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
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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
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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
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188 Community
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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
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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.
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190 Community
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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
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Community 191
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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
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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
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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
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194 Community
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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
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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
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196 Community
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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:
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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.
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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.
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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
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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
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