One autumn afternoon, the kids who normally rush inside 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 Maria 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.
Maria 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.
Maria is an innovative math educator. She is an expert in building natural mathematical understanding from the earliest years on up through hands-on, open-ended activities. The collaborative site she founded, Natural Math, is dedicated to sharing play-based, deep-inquiry math endeavors through all sorts of resources that empower parents, teachers, and kids to make their own mathematics. Foremost is a series of project-based books for families and math circles. The first title is Moebius Noodles: Adventurous Math for the Playground Crowd, aimed at children from toddlerhood to five years old. The book’s delights include robot commands and mirror books.
Now with eight titles published and more in progress, the newest book offered is Funville Adventures by A.O Fradkin and A.B. Bishop. Funville Adventures takes readers along with nine-year-old Emmy and her five-year-old brother Leo to a magical place where beings have the power to transform objects. One never knows when something will be shrunk, copied, erased, even turned into an elephant. The sibling have fun creatively solving problems and learning a thing or two about themselves in the process. The book seems like a fairy tale, yet the powers of the Funvillians are a vehicle for introducing children to the concept of functions. Each power corresponds to a transformation such as doubling in size, rotating, copying, or changing color. The authors bring their own math “powers” to the story. Here’s a little about the co-authors.
Dr. Sasha Fradkin has loved math from an early age, and seeks to share that love of math with others. After receiving her PhD in mathematics from Princeton University, she worked for several years as a professional mathematician and taught enrichment math at the Golden Key Russian School to children ages 4-10. Last year, Sasha became the Head of Math at the Main Line Classical Academy, an elementary school in Bryn Mawr, PA. She develops their math curriculum and teaches children in grades K-5. She writes a blog about her teaching as well as various math adventures with her two daughters, and enjoys pondering about exciting and engaging ways to present the beauty of mathematics to young children.
Dr. Allison Bishop grew up with a passion for writing, and initially disliked math because it was presented as formulaic. She belatedly discovered the creative side of mathematics and science, and now sees it as a vital component of the curiosity that drives her life. She is currently a professor of computer science at Columbia University as well as a quantitative researcher at the Investors Exchange. She remains an avid fiction enthusiast and writer, and is always seeking new ways to expose young minds to creative mathematical thinking and fuel their scientific curiosity.
The paradigm in math education is shifting.
Let’s find out more in an interview with Sasha, Allison, and Maria.
Laura: Can you tell us a bit of your own story and what led you to this work?
Maria: My story keeps changing. Growth requires better stories, right? It used to be about me, a little girl from a little Ukrainian town who wanted to be a scientist like the cool sci-fi characters, when she grows up. Now I am also a parent, a teacher, and a community organizer, and my story is about many people. It is a story about people’s access to real math and science.
I work on helping my young friends and their adults be mathematicians – not when they grow up, but here, now, in their own ways. Let’s say we make functions and functionals into fantastic creatures that five-year-olds find friendly enough. That’s what Funville Adventures is all about. What other groups of people now gain access to this abstract algebra? Maybe math-phobic adults, or those working in their second language, or people with learning disabilities? Maybe tired people who work long hours and only have a bit of time late at night? That dream of radical access to math is what’s guiding my projects.
Sasha: Growing up, I loved the math puzzles that my dad shared with me but found most of my math classes in school dry and repetitive. I was determined to share the exciting and creative side of math with my children and their friends from an early age. My older daughter, who loves turning everything into a story, inspired me to think about presenting math through storytelling and that is how the idea for Funville Adventures was born.
Allison: As a young student, I loved creative writing and hated math because it seemed too formulaic. I want to help kids discover the creative side of mathematics and science at an earlier age than I did.
Laura: Let’s start with Moebius Noodles. In the introduction, math is described as an 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: 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.
Laura: Maria, you were featured in a popular article in The Atlantic titled “5-Year-Olds Can learn Calculus.” In it you explain that math instruction traditionally follows a hierarchical progression that, as you say, “Has nothing to do with how people think, how children grow and learn, or how mathematics is built.” You point out that the standard curriculum starts out with arithmetic which is actually more difficult for children than play-based activities based on more advanced fields of mathematics. You’re quoted as saying, “Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture.” How do books like Funville Adventures approach math differently?
Maria: Stories, pretend-play, and imagination! These are keys to growth. Let’s hear more from Funville authors.
Sasha and Allison: In Funville, kids will encounter math under the surface of an engaging story, which will naturally appeal to some kids who might not connect with the more traditional way that mathematics is often taught. Readers will see examples of problem-solving throughout the narrative, and will have plenty of material as a jumping off point to invent their own characters and stories. Since many kids love coming up with stories already, linking mathematical functions to “powers” that characters can have presents them with a new opportunity to interact with math through storytelling.
Laura: Bringing autonomy and fun to math is revolutionary in an era when parents feel pressured to push math on even the smallest kids via apps, educational toys, and academic preschools. Your books and Pinterest page offer wonderful ideas. Please give us a few examples of advanced yet playful math for kids of different ages.
Maria: 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.
Sasha and Allison: The concept of functions is very fundamental and can be studies/played with on many different levels, starting at a very young age. After reading Funville Adventures, children can play games such as “Guess My Power” where one person comes up with a power and others try to guess it by asking for outputs for given inputs and/or by asking questions about the characteristics of the underlying function such as: Is it invertible? What is the domain? Is it periodic?
Here are more examples:
- Logic puzzles: Both of us really enjoyed engaging with problem-solving through logic puzzles when we were in elementary and middle school.
- Sports math: A kid who likes to watch or play a particular sport might be encouraged to discover patterns in the many numbers and statistics surrounding it. Certain point totals in football are much more common than others – why? How many ways can one reach a score like 21? If two baseball teams are evenly matched and play n games, how close to n/2 do you expect the win totals to be and why?
- Patterns in music and art: Older kids who like music can learn about the basic patterns of chords underlying popular songs. Children can learn the mathematics of juggling patterns, or how to make art based on fractals or tiling.
- Estimation: Kids of many ages can learn through experiments how to estimate quantities like Pi, or how to guess how many M&Ms are in a jar. They can then learn how to extrapolate estimations to quantities they can’t test experimentally, like how many cars are in a city, or how many workers it should take to do a census, etc.
Laura: On NaturalMath.com, 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.
Maria: 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. 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. NAMC math circle conferences, Julia Robinson festivals, or the Natural Math network called 1001 Math Circles help local leaders grow together.
Laura: Tell us about the Creative Commons nature of Natural Math books.
Maria: 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.
After Funville Adventures came out, readers started to create fan stories and art about their own Funvillians. For example, Dylan has a tall hairdo and too-long shirt because his power is dilation. You can see some of fan works in the book’s web tour.
Laura: All sorts of projects are in the works through the community incubator, where teams of authors develop books with crowd-sourced input. Tell us more about this approach and other Natural Math books we can read, use, and share.
Maria: We developed a community 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, or else an author with a developmental editor, make the first draft. That stage is intense and private: brainstorming, building, bouncing ideas. 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 Natural Math writers and readers—and the book is ready to go out to everyone. We see publishing as a gradual, participatory, ongoing process where ideas grow more and more accessible to wider and wider public.
Our newest book created with this model is Math Renaissance: Growing Math Circles, Changing Classrooms, and Creating Sustainable Math Education by Rodi and Rachel Steinig. It is for teachers and parents of children ages six and up. The authors share their insights on how math experience might be improved at home, school, and math circle.
Check out other Natural Math books at the web site.
Funville Adventures by A.O Fradkin and A.B. Bishop Avoid Hard Work! … And Other Encouraging Problem-Solving Tips for the Young, the Very Young, and the Young at Heart by Maria Droujkova, James Tanton, and Yelena McManaman Socks Are Like Pants, Cats Are Like Dogs: Games, Puzzles, and Activities for Choosing, Identifying, and Sorting Math by Malke Rosenfeld and Gordon Hamilton. Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers edited by Sue VanHattum Bright, Brave, Open Minds: Engaging Young Children in Math Inquiry by Julia Brodsky Camp Logic: A Week of Logic Games and Activities for Young People by Mark Saul and Sian Zelbo Moebius Noodles: Adventurous Math for the Playground Crowd by Yelena McManaman and Maria Droujkova
Here’s to more math adventures!
to spark your own math-fueled journey. 
offers many moving math activities for children.
, and 
or 
it’s a great time to pique their interest using almanacs and atlases.
, a group of students at the Sudbury Valley School approached him saying they wanted to learn arithmetic. He tried to dissuade them, explaining that they’d need to meet twice a week for hour and a half each session, plus do homework. The students agreed. In the school library, Greenberg found a math book written in 1898 that was perfect in its simplicity. Memorization, exercises, and quizzes were not ordinarily part of the school day for these students, but they arrived on time, did their homework, and took part eagerly. Greenberg reflects, “In twenty weeks, after twenty contact hours, they had covered it all. Six year’s worth. Every one of them knew the material cold.” A week later he described what he regarded as a miracle to a friend, Alan White, who had worked as a math specialist in public schools. White wasn’t surprised. He said, “…everyone knows that the subject matter itself isn’t that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step. The only way we have a ghost of a chance is to hammer away at the stuff bit by bit every day for years. Even then it does not work. Most of the sixth graders are mathematical illiterates. Give me a kid who wants to learn the stuff—well, twenty hours or so makes sense.”
that we’re schooled to express math in formal terms, but that’s not necessary for most of us—no matter what careers we choose. People who rely on mental math in their everyday lives are shown to have an accuracy rate around 98 percent, yet when they’re challenged to do the same math symbolically their performance is closer to 37 percent.
Laura Grace Weldon 