By Benjamin Braun, Editor-in-Chief
Two of my favorite pieces of mathematical writing are recent essays: Francis Su’s January 2017 MAA Retiring Presidential Address “Mathematics for Human Flourishing”, and Federico Ardila-Mantilla’s November 2016 AMS Notices article “Todos Cuentan: Cultivating Diversity in Combinatorics”. If you have not yet read these, stop everything you are doing and give them your undivided attention. In response to the question “Why do mathematics?”, Su argues that mathematics helps people flourish through engagement with five human desires that should influence our teaching: play, beauty, truth, justice, and love. In a similar spirit, Ardila-Mantilla lists the following four axioms upon which his educational work is built:
Axiom 1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4. Every student deserves to be treated with dignity and respect.
These essays are two of my favorites because they provide visions of teaching and learning mathematics that are rich with humanity and culture, visions that welcome and invite everyone to join our community.
The ideals and aspirations offered by Su and Ardila-Mantilla are inspiring, emotional, and profound, yet also fragile — for many mathematicians, it can be difficult to balance these with the sometimes harsh reality of our classes. An unfortunate fact is that for many of us, a significant part of teaching mathematics consists of the struggle to support students who are uninterested, frustrated, inattentive, or completely absent. We are regularly faced with the reality that large percentages of our students fail or withdraw from our courses, despite our best efforts, and often despite genuine effort on the part of our students as well. How does a concerned, thoughtful teacher navigate this conflict between the truth of the tremendous potential for our mathematical community and the truth of our honest struggle, our reality?
In my practice of teaching, I have found that the only way to resolve this conflict is to simultaneously accept both truths. This has not been, and still is not, an easy resolution to manage. In this essay, I want to share and discuss some of the mantras that I have found most helpful in my reflections on these truths.
1. Excellence is possible, perfection is not
While perfection is impossible, excellence as a teacher is achievable, though elusive. The most difficult part of teaching for me is that I deeply want all of my students to succeed. In reality, this rarely happens, for many different reasons. Nevertheless, it is possible to reach excellence in teaching and learning, even if that doesn’t translate into an idealized outcome for every student in every course. By holding ourselves to the standard of excellence rather than perfection, it also becomes easier to hold our students to more reasonable standards of excellence as well. I have found that it is easy for me to slip into a mode where I am disappointed when my students don’t reach what I feel is their full potential, but by doing so I can also miss the opportunity to recognize the successes that they do achieve. To seek excellence rather than perfection, in ourselves and others, allows us to maintain our ideals while accepting the challenge of our reality.
2. All human systems have flaws
Colleges and universities are complex institutions, many of which serve diverse communities of students and employ faculty in a broad range of positions. Like many of my friends and colleagues, I have at times become focused on specific institutional flaws that are impossible to effectively address, often at the expense of turning my energy toward reaching more tractable goals. It is common to hear people say “pick your battles,” but at an institutional level I prefer to phrase this as choosing to engage with certain challenges and to yield to other challenges. There are many times when yielding to a challenge can provide significantly more freedom than fighting “the good fight.”
Consider Richard Tapia, a mathematician at Rice University, who received the National Science Board’s 2014 Vannevar Bush Award for “his extraordinary leadership, inspiration, and advocacy to increase opportunities for underrepresented minorities in science; distinguished public service leadership in science and engineering; and exceptional contributions to mathematics in the area of computational optimization.” In a video produced by the NSF, Tapia states:
When I started, I was so naive I thought I could change my colleagues, OK. You don’t change colleagues. You get them to maybe tolerate things you’re doing. You know, “Richard Tapia does ‘this’.” But where I see things changing and things going on is through my students. Without even directly telling them “you have to do ‘this’ and ‘this’,” they see it by example. And so I am really satisfied when I see how many students of mine are doing exactly what I was doing.
What Tapia describes is not a direct confrontation with the cultural norms and reward structures that influence his colleagues, but rather a yielding to these forces and a redirection of his energy in more effective directions. While there are certainly times when we must directly challenge flawed systems, we must also recognize that for many institutional problems, we create a higher impact by yielding to them in the short term and making progress through a different approach.
3. Maximize student learning within a set of constraints
In addition to large-scale institutional flaws, there are many additional constraints on our teaching. It is important to remember that our goal is not to have perfect learning from every student, but rather to maximize student learning given these constraints. The way we do this will vary dramatically given our situations, but there is a core principle that we can and should always rely on: focus on the experiences of our students. Here is an example of what I mean.
At the University of Kentucky, our first-year courses in Calculus (for students majoring in engineering, mathematics, and the physical sciences) are taught in large lectures of ~150-180 students with ~32-student recitation sections led by graduate teaching assistants. The first three times I taught these courses, the outcome was mediocre at best. My original strategy with large lectures was to import the best methods I had developed for small-scale teaching into the large lectures, but they were not effective. The constraints for teaching large lectures are completely different from those for my small courses, and the solutions I had used to maximize student learning in my small courses were not optimal solutions for the large courses.
The fourth time I taught a large-lecture calculus course, I completely yielded to these constraints. I was not excited about this, and was not expecting the course to be enjoyable for anyone, myself or my students. I could not have been more wrong — this was one of the most memorable courses I have ever taught, and my students were both successful and happy with their experiences. In hindsight, I realized that yielding to hard constraints had led me to a profound change in my perspective about large lectures: my primary focus should be to identify positive aspects of the large-lecture environment from the perspective of my students and take advantage of these as much as possible. Previously, I had focused almost exclusively on the negative aspects of large courses from my personal perspective as a teacher. This caused me to overlook most of these potential positive aspects, such as the effectiveness of a well-organized teaching team, the vibrancy of student excitement in a large class, and the broader range of peer interactions students can have among a large group.
A concrete example of an in-class change I made is my method of presentation. Like most mathematicians, I prefer to use the chalkboard when I teach; with large lectures, this was never as effective as it is when I teach 20-30 students. I also never wanted to use a microphone, as a personal preference. I finally gave up on all of these teacher-focused preferences. I now use Crayola markers to write on blank white paper projected using a document camera, use desmos.com for all my graphing, and use the lapel microphone. With the microphone, the students hear me clearly and the class is more relaxed since I am not straining my voice. By using markers and desmos, students can see better, I can scan my actual in-class notes and post them after class, and the lectures are literally colorful. I had dramatically underestimated the impact of these simple changes — my student evaluations are now consistently full of positive comments about how my use of colorful markers and dynamic graphics are uplifting in a drab room and help students pay attention.
Do these things make every student learn perfectly? Of course not. However, by thinking more purposefully about working within constraints to maximize student learning, leaving some of my own personal preferences aside, I have developed an approach to teaching large lectures that is more successful, and which my students and I feel reasonably positive about.
4. Students can have meaningful mathematical experiences without us
In my early teaching, I had bought into a false idea that student contributions were most meaningful when I could provide feedback about them. This was one reason for my preference for whole-class inquiry-based learning courses, and my distaste for large lectures. As with many other things in life, sometimes less is more in this regard. In my courses for first-year graduate students and in my large-lecture calculus courses, students are engaged and report positive experiences when I give 7-10 minute lectures followed by a 2-3 minute pause where students can discuss any points of confusion with their neighbors. The most effective prompt that I have found is to tell the students to turn to their neighbor and ask “do you have a question, yes or no? If no, why does this make sense?” It actually does not seem to matter whether or not I hear these conversations, what matters is that the students are talking about mathematics, struggling with the ideas, and are regularly engaged in conversation about what we are doing.
Similarly, in my small courses, I am less concerned than I used to be with having every small group report on their work, or check with me. This does increase the risk that students might have a misconception that is not immediately addressed, but it gives students more agency and authority in their own learning. It also recognizes the reality that students can achieve excellence in their learning without being perfect, and have meaningful experiences in mathematics without me being intellectually present at every moment.
At a deeper level, when we recognize that students can have meaningful mathematical experiences without us, we allow ourselves to embrace our most important task, to guide and inspire students, rather than to seek a false sense of control over their learning. Our most fundamental role as teachers is not to transmit truths to our students, but to create and sustain supportive environments in which students deeply learn, to create opportunities for students to engage with mathematics at a fundamental and profound level. We balance the tension between the aspirations and ideals that Su and Ardila-Mantilla offer and the reality of teaching by honoring this fundamental role we play, while simultaneously allowing students the choice of whether or not to take advantage of the opportunities they have. This leads to my final mantra.
5. Do not be afraid of honest failure
This has been the most important mantra for me. Like most people, I want to reach my goals. I want my students to succeed in my courses. However, the dichotomy of “success versus failure” is not sufficient when we set challenging goals, and goals in the context of mathematics are almost always challenging! Instead, we should strive to succeed or fail honestly. It is debilitating to have dishonest failure, where we fail because we choose not to put in our best work, where we fail because we do not risk anything. It is also a waste to have dishonest success, through cheating or gaming the system. If we succeed in our teaching, if our students succeed in their learning, these successes are most meaningful when they are honestly earned. If we fail in our teaching, or if our students fail in a given course, that is still a meaningful experience as long as the failure is honest.
I have been fortunate that I have not yet encountered epic failures in my mathematical life. However, this is not true of my life overall; whether in mathematics or something else, each of us has stories to tell of when things went awry despite our best efforts. If our students are doing what we hope they will, are pursuing challenging goals, chasing after dreams, learning beyond what they thought they were capable of, there will be honest failure along the way. We must honor those failures, and value them, and make sure our students know this. If we as teachers are striving to realize our aspirations and ideals, we will have honest failures as well.
A final thought
These mantras and my approach to teaching have been influenced by the concept of strengths-based practice in social work, by my interest in mindfulness practice, and by my readings about history and politics. At a fundamental level, all of these are about the challenge of resolving the tension between ideals and reality. I am far from unique in having had a significant influence on my teaching come from non-mathematical sources. For example, in his essay mentioned at the beginning of this article, Francis Su describes how his teaching has been informed by the ancient Greek idea of eudaimonia, and Federico Ardila-Mantilla’s essay describes how his work with students has been informed by research in social science and psychology. While it is worthwhile and meaningful for us to look inward and see how the strengths of mathematics and our community can be used in the practical pursuit of our ideals, we should remain open to inspiration from all aspects of our lives.
Thanks to my colleague Serge Ochanine for his insightful comments about Francis Su’s article, which inspired me to put these thoughts into coherent form. Thanks to my father James Braun for introducing me to strengths-based practice in social work. Thanks to the other editors of this blog for many thought-provoking conversations and their helpful comments on a previous version of this essay.