Todxs cuentan: cultivating diversity in combinatorics
TTodxs cuentan: cultivating diversity in combinatorics
Federico Ardila–Mantilla ∗ After nine years of activity, the SFSU-ColombiaCombinatorics Initiative has helped build a strong,active community of more than 200 mathematicians,most of whom are members of underrepresentedgroups in mathematics. More than 50 of them havepursued Ph.D.s in mathematics, while others con-tinue to be mathematics users, enthusiasts, and am-bassadors in other fields, and to encourage and in-spire the next generation of scientists in their com-munities. This article tells our story, and shares somelessons we have learned about broadening and deep-ening representation in mathematics.Figure 1: The SFSU-Colombia combinatorics family. ∗ San Francisco State University, San Francisco, CA, USA, andUniversidad de Los Andes, Bogot´a, Colombia. [email protected] by NSF CAREER Award DMS-0956178 and NIH SFBUILD grant 1UL1MD009608-01.
We begin by stating our axioms in Section 0. Wethen outline our work in Section 1, and discuss ourunderlying sociopolitical framework and pedagogicalstrategies in Section 2.
0. THE AXIOMS
Let me begin with some axioms. I cannot provethem but I firmly believe in them, and I build mywork upon them.
Axiom 1.
Mathematical potential is distributedequally among different groups, irrespective of geo-graphic, demographic, and economic boundaries.
Axiom 2.
Everyone can have joyful, meaningful,and empowering mathematical experiences.
Axiom 3.
Mathematics is a powerful, malleable toolthat can be shaped and used differently by variouscommunities to serve their needs.
Axiom 4.
Every student deserves to be treated withdignity and respect.
These statements should not sound revolutionary;but considering the current practices of the mathe-matical society, they are a call to action.
1. SFSU-COLOMBIA COMBINATORICS INITIATIVE
The SFSU-Colombia Combinatorics Initiative isa research and training collaboration which seeksto offer every interested student a challenging andsupportive mathematical experience, while plantingseeds for the broader and deeper representation ofdifferent groups in mathematics.
The partners.
Los Andes is an elite private uni-versity. Its math department is one of Colombia’sstrongest, and through scholarships, it attracts someof the best prepared students in the country. Most1 a r X i v : . [ m a t h . HO ] A ug os Andes participants in the program are under-graduates with a very solid mathematics backgroundbut little understanding of what research looks like.About one fifth of them were members of the Colom-bian math olympiad team, which I coached for 15years.SFSU is a large public 4-year university with adiverse population: over 60% of the students comefrom ethnic minority groups, and almost half arefirst-generation college students. The Master’s pro-gram welcomes and serves a student body with a widerange range of demographics and academic prepara-tion. SFSU is home to an active research group incombinatorics, including about 15 students at a time.I was born and raised in Colombia, discoveringmathematics through the Olimpiadas Colombianasde Matem´aticas. I came to the US 22 years agoas an undergraduate on a scholarship at MIT andhave been here since, while remaining in close con-tact with Colombia and its mathematics. In the USI am usually counted as a minority mathematician,and I have often felt in the minority. Though I wastreated well, most of my training in the US tookplace alone, avoiding mathematical spaces where Ifelt uncomfortable. However, unlike most studentsfrom marginalized groups in mathematics, I neverhad to overcome the structural inequalities of our ed-ucational system. In particular, I never had a teacheror peer who doubted my abilities or told me that Iwas not good enough to succeed. How it all started (2005) . Felipe Rinc´on, then anundergraduate at Los Andes, had taken my AlgebraicCombinatorics course there in 2003, and was writinghis thesis under my supervision. In view of his class-mates’ interest and the relative lack of activity in thisfield in Colombia, he volunteered to teach an unoffi-cial combinatorics course for free. The course was agreat success, attracting more than 20 of the approx-imately 120 math majors.I had recently begun an Assistant Professorship atSan Francisco State University, and was very inter-ested in contributing to mathematics in Colombia.Felipe’s course exposed a great need, so I decided tooffer a followup topics course on matroid theory, of-fered jointly at SFSU in person and at Los Andes electronically. Internet courses were still not in fash-ion, and this was exactly the kind of wild, uncertainexperiment that Colombians love to embark upon.
SFSU/Los Andes, Matroid Theory (2007) . I taughtthe class at SFSU and filmed it using my colleagueArek Goetz’s wonderfully low-budget artisanal setup,modeled after the one used by his figure skating sis-ter: heat sensors across the front of the room detectedwhere I was standing, and told the camera in the backwhere to point. With practice I learned to use largeand clear handwriting, and to move around less soI would not make the viewers dizzy. Arek also im-proved his algorithm, so the camera would not followme when I walked across the room and back to getthe eraser.Los Andes students watched the lecture together ina classroom, where I was present virtually to answerquestions. We could not meet simultaneously due totime zone differences and the limits of the technologyavailable to us. I visited Bogot´a early in the semesterto meet the students in person. Ph.D. students fromUC Berkeley also took the class; some in person, andsome on video.We made great efforts for US and Colombian stu-dents to feel that they were in the same class. Stu-dents created minibios including their photos, per-sonal background, and mathematical interests; theywere not afraid to let their personality shine through.I also created an online forum where US and Colom-bian, got to ‘meet’, discuss the course material andthe assignments, and find future collaborators.Figure 2: Student profiles from the 2007 joint course.In the first half of the course I assigned homeworkwhich ranged from reasonably straighforward exer-cises, accessible to anyone in the class, to approach-able but challenging recent results in the matroidtheory literature, which required extensive (some-2imes international) teamwork. Occasionally I as-signed unannounced open problems. Posting studentsolutions – while trying to represent everyone in theclass – raised the quality of the work and the writing.Most but not all Colombians were proficient in En-glish, and some took the opportunity to write math-ematics in English for the first time. Others didn’t,and SFSU students got to read mathematics in Span-ish; this was especially exciting for some of the USLatinas/os, many of whom spoke Spanish at homebut not in their mathematical life.In the second half, students did final projects inpairs; I suggested dozens of projects, from surveysof classic topics to current open problems of interest.Most students tried to do original research. ~CI'\
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G,<- '"',," C. en eo.C Figure 3: Cesar Ceballos’s work progresses from ahomework assignment to a final project, a Master’sthesis, and a publication [2] about the triangulationsof the product of two simplices ∆ m × ∆ n . • Results.
The 11 projects in the class led to 5 papersin international journals (3 coauthored internation-ally) [1, 2, 3, 4, 10] and 8 theses. Nine years later,of the 21 students in the class, 11 have completedPh.D.s in mathematics. Currently,– 4 are university professors in mathematics,– 6 are postdoctoral researchers in mathematics, – 4 are community college faculty in mathematics,– 2 are Ph.D. students in economics, and– 5 are working in industry. • Observations and lessons learned. – Every student did deep work, especially in their fi-nal projects. Students with substantial mathematicalgaps learned what they needed along the way.– A strong sense of teamwork and collaborationhelped most students lift each other up, though a fewstruggled with having to do mathematics in groups.– Most projects were international, and those weremost productive. Students’ curiosity towards and ac-countability to strangers played a useful role.– Generally speaking, SFSU students were impressedwith the knowledge and problem-solving skills of LosAndes students. Los Andes students were impressedby the work ethic and determination of SFSU stu-dents, particularly in their research projects.– Asynchronicity was far from ideal; it made classlecture-centric. Creating a coherent shared experi-ence took hard work, and it was still not a great sub-stitute for personal interaction.Figure 4: With students presenting their work at theFPSAC 2011 combinatorics conference in Iceland.
Some guiding principles for the course design.
TheMatroid Theory class became the blueprint for 8 top-ics courses over the last 9 years; 6 of them were offeredjointly at SFSU and Los Andes. As I continued todesign these courses, I identified a few key principlesthat I always attempted to implement.– Choose a deep topic of current interest that is ac-cessible to students with different backgrounds.3 Hold students to extremely high standards, andmatch that expectation with a solid support system.– Devise challenging, interesting, and inspiring as-signments, including a final project in pairs that stu-dents have the freedom to design themselves.– Give students the time they need: allow 2 weeks foreach homework and 2 months for the final project.– Create a course structure that builds a strong com-munity through a shared mathematical purpose.Several student evaluations mentioned this sense ofcommunity, and some took it to the social media:
Online resources.
I made all materials for my 6SFSU-Colombia courses freely available, includingmore than 240 hours of videos on the YouTube Chan-nel federicoelmatematico . According to the chan-nel’s statistics, users in 155 countries have viewedmore than 10,000 hours of combinatorics videos.More meaningfully to me, I have personally heardfrom active users in places like Colombia, Germany,India, Iran, Sudan, Turkey, and the US.Technology offers exciting possibilities to increaseaccess to education, but we should be cautious ininterpreting the numbers above. For example, theaverage view duration for these 50-75 minute videoswas 8:33. Of the viewers who divulged their gender, 81% were men. The last few lectures of each coursewere watched between 60 and 300 times each, withviewers concentrated in the US and Western Europe,and including several of my students and colleagues.These online resources have certainly been helpful topeople, but I do not believe they have had a signif-icant effect in closing the gap for access to qualitymathematics resources.Figure 5: MIT professor Richard Stanley, as he isabout to zipline across the Chicaque Cloud Forestwith US and Colombian students: “I hope I don’thave a heart attack right now; I still have to go tomy Birthday Conference next week!”
Encuentro Colombiano de Combinatoria (ECCO).
It soon became clear that US and Colombian studentswanted to build closer ties and collaborate in person.I organized the Encuentro Colombiano de Combina-toria to benefit young mathematicians first and fore-most, especially those who do not have easy access tosuch an opportunity. Now organized by a committeeof former participants, ECCO meets biyearly. Wedo our best to build an atmosphere that is equallyprofessional and welcoming.ECCO features minicourses by international ex-perts, collaborative problem sessions, research talksby students, open problem presentations, mentoringsessions, a hike or two, and (inevitably, it seems) animpromptu street party.Now that ECCO has gained some international no-toriety, several combinatorics experts have asked toattend. We welcome them with care, keeping in mindthat this is a school and an encuentro , not a regu-lar conference. We ask these experts to do problemsets with the students, to present open problems thatthey would like help with, to serve as mentors, and gathering, usually of people with shared experiences Funding.
The Matroid Theory class and ECCO 2008were generously funded by seed grants from SFSUand Los Andes. With those results in hand, I ap-plied for the NSF CAREER and RUI grants that havefunded our activities since then.The cost of tuition and living in San Francisco isa real challenge for our students, most of whom havesignificant work loads while they study. We have sup-ported some of them using other NSF and NIH grantsat SFSU, such as Matt Beck’s GK-12 grant in math-ematics and Frank Bayliss’s SEO grants in science.We plan to implement a more sustainable fundinginfrastructure that will allow our students to betterfocus their efforts on their academic preparation.
Some statistics.
Approximately 200 students have officially enrolled in the classes offered by this ini-tiative, and approximately 50 of them have pursuedPh.D.s in the mathematical sciences.The initiative has had 40 theses students (4 Ph.D.,27 M.S., 12 B.S.): 28 in the US (14 women, 15URMs ) and 12 in Colombia (3 women, 12 URMs),who authored more than 20 publications, includingjournals such as Duke Math. J., Advances in Math,Int. Math. Res. Not., and J. Combin. Th. (A).
Of these 40 students, 28 have entered math Ph.D.s(19 US, 11 women, 21 URMs), and 25 have finishedor are current students, including all 21 URMs.
2. DEEPENING REPRESENTATION.
The underrepresentation of women, Latinas/os,and African-Americans in US mathematics is welldocumented. These groups constituted respectively • • • underrepresented minorities Numbers for US and women omitted for confidentiality. We focus for the moment on these three large groups forwhich data is available.
The mathematics.
To succeed in mathematics, onemust do high quality mathematics. This is especiallytrue for students without elite credentials, for whomthe bar to success is often set higher. It is crucialto involve mentors with active research programs inareas of current interest.
Todos cuentan.
Our activities are designed to serve every interested student, by building an inclusive en-vironment that everyone contributes to and benefitsfrom. We aim to increase students’ sense of belongingand responsibility to their mathematical community.Intentionally or not, most programs in highermathematics are designed to select the ‘top’ math-ematicians at each stage, and prepare them for thenext stage. Such programs have certainly played acrucial role in the careers of many mathematicians,including mine. However, they have also played a role in excluding and discouraging others with greatmathematical potential, particularly among under-represented groups.
Equitable spaces.
We try to be mindful of how differ-ent students experience the same environment, andfind ways to make sure each one of them is activelyinvolved and engaged within the spaces we provide.Language matters. Many of the standard patternsof communication of mathematicians – like calling afact ‘easy to see’ when it isn’t, or saying ’I get it’when one doesn’t – are harmless to some, but theyfeel exclusive to those who already feel like outsiders.Structure matters. Without an explicit and mind-ful effort, classrooms easily turn into conversationsbetween a professor and a handful of students. Mycurrent courses are organized so that every studentparticipates in every class, either verbally, in writing,or in small groups; I use several of Kimberly Tan-ner’s techniques for creating equitable learning envi-ronments. [22] math 850 . representation theory federico ardila san francisco state university
Agreement.
The goal of this course is to o↵er a meaningful, rigorous, and rewarding experienceto every student; you will build that rich experience by devoting your strongest available e↵ort tothe class. You will be challenged and supported. Please be prepared to take an active, patient,and generous role in your own learning and that of your classmates.
Diary.
After class n you will discuss the material with a classmate, and turn in a very briefsummary of the key points at the beginning of class n + 1. We will begin class n + 1 by discussingthese. After class n + 1, you will edit and transcribe your summary of class n into your “diary” inLaTeX. You will get extra credit if you write diary entries with > / Final Project.
This is a chance to go much deeper into a topic that interests you, in pairs. It couldbe an expository paper, the beginning of an original research project, or (why not?) the solutionto an open problem. I will suggest possible projects. For most of you, this is the first time youreceive such an open-ended assignment in a mathematics class. Have fun with it! Don’t be afraidto take a risk; this is an opportunity to try something intriguing that you don’t know how to do.
Figure 7: Language and structure: syllabus excerpts.
The support system.
A challenging experience caneasily become alienating if it is not presented mind-fully and accompanied with abundant support. Wedo not shy away from the emotional side of this work.A career in mathematics requires balancing long pe-riods of frustration with (sometimes too brief) mo-6ents of great joy, and students find it surprisingand beneficial to learn that they are not the only onesstruggling with that balance. Readings and discus-sions on the psychology of mathematics and science[21, 23], growth mindset [11], stereotype threat [20],and impostor syndrome [9] have been helpful.At SFSU we collaborate with initiatives like SFBUILD, which works to create inclusive and support-ive research environments across 6 departments, andthe Mathematistas student group for women in math,which builds community in our department throughmany informal activities. Outside of SFSU we benefitfrom interacting with organizations such as SACNAS,the Math Alliance, USTARS, and Latin@s in Math.We are aware that the crucial work of supportingstudents traditionally falls mostly on women and peo-ple of color [13, 16, 17]. We make every attempt tocounter that tendency.
People rise to high expectations.
Many of my stu-dents from marginalized groups have been told, oftenby well-meaning professors in positions of power overthem, that they cannot do something or that theyare not good enough to be mathematicians. I never say this to a student. I cannot possibly know that. I have worked with students whose mathematicalpotential is not immediately apparent to me, but Iknow they are here for a reason. My approach is toalways treat them with respect, give them an intrigu-ing project that suits their experience and interests,and support them along the way; I have seen most ofthem rise to that challenge.
Unconscious bias and discrimination.
The math-ematical society at large is relatively homogeneous,and students who do not fit neatly within its domi-nant cultures are often faced with unspoken but realobstacles. This is especially true for many underrep-resented minorities, women and gender minorities,parents, disabled, low income, first-generation, andreturning students among others. On a professionallevel, these students face skepticism and their workis undervalued by professors and peers.[8, 14, 18, 26] Upon meeting Endre Szemer´edi, Israel Gelfand told him“Just try to find another profession; there are plenty in theworld where you may be successful.” [19] Szemer´edi went onto write his Ph.D. thesis under Gelfand, and win the 2012 AbelPrize among many other honors.
Figure 8: Celebrating Anastasia Chavez and NicoleYamzon’s new theorem on the Dehn–Sommerville re-lations [7], Carolina Benedetti’s visit to San Fran-cisco, and Nicole’s birthday.On a personal level, many feel pressured to leave theirtrue selves at the door to try to fit in.These are deep problems that we are far from solv-ing. When they do arise, we talk about them openly,learn from them, and try to improve. We also empha-size that this is not just a problem for marginalizedgroups to address; every member of the mathematicalsociety plays a role.
The bigger picture.
The SFSU-Colombia Combina-torics Initiative tries to instill an awareness of thetremendous power that mathematicians hold in to-day’s society, and a collective belief in using our partof that power to do positive work. Each individualgets to decide what this means for them, whether ornot they pursue careers in mathematics. For many ofthem, it means planting the seeds for the next gener-ation of scientists in their communities. Today’s un-even representation in mathematics is largely, thoughnot exclusively, a consequence of the uneven accessto opportunities before students arrive to our collegeclassrooms. Several of our students and alumni aredoing powerful work with young people at variousstages of their education. We are slowly integratingtheir initiatives through a network of mentorship that7veryone contributes to and benefits from.In Colombia, several participants in the initia-tive supervise undergraduate research projects andlead math olympiad programs. Others partner withthe Clubes de Ciencia Colombia and the interactivescience museum Parque Explora; thanks to them,ECCO 2016 will include a week-long workshop forstudents in Medell´ın public high schools, and a pub-lic math talk featuring the students’ work which willbe broadcast by television and streaming.In the U.S., participants direct the San FranciscoMath Circles, making great efforts to reach the (88%ethnic minority) populations of the San FranciscoUnified School District (SFUSD). Through the GK-12 program, 50 Ph.D.-bound students spent 10 hoursa week supporting the mathematics departments ofvarious local public high schools. Other alumni teachmathematics full-time in local community collegesand high schools; the SFUSD claims that 70% of itsteachers have received training at SFSU.In any initiative of this sort, one should share thedecision making power and make oneself replaceableas soon as possible; I am inspired to be surrounded bya community of mathematicians who see themselvesas agents of change in our scientific culture and inour societies. They are doing extremely interestingmathematics, they are working hard to train the nextgeneration of scientists – a diverse, engaged, dynamiccommunity that works to serve the needs of all – andthey are having a lot of fun in the process.
More information.
Our webpage, containing addi-tional information and resources, can be found at: http://math.sfsu.edu/federico/sfsucolombia.html.
Acknowledgments.
I thank the AMS Committee onEducation, AMS President Robert Bryant, and theeditors of the Notices of the AMS for inviting me toorganize and share my ideas around a topic that isvery challenging to me, partly because it is also verypersonal to me.I am just one of many people doing this work at SanFrancisco State University and the in Colombia. I amindebted to my colleagues for their invaluable supportof this project. The financial support by the NSF, theNIH, and CIMPA have also been instrumental.If I have learned one thing, it is that I still have alot to learn. I am extremely grateful to the teachers,colleagues, family members, friends, organizers, andfutbolistas who have shaped this initiative; I mustmention Natalia Ardila, Matthias Beck, Ben Braun,Dania Cabello, Jeff Duncan-Andrade, May-Li Khoe,Mar´ıa de Losada, Amparo Mantilla de Ardila, Leti-cia M´arquez-Maga˜na, Bob Moses, Ali Nesin, GustavoSalazar, Kimberly Tanner, and Nicole Yamzon. Mostimportantly, I would like to thank my students, whoteach me something new every day and give meaningto my mathematical career.
References [1] Federico Ardila, Carolina Benedetti and Jeff Doker. Matroidpolytopes and their volumes.
Discrete and ComputationalGeometry (2010) 841-854.[2] Federico Ardila and Cesar Ceballos. Acyclic systems of per-mutations and fine mixed subdivisions of simplices. Discreteand Computational Geometry. (2013) 485-510.[3] Federico Ardila, Alex Fink and Felipe Rincon. Valuations formatroid polytope subdivisions. Canadian Journal of Math-ematics (2010) 1228-1245.[4] Federico Ardila and Amanda Ruiz. When do two plantedgraphs have the same cotransversal matroid? Boletin de laSociedad Matematica Mexicana (2010) 63-73.[5] Joan Burrelli. Academic institutions of minority facultywith science, engineering, and health doctorates. NationalScience Foundation InfoBrief, NSF 11-320, October 2011.
6] Richelle Blair, Ellen E. Kirkman, and James W. Maxwell.Statistical Abstract of Undergraduate Programs in the Math-ematical Sciences in the United States. American Mathemat-ical Society, 2013.[7] Anastasia Chavez and Nicole Yamzon. The Dehn-SommervilleRelations and the Catalan Matroid. Preprint, 2015. arXiv:1512.04513 [8] Dolly Chugh, Katherine Milkman, and Modupe Akinola. Pro-fessors are prejudiced too.
The New York Times Sunday Re-view , pg. SR14, May 9, 2014.[9] Pauline Rose Clance and Suzanne Imes. The Imposter Phe-nomenon in High Achieving Women: Dynamics and Thera-peutic Intervention.
Psychotherapy: Theory, Research andPractice . (1978) 241-247[10] Harm Derksen and Alex Fink. Valuative invariants for poly-matroids. Advances in Mathematics (2010), 1840–1892.[11] Carol Dweck. The secret to raising smart kids.
ScientificAmerican Mind (2007) 36-43.[12] Daniel Z. Grunspan, Sarah L. Eddy, Sara E. Brownell, Ben-jamin L. Wiggins, Alison J. Crowe, and Steven M. Goodreau.Males Under-Estimate Academic Performance of Their Fe-male Peers in Undergraduate Biology Classrooms. PLOSONE, (2016), e0148405.[13] Audrey Williams June. The invisible labor of minority pro-fessors. Chronicle of Higher Education . Nov. 8, 2015.[14] Sarah-Jane Leslie, Andrei Cimpian, Meredith Meyer, EdwardFreeland. Expectations of brilliance underlie gender distribu-tions across academic disciplines.
Science (2015) 22–265.[15] Audre Lorde. Sister outsider: Essays and speeches. CrossingPress, 2012.[16] Joya Misra, Jennifer Hickes Lundquist, Elissa Holmes, andStephanie Agiomavritis. The ivory ceiling of service work.
Academe (2011): 22.[17] Kristen Monroe, Saba Ozyurt, Ted Wrigley, and Amy Alexan-der. Gender Equality in Academia: Bad News from theTrenches, and Some Possible Solutions. Perspectives on Pol-itics (2008) 215-233[18] Corinne A. Moss-Racusin, John F. Dovidio, Victoria L.Brescoll, Mark J. Graham, and Jo Handelsman. Science fac-ulty’s subtle gender biases favor male students Proceedings ofthe National Academy of Science
109 (2012) 16474–16479.[19] Martin Raussen and Christian Skau, Interview with EndreSzemerdi. Notices of the American Mathematical Society (2013) 221 – 231.[20] Claude Steele. Whistling Vivaldi: And Other Clues To HowStereotypes Affect Us. New York : W.W. Norton and Com-pany, 2010.[21] Martin A. Schwartz. The importance of stupidity in scientificresearch. Journal of Cell Science (2008) 1771.[22] Kimberly Tanner. Structure matters: twenty-one teachingstrategies to promote student engagement and cultivate class-room equity.
CBE-Life Sciences Education (2013) 322-331. [23] William Thurston. On proof and progress in mathematics. Bulletin of the American Mathematical Society [25] William Yslas V´elez, James Maxwell, and Colleen Rose. Re-port on the 2013-2014 new doctoral recipients. Notices of theAmerican Mathematical Society (2015) 771-781.[26] Ed Yong. XY Bias: How Male Biology Students See TheirFemale Peers. The Atlantic [27] Estanislao Zuleta. Educaci´on y democracia: un campo decombate. Corp. Tercer Milenio, Bogot´a, 1995.[27] Estanislao Zuleta. Educaci´on y democracia: un campo decombate. Corp. Tercer Milenio, Bogot´a, 1995.