Helping students to learn to do mathematics, rather than just calculate and determine solutions, requires students to adopt the identity of a student mathematician. This includes adopting the academic behaviours, habits of mind, and ways of interacting with the content in order conceptually learn and apply mathematics even when faced with unfamiliar or new situations. Essentially, learning how to struggle in productive ways—ways that leverage a growth mindset, value persistence, and view mistakes as opportunities—requires students to have particular learning conditions met prior to being able to engage in productive struggle.
Common approaches often focus on the nature of questions asked to students, asking students to “think more” on the problem, or “try a different way.” While such suggestions and actions can be helpful, they are only helpful if other conditions are in place. Three of these conditions include Students’ mathematical identity, mathematical and learning norms, and beliefs about doing mathematics. To avoid your own struggles in helping students learn how to be a mathematical problem solver who can and does struggle in productive ways, attending to these conditions can help tremendously.
A student’s mathematical identity is paramount to their overall success in mathematics. Recently I asked a group of elementary and middle school students to draw a picture of a mathematician. In not one drawing did they draw a student. All of the drawings were male (except for the drawing of a robot) and nearly all were older wearing eyeglasses and lab coats with scraggly hair (anyone come to mind?). Unless students understand that they can be a mathematician, they will struggle to relate to the content.
A few tips to help strengthen students’ mathematical identity include referring to them as mathematicians and the work they do as work of mathematicians. Also, point out and commend them when they use mathematical behaviours (like those found in the Standards of Mathematical Practices link below) and demonstrate persistence. Also, remind them that mistakes are expected; anything worth learning will fight back, paraphrasing one of my favourite quotes from the Danish mathematician Piet Hein.
Mathematical and Learning Norms
There are various acceptable ways to establish classroom learning norms but one that helps with productive struggle centres on the importance of quality work over completed work. This learning norm also strengthens an important mathematical norm that values the process of doing mathematics over the answer or the completed product. There is no doubt that answers matter in mathematics but they are the last thing that matters. The way students think about, conceptualize, and engage with the content matters far more as the answer will not mean anything if their thinking is not sound.
A few tips to help build strong mathematical and learning norms begin with altering some grading/assessment practices. Instead of assigning a grade or score to an early learning attempt, provide focused feedback with the expectation that revisions are expected. Even then, make the classroom discussion on the process, and what they learn from the process, rather than on just the answer. The norms we set for mathematics should reflect the kinds of behaviours and habits of mind we expect to see from students. A good resource and list of mathematical norms you might want to use with your students can be found at YouCubed link below.
Students’ mathematical identity, the mathematical and learning norms in the classroom, as well as students’ and teachers’ beliefs about doing mathematics are all prerequisite conditions to supporting the productive struggle and developing students’ Mathematics identity.
By Cory A. Bennett, Ph.D.