I just read this great paper by Karen Bogard Givvin and James Stigler, “Removing Opportunities to Calculate Improve Students’ Performance on Subsequent Word Problems.”They ask one group of students to explain typical word problems, and ask another group to explain similar word problems but with the numbers removed so that calculating an answer is impossible. Students who solved problems with numbers removed both solved similar problems more accurately, and transferred their knowledge to new and different problems more successfully than the control group. Cool!
Brian Bushart calls these numberless word problems and has done awesome work writing, collecting, and thinking about them on his blog. Both Brian and the authors of the paper describe the strategy as a way to help students slow down rather than immediately applying procedures without thinking about the relationships in a problem. Givvin and Stigler reference this as the “compulsion to calculate.” They write:
They found that when students are presented with a mathematics word problem, their first response often is to try to compute an answer, even before they have tried to understand the problem. The description offered by Stacey and McGregor (1999) reminds us of the community college students we interviewed, who appeared not to think long about the problem posed, but instead to search their memory for a procedure that some teacher, at some point, had told them to use.
Something really interesting: almost half of the students in the study made up numbers to calculate with, seemingly feeling that number are always necessary to solve a math problem.
I’ve never used numberless word problems with my students (though now I want to). But the strategy reminds me of other activities I’ve come to find useful. I’m interested in better understanding this idea of the “compulsion to calculate” in this context. I worry that, as I experiment in my teaching, I’m drawn to activities that feel fun and new and clever, but might not actually have much value for student learning. The online math education world can lend itself to style over substance. I think that the “compulsion to calculate” articulates something useful that goes beyond clever ideas. If students are prevented from jumping into a procedure, but instead think about the relationships in a problem, focusing on where a certain procedure is useful and why, they are making important connections that are often lost in the race to calculate.
I think calculation is important! Students should calculate things in math class. But if students are only calculating, they are missing opportunities to make broader connections. Is a problem about multiplying numbers, or knowing when a problem requires multiplication? Is a problem about solving a logarithm, or considering what a logarithm represents? Is a problem about graphing a function, or thinking about the structure of the expression that determines the graph? As the study demonstrates, students are often predisposed to lean on calculation as the central component of what it means to “do math.” Finding ways to mitigate the compulsion to calculate acts as a corrective, balancing calculation with relational thinking that is otherwise lost.
Having this lens helps me to think about which new activities I want to find ways to bring more regularly into my class. I don’t want to try new things for the sake of trying new things, but ideas like menu math, connecting representations, and which one doesn’t belong? provide opportunities to facilitate this type of thinking. I also appreciate having a new lens through which to understand student thinking and diagnose when things are going wrong. When students all struggle with a certain type of problem, maybe the issue is over-reliance on a procedure without understanding where that procedure applies. Strategies to mitigate the compulsion to calculate are particularly useful in those moments to create opportunities for thinking that is missing.