{"id":903,"date":"2015-09-01T00:01:56","date_gmt":"2015-09-01T04:01:56","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=903"},"modified":"2015-08-23T10:48:54","modified_gmt":"2015-08-23T14:48:54","slug":"the-secret-question-are-we-actually-good-at-math","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2015\/09\/01\/the-secret-question-are-we-actually-good-at-math\/","title":{"rendered":"The Secret Question (Are We Actually Good at Math?)"},"content":{"rendered":"

By Benjamin Braun, Editor-in-Chief<\/a>, University of Kentucky<\/span><\/i><\/p>\n

\u201cHow many of you feel, deep down in your most private thoughts, that you aren\u2019t actually any good at math? That by some miracle, you\u2019ve managed to fake your way to this point, but you\u2019re always at least a little worried that your secret will be revealed? That you\u2019ll be found out?\u201d<\/span><\/p>\n

Over half of my students\u2019 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. \u00a0Self-conscious chuckling darted through the room from some students, the laughter of relief, while other students whose hands weren\u2019t raised looked around in surprised confusion at the general response. \u00a0<\/span><\/p>\n

\u201cI want you to discuss the following question with your groups,\u201d I said. \u00a0\u201cHow 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?\u201d<\/span><\/p>\n

If this interaction took place in a math course satisfying a general education requirement, I don\u2019t think anyone would be surprised. \u00a0Yet 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. \u00a0I\u2019ve 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. \u00a0These 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. \u00a0At every level of achievement, no matter what we\u2019ve done, no matter how much we\u2019ve accomplished, many of us believe that we\u2019re simply not good at math.<\/span><\/p>\n

**************<\/span><\/p>\n

I first discovered the work of Carol Dweck from a <\/span>link<\/span><\/a> on Terry Tao\u2019s <\/span>blog<\/span><\/a>. \u00a0Dweck is a psychologist at Stanford whose <\/span>studies on the relationship between self-beliefs and achievement<\/span><\/a> have had a tremendous impact in education and beyond. \u00a0Her 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. \u00a0In 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. \u00a0On 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.\u00a0 The former belief is referred to as a \u201cgrowth\u201d mindset, while the latter is called a \u201cfixed\u201d mindset.<\/span><\/p>\n

Dweck\u2019s work <\/span>fit naturally<\/span><\/a> 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. \u00a0As an undergraduate, I majored in English composition and Mathematics, originally intending to be a high school teacher. \u00a0While I was interested and active in the math club and math competitions, I wasn\u2019t a particularly strong math student, earning a mixture of A\u2019s, B\u2019s and a C in math major courses. \u00a0After completing my degree, I took a job at a planetarium as a low-level manager. \u00a0I had previously considered going on to graduate school in mathematics, so in my spare time I read about math and science. \u00a0Slowly, through a series of fortunate moments, I came to understand the depth of my lack of mathematical knowledge. \u00a0I found out that \\(x^2+y^2=1\\) \u00a0 is <\/span>the equation for the unit circle because of the Pythagorean theorem<\/span><\/a>; in high school, this had simply been presented as a fact. \u00a0I 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 <\/span>connection between sine, cosine, tangent, their graphs, and the unit circle<\/span><\/a>, which I had never seen.<\/span><\/p>\n

The next year I started graduate school. \u00a0On 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. \u00a0On 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. \u00a0I spent a lot of time in the math library that first year in graduate school, reading books like Serge Lang\u2019s <\/span>Basic Mathematics<\/span><\/i><\/a> (a high-school text) and Liping Ma\u2019s <\/span>Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States<\/span><\/i><\/a> (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. \u00a0It 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. \u00a0Only as time passed did I realize how many students of mathematics, even successful students, operate in this way.<\/span><\/p>\n

**************<\/span><\/p>\n

When I first read Dweck\u2019s 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. \u00a0While students responded well to this, I was never satisfied at the end of the course. \u00a0There were too many students who didn\u2019t develop their understanding of mathematics, who were hesitant to fully engage in the course. \u00a0I decided to directly intervene, using Dweck\u2019s work as the basis for an explicit discussion of the role of beliefs in learning and achievement.<\/span><\/p>\n

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. \u00a0On the first day of class, I assigned Dweck\u2019s survey article <\/span>\u201cIs Math A Gift?: Beliefs That Put Females at Risk\u201d<\/span><\/a>; recently I have instead used her Scientific American article <\/span>\u201cThe Secret to Raising Smart Kids.\u201d<\/span><\/a> \u00a0I also had students write a one-page autobiographical statement about their previous experiences in math courses. \u00a0The second day of class was devoted entirely to psychological aspects of mathematics: How do you feel about mathematics? \u00a0Do you actually like it? \u00a0Do you feel you are good at it? \u00a0What are the reasons you have succeeded to this point? \u00a0At 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\u2019t have to hear everything they say in order for the discussion to be meaningful. \u00a0I found that starting the class this way completely changed the tone of my courses for the better. \u00a0It was surprising and refreshing\u00a0to 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.<\/span><\/p>\n

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. \u00a0Cultural 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\u2019s understanding. \u00a0Students frequently compared doing math to training to be an elite athlete. \u00a0In more than one small group, in more than one class, I heard statements such as \u201cno amount of hard work will make someone <\/span>play basketball like Michael Jordan<\/span><\/a>.\u201d \u00a0The 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\u2019t occur to them until I raised the point explicitly.<\/span><\/p>\n

**************<\/span><\/p>\n

Ten months ago, I was grading homework that had asked students to work on a <\/span>particularly challenging open-ended problem<\/span><\/a>. \u00a0I was struck by the fact that almost a third of the students made a negative remark such as \u201cthis is wrong, I\u2019m an idiot\u201d in their solutions. \u00a0This type of language is so culturally embedded in our mathematical discourse that we often don\u2019t realize the level of negative self-talk we use. \u00a0I frequently hear mathematicians make comments like \u201cOh, I see that now, I should have realized that before\u201d when the reality is that we earn our realizations through effort and persistence, because real mathematical understanding requires time. \u00a0The day after I graded that homework assignment, I implemented the following course policy, which is now part of every course I teach.<\/span><\/p>\n

Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. \u00a0Here are example statements that are now banned, along with acceptable replacement phrases.<\/span><\/p>\n