By Cody L. Patterson, University of Texas at San Antonio
Several years ago, I took up running. At first, I wasn’t particularly good at it, but I persisted: about two or three times each week, I would go for a jog, increasing my pace or distance in small increments. This measurable growth in my running ability and physical fitness was a great motivator for me, and I increased the frequency of my workouts. After about a year, I was able to complete a local 5K race; this remains among the proudest achievements of my life to date. This was the most authentic experience I’ve had of putting sustained effort into a domain in which I had little natural ability, observing my own growth, and working toward a specific, achievable goal. I attribute my success to two factors:
- I didn’t measure my own performance against others’. I knew that many people were more accomplished at running than I was when I got started. I set this thought aside and enjoyed the fresh air and the feel of the pavement under my feet.
- I took notice of any growth in my distance or speed, no matter how small. I took pleasure in being able to observe so many improvements in such a short time.
I have often wondered how I can create a similar experience for students in my mathematics classes, especially for those students who lack confidence in their mathematical knowledge and skills. These are the students who are in danger of developing the mindset that the sustained effort they need to master challenging topics indicates that they are not qualified for advanced study in mathematics. Therefore, one goal of every class I teach is to help students let go of concerns about how they are performing relative to their peers, and enjoy observing their own growth and learning. In his September 2015 article in this blog, Benjamin Braun described some of the mindset interventions he uses to help focus students’ attention on their mathematical growth. In this article, I’ll describe how the recent work on growth mindset has influenced assessment practices in my own courses.
In her research, Carol Dweck describes implicit theories of intellectual and social traits that influence how and whether people choose to invest effort in developing skills (see, for example, Dweck, 2008). Dweck uses the term entity theory to refer to the idea that traits such as mathematical skill are innate, and that adversity and failure are indications that one does not possess these traits. She uses the term incremental theory to refer to the idea that traits such as mathematical skill are malleable and can be developed through sustained effort. Students who have an entity theory of mathematical intelligence often demonstrate a “fixed mindset” in mathematics classes, interpreting challenges as opportunities to display their innate abilities in mathematics, or as threats to their mathematical identity. On the other hand, students who have an incremental theory often demonstrate a “growth mindset,” embracing challenging and open-ended tasks as opportunities to discover and develop new ideas.These mindsets are consequential for achievement in mathematics. In one study, seventh graders with both mindsets began the year with comparable grades, but students with growth mindsets outperformed those with fixed mindsets by the end of the year; however, growth mindset interventions were shown to mitigate this effect (Blackwell, Trzesniewski, & Dweck, 2007). In higher education, disciplines where fixed mindsets are prevalent among faculty tend to see underrepresentation of women among Ph.D. recipients (Leslie et al., 2015). On the other hand, orienting students toward incremental theories of mathematical ability and creating learning environments consistent with growth mindsets can reduce students’ vulnerability to stereotype threat (Aronson, Fried, & Good, 2002; Good, Rattan, & Dweck, 2012).
Jo Boaler’s recent book Mathematical Mindsets (2015) provides a wealth of advice on how to structure mathematics instruction to promote the development of growth mindsets. She recommends practices that recent research has proven successful, such as the use of low-threshold-high-ceiling problems that are accessible to all students but require extended effort to solve completely, and strategies for managing groupwork that are consistent with Complex Instruction (Cohen et al., 1999).
Integrating the growth mindset into assessment and grading
I have taken some steps to reframe assessment and grading in my courses as a way of stimulating growth and providing guidance for learning, rather than rewarding success or punishing failure.
Specifications grading. Specifications (specs) grading (Nilson, 2015) is a system in which students earn course grades by meeting a set of clearly defined criteria rather than by achieving a certain weighted average across exams, homework, and other assignments. I now use specs grading in all of my courses; in most cases, to earn a grade of A, students must pass exams with specified scores, give a successful presentation in class, and earn a passing score on homework problem sets. I also include class attendance and participation in my specs grading scheme; since I started doing this, I have had over 90% attendance in my classes. I have an “exception clause” in which students who fall short of a standard in one category can compensate by exceeding standards in another category. This provides flexibility and sets the tone that there are many ways to demonstrate mastery. In November 2015, Kate Owens wrote about a similar system called standards-based grading (SBG); while SBG is generally organized around learning goals rather than assignment types, both systems have the essential feature of providing opportunities for students to deepen their own mastery of course content.
I’ve found that specs grading provides greater clarity for students; at the end of the semester, there is little mystery about what students must do in order to earn a desired grade. The grading scheme also allows me to be serious about things that matter: since I’ve adopted this system, I’ve never had to award a course grade of B to a student who did extremely well on homework (by getting external assistance) but did not demonstrate any mastery of the course material on exams.
Revision policy. I knew that if I implemented a specifications grading scheme and did nothing else, I would only end up being stingier with grades. I wanted to reshape my course policy into one that embraces mistakes as opportunities for growth and learning. Therefore, I have the policy that any written homework assignment in my course can be revised. Students get constructive feedback on problem sets; if they read the feedback and submit revisions, I replace the old grade with a better one. I used to impose a nominal penalty (say, one point out of ten) on revisions, but I stopped doing this because I could no longer defend a practice that punished students for making mistakes. The revision policy allows me to be much more consistent in holding students accountable for producing high-quality work. This does not cause too great an increase in my overall grading load, because students’ revised work is usually of higher quality and therefore easier to grade.
Exam scores as “work in progress.” Exams have a way of bringing students’ sense of non-belonging in mathematics into sharp relief. I try to manage exams in my classes in ways that encourage growth and do not position students as competing with one another. First, I set cut scores for each exam based on the difficulty of the test itself, not based on a “curve” nor in response to student performance on the test. I don’t use a 90-80-70-60 scale to interpret exam scores; there is nothing mathematically natural about this scale (Reeves, 2006), and it offers little hope for students who earn a score in the 20s or 30s on a test.
I make it clear that exam scores are “in progress” until the end of the semester, as each student earns a number of “extra lives” that can be used to retry exam questions at the end of the course. Students earn “extra lives” by doing things that will help them succeed in the course, such as completing the homework, doing practice problems, and completing short “Lesson Launch” assignments in which they watch video examples prior to class and write summaries (as in some “flipped” instruction models). On the last day of class, students take a customized test with questions covering topics on which they didn’t demonstrate mastery during the midterm exams. Their scores on these questions replace their old midterm question scores.
Finally, I try to make sure my own messaging about exams is consistent. After each exam, I send a brief e-mail summarizing a few places on the exam where I thought the class as a whole did well, and reinforcing the “growth mindset” message. My most recent e-mail contained the following:
I don’t make it a practice to give class-wide statistics from the exam … My purpose in giving exams (as with all the other work) is to give you opportunities to discover where your knowledge is already strong, and where you still have room to grow. I’d sooner see you spend your energy and effort on learning the material you haven’t mastered yet, rather than positioning yourself with respect to your classmates. My view is that this class has 19 terrific mathematical thinkers, and your current score on this exam is an indicator of your current level of mastery of this material, not of how smart you are in mathematics. I say “current score” because as you know, under the Extra Life system your score on this exam may well improve at the end of the semester if you do a good job of learning the material you haven’t mastered yet.
Impact on students
Students in my courses seem to appreciate the various opportunities to revisit and improve their work. In a typical semester, I will receive homework revisions from about 75% of my students, with some students submitting revisions for as many as 50% of the problems. The revised solutions that students submit are usually substantial improvements; the majority of revisions earn at least two additional points on a five-point scale.
I asked students in my Fall 2016 capstone course for preservice secondary teachers for feedback on how the course policy influenced their learning and their identity as mathematics learners. One student responded,
A huge benefit was that we could correct our assessments after being graded. It made us go back and actually think about every problem and how we could correct it. You gave great feedback and showed that you were willing to help us.
Another student commented not only on the revision policy, but on the overall tone of collaboration and personal growth it set:
Feedback – Fantastic. No other course has allowed me to continuously correct my work. Although grading must be time consuming, it’s greatly appreciated.
Engagement/Involvement/Interpersonal Connection – I felt a unique atmosphere of collaboration in MAT 4303. The trichotomy of student, instructor, and group motivated me to consistently work to the best of my ability. This is especially true in homework. In most courses, if I have an imperfect solution to a problem that I can’t resolve, it remains imperfect. This was not true for MAT 4303.
Above all else, MAT 4303 helped me to mature as a student and take a collaborative, selfless approach to courses. I enjoyed helping other students learn just as much as I enjoyed learning.
In the future, I also hope to research the use of growth mindset assessment practices and investigate their effects on students’ learning and mathematical identity. Every program in which I have taught has wanted students to be more confident in their potential to solve challenging problems and more motivated to pursue learning opportunities on their own. I believe that an indispensable step in this direction is to help students develop the mindset that even when initial efforts fail, their hard work will result in powerful and long-lasting intellectual growth.
Acknowledgement: I would like to thank my colleague Dr. Priya V. Prasad, who uses a similar system in her courses at UTSA and whose feedback led to improvements in my own implementation.
Aronson, J., Fried, C. B., & Good, C. (2002). Reducing the effects of stereotype threat on African American college students by shaping theories of intelligence. Journal of Experimental Social Psychology, 38(2), 113-125.
Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S. (2007). Implicit theories of intelligence predict achievement across an adolescent transition: A longitudinal study and an intervention. Child development, 78(1), 246-263.
Boaler, J. (2015). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. John Wiley & Sons.
Butler, R. (1987). Task-involving and ego-involving properties of evaluation: Effects of different feedback conditions on motivational perceptions, interest, and performance. Journal of educational psychology, 79(4), 474.
Cohen, E. G., Lotan, R. A., Scarloss, B. A., & Arellano, A. R. (1999). Complex instruction: Equity in cooperative learning classrooms. Theory into practice, 38(2), 80-86.
Dweck, C. S. (2008). Mindset: The new psychology of success. Random House Digital, Inc.
Good, C., Rattan, A., & Dweck, C. S. (2012). Why do women opt out? Sense of belonging and women’s representation in mathematics. Journal of personality and social psychology, 102(4), 700.
Leslie, S. J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. Science, 347(6219), 262-265.
Nilson, L. (2015). Specifications grading: Restoring rigor, motivating students, and saving faculty time. Stylus Publishing, LLC.
Reeves, D. B. (2006). The learning leader: How to focus school improvement for better results. ASCD.
Cody, you continue to be inspired and inspiring! Thank you for your work with assessment that truly guides and motivates authentic learning.
I believe the specifications grading policy is extremely helpful to students. As a mathematics student who is pursuing his bachelors and eventually my masters in Data Science I find it extremely helpful when professors break down how to earn each grade. I always aim to earn an A in my classes especially my mathematics course so my gpa is as high as it possibly can. When a professors tells us what we have to do to get an A in the class it makes it so much easier for me. I know exactly what I had to do to earn that A and I do it to earn that A. It makes things so much easier than an arbitrary grade scheme that tells us just the break down of how each component is weighted towards our grade. I also believe the idea of homework revisions is extremely important to academic pursuits. Sometimes the professors assign homework on material before it is covered and then grade it before we have finished the material. I believe this is extremely detrimental. Homework is meant to reinforce the ideas we learn in class not introduce us to those ideas. Some students have issues with homework that they are covering and the ability to go back and do revisions on problems they haven’t fully understood is extremely important. It allows them to master the material and fully understand it. Allowing them to do better on exams and other material that require the material. I also really love your idea of “extra lives”. The ability to go back and redo exam questions they didn’t fully understand at the time of the exam and replace those points lost is crucial. It promotes students to once again fully master all the material at the end of the class which prepares them better for the following courses and more advanced mathematics that use that material as the basis for the content. At the end mathematics courses are meant to reinforce and improve the student’s mathematical skills and promote the mathematical thinking process. I believe your classroom policy is one of the best I’ve seen so far and I commend you for teaching your class as such. Keep up the good work.
Jacob – Thanks so much for your comment. It’s great to hear student perspectives on grading and assessment policies, since those perspectives so often get lost in these discussions. I almost want to print and frame your statement that “mathematics courses are meant to reinforce and improve the student’s mathematical skills and promote the mathematical thinking process,” because it’s a beautifully succinct articulation of what we should be thinking about when we are teaching (and also of what students should think about when they participate in a course!). I believe that if we kept those two goals at the forefront of our thinking, it would improve much of our decision-making about how we teach, how we choose tasks for our students, and how we manage the interactions between students and their mathematics courses.