Advanced Math Is Child’s Play: An Interview With Maria Droujkova and Yelena McManaman

math circles, math play,
Increase the number of sides in a polygon to get “almost circles,” the idea of limits, and see how the ancients obtained the value of Pi. (

One autumn afternoon, the kids who normally rush inside eager to participate in math circle activities with Maria Droujkova lingered outdoors instead. She discovered them sitting in a large pile of leaves under an oak tree. There the 5- to 7-year-olds were speculating how many leaves were on the ground. Counting them one by one proved futile. So Droujkova helped the children pile leaves into groups of ten, then measure out 100 piles of 10,  fitting them into a small box. Filling that box ten times and then emptying the leaves in a pot gave them approximately 10,000. Ten of those pots filled with leaves fit into a recycling container, for an approximate count of 100,000 leaves. After the kids filled the recycling container 10 times (handily emptying it into a compost pile) they could reasonably estimate that about a million oak leaves had been on the ground.

Counting a million leaves. (pixabay)
Counting a million leaves. (pixabay)

Droujkova says the kids were expansive throughout, full of questions and theories, and “kept that first charge of joy from the sun and the leaves for the whole hour.” They never did get back indoors to take part in the activity she’d planned.

Current research shows that preschoolers naturally grasp principles of basic algebra.  That doesn’t mean formal instruction is a good idea for this age group. In fact Droujkova, an innovative math educator, is an expert in building natural mathematical understanding from babyhood up through hands-on and open-ended activities.

The collaborative site she founded, Natural Math is dedicated to sharing math-rich, play-based endeavors. It offers a Q&A hub, webinars, and mini-courses. Now she’s developing a series of project-based books for families and math circles. The first title is Moebius Noodles: Adventures in Math for the Playground Crowd. Like the resources on, the book’s content is Creative Commons, and the PDF is priced “from zero to infinity.” (Here’s a review of the book.)

We wanted to hear more. We’re delighted to bring you our paradigm-shifting interview with Droujkova and her Moebius Noodles co-author Yelena McManaman.

Advanced math through play. (
Advanced math through play. (

GeekMom: At the start of Moebius Noodles, math is described as a exciting and enticingly exotic adventure that’s too often simplified into rote busy work. “It is as tragic as if parents were to read nothing but the alphabet to children, until they are ‘ready’ for something more complex. Or if kids had to learn ‘The Itsy-Bitsy Spider’ by heart before being allowed to listen to any more involved music.”  Tell us more about natural math.

Maria Droujkova and Yelena McManaman: Natural Math is about people making mathematics their own, by posing their own problems, pursuing their own projects, and remixing other people’s activities in personally meaningful ways. We believe that “learning math” means two things—developing mathematical state of mind and acquiring mathematical skills. The question of how to mix skills and concepts in learning programs is very complex, and the debates are hot among researchers, parents, and curriculum developers. The Natural Math path integrates the two in the following ways.

Within each context of mathematics, we start with open free play, with inspiring prompts and ideas that gently help children make patterns and rules. This is the stage where concepts are born, grounded in embodied experiences. When kids doodle fractal hands or stick their noses inside mirror books to peek into kaleidoscope wonderlands, they are playing freely at first. Then children begin to notice, tweak, remix mathematical patterns, and we help them formulate and name their math. Fractals have levels, and the number of objects at the third level is traditionally called “the third power”—but kids often name these tiny objects “grandchildren” of the first-level object. At this stage of “patterning” children hone their skills, because they need more precision and structure to carry on the patterns. You could ask a kid at this stage to show you 3 x 4 with the mirror book (possibly using kid’s own terms), and you’ll see mirrors at the 90-degree angle with 3 action figures inside. The infinite road to mathematical mastery is in comparing, contrasting, and organizing these mathematical patterns, and building structures out of patterns. For example, could you connect fractal with mirror book patterns? You can, if you used two mirror books in front of one another to introduce scale into reflections.

fractal play, math play,
Fractals with mirror books. (

GM: Moebius Noodles is a downright alien concept when parents feel pressured to push math on even the smallest kids via apps, educational toys, and academic preschools. Please give us an example (or two, or three) of advanced yet playful math for kids birth to five.

MD and YM: Most parents we talk to, including the ones who work in STEM fields, tell us that their math education wasn’t satisfying. They want their kids to have something better: to see mathematics as beautiful, meaningful, and useful, and not to suffer from math anxiety and defeat. The two major ways the markets respond to these worries and dreams are via edutainment toys and games, and private early teaching in academic settings.

We suggest a different approach, centered on families and communities. We introduce advanced math through free play. Formal academic environments or skill-training software can’t support free play, but friends and family can. Mathematics is about noticing patterns and making rules that describe and predict these patterns. Observe children playing in a sandbox. At first it doesn’t look meaningful. But in a little while kids make up elaborate stories, develop a set of rules, and plan for what’s going to happen next. In a sense, what we do with math is setting up sandboxes where particular types of mathematical play can grow and emerge.

The mirror book activity [found in Moebius Noodles] is an example of emergent play. Set it up on a table or on the floor and watch what happens. It might start in a very chaotic way, for example, your kid may pile up lots of toys between the mirrors and walk away. But keep the mirror book available, and you will soon notice more purposeful, diverse and complex investigations. Kids come up with symmetric mosaic patterns, intricate designs that involve drawings and reflections, magic tricks, puzzles for other kids, and many ways to show infinity, such as scaling-down of mirrors within mirrors within mirrors. Kids start with free and open play, then develop a set of rules that describe reflections and rotations, or predict results of these transformations. And symmetry is one of those fundamental math concepts that appears over and over again in all areas of math, such as the binomial formulas in algebra or crystal groups in 3D geometry.

When we talk to parents, one of the biggest problems they mention is the inability to see math, beyond quantities and basic shapes. That’s why many of the activities we offer encourage children and adults to build collections, to take photographs and to go on scavenger hunts and math treks. As children build up their math collections, they start noticing patterns and relationships between objects. The adults, who collect and notice patterns alongside kids, often are surprised and delighted that these connect to the formal rules they may have studied in school or college. For example, we often hunt for fractals. They are self-similar structures: tree branches or delta streams or lung air passages that split and split and split again, reflections within reflections within reflections, or boxes within boxes within other boxes. Once you figure out fractals, you’ll realize they are everywhere, because it’s such a smart method of structuring space.

Fractals in nature. (
Fractals in nature. (

GM: On, you write about a community of people sharing naturally math-rich and meaningful activities for children from babyhood on. We’d love to hear about math circles and what you mean by math communities.

MD and YM: It takes friendly local people to support mathematical free play: to provide inspiring prompts, to get the action going, and to know when to stand aside and let kids explore on their own. Making, collecting, and remixing patterns depends on other pattern-drafters even more. Parents and teachers need to meet like-minded people to share ideas and encouragement. That brings us to math playdates and math circles.

There are quite a few math circles for middle and high school students, for example, in the National Association of Math Circles. (Maria is a board member.) It’s harder to find math circles for younger kids, or toddler and parent playgroups. Each circle develops its own flavor, and its own lore—the little patterns of play, sayings, and favorite activities. Some of these treasures have to stay local and intimate, but we believe the ideas, experiences, questions and answers could be shared more broadly. We are working on a project called “1001 circles” that will help everyone involved in circles for young children to connect and to share.

Function machines and Lego graphs. (
Function machines and Lego graphs. (

GM: Tell us about the Creative Commons nature of  Moebius Noodles.

MD and YM: We need this openness, because families, math circles, and other groups in our community are very diverse. Some use the activities as is, but the point is to change, remix, translate, and modify everything to better fit each unique situation.

Storytelling and pretend-play are modifications almost everyone uses. We believe in compelling reasons behind each math activity, but what story is compelling depends on the child. Parents and caregivers change settings and characters: a function machine can be used to magically grow and shrink heroes in a fairy tale, or it can provide enough feed for animals of different sizes at a zoo, or it can fuel starships in a sci-fi setting.

Another modification is about tools and media. Our original activity might call for painting, but kids who don’t like to paint can use clay, or building blocks, or flower arrangements. We try to give specific hints for different media, for example that a symmetry activity requires a lot of folds, so you are better off with thin paper. But we want everyone to experiment on their own, like in this large crowd-sourced collection of multiplication towers.

Of course, the most substantial way to change an activity is to go for its rules. One of our most popular activities is based on a mirror book: two small mirrors taped together. You can use it to explore rotational symmetry, reflections, and multiplication. But what if you change what the mirror book is? Kids set up several “books” at different distances and angles. They place mirrors over or under the mirror book to build mirror chambers and kaleidoscopes. They move mirrors and objects to produce animations…

Grown-ups send us photos and stories of such modifications or original designs via blog comments or the Q&A forum or by email. We aggregate neat ideas into collections, for example, about toddlers and young kids who draw grids, or publish them as separate guest blog posts. Another important contribution is questions, and sometimes we call for thematic questions as well, such as this collection about multiplication.

One excellent way to get involved is to help translate our book into other languages. Contact us if you want to join crowd-translating efforts! Russian and Persian translations are finished, Hebrew and Turkish are on the way, and the teams are starting on several more languages.

Makers of all kinds contribute their creations. People send us beautiful photos of math projects they made, and art pieces in physical or digital media. Writers celebrate Math Storytelling Day by making new tales. Software developers make apps, mini-games, and interactive toys. We are working on better ways of sharing, curating, and aggregating that math renaissance under the umbrella project “1001 Circles.”

Learning together. (
Learning together. (

GM: All sorts of projects are in the works through the community incubator, Delta Stream Media, where teams of authors develop books with crowd-sourced input. Tell us more about the books and when we can anticipate reading/using/sharing them.

MD and YM: We developed Delta Stream idea at first as the support mechanism for producing Moebius Noodles. It boosted the book’s quality, and was a source of morale to us, so we kept it going to help other authors with their projects. The idea is to grow books in the nurturing ecosystem of people who care. Two to three coauthors make the first draft, bouncing ideas and brainstorming during this intense and private stage of work. Then a few more like-minded colleagues, who work on similar ideas themselves, join as advisors and reviewers. With their feedback, the draft is ready for “beta reader circle”—a more open field test of activities from the book by parents and teachers, sometimes combined with crowd-funding. More revisions, more discussions with other writers in the Delta Stream Guild—and the book is ready to go out to everyone. We see publishing as a gradual, participatory, ongoing process of making ideas more and more accessible to wider and wider public.

These projects are close to being available, going through copy-editing, illustrations, and layout:

Playing with Math , a forty author anthology edited by Sue VanHattum

Problem-Solving for the Young, the Very Young, and the Young at Heart by James Tanton, Maria Droujkova, and Yelena McManaman

Bright, Brave, Open Minds by Julia Brodsky (insight problems for young kids)

These projects are almost ready for their beta reader circles:

Socks Are Like Pants, Cats Are Like Dogs by Malke Rosenfeld and Gordon Hamilton (art activities and puzzles about variables and attributes)

CampLogic by Mark Saul and Sian Zelbo

These projects are in draft stages, so these are working titles:

SubQuan by Cooper Patterson and Rebecca Lynne Patterson

Stick Math Circle by Rodi Steinig and Rachel Steinig

Inspired by Calculus by Maria Droujkova, Yelena McManaman and Kalid Azad

We are very excited about meaningful and joyful mathematics in these books. Here’s to more math adventures!

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Laura is the author of a poetry collection titled Tending and Free Range Learning, a handbook of natural learning. She lives on a small farm notable only for its lovestruck goose.