The Way We Teach Math Is All Wrong

January 23, 2023 / Laura Grace Weldon / 7 Comments

“What children need is not new and better curricula but access to more and more of the real world; plenty of time and space to think over their experiences, and to use fantasy and play to make meaning out of them; and advice, road maps, guidebooks, to make it easier for them to get where they want to go (not where we think they ought to go), and to find out what they want to find out.” –John Holt

Miss Gribbon set up a new teaching prop at the front of our first grade classroom — three stick figures made of metal with round blank faces and oversized magnetic hands. Each figure was about the size of a toddler, although she referred to them as “men.” She said the first figure’s name was Ones. The next, to our right, she named Tens. The last in the row she named Hundreds. She added two bright red magnetic fingers to each figures’ hands. Then she announced that One’s fingers were worth two, Ten’s were worth 20, and Hundred’s were worth 200.

I could NOT understand how identical magnetic people could have fingers worth different amounts. The hundreds man wasn’t taller than the tens man or the ones man. Their fingers were the same size. So I watched carefully as she stood them up the next day, hoping to figure out what distinguished them. Nothing. The Ones man from yesterday might be today’s Hundreds man. Their value wasn’t intrinsic to who they were. I struggled mightily to understand how one man could be worth more than another. (Story of my political confusion, even now.)

Each time Miss Gribbon rearranged the characters’ fingers she asked a different student, “What number do you see?” If they got it wrong, she asked again in a louder voice before reluctantly providing the answer. To me, math lessons seemed very similar to playing an unfamiliar game with kids who owned the game. They’d always say, “You’ll figure out the rules as we play.” By the time I did, they always won.

We start out in life equipped to pick up mathematical concepts easily. Well-designed studies reveal even babies demonstrate strong understanding of certain mathematical principles.

We continue to advance in our comprehension almost entirely through hands-on experience. Math is implicit in play, music, art, dancing, make-believe, building and taking apart, cooking, and other everyday activities. Only after a child has a strong storehouse of direct experience, which includes the ability to visualize, can he or she readily grasp more abstract mathematical concepts. As Einstein said, “If I can’t picture it, I can’t understand it.”

Yet right around the time formal instruction starts, children increasingly report that they worry about and fear math. Math anxiety, even in first and second graders, disproportionally affects children who have the most working memory. These are the very children most likely to show the highest achievement in math. But stress can disrupt working memory and undermine performance. Otherwise successful children with high degrees of math anxiety fall about half a school year behind less anxious students. In a study of 154 young students, about half had medium to high math anxiety.

Early math anxiety can intensify, leading to increased math avoidance and lowered competence. Over 60 years of research show that positive attitudes toward math tend to deteriorate as students move through school. More than half the adult population in the U.S. is said to suffer from math anxiety, some with math avoidance so extreme that it has the potential to damage financial decisions and careers.

Is math instruction to blame?

Innovative math educator Maria Droujkova says, in an Atlantic article titled “5-Year-Olds Can Learn Calculus,” that math instruction typically follows a hierarchical progression starting with counting, then addition and subtraction, then multiplication and division, onward to fractions, algebra, and so on. Unfortunately, she says, this approach has “… nothing to do with how people think, how children grow and learn, or how mathematics is built.” She and other math educators around the world say the standard curriculum that begins with arithmetic is actually more difficult for children than play-based activities related to more advanced fields of mathematics. As Dr.Droujkova writes, “Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture.”

That torture is compounded by the way math is taught. Extensive research demonstrates that kids readily understand math when they develop the ability to use numbers flexibly, what’s called a “number sense.” Number sense is fundamental to all higher-level mathematics. This does not develop through memorization but instead from relaxed, enjoyable exploratory work with math concepts. In fact, math experts repeatedly point out that math education standbys — flash cards, repetitive worksheets, and timed tests — are not only unhelpful but actually damaging. These common methods discourage number sense, setting young people off in the wrong direction. In fact doing math under pressure impairs the working memory students need to access what they already know. Pressure also leads to math anxiety. There’s no educational reason to use these tactics in the classroom or at home. Greater math ability has nothing to do with working quickly nor does quick recall of math facts relate to fluency with numbers.

Add to this the burden of grades and test scores. Students today deal with a heavy load of standardized tests across all major subjects, plus tests in math class as often as every few days. They quickly learn math has to do with performance, not with usefulness and certainly not with beauty or mystery.

As mathematics educator Jo Boaler writes in Mathematical Mindsets, it’s well known that grades and test scores damage motivation and result in limiting self-labels in high, middle, and low-achieving students. Research consistently shows that alternatives to grading are far more beneficial. One study compared the way teachers responded to math homework in sixth grade. Half the students were graded, the other half were given diagnostic comments without a grade. Students who got only comments learned twice as fast as the graded group, attitudes improved, and any achievement gap between male and female students disappeared.

Dr. Boaler writes about another study in which fifth and sixth grade students were assessed three different ways. Some students received only grades, some only comments, and some both grades and comments. The students who achieved at significantly higher levels were those who were given comments only. Those who got any grade at all, with or without comments, did poorly. This was true for students across the spectrum of ability. Further research found that students only needed to believe they were being graded to lose motivation and achieve less.

Studies continue to show that students given positive feedback and no grades are more successful as they continue through school. There’s a strong relationship between teachers’ assessment practices and students’ attitude about their own potential. Unfortunately teachers give less constructive feedback as students get older and students’ belief in their own chance of improving also declines steadily from upper elementary grades through high school and beyond. Even at the university level, teaching and testing has a tendency to undermine sense-making. Students are likely to limit themselves to rigid sets of rules and procedures while lacking the relational understanding to correctly apply or adapt those algorithms to the problem at hand.

What happens when students aren’t assessed?

Dr. Boaler followed teenagers in England who worked on open-ended math projects for three years. These students were not graded or tested, and only given information about their own learning, even though they faced national standardized tests at the end of that period. A few weeks before the test they were given practice exams to work through. Although they were largely unfamiliar with exam questions or timed conditions, when tested these students scored at a significantly higher level than students who had gone through standard math classes with frequent tests similar to the national exam questions.

What happens when math instruction is even more limited?

Back in 1929, pioneering educator Louis P. Benezet, superintendent of the Manchester, New Hampshire schools, wrote, “The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.” He began an experiment. In five classrooms, children were exposed only to naturally occurring math like telling time and playing games, while in other classrooms children received typical math lessons.

At the end of the first year differences were already apparent between students exposed to these two different approaches. When children were asked the same mathematical story problem, the traditionally taught students grabbed at numbers but came up with few correct results, while the experimental students reasoned out correct answers eagerly, despite having minimal exposure to formal math. Based on these successes, the experiment expanded. By 1932, half of the third- to fifth-grade classes in the city operated under the experimental program. After several years, the experiment ended due to pressure from some principals. Children in the experimental classrooms went back to learning from a math book in the second half of sixth grade. All sixth-grade children in the district were tested and in the spring of that year all the classes tested equally. When the final tests were given at the end of the school year, one of the experimental groups led the city. In other words, those children exposed to traditional math curricula for only part of the sixth-grade year had mastered the same skills as those who had spent years on drills, times tables, and exams. Even more remarkably, the students in the experimental classrooms were from the most poverty-stricken neighborhoods where poor school performance was common. The Journal of the National Education Association published the last of Mr.Benezet’s articles in 1936, calling on educators to replace formal math instruction with naturally occurring math.

What happens when there’s no math instruction by trained educators?

Homeschooling and unschooling families around the world devote much less, if any, time to formal mathematics instruction. There are significant limitations to research of homeschooled and unschooled youth for a variety of reasons, including a self-selecting population, so findings are interesting but inconclusive.

Multiple studies indicate homeschooling offers significant academic advantages, regardless of the parent’s educational attainment. Those tested in the last two years of homeschooling, what would be a schooled student’s junior and senior years, statistically score in the 86^th to 92^nd percentile. The percentage of homeschooled students who complete college far exceeds the rate of public school students.

Studies show homeschoolers taking the SAT tend to score significantly above average in all areas except math where their scores are still above average. The most recent College Board stats show mean scores for all college-bound seniors were 497 in critical reading, 487 in writing, and 513 in mathematics. For the 13,549 homeschooled seniors who took the test that year, means scores were 567 in critical reading, 535 in writing, and 521 in mathematics.

It’s hard to wade through research comparing math achievement of homeschooled versus conventionally schooled young people because much of the research includes as “homeschooled” those students who are educated using district or state sponsored programs which provide conventional-style math instruction to be done at home, which largely replicates the problems of conventional classroom instruction.

Still, several informal surveys show disproportionate number of homeschool and unschool adults working in STEM careers. And it seems that a significant number of today’s high-achievers in technology, science, and math have emerged from the homeschooling community. Their numbers include:

Erik Demain — professor of theoretical computer science at MIT and named “one of the most brilliant scientists in America” by Popular Science
Ruth Elke Lawrence-Naimark — researcher in knot theory and algebraic topology,
Francis Collins — geneticist and director of the National Institutes of Health, Samuel Chao Chung Ting — physicist and Nobel Prize recipient,
Phillip Streich — holder of numerous patents and co-founder of nanotechnology company making him a multimillionaire by the time he entered Harvard,
Arran Fernandez — youngest mathematician with sequences published in Encyclopedia of Integer Sequences,
Willard Boyle — physicist, co-inventor of charge-coupled device and Nobel Prize winner.

What happens when there’s no math instruction other than what young people request?

Democratic schools exist at the opposite end of the spectrum from conventional schooling. Students are not segregated by age and each student has one vote, just as staff members do, to democratically run the community. All young people are trusted to choose their own activities and no classes are mandatory, making these schools a collectively managed and open setting for self-directed learning.

Psychologist Peter Gray surveyed graduates of one such school, Sudbury Valley School (SVS) in Framingham Massachusetts. He found that young people who were not mandated to follow curricula, take tests, and receive grades “…have gone on to good colleges and good jobs…They are taking responsible positions in business, music and art, science and technology, social services, skilled crafts, and academia.” Dr. Gray notes that employers are rarely concerned about a prospective employee’s grades in algebra. Instead the traits for career success are those that graduates say were fostered by their time at SVS, such as “…a strong sense of responsibility, an ability to take initiative and solve problems, a desire and ability to learn on the job, an ability to communicate effectively, and perhaps most of all, a high interest in and commitment to the field..”

And there’s this anecdote, shared by teacher Daniel Greenberg in his book Free At Last. 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 regularly and do homework. The students agreed to do so. 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 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.”

These examples aren’t meant to be anti-teaching, they are meant to broaden our understanding about when instruction is most useful and effective. That happens less often than we’d think — when the learner seeks guidance, demonstration, resources, or help. Learning that’s sought out sticks with the learner. It promotes curiosity, persistence, passion, and deep inquiry — exactly what’s needed to dig into the fathomless depths of mathematics or any other pursuit.

Math as it’s used by the vast majority of people around the world is actually applied math. It’s directly related to how we work and play in our everyday lives. In other words it’s useful, captivating, and often fun.

Interestingly, people who rely on mental computation every day demonstrate the sort of adroitness that doesn’t fit into conventional models of math competence. In a New York Times article titled “Why Do Americans Stink at Math,” author Elizabeth Green (who defines the term “unschooled” as people who have little formal education) writes,

Observing workers at a Baltimore dairy factory in the ’80s, the psychologist Sylvia Scribner noted that even basic tasks required an extensive amount of math. For instance, many of the workers charged with loading quarts and gallons of milk into crates had no more than a sixth-grade education. But they were able to do math, in order to assemble their loads efficiently, that was “equivalent to shifting between different base systems of numbers.” Throughout these mental calculations, errors were “virtually nonexistent.” And yet when these workers were out sick and the dairy’s better-educated office workers filled in for them, productivity declined.

The unschooled may have been more capable of complex math than people who were specifically taught it, but in the context of school, they were stymied by math they already knew. Studies of children in Brazil, who helped support their families by roaming the streets selling roasted peanuts and coconuts, showed that the children routinely solved complex problems in their heads to calculate a bill or make change. When cognitive scientists presented the children with the very same problem, however, this time with pen and paper, they stumbled. A 12-year-old boy who accurately computed the price of four coconuts at 35 cruzeiros each was later given the problem on paper. Incorrectly using the multiplication method he was taught in school, he came up with the wrong answer. Similarly, when Scribner gave her dairy workers tests using the language of math class, their scores averaged around 64 percent. The cognitive-science research suggested a startling cause of Americans’ innumeracy: school.

And Stanford mathematician Keith Devlin explains in The Math Gene 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 (as in standardized tests) their performance is closer to 37 percent.

Conventional math education may also limit our concept of what math can do. As Dr. Devlin notes in a post titled “Most Math Problems Do Not Have a Unique Right Answer:”

One of the most widely held misconceptions about mathematics is that a math problem has a unique correct answer…

Having earned my living as a mathematician for over 40 years, I can assure you that the belief is false. In addition to my university research, I have done mathematical work for the U. S. Intelligence Community, the U.S. Army, private defense contractors, and a number of for-profit companies. In not one of those projects was I paid to find ‘the right answer.’ No one thought for one moment that there could be such a thing.

So what is the origin of those false beliefs? It’s hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that

are well defined,

have a unique correct answer, and

whose answer can be obtained with a few lines of calculation.

How can we translate all these findings into math education?

We not only need to drop flashcards, timed tests, and rote worksheets. We need to emphasize math as meaningful, useful, and connected.

A. The most statistically significant predictors of long-term math achievement, according to a study that tracked children from age three to age 10, had very little to do with instruction. Instead the top factors were the mother’s own educational achievements and a high quality home learning environment. That sort of home environment included activities like being read to, going to the library, playing with numbers, painting and drawing, learning letters and numbers, singing and chanting rhymes. These positive effects were as significant for low-income children as they were for high income children. Children who attended highly effective preschools (but not moderately effective programs) also benefited. Understanding numbers as meaningful and fun is important from the earliest years.

B. Technology innovator Conrad Wolfram says we need to go beyond computation. He suggests these four steps:

Pose the right question about an issue
Change that real world scenario into a math formulation
Compute
Turn the math formulation back into a real world scenario to verify it

C. Barnard College president Sian Beilock president says math is best learned as storytelling and done so by incorporating the body, the way children naturally absorb real world math. As neuroscientists map the brain, they find humanity evolved skills that overlaid onto areas of the brain that control the body. Math doesn’t sink in when confined to the intellect. It is drawn in through the body. We see this in studies showing babies who are able to move and explore more freely learn more quickly. “Math, Dr. Beilock says, “is a very recent cultural invention.” The part of the brain used for numerical representation is related to finger motion, demonstrating exactly why children best learn by counting on their fingers. Hand movement all the way up to full body engagement, such as walking while thinking, are actually more valuable than speech in comprehending everything from early computation to abstract concepts in physics. Dr. Beilock also emphasizes the benefits of time in nature to refresh one’s attention, leading to greater focus and comprehension.

D. Dr. Droujkova adds to this by emphasizing richly social math experiences that are both complex (able to go in a variety of directions) and simple (open to immediate play). She says any branch of mathematics offers both complex and simple ways in. It is best, she explains, to keep from chaining kids into formal equations early on. There’s an informal level where kids play with ideas and notice patterns. Then comes a more formal level where kids can use abstract words, graphs, and formulas. But it’s best if a playful attitude is kept alive, because what mathematicians do at the highest level is play with abstract ideas.

Dr. Droujkova notes that the community she founded called Natural Math is essentially a “freedom movement.” She explains: “We work toward freedom at many levels — the free play of little kids, the agency of families and local groups in organizing math activities, the autonomy of artists and makers, and even liberty for us curriculum designers…. No single piece of mathematics is right for everyone. People are different, and people need to approach mathematics differently.” Although we’ve been schooled to believe that math must be taught in a structured way by professionals, Dr. Droujkova continues to establish lively and engaging community-based, open-learning math circles that can be led by any adult. She and her colleagues make their materials open under Creative Commons license and offer online hubs with courses and resources for parents, teachers and teenagers who want to lead local groups. (See naturalmath.com) As Dr. Droujkova says in a recent interview, “math circles are magic circles.”

School-like instruction has been around less than a fraction of one percent of the time we humans have been on earth. Yet humanity has thrived. That’s because we evolved as free range learners gaining mastery as we explore, play, emulate role models, challenge ourselves, make mistakes and try again. That’s how everyone learns to walk and talk. That’s how young people have become capable adults throughout history. And that’s how innovation happens in the arts, sciences, and technology. In the long view, school is the experiment.

For many it’s hard to see beyond the school mindset because most of us went to school. So when we think of education, we view school as the standard even if we simultaneously realize that many parts of that model (also found in daycare, preschool, kids’ clubs, sports, and enrichment programs) aren’t necessarily beneficial. Narrowing the innate way we learn can unintentionally narrow enthusiasm, creativity, persistence, and the desire to dive deeply into any pursuit. It can interfere with the full development of our abilities.

My first grade math lessons taught me to equate math with fear. I went on to get good grades in the subject, but by high school my math anxiety led me to give up hopes of working in a science field. Math misery doesn’t have to be imposed on the next generation.

It’s time to free ourselves from the assumption that math instruction is a painful necessity. Approaching math in ways that are disconnected from a child’s life subtracts the meaning and the joy. It multiplies fear. Data shows and experience proves that real learning flows from the learner’s consent and the learner’s interest. We can offer math as an enlivening, beautiful tool to the next generation as soon as we free ourselves from the limitations of the school mindset.

For more information:

Math Instruction versus Natural Math: Benezet’s Example
Natural Math: 100 Plus Activities & Resources

Published in Tipping Points, originally adapted from the author’s book Free Range Learning.

Interview: Math Is Child’s Play

March 30, 2018 / Laura Grace Weldon / 4 Comments

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