Math Teachers’ Circles:Partnerships between

Mathematicians andTeachers

B. Donaldson, M. Nakamaye, K. Umland, and D. White

Mathematicians depend on primaryand secondary teachers to educateyoung students and inspire theirmathematical minds. Teachers inturn benefit from mathematical ex-

periences led by mathematicians that kindle theircreative energy and broaden their intellectual hori-zons. Mathematics departments, as part of theirmission, provide mathematical training for currentand future teachers. Many mathematics facultyhave a natural desire to share their excitementfor mathematical inquiry in ways that can reachbeyond their own classrooms and affect K–12education. Unfortunately, mathematicians who arelooking for opportunities to interact with teachersin a productive, collaborative spirit do not alwaysknow where to find them.

Math Teachers’ Circles (MTCs) are a uniqueform of professional development that tap into

Brianna Donaldson is director of special projects at theAmerican Institute of Mathematics. Her email address isbrianna@aimath.org.

Michael Nakamaye is professor of mathematics and statis-tics at the University of New Mexico. His email address isnakamaye@math.unm.edu.

Kristin Umland is associate professor of mathematics andstatistics at the University of New Mexico. Her email addressis umland@math.unm.edu.

Diana White is associate professor of mathematics at theUniversity of Colorado Denver. Her email address is diana.white@ucdenver.edu.

All four authors are partially supported by NSF GrantDiscovery Research K–12 1119342.

DOI: http://dx.doi.org/10.1090/noti1188

this instinct that mathematicians have to sharetheir love of mathematics with others. In contrastto other important forms of professional devel-opment for mathematics teachers that requirespecialized educational knowledge, MTCs leveragemathematicians’ disciplinary knowledge and skill.That is, as mathematicians facilitate an MTC session,they ask probing questions, model mathematicalthinking, and encourage mathematical conjectur-ing, exploration, communication, and discovery.This is similar in many ways to other activitiesthat mathematicians regularly engage in, such astraining Ph.D. students, guiding research experi-ences for undergraduates, or leading a capstoneexperience for mathematics majors. This makesMTCs an accessible entry point for mathematiciansinterested in working with teachers.

In this article we describe the history of MTCs,the MTC model, a sample session, and whatwe know about the impact of MTCs on teach-ers, mathematicians, and the K–20 mathematicalcommunity.

The Rise of MTCsThe idea for Math Teachers’ Circles originatedfrom a middle school mathematics teacher, MaryFay-Zenk, at the time a teacher at CupertinoMiddle School in Cupertino, CA. Intrigued by themathematics at the San Jose (CA) Math Circle forstudents, Fay-Zenk thought that teachers shouldhave their own outlet for mathematical exploration.She and a small group of other teachers andmathematicians, including Tom Davis (co-founder

December 2014 Notices of the AMS 1335

of Silicon Graphics), Wade Ellis (emeritus fromWest Valley College), Tatiana Shubin (San JoseState University), Sam Vandervelde (now at St.Lawrence University), and Joshua Zucker (now atthe American Institute of Mathematics), developedthis idea over the course of about a year. Theysecured sponsorship for a summer workshop atthe American Institute of Mathematics (AIM) in2006, thereby launching the first MTC. It continuesto thrive, with some participants having attendedthroughout the eight years since its inception.

In 2007 AIM began to organize annual summertraining workshops, entitled “How to Run aMath Teachers’ Circle,” to support the spreadof the model. The fifteen workshops held todate have resulted in the formation of almostseventy new MTCs in thirty-seven states. Aseach MTC typically involves between 15 and 20teachers during a normal year, most at the middleschool level, collectively these MTCs reach 1,000to 1,500 teachers per year. The MTC Network(http://www.mathteacherscircle.org) linkstogether MTCs around the U.S., providing ongoingsupport in the form of materials from successfulmath sessions, discussion groups, and a semi-annual newsletter. The program has garneredsupport from the National Science Foundation(NSF), National Security Agency, American Mathe-matical Society (AMS), Mathematical Associationof America (MAA), Math for America, EducationalAdvancement Foundation, and private donors. Inits recent Mathematical Education of Teachers II(MET2) document [C], the Conference Board ofthe Mathematical Sciences (CBMS) recommendedthe MTC model among a handful of professionaldevelopment models for middle and high schoolteachers.

The MTC ModelMath Teachers’ Circles focus on highly interactivesessions in which teachers work with mathemati-cians on rich mathematical problems. Typically, anew MTC group begins with an intensive summerworkshop for between fifteen and twenty-five mid-dle school teachers who immerse themselves inmathematics full-time for a week guided by three orfour mathematicians. During subsequent academicyears, the group continues to meet approximatelyonce a month, with summer workshops held asnecessary to involve new teachers. Needs of localteachers drive the logistics of how the model isimplemented at different sites, but all MTCs focuson problem solving in the context of significantmathematical content, draw on the content ex-pertise of mathematicians, and have high levelsof interactivity among participants and sessionleaders.

Two primary goals of the MTC model are toencourage teachers to develop as mathematiciansby engaging them in the process of doing math-ematics and to create an ongoing professionalK–20 mathematics community. The core modeldoes not explicitly address enhancing instruction,although many MTC groups choose to includeformal discussions of pedagogy as well as informaltime to discuss classroom ideas and concerns.Thus, MTCs are not intended as a comprehensiveform of teacher professional development. Rather,they fall on the mathematical end of the spectrum,and they complement other forms of professionaldevelopment that are more focused on directclassroom applications. As a result, it is naturalto pair MTCs with other kinds of professionaldevelopment, such as classroom coaching, lessonstudy, or Professional Learning Communities.

The problems that form the basis of MTC activi-ties are mathematically rich and have multiple entrypoints or methods of attack, thereby encouragingcollaboration and communication among groupmembers with varying skill levels and backgrounds.Since participants do not know the method ofsolution in advance, the problems require cre-ativity and perseverance. To draw participants in,session leaders try to choose problems that areimmediately appealing in some way. The workthat participants do to solve the problems leadsthem to methods and techniques that are broadlyapplicable in other mathematical endeavors, suchas considering a special case or a more generalcase or drawing a picture or thinking abstractlyor one of the many other techniques familiar togood problem solvers. The pursuit of the solutionalso reveals specific mathematical content. Thosereaders familiar with the Common Core StateStandards for Mathematical Practice will recog-nize that this focus of the MTC model on doinginteresting mathematics presents an opportunityto help teachers develop a deeper understandingof how the mathematical practices interact withmathematical content.

The initial leadership team for each MTC gener-ally consists of two mathematicians, two middleschool math teachers, and an additional organizerwith expertise such as well-developed connectionsto the local teaching community or grant-writingexperience. These leaders are genuine partnersin planning and developing their MTC to meetthe needs of the local mathematical and teachingcommunity, with each person bringing his or herdistinct skills to the table. In a typical case, themathematicians provide leadership in choosingrich material and facilitating the teachers’ workduring the sessions, the teachers help design theprogram so that it is an appealing and meaningfulexperience for their colleagues, and the additional

1336 Notices of the AMS Volume 61, Number 11

Four Squares Seven Squares Nine Squares

Ten Squares Thirteen Squares

organizer helps connect the team with resourcesfor recruiting participants and raising funding.They all work collaboratively on developing avision for their MTC, planning workshops andmeetings, and fostering a positive atmosphere forall involved.

To provide a better understanding of what anMTC session entails and the kind of mathematicalexploration and dialogue MTCs can support, wenow turn our attention to an example of thetype of problem that teachers and mathematicianscollaborate on in a typical meeting.

An MTC Session: Dividing SquaresThe session begins by showing teachers somepictures of a square divided into smaller squares,not necessarily of the same size. Teachers work forabout fifteen minutes creating different divisionsof a square and recording their results. Aftersharing, two natural questions about this divisionarise:

1. For which whole numbers n is it possible tosubdivide a square into n smaller squares?

2. For which whole numbers n is it not possibleto subdivide a square into n smaller squares?

For the first question, the examples teachersproduce seem to indicate that when n ≥ 6 there isa division into n squares. Teachers then identify,possibly with some guidance from the sessionleader, some nice relationships between theirdifferent pictures. For example, the pictures above

with four and seven squares are closely linked:these are the same except that one square hasbeen further subdivided into four smaller squares.This is a process which can be repeated, andeach time this is done, three new squares areadded—since a single square is replaced with foursmaller ones. Above are pictures showing twofurther divisions starting from the decompositioninto seven squares.

Teachers observe that these divisions give4,7,10,13,16, . . . squares. This is an example ofan arithmetic sequence generated by a geometricprocess. The same operation, applied to the “tic tactoe” division into nine squares, produces exampleswith 9,12,15,18, . . . squares. At this point, most ofthe missing numbers belong to another arithmeticsequence, namely 5,8,11,14, . . . . Teachers cannow apply this technique to a division into eightsquares, as shown below:

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Putting all of these examples together gives divi-sions of the square into 4,7,8,9,10,11, . . . smallersquares. The original square itself, without division,is one square. In our experience, teachers comeup with divisions into these numbers, up to aboutthirty, and also identify patterns which can be usedto generate sequences. The session leader oftenneeds to provide some guidance putting togetherthe different ideas and constructions if a formalargument, as presented here, is a major goal of thesession.

The only remaining possibilities are 2,3,5, and6 squares. Dealing with these special cases canrequire some prompting from the session leader,particularly the cases of 2 and 3, which appearvery simple. A good argument to eliminate these isto note that, as soon as a division is made, each ofthe four corners of the initial square must belongto a different smaller square. The session leadermust decide how much to stress this argument,as it does not always come naturally to teachersand is not the main focus of the problem. Fivesquares is also not possible, but the reason is morechallenging to explain: the four squares containingthe corners of the original square cannot be thesame size in this case, but it is impossible forthe complement of these four squares to be asquare, since this complement is not a convexset. This is subtle and necessarily so because itestablishes that regardless of how we cut, we willnever arrive at exactly five smaller squares. Heretoo the session leader must decide how much rigorto pursue in the argument. Cutting into six squaresis possible and is left to the reader. The conclusionof this first investigation answers questions 1 and2: it is possible to cut the large square into anynumber of smaller squares with the exception of2,3, and 5. This first part of the session can takeanywhere from forty-five minutes to an hour anda half depending on the level of precision andthoroughness of the arguments.

One of the hallmarks of Math Teachers’ Circleinvestigations is beginning with a simple, readilyaccessible question and then heading as far as themind and imagination are willing to go. In thisexample, there are many further questions whichcan be asked about the process of subdividing asquare into smaller squares. In the AlbuquerqueMath Teachers’ Circle session described here, theleader posed the following question:

3. Is it possible to perform the division so thatthere is a unique square of smallest size?

The open-endedness makes this a challengingquestion, and intuitions for the answer vary. In thesession described here, one of the teachers wasable to successfully answer this question, but, ingeneral, the session leader needs to decide whether

to provide guidance or to leave teachers with anopen question. This line of inquiry can be carriedeven further:

4. Is it possible to perform the division in sucha way that no two squares have the samesize?

Another direction of inquiry would go in theopposite direction: given a certain number ofsquares (of various sizes) is it possible to assemblethem to form a single square? The question aboutsquares can also naturally be adapted for differentshapes. One very famous example of decomposingan equilateral triangle into smaller equilateraltriangles is Sierpinski’s triangle.

A substantial amount of research1 has been doneon the question of decomposing a rectangle intosquares [D], [BSST], [G]. However, the teachers arenot familiar with the details of that mathematics,and unless this happens to be directly in theresearch area of the mathematicians, they will notbe familiar with it either. Thus, together they areparticipating in their own miniresearch experienceas they explore this topic.

This problem illustrates several features oftypical MTC problems. They are easy to state andrequire minimal background knowledge, allow-ing everyone to immediately engage in a searchfor an answer. The questions considered are of-ten open ended, providing teachers an invaluableopportunity to experience problem solving as math-ematicians. They also lead to rich and interestinginvestigations, leaving the door open for teachersto continue their mathematical explorations longafter the MTC session has ended.

MTC Problems Can Lead AnywhereOne session of a Math Teachers’ Circle in the SanFrancisco Bay Area led to sustained collaborationamong mathematicians, teachers, and students. Inthis session, teachers were given a few minutes tomake their best guess as to the size of 100! withthe answer expressed in scientific notation. Afterdiscussion, the session continued with a goal ofproviding an estimate that is accurate to withina factor of 2. A good starting point for makingthese estimates is that 210 = 1024, which is prettyclose to 103; from here it follows that 2 is about100.3. We can deduce from here that 5 is about100.7. Using these estimates and the fact that 80and 81 are neighboring integers whose only primefactors are 2, 3, and 5, we can deduce that 3 isabout 100.48. These estimates for 2, 3, and 5 canbe used directly to give estimates, as a power of10, for many of the numbers between 1 and 100.

1The authors are grateful to Jim Madden for sharing therich history of this problem with them.

1338 Notices of the AMS Volume 61, Number 11

More work is then needed to fill in numbers withprime factors other than 2, 3, or 5.

Critical in this method was finding the numbers80 and 81, which are adjacent integers with smallprime factors. Intrigued, the session leader had twohigh school students from Morgan Hill, CA, MarkHolmstrom and Tara McLaughlin, investigate thesespecial numbers further. This led to a method ofconstructing large neighbors out of sets of smallneighbors. In the end, around 345,000 neighborswere identified with prime factors less than 200,the largest having twenty-three digits! Holmstromand McLaughlin’s project won first prize in theregional science fair and a trip to Pittsburgh for theIntel International Science and Engineering Fair,where they received an honorable mention fromthe AMS. The two students and the session leaderwrote up these results, and they will be publishedin Experimental Mathematics.

On at least two other occasions, original mathe-matical research has stemmed from MTC sessionsand been published. One of these cases [A] involvedresearch developed by the mathematical facilitatorand a collaborator. The other example, involvingthe geometry of the card game Set, recently ap-peared in an article [B] in the College MathematicsJournal of the Mathematical Association of Amer-ica. It is coauthored by ten teachers and threefacilitators!

Imagine the impact on these teachers’ self-confidence and mathematical self-identities thatcomes from contributing in a bona fide way togenuine mathematical research!

Focus on PartnershipsWhereas the organization of each local MTC variesconsiderably, core principles persist across locales.One of the key principles involves cultivating cross-community partnerships. As noted earlier, theleadership team should consist of mathematicians,teachers, and administrators. Additionally, MTCsessions themselves often find a variety of partnerscoming together to engage in mathematics. Indeed,depending on the local chapter, MTCs can includenot only the mathematician facilitators and middleschool teachers but also some elementary or highschool teachers, other mathematics or sciencefaculty, and undergraduate or graduate studentstraining to be teachers.

The CBMS MET2 document cited this verticalintegration as follows: “A substantial benefit of[Math Teachers’ Circles] is that they addressthe isolation of both teachers and practicingmathematicians: they establish communities ofmathematical practice in which teachers andmathematicians can learn about each other’sprofession, culture, and work” [C].

Impact on TeachersOngoing research demonstrates the benefits ofMTCs for teachers’ confidence, knowledge, andteaching of mathematics. A team led by DianaWhite has recently reported [W] that teachers’mathematical knowledge for teaching (MKT) in-creases significantly over the course of a four- tofive-day intensive summer MTC workshop. Pre-vious research has found a positive correlationbetween teachers’ MKT and student achievementscores [H]. Additionally, researchers from theUniversity of Colorado at Colorado Springs havereported evidence from classroom observationsthat MTC participants significantly increase the useof inquiry-based teaching practices over a one-yearperiod [M].

The coauthors of this article are members of aresearch team that is investigating the impact ofMTCs with support from the NSF. Our research hasthree primary components: a further investigationinto the effects of MTC participation on teachers’mathematical knowledge for teaching, a series ofannual surveys aimed at learning more about theteachers who participate and their motivations fordoing so, and the development of in-depth, mul-tiyear case studies that include classroom videoobservations and interviews. The case study resultsare too preliminary to report here, but the surveysare providing an interesting picture of teachers’experience with MTCs. For example, on the 2012survey, 75 percent of the 169 respondents saidthat MTC participation has affected their teaching.Examples ranged from specific topics that MTCsessions have helped them teach more effectively(e.g., “I’ve incorporated number theory ideas Ilearned with respect to comparing fractions”) tobroader changes in how they ask students toapproach mathematics (e.g., “I give students fewersteps and ask them to tackle complex problemswith less scaffolding of the problem itself. I alsohave been helping students reflect on what theirproblem-solving process is and how to optimizetheir problem-solving process”). Over half of therespondents told us that MTC participation haschanged their view of mathematics. They citethat MTCs enable them to see deeper connectionsbetween mathematical topics, expose them tofields of mathematics that they were previouslyunfamiliar with, and help them envision doingmathematics as a more collaborative and enjoyableenterprise. Approximately half of the respondentsalso wrote that participating in MTCs has affectedtheir other professional activities, for example,leading them to become more involved in collab-orations, leadership, and student extracurricularmathematics activities.

Perhaps the most rewarding result we have seen,and the one that we believe will ultimately have the

December 2014 Notices of the AMS 1339

biggest impact, is that participating teachers reportbeginning to see themselves as mathematicians.For example, here are a few quotations from middleschool teacher participants:

• “You encouraged me as a mathematician. I havenever actually seen myself as one before.”

• “I’ve become much more of a recreationalproblem solver—I always have my problemnotebook with me for down times.”

• “MTC grows teachers who are also mathemati-cians.”

Impact on MathematiciansThere has been considerably more research intothe benefits of MTC participation for teachers, butwe now turn our attention to how mathematiciansbenefit from MTCs. Beyond the opportunity to getinvolved in K–12 education while sharing one’slove of mathematics, MTCs can be synergisticwith other efforts, such as mentoring studentresearch, as illustrated in the example above.MTCs can also provide an excellent avenue forcontributing to one’s departmental or institutionalmission, especially at a state-supported institutionor an institution with a large teacher-trainingprogram. For those involved in teacher trainingor professional development, MTCs can providea way to maintain a professional connection withthe local teaching community and to offer ongoingsupport to teachers who have participated in otheruniversity programs. One mathematician chose tobecome involved because “we have relationshipswith many teachers in our surrounding districts.Our teachers are eager for any help we canprovide. The Math Teachers’ Circle will providea venue to address their needs.” Another writesthat “offering Math Teachers’ Circles would bea perfect opportunity for our master’s programto continue to support the teachers graduatingfrom it, and providing leadership opportunitiesfor them.” Having an impact on the preparation oflocal K–12 students who will eventually attend theuniversity is another potential long-term benefit.For example, another mathematician writes that theprimary reason for becoming involved was “to formrelationships with local schools while enhancingthe instruction provided to future undergraduatestudents in our target recruitment populations.”

Increasingly, MTCs are becoming a profession-ally recognized forum for mathematicians tocontribute to education. The AMS and MAA havesupported the effort from early on, and the MAAcontinues to partner with AIM to host one of the“How to Run a Math Teachers’ Circle” trainingworkshops each summer. Anecdotally, more thana handful of MTC leaders have noted that theirMTC involvement was a contributing factor in a

positive job search experience or tenure or pro-motion decision. Several MTC leaders have alsoreceived partial support for their MTCs as part ofthe broader impacts of their NSF research grants,indicating a wider recognition that MTCs constitutea valuable contribution that mathematicians canmake to society.

Final ThoughtsBy their very nature, Math Teachers’ Circles are anatural forum for mathematicians and mathemat-ics departments to reach out to practicing teachersand form lasting relationships that recognize thecommon purpose which unites K–20 educators.Some faculty who participate in MTC meetings areinspired to seek other ways to contribute to thelarger teaching community. This may include work-ing with preservice teachers at their institutionor becoming more involved with state or nationaleducational efforts. Whether it serves as a launch-ing point for deeper involvement with teachereducation or as an end in itself, participation inan MTC can impact both teachers and studentsand can play an important and fulfilling role in theprofessional lives of many mathematicians.

AcknowledgmentWe would like to thank Brian Conrey for his insightsregarding the origins of MTCs and his account ofthe MTC session that led to the student researchproject.

References[A] G. Abrams and J. Skylar, The graph menagerie:

Abstract algebra and the mad veterinarian,Mathematics Magazine 83 (3) (2010), pp. 168–179.

[B] M. Baker, J. Beltran, J. Buell, B. Conrey, T.Davis, B. Donaldson, J. Detorre-Ozeki, L. Dib-ble, T. Freeman, R. Hammie, J. Montgomery, A.Pickford, and J. Wong, Sets, planets, and comets,The College Mathematics Journal 44 (4) (2013), pp.258–264.

[BSST] R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T.Tutte, The dissection of rectangles into squares,Duke Mathematical Journal 7, (1) (1940), pp. 312–340.

[C] Conference Board of the Mathematical Sciences, TheMathematical Education of Teachers II, AmericanMathematical Society, Providence, RI; and Mathe-matical Association of America, Washington, DC,2012.

[D] M. Dehn, Über Zerlegung von Rechtecken inRechtecke, Mathematische Annalen 57 (1903), pp.314–332.

[G] D. Gale, More on squaring squares and rectangles,Mathematical Intelligencer 15 (4) (1993), pp. 60–61.

[H] H. C. Hill, D. L. Ball, M. Blunk, I. M. Goffney, andB. Rowan, Validating the ecological assumption:The relationship of measure scores to classroom

1340 Notices of the AMS Volume 61, Number 11

teaching and student learning, Measurement: Inter-disciplinary Research and Perspectives 5 (2-3) (2007),pp. 107–117.

[M] P. D. Marle, L. L. Decker, and D. H. Khaliqi,An inquiry into Math Teachers’ Circle: Findingsfrom two year-long cohorts, paper presented atthe School Science and Mathematics AssociationAnnual Convention, Birmingham, AL, 2012.

[S] T. Shubin, Math circles for students and teachers,Mathematics Competitions, Journal of the WorldFederation of National Mathematics Competitions19 (2) (2006), pp. 60–67.

[W] D. White, B. Donaldson, A. Hodge, andA. Ruff, Examining the effects of Math Teach-ers’ Circles on aspects of teachers’ mathe-matical knowledge for teaching, InternationalJournal of Mathematics Teaching and Learn-ing, published online September 26, 2013, athttp://www.cimt.plymouth.ac.uk/journal/.

Research topic: A three-week summer program for Geometry of moduli spaces graduate students and representation theory undergraduate students mathematics researchers Education Theme: undergraduate faculty Making Mathematical K-12 Teachers Connections math education researchers

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Organizers: Roman Bezrukavnikov, Massachusetts Institute of Technology, Alexander Braverman, Brown University, and Zhiwei Yun, Stanford University Graduate Summer School Lecturers: Mark de Cataldo, Stony Brook University; Victor Ginzburg, University of Chicago; Davesh Maulik, Columbia University; Hiraku Nakajima, Kyoto University; and Xinwen Zhu, Northwestern University Clay Senior Scholars in Residence: Ngô Bao Châu, University of Chicago and Andrei Okounkov, Columbia University Other Organizers: Undergraduate Summer School and Undergraduate Faculty Program: Steve Cox, Rice University; and Tom Garrity, Williams College. Teacher Leadership Program: Gail Burrill, Michigan State University; Carol Hattan, Skyview High School, Vancouver, WA; and James King, University of Washington. Workshop for Mentors: Dennis Davenport, Howard University

Applications: pcmi.ias.edu Deadline: January 31, 2015

IAS/Park City Mathematics Institute Institute for Advanced Study, Princeton, NJ 08540

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