The idea of productive struggle is one that is widely explored in the literature on learning mathematics. One way of thinking about it is that learners may experience some degree of discomfort when they are pushing the boundaries of what they know, to venture into what they do not. This is a familiar notion: we all have firsthand experience in feeling discomfort in the context of learning anything new, whether it’s a smartphone update, a new TV remote control, or the route to an unfamiliar destination.
One very curious observation is the way most people react to other people’s discomfort with ‘learning’. Notice what others do if it looks like we can’t figure out the remote control of a new TV. We would not have to wait long before those in our company jump in, and tell us what to do; or even grab the remote control from our hands and do it for us. This phenomenon can be confirmed using a heuristic approach. Next time you enter a café or restaurant, pretend to struggle with opening the door —most likely, someone will jump in and do it for you.
It seems almost to be human nature to ‘help’ when we see others struggle. Or perhaps it’s that other people’s discomfort makes us feel uncomfortable ourselves. Whatever the reasons or motivations, it is the case that it is hard for people to watch others struggle, especially when learning something new. Not surprisingly, math teachers are no exception: it is thus natural for them to want to jump in, to aid students when they are struggling with a math task.
So what do we mean by a productive struggle, and why does it matter? Simply put, a productive struggle in the context of learning mathematics means that the learner grapples with the complexity of mathematical concepts that are within their grasp, but not yet fully developed (Hiebert et al. 1996). Most importantly, it is the exact opposite of students watching the teacher demonstrate a mathematical procedure on the board, then following with imitation and quiet practice on their own, and where the teacher signals whether an answer is right or wrong (i.e., show-and-tell). Instead, ‘struggle’ means that students must put in an effort to figure out how to approach a mathematical task. They may have some understanding and a possible point of entry, but do not yet know exactly how to proceed. Crucially, the struggle must be optimized: it does not mean to engage in needless frustration, where the learners are completely out of their depth, or the mathematics is far beyond their current stage of development (Hiebert & Grouws, 2007, 371-404).
This last point begs the question: how much struggle is productive? Although we might suggest an obvious answer: an effort is productive if it leads to learning and brings a sense of joy and feeling of accomplishment, it is wise to defer to the seminal work of Lev Vygotsky. His concept of the Zone of Proximal Development (ZPD) clarifies the optimal amount of effort that constitutes productive struggle. Vygotsky proposed that there are cognitive tasks we can do on our own, as well as others which we cannot do at all. Between these two extremes lies a cognitive space where we can perform tasks beyond our current development with the support and guidance of a more expert thought partner (Vygotsky & Cole, 1978, p.86).
In 2010, the introduction of the Common Core standards popularized the idea of productive struggle as an important feature of learning mathematics. However, the concept of productive struggle can be traced through the literature back for more than a century.
Dewey (1910, 1926, 1929) introduced as part of his theory of learning the idea that the path to understanding and making sense of complex ideas involves some productive struggle. According to him, “students learn by doing rather than by memorizing” (Dewey, 1910) —an idea that would gain even more prominence a century later, as today we aim to transform students into ‘doers of mathematics’ rather than mere answer-getters.
But it wasn’t only John Dewey; the concept of productive struggle as a theme for learning continued to resonate throughout the literature. In 1946, the phrase “to muddle through” was referenced as a metaphor for the process of developing understanding by Brownell and Sims. They claimed that the process of understanding required that the learner be allowed to struggle with schema which he or she does not yet completely understand (Brownell & Sims, 1946).
In 1957, American cognitive psychologist Leon Festinger published his theory of cognitive dissonance, claiming that contradictions in thought motivate actions: to think and find resolutions, connecting the process of making sense with struggle (Festinger, 1957). Contributing to this theme, the mathematician George Polya, repeatedly alluded to the idea of struggle as a core element of problem solving and sense making. He wrote:
“A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension, and enjoy the triumph, of discovery.” (Polya, 1957, v)
During his nearly century-long contributions to understanding how children learn, the great Jean Piaget defined understanding as the process of discovering. He wrote:
“To understand is to discover, or reconstruct by rediscovery, and such conditions must be complied with, if in the future individuals are to be formed who are capable of production and creativity and not simply repetition” (Piaget, 1975, p.20)
There is a vast body of literature supporting a strong connection between ‘understanding’ and productive struggle. However, we can also look into our own experiences to see this connection. Reflecting on our own experiences as learners of mathematics reveals what we already sense to be true about learning: that is, not only is learning one of the most difficult tasks our brain undertakes, but it also reveals how it feels. It feels like effort!
The process of ‘understanding’ (or making sense) is very separate and distinct from other activities we do, in that no one else can do it for us. Presumably, if we could not walk, others could carry us. Likewise, if we couldn’t breathe, others could assist with cardiopulmonary resuscitation (CPR). Similar arguments could be made for countless other human activities that others could (at least in theory) do for us. But when it comes to making sense, to understand, no one else can ‘understand’ for us. It follows that we, and we alone, are the ultimate agents of our own ‘understanding’ and ‘sense making’. And given that the process of ‘understanding’ involves some degree of productive struggle, then denying us such struggle would be equivalent to denying us the full opportunity to ‘understand’.
What parent would not want (in theory) to be operating a bulldozer in front of their children, leveling the ground so that they never stumble or fall? But we know more deeply that that is not how they will learn to cope with life, and likewise it is not how we learn mathematics either. Instead of operating a bulldozer that removes all obstacles in front of us, a good math teacher acts more like a coach who accompanies us to an obstacle course, and gives us practice in overcoming such obstacles as we will inevitably face. Such a teacher should embrace, and learn how to facilitate, productive struggle by students.
What curricular materials best promote productive struggle? Clearly not all of them do so: those that are open-ended, interesting, and provide the right level of challenge are better positioned to succeed. Specifically, examining which features of teaching and curriculum best support productive struggle is well worth the attention of school communities.
If you’d like to learn more about how productive struggle could be incorporated into your math curriculum, please contact us here.
References
Brownell, W. A., & Sims, V. M. (1946). The Nature of Understanding (Vol. 47). Teachers College Records. https://doi.org/10.1177/016146814604700903
Dewey, J. (1910). How We Think. D.C. Heath & Co.
Festinger, L. (1957). A Theory of Cognitive Dissonance. Evanston, Illinois: Row, Peterson. https://doi.org/10.1177/001316445801800424
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics. Educational Researcher, 25(4), 12-21. https://doi.org/10.2307/1176776
Hiebert, J., & Grouws, D. A. (2007). The Effects of Classroom Mathematics Teaching on Students’ Learning. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics (pp. 371-404). Information Age Pub.
National Council of Teachers of Mathematics (Ed.). (2014). Principles to Actions: Ensuring Mathematical Success for All. NCTM, National Council of Teachers of Mathematics.
Piaget, J. (1975). To Understand is to Invent. Viking Press.
Polya, G. (1957). How to Solve It (2nd ed.). A Doubleday Anchor Book.Vygotsky, L. S., & Cole, M. (1978). Mind in Society: Development of Higher Psychological Processes (M. Cole, Ed.; M. Cole, Trans.). Harvard University Press.