Productive Struggle: An Opportunity for In-Depth Mathematics Learning

By: Nitchada Kamlue, M.A., Doctoral Student, Western Michigan University
Laura R. Van Zoest, Ph.D., Professor of Mathematics Education, Western Michigan University

Struggle is a regular part of mathematics class: students struggle to get started, to carry out a process, to express misconceptions and errors, and with “uncertainty in explaining and sense-making” (Warshauer, 2015, p. 385). We have been investigating the struggles of prospective mathematics teachers (PTs) in a middle school methods course (Kamlue & Van Zoest, 2021; Kamlue & Van Zoest, 2022). In this post we draw on our research to provide insights into what we call mathematically productive struggle—the type of struggle that occurs when students “delv[e] more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions” (National Council of Teachers of Mathematics, 2014, p. 48). This type of struggle has been correlated with high achievement (see Peterson & Viramontes, 2017) and described as a way to increase equity in mathematics classrooms (Berry, 2021; SanGiovanni et al., 2020). In the following, we provide insight into (1) generating mathematically productive struggle, (2) engaging students with their struggles, and (3) focusing on the mathematics.

 Generating Mathematically Productive Struggle

An important way to increase opportunities for mathematically productive struggle is to use tasks that are at the doing mathematics level of cognitive demand (Stein & Smith, 1998; Stein et al., 1996). Doing mathematics tasks have the potential to support equity because they have multiple access points and provide freedom for students to follow their own solution path, rather than requiring everyone to approach the task in the same way. Meeting that potential requires teachers to intentionally look for and check any biases we might have about who in our classroom is—and is not—capable of making sense of mathematics (see, for example, Calvin’s Story in Berry, 2004). Recognizing that all students are capable of making sense of mathematics is crucial to ensuring that all our students are included in the rich mathematical learning opportunities that doing mathematics tasks provide. In our research, we used the Frog Problem (Andrews, 2000; Dixon & Watkinson, 1998) to generate mathematically productive struggle. This doing mathematics task requires students to predict the fewest number of moves for two equal-size teams of frogs to switch sides (see Figure 1 for a representation of frog teams of size three—the size the PTs started with). Each frog is allowed to jump over one frog to an empty spot or slide to an adjacent empty spot.

Figure 1: A Representation of Two Frog Teams of Size Three

The Frog Problem is a doing mathematics task because it requires the PTs to access their prior mathematical knowledge and experiences and effectively use them to solve the task. It required the PTs to explore and develop a mathematical model for the situation, a model which involved a quadratic relationship. The task also required the PTs to analyze what given information was helpful for finding possible strategies that they could use to solve the task. Since the task provides no clear solution path, the PTs approached it in a variety of ways. The Frog Problem generated several sites for struggle that had the potential to support PTs’ learning, including: when they discussed their different possible solutions for the fewest number of moves, when one PT asked others whether something made sense, and as the PTs tried to generate an equation to represent the fewest number of moves for any size team (Kamlue & Van Zoest, 2021).

Engaging Students with Their Struggles

A doing mathematics task lays the foundation for mathematically productive struggle, but once struggles emerge, there is work to be done to engage the students in a way that makes their struggle mathematically productive. Teachers need to maintain the task’s level of cognitive demand while scaffolding students’ sense making. Maintaining high cognitive demand is important because doing so “allows students to perform mathematics, which means that they engage in activities that involve exploring, representing, observing, conjecturing and detecting relations and constants as well as justifying and communicating results” (Estrella et al., 2020, p. 295). What it looks like to scaffold students’ sense making, however, varies across students. Here, we describe productive responses to three different ways students might respond as they engage with the task: (a) give up when they first encounter struggle, (b) encounter struggle and persevere towards solving the task, and (c) solve the task without encountering struggle.

First, it is important to take action with students who give up when they first encounter a struggle because we want to cultivate the mathematical practice of making sense of problems and persevering in solving them (NGACBP & CCSSO, 2010). Productive moves to support “stuck” students to persevere include asking them what they have tried and found out about the task, reminding them that mistakes are sites for learning, pointing out their progress, and providing time for them to struggle (Peterson & Viramontes, 2017; Warshauer, 2015).

Second, we want students who encounter struggle to persevere and, with our support, many will. The question is, are they engaged in a mathematically productive struggle or are they exerting a lot of effort that is not supporting their learning? In our study (Kamlue & Van Zoest, 2021), we identified the PTs’ struggle as productive if they explained how they thought about and solved the task, asked peers questions to make sense of the mathematics, or used hands-on materials to help make sense of the task. As a teacher, if students are productively struggling, we can enhance their sense making by asking them for mathematically relevant details about their explanation, building on their mathematical thinking, pressing them for justification, and affording time for them to work through their struggle (Warshauer, 2015). In contrast, if students are persisting with a struggle that does not seem likely to help them better understand mathematics—such as trying things haphazardly—the most productive response is to redirect them toward sense-making activities.

Finally, for those who are able to solve the task without encountering struggle, it is important to have questions planned (e.g., “Can you solve the task in a different way?”) to allow them to continue their learning and not be tempted to engage in unproductive behaviors (e.g., telling other students how to solve the task). In the Frog Problem, our students were able to extend their thinking to different sized teams and look for an equation that would work for a team of any size. Moreover, we have found that engaging the class in discussing their strategies and the struggles they encountered can generate additional sites for mathematically productive struggles for students who did not struggle with the task itself; such as being puzzled by—and needing to make sense of—alternative approaches to the task, and being challenged by needing to explain their solution to peers who were thinking about the task differently. Shifting emphasis away from the solution itself to what was learned from pursuing that solution creates rich learning opportunities for all the students in the class.

Focusing on the Mathematics

Often when we give students a task, we focus on getting them to the answer as if getting the answer is the goal, rather than learning the mathematics embedded in the task. Through the lens of mathematically productive struggle, the learning occurs in the process of getting to the answer. We have found that mathematically productive struggles can occur even when students are not making progress towards getting the answer! To illustrate this, we return to the Frog Problem. As reported in Kamlue & Van Zoest (in press), a group of PTs (PT1, PT2, PT3, and PT4) remembered learning about the arithmetic sequence and wondered if it could be used to solve the Frog Problem. At this point in their discussion, the group knew the actual numbers for frog teams of size 1, 2, 3, and 4. PT3 suggested an arithmetic sequence [a_n = a₁ + (n – 1)d] as a template for finding an equation that represented the fewest number of moves for any size team of frogs and the other PTs in their group agreed. (Note that the difference of the fewest moves from each consecutive case is not constant, thus an arithmetic sequence does not apply to this situation.) Figure 2 shows part of their struggle while working on solving this task as a small group.

Figure 2: Excerpt of PTs struggling while solving the Frog Problem

This struggle was not going to help them solve the task, but it provided the PTs the opportunity to better understand this mathematical point (Van Zoest et al., 2016): d is a constant that represents a common difference between each consecutive term in an arithmetic sequence.

Recognizing this, the teacher helped them struggle productively by addressing their current thoughts and asking whether it was some other pattern they could identify. The teacher also provided time for them to figure it out themselves by walking to check in with another group immediately after asking that advancing question (Freeburn & Arbaugh, 2017). These teacher responses aligned with Warshauer’s (2015) suggestions that teachers use probing guidance and affordance approaches to either maintain or raise a level of cognitive demand of the task to make the struggle productive.

Closing

Positive outcomes can come out of students’ struggles. Rather than struggle being something to help students avoid, we can use it as a tool to enhance the learning of all students, including those who traditionally have been marginalized in mathematics classrooms. We can support mathematically productive struggle in our classrooms by: (a) generating mathematically productive struggle by using a doing mathematics task; (b) engaging students in their struggles by scaffolding their sense-making while maintaining the high cognitive demand of the task; and (c) focusing on the mathematics that students are able to gain from their struggle. Not only will this type of support increase students’ learning, it will also provide a window into our students’ thinking and how our actions as teachers affect their developing mathematical understanding. That means that engaging our students in mathematically productive struggles affords us the opportunity to have mathematically and pedagogically productive struggles ourselves—a powerful form of professional development.

Acknowledgement

Thanks to ARISE Blog Editor, Dr. Ruthmae Sears, for inviting the authors to share their research on “mathematically productive struggle”.  Please read the ARISE blog by Ruthmae and her colleagues, “Using the T.R.U.T.H. Framework to Advance Inclusive and Equitable Pedagogy in Education.”