By Benjamin Braun, Editor-in-Chief, University of Kentucky
“How many of you feel, deep down in your most private thoughts, that you aren’t actually any good at math? That by some miracle, you’ve managed to fake your way to this point, but you’re always at least a little worried that your secret will be revealed? That you’ll be found out?”
Over half of my students’ hands went into the air in response to my question, some shooting up decisively from eagerness, others hesitantly, gingerly, eyes glancing around to check the responses of their peers before fully extending their reach. Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren’t raised looked around in surprised confusion at the general response.
“I want you to discuss the following question with your groups,” I said. “How is it that so many of you have developed negative feelings about your own abilities, despite the fact that you are all in a mathematics course at a well-respected university?”
If this interaction took place in a math course satisfying a general education requirement, I don’t think anyone would be surprised. Yet this discussion repeats itself semester after semester in my upper-level undergraduate courses, for which the prerequisites are at least two semesters of calculus and in which almost every student is either a mathematics major or minor. I’ve had similar interactions with students taking first-semester calculus, with experienced elementary school teachers in professional development workshops, with doctoral students in pure mathematics research seminars, and with fellow research mathematicians over drinks after dinner. These conversations are about a secret we rarely discuss, an invisible undercurrent of embarrassment and self-doubt that flows through American mathematical culture, shared by many but revealed by few. At every level of achievement, no matter what we’ve done, no matter how much we’ve accomplished, many of us believe that we’re simply not good at math.
I first discovered the work of Carol Dweck from a link on Terry Tao’s blog. Dweck is a psychologist at Stanford whose studies on the relationship between self-beliefs and achievement have had a tremendous impact in education and beyond. Her message is simple: when people, whether students or otherwise, believe that they are capable of improving their abilities through hard work and sustained effort, then they achieve more than when they believe they have innate abilities that will at some point be reached. In other words, if you believe that failure is a natural part of growth and development, then you are more likely to persist through failure and setbacks. On the other hand, if you believe you succeed because you are smart, then when you experience a failure, even a small one, you likely conclude that you are not actually smart and give up as a result. The former belief is referred to as a “growth” mindset, while the latter is called a “fixed” mindset.
Dweck’s work fit naturally in the context of math education research I had been reading regarding the use of active learning, and it also fit with my personal experience. As an undergraduate, I majored in English composition and Mathematics, originally intending to be a high school teacher. While I was interested and active in the math club and math competitions, I wasn’t a particularly strong math student, earning a mixture of A’s, B’s and a C in math major courses. After completing my degree, I took a job at a planetarium as a low-level manager. I had previously considered going on to graduate school in mathematics, so in my spare time I read about math and science. Slowly, through a series of fortunate moments, I came to understand the depth of my lack of mathematical knowledge. I found out that \(x^2+y^2=1\) is the equation for the unit circle because of the Pythagorean theorem; in high school, this had simply been presented as a fact. I was cleaning up the science demo room in the planetarium one day when a NASA video about trigonometric functions came on — I quietly closed the doors and watched for 45 minutes, taken aback by the beautiful connection between sine, cosine, tangent, their graphs, and the unit circle, which I had never seen.
The next year I started graduate school. On a regular basis I told my wife that I was definitely the dumbest person in my complex analysis course, but that I was doing my best anyway. On more than one occasion I sat on the living room floor and burst into tears, overwhelmed by the stress of trying to understand the barrage of ideas one encounters in graduate mathematics courses. I spent a lot of time in the math library that first year in graduate school, reading books like Serge Lang’s Basic Mathematics (a high-school text) and Liping Ma’s Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (about elementary school mathematics); it was at this time that I came to understand the importance of the distributive law and its role in the multi-digit multiplication algorithm. It was also at this time that I became deeply aware of how I had been doing math without real understanding, demonstrating high-level mathematical knowledge without substance. Only as time passed did I realize how many students of mathematics, even successful students, operate in this way.
When I first read Dweck’s work on mindsets, I had already begun using various pedagogical and assessment techniques: active learning through small group work, reflective essays as homework, semester-long individual projects. While students responded well to this, I was never satisfied at the end of the course. There were too many students who didn’t develop their understanding of mathematics, who were hesitant to fully engage in the course. I decided to directly intervene, using Dweck’s work as the basis for an explicit discussion of the role of beliefs in learning and achievement.
Because our first-year calculus courses are taught using a large-lecture/recitation, highly coordinated structure, I only felt free to experiment in my smaller upper-level courses for math majors and minors. On the first day of class, I assigned Dweck’s survey article “Is Math A Gift?: Beliefs That Put Females at Risk”; recently I have instead used her Scientific American article “The Secret to Raising Smart Kids.” I also had students write a one-page autobiographical statement about their previous experiences in math courses. The second day of class was devoted entirely to psychological aspects of mathematics: How do you feel about mathematics? Do you actually like it? Do you feel you are good at it? What are the reasons you have succeeded to this point? At first I tried to have discussions with the entire class sitting in a circle, but found that it is much more effective to assign students to groups and have them talk with their peers — I don’t have to hear everything they say in order for the discussion to be meaningful. I found that starting the class this way completely changed the tone of my courses for the better. It was surprising and refreshing to the students for a math course to start in this manner, and it set the stage for our classroom discussions about mathematics to include both technical and psychological aspects.
The biggest surprise I had, and a challenge I still struggle with as a teacher, is the remarkable ability of students to argue in favor of the dominance of innate talent in mathematics. Cultural conditioning regarding the myth of genius is strong and embedded; for many of my students, this had developed into the false belief that the goal of doing math is to be brilliant, rather than to gain reasonable mastery and improve one’s understanding. Students frequently compared doing math to training to be an elite athlete. In more than one small group, in more than one class, I heard statements such as “no amount of hard work will make someone play basketball like Michael Jordan.” The fact that these same students enjoyed playing basketball for the purpose of honing their skills and enjoying the company of friends, rather than becoming a legendary athlete, usually didn’t occur to them until I raised the point explicitly.
Ten months ago, I was grading homework that had asked students to work on a particularly challenging open-ended problem. I was struck by the fact that almost a third of the students made a negative remark such as “this is wrong, I’m an idiot” in their solutions. This type of language is so culturally embedded in our mathematical discourse that we often don’t realize the level of negative self-talk we use. I frequently hear mathematicians make comments like “Oh, I see that now, I should have realized that before” when the reality is that we earn our realizations through effort and persistence, because real mathematical understanding requires time. The day after I graded that homework assignment, I implemented the following course policy, which is now part of every course I teach.
Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. Here are example statements that are now banned, along with acceptable replacement phrases.
- I can’t do this –> I am still learning how to do this
- That was stupid –> That was a productive mistake
- This is impossible –> There is something interesting and subtle in this problem
- I’m an idiot –> This is going to take careful thought
- I’ll never understand this –> This might take me a long time and a lot of work to figure out
- This is terrible –> I think I’ve done something incorrectly, let me check it again
Please keep in mind the article we read by Carol Dweck. The banned phrases represent having a fixed view of your own intelligence, which does not reflect the reality that you are all capable of dynamic, continued learning. The suggested replacement phrases support and represent having a growth mindset regarding your abilities and your capacity for improvement.
In my most recent courses, I introduced this policy on the second day of class, following our small group discussions of Dweck’s article, and I subsequently enforced it vigorously. Doing so has revealed even further for me the depth of the challenge math teachers face — everything operates against our goal of student learning, even the words and phrases we are subconsciously trained to use. How can I hope to have my students believe in their own abilities, when their default descriptions of their work are derogatory?
I frequently teach courses for pre-service teachers, and one remarkable aspect of building a classroom environment around growth mindsets is the connection to the Standards for Mathematical Practice in the Common Core State Standards for Mathematics. At some point in time during every course that serves preservice teachers, I show students these standards — their typical response is to be shocked that these are required of K-12 students, and also to feel uncertain of how to interpret some of them. These standards both implicitly and explicitly reflect the fact that authentically doing mathematics involves trying, failing, trying again, making mistakes, correcting, and shifting perspective. I have students work on a tough problem in small groups, one I don’t expect them to solve during class or at all, and stop every few minutes to reflect on which of the practice standards they have used, and whether or not there were missed opportunities to bring others into play. I insist that the students not criticize themselves for their missed opportunities, simply acknowledge them and, from that recognition, improve.
It is reasonable to wonder if any of these activities have a meaningful effect on students. Mindset interventions, as they are often called, are being included as part of academic orientations at various universities. I often reflect on the ability my students have demonstrated to resist these messages regarding mathematics, even though various studies provide strong evidence that such interventions improve academic achievement. I wonder: while students’ performance improves after a brief mindset intervention, how much does it change what they believe about the nature of mathematical ability?
I’ve found that the hardest questions to ask students are the ones I most want to know the answer to: “What are you really thinking? What do you truly understand? What do you believe you are capable of accomplishing?” To obtain reasonably deep answers to these questions, I decided early in my teaching career that I need to have students write reflective essays in my courses. Here are excerpts from end-of-semester essays that four of my students have allowed me to share.
Student 1: I had always conceived of mathematics — and, by extension, science and engineering — as a field advanced by sheer brilliance. Yes, I realized that these fields were more parts failure than success, but nothing has contributed to cementing in my mind that anyone can succeed in any field through hard work and dedication than the Dweck article presented near the beginning of the semester. I have made this an integral message in my private chemistry tutoring; no regular client of mine this semester has managed to escape my spiel about how they can’t allow their fears and lack of confidence to hold them back from working hard to succeed.
Student 2: Speaking of teaching, the [Dweck] article that we read at the very beginning of the semester has stuck with me this whole time and is something I want to be sure I keep in mind when I have a classroom of my own. While I would still say that most of the mathematicians we have read about are super geniuses, they did work diligently towards what they wanted to achieve. It stood out to me that when you present these geniuses as people who worked really hard it influenced the students’ thought processes in a positive way, making them more likely to try, whereas when presented with material that said they were geniuses the students took a more negative approach of “I don’t have the gift therefore I can’t do the math.” It will definitely influence the way that I present material in my own future classroom, making sure to focus on working hard rather than “just being good at math.”
Student 3: One of the misconceptions I held when I came into this class at the beginning of the semester was that if I had to spend a large amount of time on a problem that meant I was dumb. I don’t believe that I ever voiced that opinion to anyone, but I now know that it was there. And it was because of this and my other math courses this semester, that I only slightly have that thought. I learned the very hard way that good mathematics takes time. I can no longer just plug and chug like I could with calculus or matrix; now I actually have to think about what I’m doing. It was, and still is, a very frustrating feeling, but underneath that feeling is the understanding that sometimes this is just what solving problems is: it’s time, and frustration, and sometimes having a tantrum before the problem can be solved.
Student 4: I had always thought that mathematics was a gift. You were either good at math or you were not. It was this reasoning that caused me to believe that some people were born to be mathematicians, while others were doomed to always struggle in mathematics. However, I have seen several people (including myself) in this class go from struggling in math class to having an impressive mathematical skill set. I now see mathematics as more like athletics. While some people are more naturally gifted than others, hard work will pay off in the end. This does not mean that I will not have more challenges. It does mean that I can face those challenges, and that in most cases I can learn the mathematics in order to do what is required.
For reasons that I don’t fully understand, our mathematical culture encourages us to define our mathematical ability by what we don’t know, what we aren’t able to do, rather than by what we do know and have learned how to do. The power of culture is strong, with deep roots — I don’t truly believe that the ripple effect from my teaching will spread very far. Yet I cannot help but think of all the students who persist in mathematics. In spite of so many unspoken doubts, so many negative influences, these students have made their way through the doors to our classrooms. And I cannot help but think of the many thoughtful, capable students who turn away from mathematics and give up hope. We are surrounded by potential, by possibility, by self-inspiration yearning for a spark. I believe that the brightest sparks come from people rather than mathematics. That our thoughts, emotions, and beliefs are the gateway toward a more diverse, equitable, proficient, and beautiful mathematical culture. The key is allowing time for these alongside technical mathematics in our classrooms; real mathematical understanding requires time.