By Keith Devlin [Twitter @profkeithdevlin; Mastadon https://fediscience.org/@KeithDevlin]

Google “playful math” and the world’s most famous mathematical algorithm will return over 30,000 hits. Many of them take you to websites or blogs created by teachers, home-schooling-parents, and pre-schooling providers. Whatever “Playful Math” is, a lot of parents and at least some teachers are all for it.

In addition, a few links lead to scholarly articles on the role of play in Kindergarten and K-12 mathematics education, particularly the earlier years. Noticing that “playful math” is a thing, education researchers have observed it, studied it, and written professional blogs and even scholarly papers about it.

Browsing through a few such that I selected at random, which describe playful math activities with groups of children (for example, this one coming from UCLA’s Graduate School of Education and Information Studies), what struck me were strong similarities between kids engaged in playful learning and (from my own observations and experiences) a small group of professional mathematicians working together on a problem, particularly a relatively new problem, where no one is really sure how best to approach it.

The similarities with my own experiences were strongest for my early days as a graduate student, where I was one of maybe seven or eight other fledgling researchers (including a couple of postdocs) all working in the same general area of mathematics. I suspect the similarities are more striking for that period of my mathematical journey because it is at graduate school that I (like most professional mathematicians) first came up against (and sometimes formulated) new questions that no one knew how to solve.

What got me (everyone?) into graduate school was becoming good at learning, practicing, and performing tricks that others had developed before me. At high school and then as an undergraduate, I was on a much-traveled, linear track leading towards a well-defined goal. Then suddenly, I was in graduate school and there was no clear path ahead of me, save the goal of “do something original” (within three years, in the UK system).

“Playing with the new ideas” was the only option. Try this, try that, fail, start again – that became the daily routine. I was back in the pre-school sandbox. In place of parents or teachers to provide encouragement, advice, feedback, and instruction, there was the occasional “Maybe try this …” directional guidance of my research advisor, whose role was not so much to teach as to guide and follow along as a more experienced practitioner. Looking back, most of the progress I made came from group play with my fellow graduate students. And “play” is the right word, including a fair amount of goofing off.

Since other research mathematicians will surely have had a similar experience as doctoral students, I suspect that when they too observe, or read accounts of, “playful math” activities, they will likewise recognize their educational value. Particularly when it comes to starting out on a new, unfamiliar problem.

Of course, graduate students – and professional research mathematicians (“graduate students who get paid more” when actually engaged in their research, and not teaching or engaged in administrative tasks) – already know a fair amount of mathematics and have made a decision to pursue the subject. So, they are able to keep the “play” from veering too far off course for too long. They love the subject, they always have an immediate goal in mind (solve the problem at hand, or at least make recognizable progress), and they are motivated to complete their doctorate and enter the world of professional mathematics research. (To do more of the same.)

For early learners, the situation is different. They do not yet know where they are heading; everything is new. They do, however, have a seemingly endless amount of curiosity – about almost anything and everything. We are born that way. So too are other living creatures, or at least those that move around on legs. As the legendary children’s television presenter Fred Rogers (“Mister Rogers”) correctly observed, “play is the work of childhood.” And we see the same in the puppies and kittens we bring into our homes.

With suitable planning, a good teacher can utilize children’s play to achieve good learning. But it can look very different from a traditional, instruction-based math classroom. In the UCLA article cited earlier, the authors write:

“Over the past several years, we have been fortunate enough to observe and learn from several Los Angeles teachers and students. Throughout these experiences, we witnessed and appreciated how learning is almost never a linear experience; it is messy and can be a curiously challenging process before we understand something in the world. This is one of the many reasons why we believe play should never be pitted against work. Play can help students make sense of the messiness that comes with learning new things. It can provide students with a space to engage in creative and inventive ways, including opportunities to explore, apply, share, and reflect on misconceptions and problem-solving strategies.”

Since play is how young children learn, and because some of us continue to “play math” and enjoy math throughout our lives, particularly those of us who read and write articles published by organizations like the MAA, it’s tempting to try to utilize mathematical play as a pedagogic tool in Kindergarten through at least K-8 school education. But how do we do so?

Answering that question is particularly challenging in cultures like ours, that are organized in a way that requires young people to “progress” through a stipulated sequence of “learning gates” in order to “graduate” to take their place – and survive – in society. (I say “cultures like ours”, but that’s pretty well all of the developed world.) Challenging, but not, it seems, impossible. The secret is to find a way to thread the needle between the systemic need to follow a curriculum and the inescapable fact that, by its very nature, learning-play, as play, has to be open-ended and driven by the learner.

And already I need to add a caveat: all play results in learning. That’s its purpose (as an instinctive behavior), courtesy of evolution by natural selection. In writing “learning-play” I was focusing on the use of play as a pedagogic tool to aid learning a particular set of skills that society has deemed important (for life in that society); in the case at hand, mathematical skills.

So how do you get a class of early learners to engage in play that leads to a mathematical goal? One way is to give the class an open-ended mathematical task and then set them loose to see how they do. There are two things to bear in mind. First, pick a task that is doable and allows for a lot of freedom to explore. Second, monitor the students’ progress (“activities” is a less loaded term), but avoid stepping in with any kind of instruction. The moment you start to teach, it stops being play. Most teachers and parents (myself included) find this difficult; we instinctively want to help. Far better though, is to give the students time to come up with various ideas and approaches of their own. Let them explore the task, i.e., play with it.

If a student gets into difficulty where you can see there is little possibility of going further, ask the rest of the class to try to help. This can lead to some very productive exchanges. More to the point, it leaves the students in charge of their play. The story of math teacher Nancy and her 5th Grade class recounted in the UCLA article now cited a third time provides an excellent example of how this can turn out.

The point is, provided the teacher has sufficient deep understanding of the concepts, and selects a doable mathematical task that has real mathematical content, minimal guidance can help the students make valuable discoveries that result in understanding. As the authors of the article observe at the end:

“[S]tudents displayed knowledge of problem solving, division, place value, money, cultural knowledge, and even fractions … in their own meaningful ways. Learning is a process and making room for these messy and playful interactions as a valid and valuable part of school interactions can cultivate a love for mathematical thinking that leads students to be active in learning and doing math. Math is beautiful, artistic, and varied, and should not be a rigid set of rules for students to follow. Math is a subject that should be fun and playful because when you play with math you create conceptual understanding that is yours to keep.”

To which I would add that experiencing mathematics as a creative, explorative, and often messy, way of thinking that often intermingles with other walks of life and other ways of thinking, is at least as valuable as acquiring rote mastery of formal procedures for computation (numeric or symbolic); and, I would argue, in the early years is more valuable.

Of course, there is a need to master those formal procedures (at least to the point of understanding – actual application of those procedures in any real-world situation is today done by technology). How can playful math learning occur in an educational context where a curriculum must be adhered to?

This is the question addressed by my former Dean of Graduate Education at Stanford, Deborah Stipek, in a 2017 article titled Playful Math Instruction in the Context of Standards and Accountability. As she explains, given sufficient planning by a skillful, knowledgeable teacher, playful math activities like the one described in the UCLA article can be weaved into a standards-aligned course.

As she says in her Conclusion:

“Standards and accountability have value, but we must make sure they do not get in the way of child-centered, developmentally appropriate, playful learning. The kind of teaching described here requires teachers to be intentional, to plan lessons carefully, and to be somewhat directive—at least for some math activities.”

To my mind, the more these kinds of playful mathematical endeavors can be brought into the early-learning math classroom the better. But there is no denying it requires a compromise; the very term “playful math instruction” used in the article’s title is oxymoronic, since the “context of standards and accountability” that makes up the rest of the title requires the teacher provides a lot of fairly traditional (i.e., non-playful) classroom instruction.

Where playful math can really come into its own is the area of informal learning. Many of the web resources you get when you google “playful math learning” are aimed at that audience. Free of the necessity of following a curriculum, providers of playful math resources can go full-out on the play aspect. Resources and activities are chosen not because of their mathematical content (though they necessarily have math content) but the degree to which they can be presented as pure play. In general, that means in particular no instruction and no assessment or evaluation. The focus is on the activity, and the progress kids make on it, not a resulting product.

Which brings me to Christopher Danielson, a prolific creator of playful math artifacts and activities, author of several excellent math books for young children, developer of math learning resources for the online educational platform Desmos, and the author of the excellent parents (and teachers) blog Talking Math With Your Kids.

Starting in 2015, Danielson has designed and mounted an interactive play exhibit at the huge Minnesota State Fair, called “Math On-A-Stick”, the name being a nod to the kind of eat-as-you-walk food offering that is a staple of American state fairs, and such a part of American culture that politicians regularly attend and make sure the press take photographs of them consuming a corn dog or similar snack on-a-stick.

Apart from 2020, when the fair was cancelled due to the COVID-19 pandemic, Danielson has put on Math On-A-Stick every year since, so this year was the seventh iteration.

In a blog post to promote this year’s MOAS, he described the exhibit this way: “The Minnesota State Fair has everything on a stick … even math! In this exhibit math comes to life in a photo scavenger hunt called The Number Game. Children and caregivers count their way through the fair. You’ll make shapes and patterns, look for similarities and differences, and meet visiting math artists sharing their beautiful, mathematical creations and helping you to make your own. Activities are brought to you by the Minnesota State Fair Foundation.”

To play the  Number Game, visitors pick up a card from any State Fair Information Booth (or download one in advance), and then, as they explore the fairgrounds, the are asked to notice sets of things: a corn dog has 1 stick, a cow has 4 legs, the Ferris Wheel has 20 carts, and so on. They keep track of the sets they find on their game card. They can also take pictures and share them with the appropriate #numbersatthefair hashtag on Facebook, Instagram, or Twitter. Before they leave, they can take their completed card to the MOAS exhibit to claim their prize.

In addition to the Number Game, the exhibit has Activity Tables, where visitors can have hands-on shape, number and patterning fun by playing together. There are colorful rectangular tiles, tiles shaped like turtles and lizards, and colorful eggs to arrange in trays. They can play with Pattern Machines or hop onto the Stepping Stones. See the images below to get a flavor of the exhibit.

Danielson emphasizes that there are no instructions to be followed. The idea is that the children simply start playing. The trick is to design the artifact or the activity in ways that make particular activities more likely. The tiling turtles, for example. They are two colors, and that encourages using those colors for patterning — but usually not until they’ve worked out some basic ideas about how they go together (e.g. nose to nose and butt to butt). So, patterning with color is a process he designed for. But as he points out, that comes out in lots of different ways, not only as a checkerboard.

In a blog post, Danielson describes how children interact with the colored eggs. When children come to the egg table at Math On-A-Stick, they know right away what to do. There are plastic eggs, and there are large empty egg cartons. The eggs go in the cartons. No one needs to give them instructions.

A typical three- or four-year old will fill the cartons haphazardly, unconcerned about the order of filling, or the colors selected, or anything else. It’s just put eggs into the carton one at a time in a seemingly random order. But when that child plays a second or third time, emptying and filling the egg carton—without being told to do so—they usually begin to see new possibilities. After five or ten minutes of playing eggs, they are filling the carton in rows or columns. Or perhaps making patterns such as pink-yellow, pink-yellow, … Or counting the eggs as they put them in the carton. Or orienting all of the eggs pointy side up.

The longer the child plays, the richer the mathematical activity they engage in. This occurs because the materials themselves have math built into them. The rows and columns of the egg crate; the colors and shape of the eggs; the fact that the eggs can separate into halves—all of these are mathematical features that kids notice and begin to play with as they spend time at the table. Danielson reports that he’s seen four-year-old’s spend an hour playing with the eggs.

He says he’s also observed that the children who receive the least instruction from parents, volunteers, or from him, are the most likely to persist. Those are the children who will spend 20 minutes or more exploring the possibilities in the eggs.

The children who receive instructions from adults are least likely to persist. When a parent or volunteer says, “Make a pattern,” kids are likely to do one of two things: (1) make a pattern, quit, and move to something else, or (2) stop playing without making a pattern

As Danielson observes, adults have a responsibility to let the children play. We can be there to listen to their ideas as they do. We can play in parallel by getting our own egg cartons out and filling these cartons with our own ideas. But when we tell kids to “make a pattern” or “use the colors”, we are asking the children to fill that carton with our ideas, rather than allowing them to explore their own. On-A-Stick food for thought, no?

Finally, I’ll end with a quiz. (This would NOT occur at Math On-A-Stick!) MOAS has a puzzle table for the youngest fairgoers. The table is low to the ground, near an entrance to the space, and filled with colorful, chunky shapes—durable and sized for chunky toddler hands. The image below shows three typical states of the puzzle table during the day: neat and tidy, stacked and tidy, stacked and messy.

Which one of these generates the most engagement? (In terms of percentage of children who play with them and in terms of the average length of time they play.) What do you think?

After many hours of observation, Danielson knows.

Stacked and messy wins the day. Stacking a few puzzles makes space for children to maneuver the blocks, rearrange them, and to play with them in novel ways. Also, stacked gives the impression of plenty. But beyond that, messy is an invitation to play, to be creative. Neat and tidy means it’s someone else’s creation. To play with it, you have to start out by breaking/destroying their game/work. But with messy, there’s nothing to mess up. Messy is essential for play.

For the same reason, in the turtle game, Danielson and his team deliberately break up a big turtle tiling after the children who made it leave (after first making sure they didn’t just go off to find Mom or Dad to show off their work!)

PLACES TO LOOK NEXT: If you want to find out more about Danielson’s work on playful math and some of the pedagogic thinking behind it, his post about the 2022 Fair has a number of references at the end – together with links to many more photos of kids at Math On-A-Stick. For research into playful learning, he recommended taking a look at the work of Ilana Horn’s Playful Math Learning group, which he says has been doing some really interesting stuff, including conducting research at MOAS. (See her description of her project on the 2016 Math On-A-Stick exhibit.) He also provided me with two book references to pass on: Exploring Mathematics through Play in the Early Childhood Classroom by Amy Parks (more practitioner focused than theoretical), and Joyful Math by Deanna McLennan. Enjoy!