12 community The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. Bill Thurston What is belonging? The extent to which an individual feels accepted, valued, and legitimate within the community. Deanna Haunsperger Copyright © 2020. Yale University Press. All rights reserved. Ricardo Gutierrez is a New York City native, the son of im­ migrants in a working-­class neighborhood. His father never finished high school, and his mother didn’t make it past the Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Community 187 eighth grade. He wrote to me in 2017 after reading a speech I gave about flourishing in mathematics. He had shown an apti­ tude for math at an early age, but lacked a mentor to guide him. So he pursued another interest in college, and for the past nine­ teen years has had a successful career as an audio engineer—not really a technical job, but, as he puts it, “I work on music and make it sound better.” He loves his career—in fact, one of the projects he worked on was up for a Grammy—but he felt that something was missing: I just yearn for more, and it felt like there was a hole in my life. I yearn for more knowledge and more thinking about the types of things that mathematics and computer science gives me. Maybe it is the drilling down on logical problems that I find appealing, even meditative. So maybe it is that I have outgrown the boundaries of my career and I wanted to supplement it with mathematics and program­ ming. . . . Maybe it is more appropriate to say that having mastered my career, it has become rote. Copyright © 2020. Yale University Press. All rights reserved. He took the brave step of returning to school at age forty, in a program for nontraditional students. He says: While the rigor and pressure of being in a really tough academic environment have been almost impossibly dif­ ficult—especially after I’ve been out of practice for so long—it is the entrenched feeling that I don’t belong in those Math and CS classes that has sometimes been the most harmful. Those feelings are most probably tied to my early life and the fact that any dreams I may have had then were in discord with the cold realities of my neighborhood and life, and that I didn’t have any mentor then to disabuse me of that great untruth. That distorted reality, the “I’m Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. 188 Community Copyright © 2020. Yale University Press. All rights reserved. not supposed to be here,” runs as an infinite loop process in even unseen ways. It is a constant struggle. This feeling—I don’t belong—can be quite crippling. And this is where community is terribly important: for us to feel belong­ ing. Because if, as Parker Palmer says, “To teach is to create a space where the community of truth is practiced,” then we are called to speak truths that others cannot see for themselves.1 We can reassure them that they do belong. None of us can flourish without a supportive community— people with whom we can share joys and sorrows, hopes and fears. A community helps us normalize struggle, and realize “I’m not alone in my struggle.” Community is a deep human desire. Thus, it serves as an entryway for many people to flourish in mathematical pursuits, whether recreational, educational, professional, or in the home. By “mathematical community,” I mean any group of people who gather together over common mathematical experiences. You are forming a mathematical community in your home when you share math jokes, show enthusiasm for mathematics, make geo­ metric objects, read stories together that involve math, or cook together—adding fractions for a recipe and talking about it. You are entering a mathematical community when you walk into a math class or if you join a game of strategic thinking. Community is not a word that most people associate with mathematics. Instead, a popular view of a mathematician is a lone person toiling for years on a problem in solitude. Indeed, several recent solvers of famous problems match this descrip­ tion. In 1993, Andrew Wiles announced a (then slightly flawed) proof of the 350-­year-­old Fermat’s last theorem, which is tanta­ lizingly simple to state: the equation x n + y n = z n has no solution in whole numbers for n > 2. Wiles toiled in secret for seven years Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Copyright © 2020. Yale University Press. All rights reserved. Community 189 on this problem.2 In 2003, Grigori Perelman proved the century-­ old Poincaré conjecture in topology, which, roughly speaking, says that a bounded three-­dimensional object without any holes must be spherical. No one knew that he was working on it.3 And in 2013, when Yitang Zhang proved that gaps between prime numbers are bounded—a major breakthrough in attacking the famous twin primes conjecture—no one in the field knew who he was.4 These examples, which made news for their exception­ ality, feed the myth that math should be a solitary endeavor. In reality, mathematics has always been collaborative, be­ cause people gather together over many activities that are mathe­ matical—learning, reading, games, research. We spend time in communities to enjoy math because, as Bill Thurston said (in response to a person who worried that they would never do any­ thing original in math), the real satisfaction is in the learning and sharing. Professionally, mathematics is becoming even more collabo­ rative than it was in the past. A 2002 study showed that the pro­ portion of authors doing joint research in math was 81 percent in the 1990s, up from 28 percent in the 1940s.5 In 2009, the mathe­ matician Tim Gowers famously enlisted the Internet to collabo­ rate to find an elementary proof of the Hales-­Jewett theorem, a result that roughly speaking asserts that versions of tic-­tac-­toe with many more dimensions and any number of players must always have a winner. A growing number of classroom math teachers are employing active learning techniques, which use class time for student participation and collaboration. With the rise of social media, math teachers are also connecting in novel ways to share ideas and form new interest groups. Working in teams is central to the way that math explorers interact today and an important skill for careers in business, industry, or gov­ ernment. Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. 190 Community Copyright © 2020. Yale University Press. All rights reserved. Community serves an important function in bringing people together in mathematical exploration—helping them to grow in the virtues promoted by socialization. The most engaging math-­ themed programs center on fostering community among par­ ticipants, and you’ll find a group to serve almost any niche, from kids to teachers to researchers.6 However, building community should be more than just bringing people together over mathematics. We must be vigi­ lant to overcome obstacles that are sometimes more prevalent in mathematical communities. Mathematical communities are often too focused on achieve­ ment—frequently, a narrow kind of achievement. Hierarchies get reinforced when we rank people according to a singular “ability.” We do this mentally when we evaluate who is “good at math.” We often signal to others that there’s only one way to be successful in mathematics—by forcing kids to do math quickly, or rushing students into calculus in high school, or telling pro­ fessionals that they aren’t “real mathematicians” if they don’t do research. In reality, there are multiple ways to be success­ ful. Mathematical achievement is not one-­dimensional, and we must stop treating it like it is. Too often we think of math like a pole in the ground: there’s only one way for a vine to grow up the pole. In reality, math is like a trellis: as a vine, you can find your way up at multiple places where the trellis meets the ground, and you can grow in multiple directions along the trellis. Thus, those who desire community in mathematics must develop strategies for combating the one-­dimensional view of math. In the classroom or the home, we can praise others for showing the virtues cultivated by mathematics and remind them that these are part of mathematics too. Persistence, curiosity, habits of generalization, disposition toward beauty, thirst for Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Community 191 Copyright © 2020. Yale University Press. All rights reserved. deep investigation—and the many other virtues I’ve discussed in this book—are all ways to evidence growth in mathematics. In high schools and colleges we can develop multiple pathways into mathematics rather than funneling everyone through cal­ culus. We can create math clubs based on enjoyment rather than math clubs based on elite performance. At the professional level, we can value the diverse ways that math teachers and research­ ers contribute to advancing mathematical understanding. We can hold up a wide array of role models in an exciting collection of mathematical careers.7 Mathematical communities can be very hierarchical, even if they don’t wish to be. At my local hiking club, there’s bonding over a shared love of hiking. When it’s time to go, we split into different skill levels to do the hike at various speeds. I have no trouble admitting that I’m a slow hiker, and I feel no shame in joining the beginners’ group. The joys of hiking—the scenery, the camaraderie, the quiet reflection—are separate from the skill of hiking. Similarly, if I attend a piano concert or a baseball game, the joy of watching is separate from the skill of ­playing. By contrast, in mathematics the joy often requires the skill. For instance, if I’m going to a math lecture, it’s not exciting to me unless I understand what’s going on. The joy of the talk is not just knowing the theorem statements, but also following the proofs. Unfortunately, proofs are hard to absorb in a short time frame, and speakers are often not challenged to make their talks accessible to their desired audiences. By now, I’ve gotten used to the discomfort of not understanding everything, and I know it’s normal, but it’s easy for a newcomer to feel out of place. Simi­ larly, in a classroom setting, learning math can be a challenge in community because the skill is at the center of instruction. So if group work isn’t designed well, students who need time to think Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Copyright © 2020. Yale University Press. All rights reserved. 192 Community will be dismayed by those who finish first. The singular focus on the skill, even when it’s warranted, leads people to idolize those perceived to be stronger in that skill, and creates unnecessary hierarchies in any mathematical community. I’ve heard many people express versions of Simone Weil’s despair at “the idea of being excluded from that transcendent kingdom to which only the truly great have access.”8 Therefore, those who desire community in mathematics must develop the virtue of hospitality, which includes excellence in teaching, excellence in mentoring, and a disposition to affirm others. Hospitable math explorers will bend over backward to reassure newcomers that they are welcome at any stage of de­ velopment. They will show newcomers the manual of hidden knowledge—which includes the fact that even seasoned veter­ ans don’t understand everything in a talk—and mentor them in skills, like how to read math for the big ideas. They will publicly assign competence to others, by recognizing the things they’ve done well. Those who wield power in mathematical communi­ ties must remember that they have great responsibility in setting norms for how to welcome newcomers. Hospitable math ex­ plorers will strive to become excellent teachers of mathematics, able to reveal the joy of mathematics even to those who are new to the community. There is so much evidence-­based knowledge about good teaching, and we should avail ourselves of it.9 We can help people gain access to this kingdom through excellent communication. Those who lead mathematical communities must be prac­ ticed at managing group dynamics, with attention to student agency, identity, and power. Skilled teachers know the impor­ tance of setting good norms for how people treat one another. They know that group work can be ineffective if one person dominates, and damaging if not every person in the group has Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Community 193 Copyright © 2020. Yale University Press. All rights reserved. a meaningful way to participate in the work. For that reason, math educators stress the importance of designing group-­worthy tasks: activities in which there are multiple important roles, real collaboration is required, and everyone’s contributions are needed for the group to succeed.10 Effective teachers know how to invite students to share their thinking, and find ways to lower the social risk of participation.11 Building community in mathematics involves developing collaborative skills that diminish hierarchies. Successful col­ laborations are inclusive and benefit from diverse viewpoints. They aren’t just a division of labor; rather, the best math collabo­ rations are synergistic, require preparation, and generate deeper understanding when participants push one another toward fur­ ther growth. Mathematical communities, like all communities, are prone to implicit bias: unintended, unconscious stereotypes that all of us possess. We make erroneous assumptions about others that affect the power dynamics of the group and limit whose voices are heard. In school settings, we must ask: Who hasn’t spoken yet? Whose contributions are often overlooked? At the professional level, we have to recognize how bias leads to deci­ sions that work against community. For instance, when women collaborate with men on research papers, they are less likely to get credit for the collaboration—people think the men did the work. A 2016 study in the similar field of economics shows that while women publish just as much as men, they are twice as likely to be denied tenure, except if they always publish alone, in which case there is no difference.12 Therefore, those who desire to build community in mathe­ matics must have a posture of self-­reflection. We should be mindful of our potential biases so we can mitigate them, and Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. 194 Community Copyright © 2020. Yale University Press. All rights reserved. within our communities we must establish good practices and structures that help alleviate bias.13 Mathematical communities are deeply plagued by feelings of not belonging. These can take many forms: I hope no one finds out I don’t know very much (underlying thought: I don’t deserve to be here); no one else here is like me (underlying thought: therefore no one can truly understand me); I’m never going to be good enough (underlying thought: to be like the people I idolize). These feelings can be exacerbated by the hierarchical nature of many communities. Ricardo, as a forty-­year-­old college student, may feel a mixture of all these. He’s from an underrepresented back­ ground, in both race and class. He’s been out of school for so long that reorienting himself is an adjustment. He feels handi­ capped by his past. It’s no wonder he constantly feels I’m not supposed to be here. In fact, many of us feel this way, for one reason or another. I have often felt isolated in mathematical commu­ nities, and I still do, even though I’m a well-­established mathe­ matician. When I changed research areas midcareer, I spent a semester at a research institute to try to forge connections in the new community. I often felt out of place, because I didn’t know much about the new area, and I come from an institution that’s quite different from where other research mathematicians come from—because my college prioritizes teaching. No one knew me well, and I was less likely than others to be invited to social gatherings. To be fair, if others had known how I felt, I’m sure they would have reached out more. That’s why being welcoming requires active attention. So, in addition to hospitality, those who value community in mathematics must develop the virtue of attention to people. It means looking at others and seeing beyond who they are just Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Community 195 Copyright © 2020. Yale University Press. All rights reserved. mathematically, especially the young, the new, or the forgotten. This is a virtue you must develop even if you are the newcomer. It took some reflection for me to realize, when I felt unnoticed, that there were probably others who felt the same way. Indeed, we were all newcomers at this institute, where people stayed for only a short duration. As the newcomer, you can notice the other newcomers around you, and you can extend a welcome too. And leaders of any mathematical community should con­ sider the virtue of vulnerability. Leaders who can share their journey and its difficulties can help set the tone for others to do so as well. Teachers who ask students to write their “matho­ graphies”—their autobiographical journeys in math experi­ ences—often begin by sharing their own. Leaders who are vul­ nerable can help others overcome their feelings of being an impostor. Karen Uhlenbeck, a winner of the Abel Prize (a Nobel-­ like prize in mathematics), acknowledged: “It’s hard to be a role model . . . because what you really need to do is show students how imperfect people can be and still succeed.”14 One of the great joys for me when I talk about human flour­ ishing in mathematics is that I often hear from people about their own deep experiences. The mathematics professor Erin McNicholas recounted a time when she was feeling anxious about an external event and was pulled into a moment of joy with several students and another professor: It was hard to imagine how I was going to break out of the swirl of worry, fear, and anger that jumbled my thoughts. And then . . . I happened to pass a student in another pro­ fessor’s Real Analysis class. I had been talking to an advisee about a problem on that week’s analysis assignment. She Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. 196 Community Copyright © 2020. Yale University Press. All rights reserved. had found a hole in her approach to one of the problems that neither she, nor I, could find a way to fill. So I asked this student if he had solved the problem. He had, and in the same way as my advisee, only he had not noticed the hole. So I asked him about it. Within 20 minutes, a group of five analysis students, the other professor, and I had all gathered and were working together to find a solution to this one problem. We would almost have it resolved, when another issue would spring up. Finally, through the con­ tributions of everyone there, we figured it out. There was a rush of excitement, and the student who was writing up our comments on one of the glass boards did a little dance of joy as he hurriedly jotted down the statements that completed the proof. We laughed with him, giddy in our shared victory. She said it was the sound of the combined laughter that jolted her and made her realize that she had not thought once about her worries during the thirty minutes when they were problem solving together. Mathematics was a refuge from her worldly concerns and a source of joy because of this spontaneous com­ munity that formed. In her story you see so many of the things that exemplify a healthy mathematical community. There was a lack of hierarchy. Everyone had been fooled by this problem, and the professors were modeling that it’s okay—and even ex­ citing—to be in this posture of not knowing but wanting to find out. They were driven by a common curiosity: even though the students knew that they wouldn’t be marked down for not solv­ ing a problem that the professor had not solved, they, like their professors, needed to know the truth. They felt a shared hope of solving the problem, and when they did, they were rewarded with joy. Erin reflected on this experience: Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Copyright © 2020. Yale University Press. All rights reserved. Community 197 We were all contributors to the solution. I was attempting to contain my pride in my advisee who initially noticed the hole in the argument. While I see her unusual talent for critical introspection, I think her talent is more easily overlooked than the mathematical creativity and intuition exhibited by some of our majors. She has these talents too, but humility and an ability to recognize what you don’t know can silence these talents in the group setting. If we think of a mathematical challenge as a river to cross, some mathematicians’ approach is to bound from the shore, jumping from rock to rock, worrying only about the next foot placement. Others pause at the bank, looking for a path across, calculating currents and the likelihood of slipping, looking on Google Maps for a possible bridge somewhere up- or downstream. It is easy to marvel at the bravado and daring of those intrepid rock jumpers, but often it is the diligent planners that come to their rescue when they find themselves stranded midstream. As a community, as professors and fellow students, I think we overlook the careful, methodical work that ulti­ mately gets us to the other side. I couldn’t help but revel in the fact that two professors with PhDs and several senior majors had completely overlooked the same gap in argu­ mentation, and that this student who often does not re­ ceive the recognition she deserves spotted it. This is a picture of a flourishing mathematical community: people who have joined together in a common mission of ex­ ploration and play, bouncing ideas off each other, valuing one another’s input, getting excited about the directions their ideas are taking them in, and embodying a wide array of mathemati­ cal virtues along the way. Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Five Points on a Sphere Given any five points on a sphere, show that some four of them lie on a hemisphere that includes its boundary. This is a great problem with an elegant solution.a So it’s a nice final puzzle for this book, because it showcases what makes a beautiful problem: it’s easy to state, it has a surprising conclu­ sion, there are multiple ways to explore it, and you can think about it during idle moments of your day. Remember that math explorers are comfortable with struggle, and it’s okay to just let a problem marinate in your brain. After a long struggle, when you finally discover a solution, you’ll be delighted. Copyright © 2020. Yale University Press. All rights reserved. a. Putnam Mathematical Competition problem, 2002. Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. 198 Copyright © 2020. Yale University Press. All rights reserved. July 25, 2018 I’ve actually gone through and studied the whole sections [in Stephen Hawking’s God Created the Integers] on Euclid’s “Ele­ ments,” Archimedes’ “Methods,” “Sand Reckoner,” “Measure­ ment of a Circle,” and “On the Sphere and Cylinder,” and I’ve also gone all the way through Descartes’ “Geometry.” Since then I’ve been through Lobachevsky’s “Theory of Parallels” and now am almost through a first sweep of Bolyai’s “Science of Absolute Space.” And honestly, I didn’t know Geometry was that rich. I completely understood all that Euclid and Archimedes had to say, more than 90% of Descartes’ work, about the same with Lo­ bachevsky, but Bolyai’s is somewhat harder (but more detailed) and he is somewhat ascetic and austere with some of his nota­ tions and concepts, but I definitely understand the vast majority of what I’ve gone over so far, and I plan to go over all the parts that weren’t crystal clear when I finish. This book “God Created the Integers” is great. Do you know of another book like this? That’s a compendium of mathematics from ancient up to mod­ ern times? That would be a great help. This book here over the last 4 months has really augmented how I see mathematics. . . . Your phrase “the humanity of doing math” really made me think. Back before I got really serious about mathematics, I used to play a LOT of chess, so I tended to analogize a lot of life to chess. But now that my main focus is mathematics, a lot of mathematical concepts creep into the way that I see a lot of things. . . . I have a friend that I left at the medium security in­ stitution I was at, who used to tell me all the time that it takes tenacity to attempt what I’m trying to do. Mathematics as a class in Perseverance?? But one thing I also noticed was in this book [is that] most of them within the same time period knew each other and communed with each other and/or trained each other or were trained by mathematical descendants of one another 199 Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. Copyright © 2020. Yale University Press. All rights reserved. even throughout time periods—there was a definite human structure to their mathematics. Anything that you write is an inspiration to me and I can’t wait to read your book. Yes, it would be OK with me to tell more of my story in your book. If some of the conversations we’ve had about mathematics help you to illustrate a point, please use them. If I have my way (which I absolutely believe I will), whether it’s a couple years from now or 15 or more years from now, I intend to use my story to help someone like me 15 years ago or to help someone else help that person or people. Because I truly believe if someone like me . . . had stepped into my life when I was 17 and had just gotten my G.E.D. and was enrolled in Atlanta Tech (when I was at that crossroads) and really showed me some of the better things life had to offer (or explained these things), I would’ve had a way higher chance of doing other than I did. In my opinion (this is a radical oversim­ plification of what I’m trying to say), people don’t care enough about other people right now (and haven’t for a while), from my personal experience and the things we see in the world. Chris Su, F. (2020). Mathematics for human flourishing. Yale University Press. Created from uva on 2022-12-01 04:08:00. 200