The Mathematics of Growth Mindset
True or False: Does sugar make kids hyperactive? We’ve all been there: a little extra Halloween candy and the kids are bouncing off the walls. When a psychologist friend asked his students, they unanimously agreed: True! So they were surprised to discover that study after study shows no connection at all. Yet the idea is so embedded in our culture, our discourse, our brains that it’s surprisingly difficult to admit that it might not be true.
True or False: some students pick up math math concepts instantly while others take many more iterations before they get it. “Definitely true!” I would have said – until a recent paper plunged me into a state of confusion, getting me to question some basic assumptions about how students learn math.
Let me explain…
Online homework platforms are great at giving students instant feedback. Instead of waiting a week, they find out immediately if they’re right, get pointers if they’re wrong, and can work on similar problems to give them more opportunities to learn.
Such systems also generate reams of data, all of which can be analyzed to better understand students’ learning trajectories. Ken Koedinger and his Carnegie Mellon colleagues explored 1.3 million student interactions across 27 different learning systems (from elementary to college, across a range of subjects) with the goal of identifying differences in how quickly students learned from practice.
To their astonishment, what they found wasn't what they (and I) expected. The details were published last year in the prestigious Proceedings of the National Academy of Sciences, in the aptly named paper, An Astonishing Regularity in Student Learning Rate, which I have been mulling over for weeks. Here’s a quick overview of the results followed by some of my reflections.
Students’ progress in learning new material generally follows a logistic curve as they slowly approach 100% success. Using a logarithm to unravel the embedded exponential produces straight lines. In the graph below, the student represented in red (S1) starts out ahead. S2 (green) starts behind but catches up a bit over 8 problems, learning slightly more per attempt than the red student and quite a bit more than S3 (blue) who starts out in roughly the same place. Importantly, the horizontal axis represents not time but opportunities – the number of questions attempted (with each attempt followed by informative automated feedback.)
The authors expected to see both a wide variety of starting points (y-intercepts) and a wide variety of learning rates (slopes) among thousands of students’ paths. They were only half right – students did start with a variety of levels. The unexpected part was the uniformity of slopes. The data showed most students learned at nearly identical rates. Picking up a new component of knowledge required about seven opportunities – with amazingly little variation. That astonishing result held across different age groups and different content. Nearly every data set they examined showed an astonishing uniformity in student learning rates. Here’s a sampling of five of the 27 examples.
My reflexive reaction, truthfully, was to dismiss these results. Online homework systems are such an artificial context with fairly banal problems – nothing like my interactive classrooms where students are discussing rich problems in groups. If I’m being honest though, I would have expected the student learning differences I see in class to show up in these data too. The students I saw picking things up quickly in class should have been quick on these platforms too.
I’ve come to grips with the disconnect by focusing on the horizontal axis, which represents attempts, not time. Even if two students have lines with the same slope, one student can progress more quickly if they make those attempts in a smaller amount of time – essentially compressing the horizontal axis and increasing their learning rate. That may have been what I’ve seen so many times. Conversely a student working more slowly through examples might have the same rate of learning relative to the number of opportunities, but would have a slower learning rate with respect to time.
In so many math classrooms (including mine early in my career), classroom interactions make initial differences (different y-intercepts) worse not better. Who talks most in class? The students who are already doing well, who came in already knowing some of the material. Everytime I interact with them, I’m giving them more opportunities to learn. Their prior success breeds future success; they are more likely to speak up with me, with their peers, and they learn even faster. In contrast, a student who is behind is less likely to speak up and isn’t getting that positive feedback. It naturally takes them longer to do the problems and thus they do fewer of them, spreading out their opportunities over a longer time. They do actually learn the material slower.
Those two students might take about the same number of opportunities to learn new material, as the paper claims, but other factors would lead to differences in how long that took. This tendency for performance gaps to widen (because success leads to more opportunities which lead to more success) has been dubbed the Matthew Effect (scroll down to p. 10 in this Just Equations report for more.)
Amplifying this effect is the deeply-held (Western) idea that success in mathematics is primarily due to innate abilities – contradicted by this research which would suggest that hard work and opportunity are primary. When students walk into our classes and are doing well, we tend to think of them as “smarter” and more “brilliant”, instead of viewing them as having had more prior opportunities or having worked harder.
Too often we mistake privilege for potential.
There are systemic ways to narrow gaps rather than widen them, like corequisite support classes and Emerging Scholars classes. These programs accelerate student learning through intensification, giving students more opportunities to learn material. Active learning can be seen as giving students more opportunities to learn, relative to them passively listening to a lecture. Assessment policies like grading for growth steer us away from high stakes exams that are really better measures of prior preparation. As the paper predicts, students experiencing all of these innovations are able to catch up to their peers by compacting their learning curves into a smaller amount of time.
What would the mathematical sciences community look like if we all internalized the lessons of this paper? The ubiquitous gatekeeping would be replaced by supports that helped students catch up. The favoritism shown toward the students with the most preparation would be replaced by structures that encouraged others to persevere. And students who start a course a little behind would get the support they needed to slowly catch back up.
I’ve changed my mind and now accept that there’s no connection between sugar and hyperactivity. Are we in the mathematics community willing to change our minds about some students seemingly being better learners than others - and change our teaching practices to give everyone a more fair shot at success?
Editor’s Note: Have thoughts about this post (and the referenced paper)? Join a dialogue about this topic on MAA Connect.
Dr. Dave Kung serves as a mathematician without borders and as Director of Programs for Transforming Post-Secondary Education in Math (TPSE-Math). He has worked in the intersection of mathematics and equity as the Director of Policy at the Charles A. Dana Center at The University of Texas at Austin, and as Director of MAA Project NExT. He also works closely with K-12 and higher ed organizations, especially concentrating on equity issues in mathematics. Kung was awarded the Deborah and Franklin Tepper Haimo Award, the MAA’s highest award in college math teaching, for his work at St. Mary’s College of Maryland. He resides in the DC area, playing violin and running–never simultaneously, but sometimes alongside his partner and daughter.