How to Encourage Productive Struggle in Math
Insights from Research and Strategies for the Classroom
What is productive struggle?
Research identifies productive struggle as an essential component of effective mathematics classrooms (Boaler & Dweck, 2016; Preiss & Sternberg, 2010; Warshauer, 2014) which arises when teachers include opportunities for students to attempt solving problems that target concepts that are new to them, rather than limiting those opportunities to tasks with familiar/known skills (Hattie, 2017, p. 117).
Drawing on the idea that students need to engage in thinking that has some perplexity, confusion, or doubt (Dewey, 1933), Hiebert and Grouws (2007) describe productive struggle as “the intellectual effort students expend to make sense of mathematical concepts, to figure out something that is not immediately apparent” (p. 387).
💡 Research Spotlight: In a study of ninth-grade students, Kapur (2014) also defined the notion of productive failure, which, similarly to productive struggle, notes the importance of students’ failure to generate correct solutions. However, this failure, if generated through the activation of students’ prior knowledge, “can be productive in preparing them to learn better from the subsequent instruction that follows” (p. 1009).
What productive struggle is NOT
When defining productive struggle, it is important to note what it does not mean:
- Productive struggle should not result in unnecessary frustration derived from overly difficult tasks or challenges that are not mathematically appropriate or useful (Hiebert and Grouws, 2007; Warshauer, 2014).
- Teachers can perceive student struggle or delay in arriving at correct answers as a reason to “rescue” students and show them an effective procedure too soon, ultimately to the detriment of students’ learning and conceptual understanding of new mathematical ideas (NCTM 2014, p. 48).
Encouraging productive struggle is all about balance — and it can be tricky to identify the level of struggle that’s truly productive, especially in a classroom full of learners with different needs and abilities.
Designing Tasks to Encourage Productive Struggle
One way to ensure students are engaged in an appropriate level of struggle is by selecting or creating the correct tasks. Selected or created tasks should be within students’ zone of proximal development, “the distance between the actual developmental level as determined by independent problem-solving and the level of potential development as determined through problem-solving under adult guidance, or in collaboration with more capable peers” (Vygotsky, 1978, p. 86).
Supporting students to think and learn at levels right above their abilities facilitates increased learning and understanding. Thus, choosing tasks becomes crucial for using productive struggle as a learning tool.
Cognitive demand, a framework that describes the level of thinking processes needed to solve a task, supports teachers’ efforts to choose the appropriate type of task for students (Henningsen and Stein, 1997). Cognitive demand levels in math include:
- Memorization
- Procedures without connections
- Procedures with connections
- Doing math
Engaging in tasks at all demand levels is important in math learning. However, opportunities to develop productive struggle are most apparent in the levels of procedures with connections and doing math. Tasks at these higher levels of cognitive demand also have other features that support appropriate struggle: the tasks afford multiple solutions, strategies, and representations and activate students’ prior knowledge (Kapur, 2014).
Well-designed and cognitively demanding tasks can provide students opportunities to enhance their conceptual learning and offer up a wider view of what it means to do mathematics (Henningsen & Stein, 1997; Hiebert & Grouws, 2007).
Fostering Student Agency
Productive struggle relies not only on the cognitive level tasks can support but also on the ways in which students are expected to participate and with whom they collaborate in the learning process. By focusing on developing rich tasks with multiple avenues for engagement (and using productive struggle as part of that), teachers provide students opportunities to exercise their agency toward learning mathematics.
Through productive struggle, students learn to reason about math, fail and make mistakes, and debate ideas and solutions, all of which can allow the expression of student agency toward the goal of productively understanding and doing mathematics (Sengupta-Irving, 2015).
Looking to the Research
While research on productive struggle is relatively new in the field, some studies show the positive impact struggle affords students in learning mathematics. Here’s a quick example:
💡 Research Spotlight: Warshauer (2014) analyzed 186 episodes of middle school students’ productive struggles in proportional reasoning tasks to better understand the ways students can struggle and how teachers respond and support productive struggle in students. From this work, Warshauer developed a Productive Struggle framework that maps the progress of struggle from initiation, through student and teacher interactions, to resolution. Using the framework provides a way to categorize teachers’ responses to students’ productive struggles by mapping how teachers “maintain the task’s level of cognitive demand, address the student struggle, and build on student thinking” (Warshauer, 2014, p. 387).
Warshauer found that students’ struggles with tasks emerged in different ways, including difficulties getting started on the problem, not knowing what process to carry out, being uncertain about how to explain their work and math thinking, or trying to solve problems based on a misconception.
Warshauer’s study showed that teachers who responded to student struggles by telling students what steps to take or by directing students through a narrowed solution procedure lowered or minimally maintained the cognitive demand of the task. However, if the teacher supported student struggle by probing their thinking and asking questions with limited intervention, the cognitive demand of the task was maintained, or in some cases, raised.
This study points to the need for teachers to be thoughtful in how they encourage and respond to productive struggle in students, so the struggle can be productive and not frustrating.
How can math teachers encourage productive struggle?
Here are some actionable strategies you can take back to the classroom today to promote productive struggle in math:
- Place student-centered activities at the center of instruction, rather than as an option, to allow opportunities for students to regularly grapple with challenging problems with support from others in the learning community. (Cowen, 2016).
- Consider situating activity-based exploration early in lessons, so that you introduce new concepts to students by inviting them to explore a problem before teaching particular methods (Boaler & Dweck, 2016, p. 81).
- Design or select activities that have rich tasks with multiple points of entry (Hattie, 2017) — or, put differently, tasks that have a “low floor and high ceiling” (Boaler & Dweck, 2016, pp. 84–85).
Here’s what’s critical: During the activity-based explorations, students must engage in productive struggle while working toward solutions — drawing on their intuitions and existing knowledge and taking opportunities to engage in reasoning about the nature of the problem.
Seely (2016) posits, “When students have some time to explore and even struggle with a problem, our role as teacher becomes one of facilitating and stimulating conversation among students to ensure that they uncover and discuss the important mathematical ideas that lie within the problem” (p.33).
This instructional method is designed to maximize engagement and set the stage for new concepts, vocabulary, and procedures later on, when students are engaged, confident, and ready to learn.
For more on the importance of productive struggle in mathematics, and how productive struggle is practiced through specific instructional elements in Reveal Math K-5, see the full Reveal Math K-5 Research Foundations.
About the Authors
Lanette Trowery, Ph.D. is the Senior Director of the McGraw Hill Learning Research and Strategy Team.
Lanette was in public education for more than 25 years, working as a university professor, site-based mathematics coach, elementary and middle school teacher, mathematics consultant, and a professional learning consultant, before coming to McGraw Hill in 2014. She earned her Master’s and Doctorate from the University of Pennsylvania.
Lanette’s team, Learning Research and Strategy, serves as the center of excellence for teaching and learning best practices. Her team conducts market, effectiveness, and efficacy research into products to provide insights and recommendations to product development. They collaborate across internal teams, external experts, and customers to establish guiding principles and frameworks to move from theory to practice.
Margaret Bowman is an Academic Designer in the Mathematics Department at McGraw Hill.
Margaret earned her Bachelor of Science in Education from Ashland University with a teaching license in Middle Grades Education, and her Master of Education from Tiffin University. She was a middle school Math and Language Arts teacher for six years before joining the middle school team at McGraw Hill in 2012, writing and designing print and digital curriculum.
Margaret is also a Research Associate in the Research Laboratory for Digital Learning at The Ohio State University. She is nearing completion of a PhD in Educational Studies with an emphasis in Learning Technologies. Her past research and journal publications have focused on teachers’ value for using technology in the classroom and technology’s impact on student learning. Her current research examines how students’ use of technology can improve the value they have for mathematics and their expectations that they can succeed.
References
Boaler, J. & Dweck, C. (2016). Mathematical mindsets: unleashing students’ potential through creative math, inspiring messages, and innovative teaching. San Francisco, CA: Jossey-Bass; a Wiley Brand.
Cowen, E. (2016). Harnessing the Power of Productive Struggle. Retrieved from https://www.edutopia.org/blog/harnessing-power-of-productive-struggle-ellie-cowen. March 2020.
Gresalfi, M., Martin, T., Hand, V., & Greeno, J. (2009). Constructing competence: An analysis of student participation in the activity systems of mathematics classrooms. Educational Studies in Mathematics, 70(1), 49–70.
Hattie, J. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning.Thousand Oaks, CA: Corwin Mathematics.
Henningsen, M. & Stein, M. (1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning. Journal for Research in Mathematics Education, 28(5). 524- 549.
Hiebert, J. & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester, Jr., (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Charlotte, NC: Information Age Publishing.
Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38, 1008–1022.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM, National Council of Teachers of Mathematics.
Preiss, D., & Sternberg, R. J. (2010). Innovations in educational psychology: Perspectives on learning, teaching, and human development. New York: Springer Pub.
Seeley, C. (2016). Making sense of math: How to help every student become a mathematical thinker and problem solver. Alexandria, Virginia, USA: ASCD.
Sengupta-Irving, T. (2016). Doing things: Organizing for agency in mathematical learning. Journal of Mathematical Behavior, 41, 210–218.
Vygotsky, L. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.
Warshauer, H. (2015). Productive struggle in middle school mathematics classrooms. Journal of Mathematics Teacher Education, 18, 375–400.
