��������

of the National Council for Teachers of Mathematics (NCTM), president Trina Wilkerson delivered a stirring call for bringing joy into mathematics classrooms. The positive language and hopeful sentiments represent a welcome shift from concerns exclusively focused on remediating math anxiety. Introducing joy into the equation prompts us to transform our approaches to the teaching and learning of mathematics in ways that not only pre-empt pathology but also fundamentally enrich everyday life experiences. As another school year comes to a close, I urge teachers everywhere to take a moment to imagine math classrooms filled with joy, wonder and the sheer delight of doing mathematics. 

, mathematician Manil Suri suggests an urgent need to rethink what math can be.  “Each time I hear someone say, “Do the math,” I grit my teeth. Invariably a reference to something mundane like addition or multiplication, the phrase reinforces how little awareness there is about the breadth and scope of the subject, how so many people identify mathematics with just one element: arithmetic. Imagine, if you will, using, “Do the lit” as an exhortation to spell correctly. …As a mathematician, I can attest that my field is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall. (Manil Suri, “How to fall in love with math”, New York Times, 2013).

�������� professor of math education, Zack Hawes, speaks of the emotional response that mathematical epiphanies can elicit in his students.

“What immediately comes to mind is the surprise and joy that comes from having students exploring circles. We walk through three brief inquiries designed to have students explore ideas related to pi, the area of a circle, and the volume of a cylinder. These three activities typically result in many aha moments as well as joy. I think the joy comes in part from the pride associated with understanding mathematical ideas that previously, for many students, were mere abstractions – memorizing the digits of pi and the formula for the area of a circle. I think these activities help ground these concepts and, in doing so, help students feel more connected to the mathematics at hand… and hopefully a little more curious about the nature of circles,” Hawes says.

What kind of math has this power to “surprise and enthrall”? I informally asked a few friends and educators for examples of math that had brought joy to either them or their students. Their memories spanned a wide range. 

• "I was always amazed by algebra, how a written equation could be transformed into a graph, a shape. It was so beautiful, like a bit of magic."
• "I remember absolutely loving the feeling of solving quadratic equations. There was just something so satisfying in breaking an equation down like that from something very complex into simpler and simpler elements. It looked so complicated when you first encountered it, but each step made its nature clearer until finally the true relationship was revealed."
• "Introducing kids to a function machine is always a joyful experience. They start off guessing randomly, then offering more reasoned hypotheses as they get more information, until finally they see the relationship between the input and the output. It’s very exciting when they get the full pattern! And then, they challenge us to figure out their own patterns. I remember two children who had invented a “times zero” function; whatever number went into the machine, the output was always zero. They were beside themselves with glee at “tricking” the adults!"
• "When I was in grade five, we went on the playground and had to blow up a map of the world and paint the giant map on the playground, keeping the proportions. That was very memorable!"
• "The four-colour problem – do you ever need more than four colours to colour in the countries of a map so that no adjacent areas have the same colour? I couldn’t believe that you’d never need more than four colours, and was obsessed with finding a counterexample, but no luck!"

Why should children learn math? Is it really so that one dreary day in a mundane adult future they can balance a checkbook (a what? they say), calculate compound interest, double a recipe or lay a carpet? Indeed, there are endless practical applications, some of which will likely be useful or meaningful as time goes on, but it doesn’t take 14 years of slogging through increasingly arduous, opaque and disconnected exercises to learn how to do these things when you need to. Even dressed up, tedium is still tedium. In his engaging and inspiring A Mathematician’s Lament, Paul Lockhart points out that, “You don’t need to make math interesting — it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!… We don’t need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience… Algebra is not about daily life, it’s about numbers and symmetry— and this is a valid pursuit in and of itself.”

Wilkerson touches on a related idea of relevance in her message, citing a 2020 NCTM publication: “Mathematics becomes joyful when children have opportunities to learn mathematics in ways they see as relevant to their identities and communities and when they are encouraged to explore, create, and make meaning in mathematics” (NCTM 2020, p. 17).

A topic doesn’t need practical utility to be relevant to a student’s life and interests. Indeed, it may be easier to find meaning and relevance in topics that aren’t so clearly tied to the practicalities of everyday life, which vary so widely across cultures and communities. Moving beyond a purely instrumental view of school mathematics opens up new worlds for everyone. It helps to build a classroom culture of math that is grounded in the interests, pleasures and aspirations of the students as they experience the joys of thinking through intriguing problems in ways that are meaningful to them.  

Rote learning can be disconnected, brain-numbing and alienating. Yet you have only to stand on a school playground watching children shooting baskets, again and again, day after day, encountering the limits of their skill, to recognize the immense satisfaction of motivated practice. The same is true of learning to ride a bike or even sometimes – at a certain stage of development – decoding words (as the world of print opens up to a child, even contentless decoding can be pure magic). Later on, the basketball shooter, bike rider and new reader will use these skills for a greater purpose (playing a game, getting to school, getting lost in a story) but for now practice in itself is its own reward. 

We see something similar in Kindergarten children who (again, at a certain stage of their learning) beg us for “just one more” pencil-and-paper arithmetic problem. Deadly, perhaps, when you’re forced to do it ad nauseum, but context is everything and agency makes all the difference. The exercise of competence, freely chosen, and sheer accumulation of math facts can open up new horizons for conceptual enlightenment with its accompanying delights. 

Note how easily the basketball tossing child shrugs off the missed shots in their self-propelled zone of practice. They don’t need our reminders to have a growth mindset; when the task is right, it rarely occurs to a child to think otherwise. If we want to promote joy-filled classrooms, we have to trust the children and take our cues from them about what is personally meaningful. Rather than expend too much effort on developing playful contexts for the same old problems, it’s worth remembering that what counts as play will depend upon its context and meaning for the child. 

Educators and researchers have made a strong case for the power of play to foster rich, engaging math in classrooms (e.g., Ginsburg, 2008; Hassinger-Das et al., 2018; Clements & Sarama, 2017; Ramani & Siegler, 2011). Building, patterning, drawing, puzzles, games, and opportunities for narrative absorption all offer generative contexts for shaping mathematical learning that honours both the child’s and the teacher’s purposes. The concept of play encompasses a wide spectrum of options offering opportunities for agency, creativity, and experimentation. 

Joy can serve as a fruitful criterion for the many pedagogic decisions teachers face, suggests Amy Noelle Parks (2020). While the context will shape the answer to important teaching questions, “the amazing thing about choosing joy in each of these cases is that the choice also leads to richer and more productive mathematics.”

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty” (Bertrand Russell, Mysticism and Logic, 1919). There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract . . .  If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.” (P. Lockhart, A Mathematician’s Lament, 2009)

Mathematicians argue that their discipline has as much in common with the arts as with science or technology, often evaluating proofs and solutions in terms of beauty or elegance. A 2014 fMRI study conducted by Zeki and his colleagues revealed that the section of a mathematician’s brain that lit up when they judged a mathematical formula “beautiful” was the same part that responded to sensory or perceptual experiences of beauty (such as in art or nature). 

The aesthetic pleasures of math are not exclusive to elite mathematics. John Dewey and his heirs stressed the aesthetic dimensions of learning experiences which are closely tied to their meaning for learners, citing the feeling of an experience that combines “emotion, satisfaction, and understanding” (Sinclair, 2009). The aesthetics of Deweyan learning emerge in the lived relationship between a learner and what they learn, not in the instructional activities or concepts themselves. Though aesthetic learning goes well beyond what are traditionally conceived of as “the arts”, the practices and approaches of effective arts education have much to teach us as we re-imagine math classrooms as places where open-ness, imagination and creativity flourish within the bounds of mathematical structures.

Recent curricula, such as , show increased focus on the social-emotional dimensions of math learning (e.g., pp 6-8), including the significance of “math identity”(e.g., p 65), the importance of growth mindsets, and a host of other factors that include things like persistence, risk-taking, and curiosity. This shift is a welcome improvement over a view of math learning as an activity of pure reason. While schools have been slower to embrace the aesthetic dimensions of mathematical activities and processes, a paragraph on the beauty of math (p. 64) begins to open our eyes to some of the possibilities.

Donovan Schaeffer of the University of Pennsylvania, in his recently published (2022) book Wild Experiment: Feeling, science and secularism after Darwin, argues that feeling and thinking are inextricably intertwined. “Math, science, history, philosophy, and all other forms of formalized knowledge-making are scaled-up versions of a micro-level delight in the subtle click of things coming together,” he writes. This profound satisfaction in making sense of an interesting problem is aesthetic as well as cognitive. Natalie Sinclair (2004, 2009) is one of the few math educators to stress the importance of aesthetics in children’s mathematical inquiry, detailing a motivational role related to “the aesthetic responses that attract mathematicians to certain problems and even to certain fields of mathematics’’ (2009, p. 264). 

In a survey of prospective elementary teachers’ relationships with math, Chen (2015) notes the aesthetic element conveyed by many participants, including a teacher candidate who responded, “I love math! Even when it gave me a hard time, I still love it. I love numbers; they’re just so logical and just so rational. I feel like it’s so essential for everything and anything can be broken down into numbers. Math is so structured and organized, and I appreciate that…I feel a sense of satisfaction when I can make sense in mathematics… I would break it [a situation] down into numbers, symbols, and relationships.” He argues for the need to develop “a local theory of aesthetics in K-12 mathematics”.

Montessori, Waldorf and Reggio Emilia approaches intentionally set out to create harmonious physical environments, carefully selecting materials for their tactile and visual qualities. Watching the intent focus and deep pleasure that children of any age bring to constructing elaborate geometric forms, patterns and symmetries with an array of pleasing materials, it isn’t difficult to see how incorporating the visual and aesthetic offers a promising starting place for bringing joyful mathematics into school. This perspective can lead to exploring the embedded mathematics in the spectacularly beautiful geometric patterns found in cultural traditions across the globe (such as Islamic architecture).

Chen, R-J. (2017). Prospective elementary teachers’ aesthetic experience and relationships to mathematics. Journal of Mathematics Teacher Education, 20, 207-230.

Clements, D. and Sarama, J. (2017). .

Ginsburg, H. (2008). Mathematical play and playful mathematics: A guide for early education.  

Singer, D., Golinkoff, R. & Hirsh-Pasek, K. (Eds.), Play = Learning: How Play Motivates and Enhances Children’s Cognitive and Social-Emotional Growth, Oxford University Press.

Hassinger-Das, B., Zosh, J. M., Hirsh-Pasek, K., & Golinkoff, R. M. (2018). Playing to learn mathematics. In R. E. Tremblay, M. Boivin, & R. D. Peters (Eds.), A. Pyle topic ed., Encyclopedia on early childhood development. ()

Lockhart, P. (2009). A mathematician’s lament: How school cheats us out of our most fascinating and imaginative art form. New York, NY: Bellevue Literary Press.

National Council of Teachers of Mathematics (NCTM). 2020. Catalyzing Change in Early Childhood and Elementary School Mathematics: Initiating Critical Conversations. Reston, VA: NCTM.

Parks, A.N. (2020). Creating Joy in PK–Grade 2. Mathematics Classrooms. Mathematics Teacher: Learning and Teaching PK–12, 113, 61–64.

Russell, B. (1919). Mysticism and Logic and Other Essays. London: Allen and Unwin.

Schaeffer, D. (2022). Wild Experiment: Feeling, science and secularism after Darwin. Duke University Press.

Sinclair, N. (2004). The roles of the aesthetic in mathematical inquiry. Mathematical Thinking and Learning, 6, 261–284.

Sinclair, N. (2009). Aesthetics as a liberating force in mathematics education. ZDM Mathematics Education, 41, 45-60.

Zeki, S., Romaya, J.P., Benincasa, D.M.T. & Atiyah, M.F. (2014). The experience of mathematical beauty and its neural correlates. Frontiers in Human Neuroscience, 8, 68.