# Prime Play on a Prime Day 3/11/43×473 and 11 are prime, and while we are not in a prime year, 2021 is the product of two consecutive primes.

When Dominique approached me with a 3D model of the proposal for Mathemalchemy at JMM 2020, I was delighted to jump on board at the chance to weave together stories, mathematics, and art. One pandemic year later, this collaboration has been a source of much light for me during many bleak hours.

As we designed and explored the vignettes of the installation (and epic the final creation will be!), the process was filled with a richness of creating, learning, and playing, all of which embody the key reasons attracting me to dedicate much of my time engaging in mathematical thinking, research, and teaching. The Chipmunks Sorting Primes Vignette in a way expresses my path in mathematics which went from a blind acceptance of facts – here’s a formula, plug and chug, and it will work – to understanding that mathematics is a human endeavor, one where we can create the rules and see how it evolves. For me mathematics went from being cold and austere to being exuberant and pliable (I am a topologist). This message is conveyed in the evolving Chipmunks Sorting Primes story. Enjoy!

## Chipmunks Sorting Primes Vignette

In our Mathemalchemy fantastical world of mathematical delights, tucked into a cozy nook under the grand sheets of sophisticated mathematical depictions, our young chipmunks engage in play, sorting primes on a grid/graph paper/papyrus/ (we’re still prototyping) playground using acorns to explore factors, while holding clay tablets with Babylonian cuneiform numerals.

A trajectory of exploring prime numbers is a fundamental theme in our garden (see Li-Mei’s and Samantha’s blog posts), amongst other deep themes such as hyperbolic planes and fractals. Chipmunks Sorting Primes is the beginning of exploration. Therefore, we want the story to have a playful theme, where our young characters build a solid foundation for mathematical progress, cultivating a positive disposition in our young learners and in you, the viewer. Determining whether a positive integer is prime or composite, while listing its proper divisors, is an early mathematical activity in which young learners engage.

### Take 1

#### Mice on a mission

In this representation of the scene, the young learners are mice, drawing rectangular arrays using acorn depictions to determine the proper factors of 12 and 13 respectively. When proper factors are found, the corresponding sheet is given a green check. When the number does not factor, the residues are circled in red and the sheet is given a red x. There is a smile on the mouse holding 12, but her less fortunate peer is shedding a tear.

What message do you see? What messages are the mice taking away? What learning is happening here?

In our group discussions, we loved the acorns and the task of determining whether a number is composite or prime, but we felt there were few messages we should refine:

• Finding a prime number should be a source of pride, not a tear inducer.
• The x vs was reminiscent of math educational anxiety.
• Young learners should be engaged in physical play with three-dimensional, tangible objects.
• Counting and factoring are one of the oldest mathematical explorations: as a history of mathematics enthusiast, the story should have historical context.

After discussing and refining our message, Bronna brilliantly drew an updated vision:

### Take 2

#### Cheerful chipmunks enter the scene

We converted our mice to chipmunks because they are adorable and we want more critters to join our fun. Gone is the assignment format of writing rectangular arrays on vertical sheets of paper. Instead we have ground play with physical 3D acorns.

We created a top-down line of vision which is ultimately more suitable for our viewers. The chipmunks are clutching clay tablets with Babylonian cuneiform numerals– a salute to the over four millennia of the recorded numerals. The 12 and 13 Hindu-Arabic numerals became ubiquitous only as recently as the seventeenth century, and we wanted something that dates further back in the annals of time. The sorted tablets placed in heptagonal baskets as heptagons are popping up in many places of our installation (how many can you find?). We replaced x and with a yellow highlighted tile to accentuate the non-trivial residues. Most importantly, our chipmunk 13 is standing tall, proud to have demonstrated that 13 is a prime number! Check out our adorable and euphoric chipmunks crafted by Liz.

When she sent me this photo, Liz wrote :

What message do you see? What messages are the chipmunks taking away? What learning is happening here?

## Changing the rules… or not?

Now that we’ve turned the mice’s task into the chipmunks’ game, we need to delineate the rules of the game.

The chipmunks have limited tools. They have cuneiform tablets which they seem to be able to understand. They have acorns to count the factors of the tablets’ numerical equivalents. They understand the abstraction and one to one correspondence of numerical representation to the corresponding number of items. They have an organizing grid (Tasha and Dominique have been awesomely prototyping; my attempt at papyrus and ink was not a success). In this simple interpretation of the scene, the rules of the game are simple: select a cuneiform tablet and show whether the number is prime or composite by brute force.

What happens, though, if we engage with the chipmunks in dialogue and change the rules of this game? What if we teach them or they discover the primality test: if $p$ doesn’t have any divisors $\leq \sqrt{p}$ other than 1, then $p$ is a prime number? What if we teach them the commutative property: $a \times b=b \times a$? What happens then? By giving them more knowledge, we inevitably change the rules of their game.

In giving our chipmunks knowledge, we empower them to win. They can accomplish their task with less time and effort. At the same time, giving them more knowledge is also inevitably exposing them to a loss of innocence. Their cherubic smugness in their ability to count out the acorns until they arrive at their solution is interrupted. They can now arrive at the same conclusion with less than half the acorn slabs. Their play is cut short.

On the other side of their loss of innocence, though, is the fantastical world of human play. In the human mind, abstraction in some way disconnects us from the physical play of the chipmunks’ universe but also opens us up to a playing field of infinite dimensions. With our rules the chipmunks don’t have to stop when their heptagonal containers of integers run out. Their play can continue until they want to move on.

In some ways, these chipmunks are looking at us, their creators, with the thrill of ironic jest. They’re telling us that as fantastical as our theoretical universe is, the satisfaction of their primary play is not lost. No matter how sophisticated our mathematics become, we are still innately pleased with the counting of physical objects as a means of solving problems. Just as the prime numbers are the building blocks of the positive integers, physical play is the birth of mathematics, no matter how formal and abstract this playing field can become.