Mathematical Culture Beyond the Classroom

Benjamin Braun, Editor-in-Chief, University of Kentucky

Mathematics is the result of human curiosity and our desire to explain, predict, and explore observed and imagined phenomena. Our shared curiosity and sense of wonder is the wellspring of our mathematical culture. Yet a common sadness is felt by those who love mathematics, as we witness people’s wonder and curiosity stilled by strong cultural and social forces. As Paul Lockhart writes in A Mathematician’s Lament:

If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done… Everyone knows that something is wrong.

Many mathematicians, mathematics teachers, and mathematics fans and ambassadors share these feelings. It is natural that for many of us, our primary responses to this arise through our teaching, in an effort to help students rediscover their natural sense of mathematical joy and curiosity. However, my belief is that this situation is actually a symptom of an issue that extends beyond teaching and learning, and beyond the confines of mathematics. I believe that at its core, this issue involves our cultural responses to three questions:

1. How do we build relationships with those around us?
2. What accomplishments do we reward and recognize?
3. What stories do we tell, especially about mathematics and mathematicians?

As powerful as classroom cultures and environments can and should be, I believe we must have an even grander vision for ourselves and our community. We need to find ways to change some core qualities of the culture of mathematics itself, qualities related to the three questions above. A central challenge is that these changes are generally orthogonal to cultural norms of society at large. In this article, I share some reflections on these questions, along with ideas for how we can work together to meet the challenge of improving the culture of mathematics both within and beyond the classroom.

How Do We Build Relationships With Those Around Us?

The practice of doing mathematics is one infused with emotional complexity. In Loving and Hating Mathematics: Challenging the Myths of Mathematical Life, Reuben Hersh and Vera John-Steiner write:

Mathematics is an artifact created by thinking creatures of flesh and blood… Mathematicians, like all people, think socially and emotionally in the categories of their time and place and culture. In any great endeavor, such as the structuring of mathematical knowledge, we bring all of our humanity to the work… It’s a challenge for everyone to achieve balance in one’s emotional life. It’s a particularly severe challenge for those working in mathematics, where the pursuit of certainty, without a clearly identified path, can sometimes lead to despair. The mathematicians’ absorption in their special, separate world of thought is central to their accomplishments and their joy in doing mathematics. Yet all creative work requires support.

Unfortunately, our mathematical culture does not encourage us to discuss and share the emotional ups and downs of our mathematical lives. While this is also true of our society at large, at least our broader culture acknowledges the role that therapy and counseling can play to improve our lives. In mathematics, our tacit prohibition on discussing the emotional aspects of mathematics has serious consequences for our community, ranging from mental health issues, especially among graduate students, to people unnecessarily working in isolation (e.g. Dusa McDuff’s reflections here), to differences in the sense of belonging and efficacy that people feel in mathematics. As Hersh and John-Steiner point out, these consequences are particularly detrimental in mathematics, due to the nature of our work.

Whether or not individuals choose to actively share their stories with colleagues, teachers, or peers is not the point here — what matters is whether or not individuals in our community feel that they are surrounded by people who are supportive and willing to listen without judgement. Creating a culture of authentic inquiry in our interpersonal relationships can provide the social and emotional support that we all require to pursue the creative work of mathematics. Because most mathematicians work in a community framed within professional organizations (colleges, universities, businesses, government entities, etc.), a natural source for inspiration and guidance in this professional context is the literature on organizational culture and leadership. One accessible and relevant resource in this area is Edgar Schein’s book Humble Inquiry: The Gentle Art of Asking Instead of Telling. Schein describes Humble Inquiry as the type of inquiry that

maximizes my curiosity and interest in the other person and minimizes bias and preconceptions about the other person. I want to access my ignorance and ask for information in the least biased and threatening way. I do not want to lead the other person or put him or her into a position of having to give a socially acceptable response. I want to inquire in the way that will best discover what is really on the other person’s mind. I want others to feel that I accept them, am interested in them, and am genuinely curious about what is on their minds regarding the particular situation we find ourselves in.

Schein is careful to distinguish this form of inquiry from others, such as diagnostic inquiry, confrontational inquiry, and process-oriented inquiry. He emphasizes that “the world is becoming more technologically complex, interdependent, and culturally diverse, which makes the building of relationships more and more necessary to get things accomplished and, at the same time, more difficult.” In other words, the challenge of building authentic interpersonal relationships is not only one for mathematical culture, but for society at large. Schein also emphasizes the importance of individuals in leadership positions learning to use and model authentic inquiry.

To give a concrete idea of how this approach might be used in practice, I share below some questions that we might not ordinarily consider when we are speaking with a student, colleague, employee, or peer. However, questions such as these can powerfully change our perspective toward those around us. I am not suggesting that we routinely ask these questions in regular conversation, but rather that we have these questions in our conscious mind, that we are open to the possibility that the people we interact with have complicated and difficult lives, especially when we are having challenging conversations.

• Does the person I am talking to have access to sufficient food and housing to meet their needs and the needs of their family?
• Does the colleague or student I am talking to have personal challenges or crises I don’t know about, such as a relative, spouse, or child with a serious health issue?
• Has the person I am speaking to been a victim of abuse or assault?
• How many hours each week does this student have to work to pay for their housing and basic expenses?
• Is this student or colleague suffering from PTSD due to military or other service?
• Who is this student or colleague responsible for supporting financially?

Unfortunately, these questions reflect realities that impact many more of our students and colleagues than we might guess. Knowing the answers to these questions would not necessarily change my expectations for student learning in a course, or for job responsibilities for an employee, etc., though it might inspire me to handle situations differently or with more humility. By honestly recognizing and affirming that we are usually ignorant of important aspects in the lives of those around us, we can be more empathetic, flexible, and ethical in our treatment of and relationships with others.

What Accomplishments Do We Reward and Recognize?

Complicating our relationships with other people is that, at least in the United States, a dominant social and cultural force is the drive to prize individual achievement over the building of relationships. This force extends throughout our society, not only in mathematics. In Humble Inquiry, Schein writes:

When we compare some of the artifacts and behaviors that we observe with some of the [social] values that we are told about, we find inconsistencies, which tell us that there is a deeper level to culture, one that includes what we can think of as tacit assumptions… The most common example of this in the United States is that we claim to value teamwork and talk about it all the time, but the artifacts — our promotional systems and rewards systems — are entirely individualistic.  We espouse equality of opportunity and freedom, but the artifacts — poorer education, little opportunity, and various forms of discrimination… — suggest that there are other assumptions having to do with pragmatism and “rugged individualism” that operate all the time and really determine our behavior.

How does this manifest itself in the mathematical community? As one example, publications are generally used as the currency of our realm, and it is typical that single-author publications are viewed as more valuable than publications resulting from a team of collaborators. Yet it is reasonable to ask what benefits mathematics more: having a person write a paper on their own, or having researchers build relationships and collaborative teams that are able to pool ideas and resources effectively?

Similarly, faculty are frequently given a higher evaluation for single-PI research grants than for leading a team of co-PIs on an infrastructure or education grant. Yet the NSF Education and Human Resources (EHR) and Undergraduate Education (DUE) divisions have funding available and have been actively seeking proposals in mathematics, as evidenced by Jim Lewis’s 2015 AMS Committee on Education presentation and this blog’s 2016 post by NSF program officers about the type of awards funded by EHR and DUE. Again, is this what we actually want to value in our mathematical culture? Is this what most strongly benefits the community? What is it that we collectively want to achieve, and do these recognition values reflect our common goals?

As a third example, consider two hypothetical students: a “smart” student who solves correctly every problem the instructor provides, or a student who sometimes makes errors yet is engaged, persistent, motivated, and dedicated. Which student is most likely to receive praise and support in a math course? Which is typically considered to be the most successful in mathematics? To have the most mathematical potential? To be the highest achieving? Which of these students do we typically provide with encouragement, awards, and recognition?

How will we make explicit, especially at the local level within departments and colleges, what type of collaborative activity we value, and how it will be rewarded? How will we go about recognizing and rewarding the type of activities that are needed to build supportive communities? The first step is one of the most difficult, in that we have to have clear and articulate discussions about these questions. This will almost certainly lead to serious disagreements among colleagues and peers, as mathematicians have strong beliefs about cultural issues; in many important ways, mathematical culture is quite conservative and deferential to tradition, though in my experience we rarely discuss this. Improving our habits of authentic interpersonal inquiry, which we have already seen is necessary for building better relationships, will be required of everyone involved in these types of discussions.

What Stories Do We Tell About Mathematics and Mathematicians?

A noteworthy quality of mathematical culture is that we frequently celebrate mathematical mythology rather than mathematical reality. For example, the stories we tell in mathematics are typically mythological, whether they are stories about “brilliant” mathematicians and their work or descriptions of “typical” career paths for mathematicians. For example, I have often overheard undergraduate and graduate students being told some variation of the story that an ordinary “successful career” in mathematics involves finishing high school, then immediately getting an undergraduate degree, then immediately completing a PhD, then obtaining a postdoc, then getting a tenure-track job, and then staying in that job until death. However, many of the mathematicians I know do not fit this simplified plotline at all. Rather, when we begin asking each other about our real stories, the stories that we usually don’t tell in public because they go against our cultural myths, we find that our realities are often much richer and more interesting than the standard narrative.

A deep commitment to the real instead of the mythological also influences our understanding of the nature and history of our discipline. We must be willing to challenge “traditional” and inherited narratives regarding the origins of mathematics, even when these narratives are strongly embedded in our culture. For example, in The Crest of the Peacock: Non-European Roots of Mathematics, George Ghevergese Joseph writes:

Evidence of [the contributions of Egyptian and Mesopotamian mathematicians] is not all hidden away in obscure journals or expressed in languages that tend to be ignored by many Western scholars: much is published in English in “respectable” journals and books… The reason for the neglect [of these contributions] was not that the relevant literature was inaccessible or “unrespectable” but something deeper — a serious flaw in Western attitudes to historical scholarship (one not confined to histories of mathematics or science). An excessive enthusiasm for everything Greek, arising from the belief that much that is desirable and worthy of emulation in Western civilization originated in ancient Greece, has led to a reluctance to allow other ancient civilizations any share in the historical heritage of mathematical discovery. The belief in a “Greek miracle” and the way of attributing any significant mathematical discoveries to Greek influences are part of this syndrome.

As an example of the mythological twisting of history that occurs in mathematics courses, in the edition of Stewart’s Calculus textbook that my department uses there are no women mathematicians listed in the index except for a reference to the “witch of Maria Agnesi”, which does not discuss Agnesi’s mathematical contributions at all. This perpetuates the mythology that “no women did math” in the past. However, this does not reflect reality, as there were several prominent women mathematicians and mathematical physicists working in the 1700’s and 1800’s, such as Maria Agnesi, Laura Bassi, Emilie du Chatelet, Mary Sommerville, Sophie Germain, and Sofia Kovalevskaya, who have certainly earned a place in our standard textbooks. We need to train ourselves to reflexively identify mythological stories, and to respond to them by actively seeking the real story.

Conclusion

These three questions are certainly not the only ones that should be asked about the culture of mathematics, but they are all of central importance. One of the common themes inspiring these questions is that we must insist that the humanity of mathematics and mathematicians be placed on an equal footing as mathematical knowledge and discovery itself, and be recognized as equally valuable. This is certainly not a new idea, but it is one which we must continue to emphasize, speak about, and share. With this observation in mind, I will end with the following passage from Rochelle Gutiérrez’s talk Rehumanizing Mathematics: A Vision for the Future:

If we think that mathematics is not political, not cultural, not any of these other things, then how do we remind ourselves that it is a human endeavor?… Why is it useful to me to say “Rehumanizing” instead of saying “equity”?… Rehumanizing for me… is to honor the fact that for centuries, humans… have been doing mathematics in ways that are humane.  It’s not that we have to invent something new for people to be doing, we have been doing it. People see themselves as mathematical, everyone is mathematical… The “Re-” part is a way of acknowledging that there are things that have been erased, and yet people persist.