope, Kathleen Lott, Whitney Janous, and David Hebert.



There were also key differences between the pro-grams. The Math Teachers’ Circle was less formal and involved more shared leadership. Often, Circle meet-ings were planned and led by teachers, giving them an opportunity for creative thinking and reflection. Although Circle meetings had a general agenda, the activities were less structured and there was signifi-cantly more flexibility in the schedule than with the Delta MSP. Participants in the Delta MSP were earning graduate credit, so they had many formal assignments and assessments to complete. This caused some par-ticipants stress and anxiety. The Circle meetings were more relaxing, filled with fun problem solving and engaging conversations.

In 2016, Delta State University was awarded another three-year grant to continue our work with the Delta MSP. One major change in the 2016-2018 programs is that the participants will include elementary as well as middle school math teachers. We also plan to launch an annual Delta MSP conference with invited speakers and a variety of peer-reviewed presentations. These annual conferences will be open to the public. We hope that the annual conferences will provide opportunities for local teachers to build connections. We also aim

to use these events to recruit participants for both the Delta MSP and the Mississippi Delta MTC.

The Math Teachers’ Circle also continues into the 2016-17 year, with some changes. We opened member-ship up to undergraduate students majoring in second-ary mathematics education and to elementary and high school teachers. We are eager to use our increased membership to help strengthen local K-12 and univer-sity partnerships. Another transition that occurred is that MTC members (instead of the leadership team) have begun to run partial or entire meetings on their own. We have been very impressed with the interesting and innovative presentations our members have led!

We plan to have another six Circle meetings this year, on Saturday afternoons at 2 p.m. This day and time was voted on by members, because it directly fol-lows the MSP professional development sessions. We are excited to continue this important work of support-ing teachers in their professional and mathematical growth through both formal and informal means.

Liza Cope is an assistant professor of mathematics at Delta State University and a leader of the Mississippi Delta Math Teachers’ Circle.

Attendees of the summer 2015 immersion workshop for the Mississippi Delta Math Teachers’ Circle.

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by Tom Clark

Spot It!Mathematical Structure in a Children’s Game

Mathematicians are fond of saying that mathematics can be found almost anywhere. Yet I was shocked to discover that a simple children’s game like Spot It!

can lead to unsolved problems in mathematics. Deep mathematical structure lies just below the surface of the game.

Whenever I lead the Northwest Iowa Math Teachers’ Circle I want to use a problem that is easy to explain, unfolds in many directions, and encourages inquiry. Spot It! is the best session topic I have ever seen for this. All you have to do is play the game for a minute or two, and questions arise spontaneously.

The game is simple. Each player is dealt one card, and an extra card is dealt to the middle of the table. Each card contains eight symbols (in the junior ver-sion, each card contains six symbols). To win, a player must be the first to match a symbol on her card with a symbol on the middle card.

The first “aha” moment occurs when you realize that anyone can win—that is, there must be a match between any two cards. This prompts the question, “How does it work?” The symbols cannot be assigned

randomly; there must be some deep structure built into the game. And where there is structure, there is mathematics.

Once the “aha” moment occurs, I ask, “What math-ematical questions do you have?” I want the teachers to be curious, to make conjectures… and they do! Here is a list of questions that we came up with:

Will every card match with every other card?How many cards are in the deck? Could there be more cards? Fewer cards?Does each card have the same number of symbols?How many unique symbols are there? Does each symbol appear the same number of times?What is the relationship between the number of symbols per card, the total number of symbols, and the total number of cards?

Now we get to spend the rest of the session trying to answer these questions. One approach is to lay out all the cards from the Spot It! Junior deck. There are 31 cards, each with six symbols. Careful counting reveals that there are 31 total symbols as well, and that each symbol appears exactly six times in the deck. Each card shares exactly one symbol with every other card. With some data on the table, I push for mathematical explanations of the patterns, while remaining flexible with directions of inquiry.

For example, participants might wonder how, with six symbols per card, we end up with a deck of 31 cards. Choose a Spot It! Junior card and set it to the side. It has six symbols on it, and every card in the deck matches one of those six. Let’s arrange the deck into six rows, based on the six symbols on our “key” card. (See Figure 1 at right.) If all the cards in the first row have the heart symbol, then every card in that row contains a heart and five other symbols. If there are m cards in the row, then there are 6 + 5(m - 1) total unique symbols (6 for the key card, and 5 new ones for each other card in the row). Note that if m = 6, then this yields 31 symbols. But why should this be? Any card not in the “heart” list must share a match

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Figure 1: The entire Spot It! Junior deck i s arranged into six “lines,” all intersecting at the “key” card.

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with every card in the heart row. But there are only six symbols on that card, so it can only match six cards in the heart row. The most possible cards the deck could hold is thus 31.

Now we can find a general formula connecting the number of cards, symbols per card, and symbols in the deck—assuming an optimal deck exists. How can we know if it does?

Enter projective geometry. The key insight here is that in geometry any two points determine a line. If you think of cards as “points,” and a set of cards that share a particular symbol as a “line,” then you’ve got



a geometry. (If you have played the game of SET with your MTC, these two games together make a beautiful pairing—SET also defines a geometry, but it is a finite four-dimensional Euclidean geometry.) You can show very easily, both with the cards and also by reasoning, that Spot It! cards determine a non-Euclidean geom-etry, because there are no parallel lines. The “heart line” and the “star line” will necessarily intersect at a card with both a heart and a star on it. In fact, the key card idea shows this: any card defines six lines that all intersect at that card. I recommend Resource [3] for a deeper investigation of the geometry of Spot It! It is amazing, but it is also harder for the teachers to “dis-cover” so it will require some direction to investigate.

Another interesting direction is to consider how to build our own Spot It! deck. For example, maybe we can build a deck with two, three, four, or five symbols per card. Is there an algorithm to assign symbols? This process will help derive all the equations relating numbers of cards and symbols. I’ve written about this in more detail in Resource [1]. It is interesting how

the construction breaks down in certain cases when a prime number condition is not satisfied, which also leads to the conclusion that this might not always be possible. This question of existence is in fact an open question in mathematics. A large supercomputer search showed that no Spot It! game exists with 11 symbols per card. The question is open for 13 symbols per card, outside the reach even of supercomputers.

In any MTC activity, I want teachers to practice making sense of problems, reasoning, constructing arguments, and looking for structure—the very things that their students should be doing, in line with the Common Core State Standards for Mathematical Practice. Spot It! provides many opportunities to do so. Combinatorial arguments can show up (see Resource [2]), as well as simple reasoning about patterns. It’s guaranteed to be a great session topic for any Circle.

Tom Clark is an assistant professor of mathematics at Dordt College, the leader of the Northwest Iowa MTC, and the Chair-Elect of the SIGMAA-MCST.

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1. Clark, Thomas J. and Jongsma, Cal. Analyzing Unique-Matching Games Using Elementary Mathematics. http://www.math-teacherscircle.org/assets/legacy/resources/materials/Clark_SpotItPuzzle.pdf

2. Spot It!...A Combinatoric Cornucopia. http://www.lews-garage.org/spot-it-a-combinatoric-cornucopia/

3. Sengupta, Deepu. A Mathematical Analysis of Spot It!. http://www.mathteacherscircle.org/assets/legacy/resources/materials/DSenguptaSpotIt.pdf

Resources

Figure 2: The geometry of a Spot It! deck with seven cards and three symbols per card.



One of the biggest challenges of leading a Math Teachers’ Circle session is finding a problem that is interesting and chal-lenging without being out-of-depth. At

the Eastern Upper Peninsula Math Teachers’ Circle in Sault Ste. Marie, Mich., we ran a great session which was inspired by a Numberphile video called “Freeing the Clones,” by Zvezdelina Stankova (UC Berkeley). You can find the YouTube video at http://tinyurl.com/clonesvideo. We also ran this session separately for a group of high school students. Following is an intro-duction to the problem, as well as strategies for making it accessible at various levels.

Here is how the game works: Imagine that you are standing at the corner of an infinite checkerboard. Three clones are imprisoned in the three tiles at the corner of the board as shown in Figure 1 at right.

The only way for a clone to move is to split itself into two clones, which will occupy the tiles above and to the right of the original clone. Only one clone can occupy a tile at a time. For example, if the clone on the right were the first to split, its move would appear as shown in Figure 2.

Note that there are now four clones and two are still in prison. Neither of the two imprisoned clones can make the next move, because this would place multiple clones on the same tile. Therefore, the next move must be made by one of the two clones on the right.

If we continue in this manner, is it possible for all of the clones to escape the prison?

We gave teachers checkerboards from a dollar store, and had them split into groups to play the game. All but one of the groups decided it was impossible to free the clones from the prison, but they were unsure how to prove it. (The high school students were more evenly split: some believed that it could be done, but were not sure how to do it, and some thought it couldn’t be done, but believed it was impossible to prove. That led to some interesting discussions.)

by Kimberly Muller

Freeing the ClonesExploring Paradox

Figure 1: Three clones are imprisoned in the corner of an infinite checkerboard.

Figure 2: A clone splits itself in two, escaping the prison.

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Spoiler alert: In the Numberphile video, Dr. Stankova proves that it is impossible to free the clones by assigning numbers to the occupied tiles as shown in Figure 3 at left.

Note that the original three clones have a total value of 2. After the first clone is split, the clones still have a total value of 2, and are, in fact, invariant. You then show that the infinite sum of all tiles outside the prison is in fact 2. In a finite number of moves, you cannot remove the clones from the prison.

This solution requires the teachers to compute infinite sums. To prepare them for this challenge, we used an “appetizer” activity. Our Circle meetings typically start off with an “appetizer” that we work on while we are waiting for dinner to arrive. This problem is typically a short, fun problem or activity like the Locker Problem or folding hexaflexagons that is usu-ally unrelated to the “main course.” In this particular case, though, the goal was for the appetizer to prepare the teachers to tackle the Freeing the Clones activity in the time allotted. To start we looked at one of Zeno’s Paradoxes, the Dichotomy Paradox:

Suppose that Homer would like to walk home. In order to do so, he must walk halfway. After he does this, he must walk half of the remaining distance, and then half that distance, etc. Since he will always have half the remaining distance to go, he will never reach home.

Since logically we know that Homer can reach home, what are some ways in which we can resolve this para-doxical situation?

Most participants used a graph or picture to explain why Homer could in fact make it home. This led nicely into discussions about “Proofs Without Words.” To that end, we looked at other infinite sums by shading a diagram (or creating a picture from scratch) and then deciding what sum it represented. For instance, Figure 4 shows that

We then made Baravelle Spirals with an empty template and some map colors. These are also fun to make in Geometer’s Sketchpad if your Circle has access to a lab.

Figure 3: Numerical values on the checkerboard.

Figure 4: What sum does this diagram represent?

Figure 5: Baravelle Spirals

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For these Baravelle Spirals, teachers must decide what portion of the spiral is colored in a single color and what infinite sum it represents. The teachers were able to use congruent figures and repeated reasoning to determine what geometric sum these figures repre-sented. Some were able to prove algebraically that the sums were correct. Others found the hint helpful that a similar strategy is used to determine the fractional form of the repeating decimal 0.09 = 0.09090909....

The high school students in particular enjoyed that 0.9999999.... = 1.

At this point, we began the Freeing the Clones activity with little direction. When it came to the point of assigning values to tiles that contain clones, the teachers were in a better position to solve the puzzle. The high school students needed more of a hint because they gave up much more quickly, but after telling them to think back to what we had done with Baravelle spirals, most groups were able to figure it out. In the debriefing discussion, the teachers noted that similar methods were used throughout the session, even though the problems seemed to have very little in common.

On the whole, the “appetizer” enriched the session, tying in many methods mathematicians employ, such as thought experiments, pictures, repeated reasoning, and proof.

Kimberly Muller is a professor of mathematics and chair of the School of Mathematics and Computer Science at Lake Superior State University. She also serves on the leadership team of the Eastern Upper Peninsula Math Teachers’ Circle. Playing “Freeing the Clones” with pennies.

NEWS • EVENTS • CALENDAR • MEMBER CIRCLES • VIDEOS • RESOURCES

Stay connected. Stay informed. Get all the latest news on the MTC website or our Facebook page.

www.mathteacherscircle.org www.facebook.com/mtcnetwork @mathteachcircle

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http://www.mathteacherscircle.org


NEWS AND VIEWS

Taton Becomes Director of Mathematics for School District of Philadelphia Joshua Taton (Philadelphia Area MTC) is the new Director of Mathematics for the School District of Philadelphia. A PhD candidate in math education at the University of Pennsylvania and co-founder of the Philadelphia Area MTC (PAMTC), Taton started work-ing at the School District in June. Since then, he helped kick off the District’s new summer professional development initiative, involving 1,000 teachers and their principals from 75 schools across Philadelphia. In the fall, Taton helped roll out new instructional resources aligned with the state standards — a “dizzying” amount of work, he cheerfully said in an interview with the Philadelphia Higher Education Network for Neighborhood Development (http://phennd.org/update/interview-with-the-new-director-of-mathematics-at-the-school-district-of-philadelphia-joshua-taton/). Taton said he had always wanted to support the District “in some way, shape, or form,” and credited his work with the PAMTC for providing him with contacts and credibili-ty in the math education community to do so. Taton noted that the PAMTC is a great model for how universities can partner with school districts in regard to math education. “I think of [the PAMTC] as a team,” Taton said.We engage with scholars and researchers, regularly, who are great thought partners for helping us move forward, but who also learn from our wonderful teachers and what they do. Among many other things, they develop an appreciation of the complexity and scale of the District, and we learn about the direction of education research. We are always looking to develop such projective working relationships with committed partners.”

Taton

Bolognese, Davis, Phelps, Ray, Sudnick Win Ohio Teaching AwardsFive Ohio Math Teachers’ Circle members were recognized with awards from the Ohio Council of Teachers of Mathematics Awards at their October annual meeting:

Chris Bolognese (Columbus MTC), a math teacher at the Columbus Academy, and Ryan Davis (SouthEast Ohio MTC), a math teacher at Southern High School in Racine, Ohio, were selected as District Outstanding Educators. Madeira High School math teacher Steve Phelps (Cincinnati MTC) received the Buck Martin Award, given to one outstanding secondary math educator in the state of Ohio each year. Berkshire Schools math teacher Jennifer Ray (Crooked River MTC) was selected as the 2016 Emerging Leader Award winner. Nina Sudnick (SouthEast Ohio MTC), a math teacher at West Elementary in Athens, OH, received the Myrtle Miller / Marijane Werner Award, given to one outstanding elementary teacher in the state of Ohio each year.

Bolognese Phelps

Ray Sudnick

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Danforth Presents at National ConferencesHeather Danforth (AIM MTC), a math teacher and co-director of curriculum at Helios School in Sunnyvale, Calif., presented on Math Teachers’ Circles at two conferences this fall. At the Expeditionary Learning Conference in Detroit, Mich., in October, Danforth presented on on math Circles as a structure for engaging students in mathematical inquiry. Danforth led teacher participants in the “1-100 problem,” and then discussed how math Circles can be used in the classroom to help students get excited about math. In November, Danforth presented at the National Association for Gifted Children conference in Orlando, again about math Circles, with a focus on how the math Circle structure can be particularly useful for gifted learners. “I know that Math Teachers’ Circles changed the

way that I think about math and, by extension, the way that I teach it,” said Danforth. “When I started as a teacher, I saw myself as an ‘English person,’ and I was a bit nervous about math. I think that many, many teachers share this feeling. Since then I have led many math circles with both students at teachers at my school, and I have seen this transformation happen many times, with both adults and students.”

Danforth

Capasso, Hill Win Presidential AwardsLiz Capasso (Fairfield County MTC), a math teacher at Jettie S. Tisdale School in Bridgeport, Conn., and Patty Hill (MTC Austin), a math teacher at Kealing Middle School in Austin, Texas, are two of 213 U.S. educators to win the 2016 Presidential Award for Excellence in Mathematics and Science Teaching,

the nation’s highest recognition for math and science teachers. Both received a $10,000 award from the National Science Foundation and a trip to Washington, D.C., for an awards ceremony, as well as educational and celebratory events, and visits with members of the Administration. “Many teachers give their heart and soul to this profession and to their students, knowing that their contributions may never be formally recognized,” said Hill, who has been teaching math at Kealing for 18 years. “This makes the honor of a Presidential Award incredibly humbling.” Capasso, who has taught at Tisdale for eight years, stated, “The Presidential Award celebrates all the incredible educators who have supported, taught, and inspired me on my journey as a teacher.” Both awardees are co-founders of their respective Math Teachers’ Circles.

Capasso Hill



CONNECT

California • TIn October, Math Teachers’ Circle leaders from across Southern California met to share their experience and ideas for possible collaborations. Hosted by Cal State Dominguez Hills in Carson, and sponsored by AIM, the group shared sample activities, described strategies for outreach among teachers, sources of support, and the design of intensive workshops. Leaders discussed possibly attending or leading workshops at other sites, and agreed to meet again in the spring semester to plan in further detail.

– Contributed by Sharon Lanaghan

Stephanie Molina and Jeremy Aikin, co-leaders of the Inland Empire MTC, gave a 75-minute presentation at the NSF-Noyce Summit in Washington, D.C., in July. The presentation was focused on involving NSF-Noyce Scholars in the organization of a Math Teachers’ Circle. For the presentation, Stephanie and Jeremy co-facilitat-ed a sample MTC session on the geometry of Spot It! and SET.

– Contributed by Cory Johnson

Illinois • DSouthwest Chicago MTC co-leader Dr. Angela Antonou will be presenting “Impact of the Southwest Chicago Math Teachers’ Circle on the Disposition of Teachers Toward Mathematics and Toward the Teaching and Learning of Mathematics” at the Joint Mathematics Meetings in January 2017 in Atlanta, Ga. In addition, planning is in progress for additional presentations at the Illinois Section of the Mathemati-cal Association of America in April 2017 and at the Association of Christians in the Mathematical Sciences (ACMS) Biennial Conference in late May/early June of 2017 in South Carolina.

– Contributed by Kristen Schreck

North Carolina • uThe Blue Ridge MTC met three times this year in vari-ous locations, including at the Asheville Art Museum and the Downtown Book Exchange. The Circle plans to meet three more times in the spring, and to do an escape room puzzle together in February.

– Contributed by Meghan Boutwell

Ohio • oThis fall, the Crooked River MTC was featured in the Number Notes newsletter of the Greater Cleve-land Council of Teachers of Mathematics (GCCTM). Crooked River MTC leaders also ran a sample Math Teachers’ Circle session at the GCCTM’s September 17 kick-off meeting.

– Contributed by Sara Good

South Carolina • fCoastal Carolina MTC co-leaders Anne Ho and Tara Craig will be presenting “Triangles, Squares, and Segregation: Introducing Social Issues Through Math” at the Joint Mathematics Meetings in January 2017. Ho and Craig adapt-ed “Parable of the Polygons” (Vi Hart and Nicky Case), an online simulation on diver-sity and segregation, into a math Circle session, which ran at the Coastal Carolina University MTC in Decem-ber. The session is interactive and offers multiple layers of content depending participants’ age and comfort level with conversations on social issues. These levels include: (1) a basic introduction to game theory, (2) an invitation for participants to think about the benefits of diverse groups, and (3) a discussion of how individual biases can lead to detrimental societal effects.

– Contributed by Anne Ho

Local Updates from Across the Country

Dispatches from the Circles

18



Opportunities

CONNECT

⊇19

Help Inspire a Million Students The Global Math Project aims to engage students and teachers around the world during a series of annual Global Math Weeks. During the first Global Math Week, beginning October 10, 2017, one million students will experience Exploding Dots, a refreshing and powerful mathematical approach to arithme-tic and algebra developed by James Tanton and regularly described by students and teachers alike as “mind blowing.” Learn more at www.globalmathproject.org, and stay tuned for ways to get involved through the MTC Network!

2017 Rosenthal Prize Applications OpenThe Rosenthal Prize for Innovation and Inspiration in Math Teaching, a $25,000 prize administered by the National Museum of Mathematics in New York City (www.momath.org), celebrates innovation and inspiration in the middle school math classroom by rewarding a teacher who creates an excep-tional, hands-on, math activity. Preliminary applications are due in May 2017. Find out more at rosenthalprize.momath.org.

2017 Presidential Award Applications OpenThe Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST) are the nation’s highest honors for teachers of mathematics and sci-ence. Awardees serve as models for their colleagues, inspiration to their com-munities, and leaders in the improvement of mathematics and science educa-tion. If you know great teachers working in grades 7-12, nominate them to join this prestigious network of professionals. Nominations close on April 1, 2017. Applications close on May 1, 2017. Find out more at www.paemst.org.

Great Hearts Academies is HiringGreat Hearts Academies is a non-profit network of 29 charter schools in Ari-zona and Texas dedicated to improving public education with a core, classical liberal arts curriculum that sets high expectations for all students. Great Hearts is now accepting applications for 2017-2018 mathematics faculty and appren-tice teachers. To learn more or apply, please visit www.greatheartsamerica.org/careers or contact [email protected].

Know of another opportunity of interest to the MTC community? Contact us at [email protected].

Special Events, Recognitions, Careers


MTC Network Advisory BoardKristen Chandler, MATHCOUNTSWade Ellis, West Valley College (Emer.)John Ewing, Math for AmericaRichard Grassl, University of Northern Colorado (Emer.)Edward M. Landesman, University of California Santa Cruz (Emer.)W. J. “Jim” Lewis, University of Nebraska-LincolnJane Porath, East Middle School, Traverse City, Mich.Richard Rusczyk, Art of Problem SolvingAlan Schoenfeld, University of California BerkeleyDenise Spangler, University of GeorgiaPeter Trapa, University of Utah and MSRIUri Treisman, University of Texas at Austin

MTCircularBrianna Donaldson, EditorHana Silverstein, Assistant EditorJessa Barniol, Art DirectorEditorial Contributors: Tom Clark, Liza Cope, Katie A. Hendrickson, Kimberly Muller

MTC Network Sponsors

To subscribe to MTCircular or to explore resources featured in this issue, please visit us online at http://www.mathteacherscircle.org/newsletter.

Comments and suggestions are always welcome at [email protected].

American Institute of Mathematics600 E. Brokaw RoadSan Jose, CA 95112

MTC Network Leadership TeamEstelle Basor, AIMBrian Conrey, AIMTom Davis, San Jose Math CircleBrianna Donaldson, AIMDavid Farmer, AIMMary Fay-Zenk, ConsultantTatiana Shubin, San Jose State UniversityJames Tanton, MAAPaul Zeitz, University of San FranciscoJoshua Zucker, AIM and Julia Robinson Mathematics Festival

desJardins/Blachman Fund


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