Mathematics is a habit of thought
CONTENT WARNING: This piece contains mentions of self-harm, depression, and anxiety.
By Leanne Merrill
Leanne Merrill
Every Friday in second grade, after recess, my class played Hot Pencils. We knew it was Hot Pencils time when the teacher began to count out the long skinny papers, row by row. We frantically cleared our desks of everything except the titular pencils and sat at attention.
The long skinny papers had several equations of the form a + b = x for some integers a, b. The x was a blank spot. The teacher would start a timer, we’d grab our pencils, and then get to work on the equations, which involved two-digit addition and subtraction. Some seconds later, the teacher would call out “Hot Pencils!” and we’d drop our (in fact room temperature) pencils. Then, we’d exchange papers and peer-correct them – a dangerous idea for such a high-stakes activity! Correct answers were worth a point, but incorrect answers subtracted a point. The person with the most points got a prize: stickers on a normal week, candy for special occasions. And, of course, praise from the teacher, along with the ability to gloat.
I wanted the stickers, candy, and praise, and I often received them. Unfortunately, I also became addicted to a toxic cocktail: extreme perfectionism combined with the need for external validation of my mathematical ability. This addiction persisted through the next 20 years of my mathematics career.
It would be untrue to say that external validation, or (at the very least) external endorsement of abilities and aptitude, is unimportant in mathematics. Progress along the traditional academic track, from course grades to graduate school entrance exams to job recommendation letters, relies entirely on the artifacts of external validation.
Moreover, perfectionism is all but required of mathematicians. What, if not perfection, is mathematics all about? The clarity of thought afforded by mathematical reasoning; the ability to get a correct, irrefutable answer to a calculation; the crispness and rigor of a well-written argument. Mathematicians esteem, celebrate, and teach these things.
Nevertheless, there is extreme danger in the conflation of external validation with self-worth, and of perfection with one’s basic value as a human being.
Starting in high school and continuing through college and graduate study, I regularly self-harmed in response to my grades. If I got anything less than an A on a test or in a course, I would cut myself with a razor blade deeply and painfully. I cut myself in places that were covered by my clothes, and I made up excuses to my doctors and romantic partners if ever they needed to see me with a bandage on an unexpected part of my skin. “I spilled boiling water and accidentally burned myself” was my go-to explanation.
I did this because I believed that I deserved it, and I thought it would help me be more perfect in the future. My entire self-worth was based on the assessments that other people made of me, so if I did worse than perfectly, I deserved reproach. My twisted logic told me that my actions would help prevent the problem in the future, and I would grow closer to perfection and consequently gain more praise and require less punishment. But if I failed in the eyes of others (or in my own eyes, compared to the standard of perfection), I was worthless, and I needed to pay for my failures.
My addiction to perfection and external validation manifested in other ways too: I systematically compared myself to other people; I felt near-constant anxiety and sporadic depression during every academic term; I had an extreme lack of resilience in the face of challenges. One of my therapists said, “instead of moving towards a goal, you are running away from failure.” This was my experience as a mathematician, including the years in which I earned my Ph.D.: deeply fearful of being imperfect in the eyes of others, and on the verge of giving up.
Unsurprisingly, this dangerous mindset did not sustain me. However, over time, I gradually learned to replace these thoughts with more productive, kind, and humane self-talk. While I am still on the path to recovery, I can share what has helped me so far.
Overarchingly, I have come to understand that mathematics is not nearly as straightforward as I was told during Hot Pencils. Mathematics, as a sector of academia, has been shaped by power dynamics involving race, class, gender, colonization, and many other societal delineations. Like all other endeavors of the human race, mathematics is created and governed by imperfect actors (humans) and is subject to all of our flaws, vagaries, and lack of foresight. We can see this when mathematical results are discovered to be false, when mathematics is used in nefarious ways, or when mathematicians face barriers to success because of their identities.
I began to realize this in my first proof-writing class, when I realized that mathematical proofs, unlike computational exercises, have shades of correctness, largely determined by the reader. My understanding of these issues has been transformed radically in the last several years by studying the mathematics rehumanization movement, which addresses and celebrates the diversity of types of mathematical thought and expression of which humans are capable.
I have also learned that the only truly sustaining way to practice mathematics is to find joy in it. Through joy, one can build resilience and grit. It is essential to embrace and enjoy the struggle, because the act of doing mathematics is composed largely of struggling. For most people, the struggle increases with the complexity and abstraction of the material. Until I accepted this (and stopped telling myself that I was stupid for having to struggle), I was unable to move forward.
I was able to shift my mindset through years of therapy and by connecting with other mathematicians who had similar experiences. More than that, I began to recognize the same misperception in most of my students, and I saw the deleterious effect it had on their mathematical persistence and enjoyment. It is no wonder that they felt the same way — most people who progress through traditional US schools face a task like Hot Pencils at some point in their mathematical lives, causing them to confuse mathematical speed with mathematical talent, and (for many of them) to believe that they are not capable mathematicians simply because they cannot perform arithmetic as quickly as their peers.
Now, I use the lessons I have learned to create what I hope to be a different mathematics learning environment for my students. I don’t mean this as a criticism of my own mathematics teachers, all of whom were well-meaning and adept, and some of whom began to embrace mathematical humanism before it was widely popular. However, on the whole, I envision a new framework for mathematical thought and expression, contrasting with the perfection-based, competitive model in which I was raised. As I gain teaching experience, I find this framework increasingly defines and improves my teaching practice.
I believe that everyone can do mathematics (something I inherited from the late Clarence F. Stephens by way of the fantastic mathematics faculty at SUNY Potsdam, where I was lucky to study as an undergraduate). I believe that everyone’s mathematical ability develops in unique, non-ordered ways; that is, there is no prescribed way in which knowledge is gained. Because of this, I believe that students deserve multiple attempts to encounter and demonstrate mastery over mathematical concepts. I also believe that inquiry, experimentation, and discovery should be at the heart of every mathematical endeavor, rather than a focus on repetition, memorization, and procedure. Finally, I believe that students must develop a personal and emotional connection to mathematics in order to become a lifelong mathematical practitioner.
Accepting and living these teaching ideals has required me to have radical trust, vulnerability, and patience, all of which I strive for each day, even though I sometimes fall short. It also requires a lot of hard conversations about the meaning of mathematical success, since most of my students initially define success exactly the same way I did when I was a student, which (it turns out) was just an extension of Hot Pencils: get as many correct answers as fast as you can. But, mathematics is not a race; rather, mathematics is a habit of thought, one that takes time and attention to develop.
On the whole, I see that my students are just as traditionally “successful” as they were when I taught in the same model in which I grew up. The main difference is that they are more resilient, more curious, and happier. And, I’m happier too.
If you are feeling as though you may harm yourself, contact the National Crisis Text Line at 741741. Alternatively, reach out to your local mental health professionals, or a friend or family member if you feel comfortable. You are not alone, and seeking help is a sign of strength.