math

The Way We Teach Math Is All Wrong

/ 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 86th to 92nd 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.

Magic Circles

/ Laura Grace Weldon / 7 Comments

Last year I had the pleasure of interviewing innovative math educator and founder of Natural Math, Maria Droujkova, in “Math is Child’s Play” where she talks about learning math through free play in the context of families and communities. More recently, she and I were talking via social media when she mentioned magic circles. I was instantly intrigued and asked her to explain. She wrote:

One of my consulting topics is game/experience design. One of my favorite design concepts is magic circle: a playspace co-created by the participants, where they suspend their disbelief and behave as if they inhabit another world. I’ve been collecting tools for building cool magic circles from all creative fields, from writing to engineering. Tools like pretend-play, problem-posing, or name-giving. Math circles are magic circles. The maker goal: learn to pop up constructive, emotionally secure, creative spaces wherever we go.

I had to know more. My questions to her turned into this interview.

 

What was your first experience with a magic circle?

That feeling when an activity is the thing and the whole of the thing? When the rest of the world and the rest of me pretty much disappears? I’ve been experiencing that for as long as I remember. Early on, at three or four, I rearranged stones to make tiny spring snowmelt creeks gurgle merrier. I made canals, dams, and waterfalls till my hands grew red and numb. I remember long pretend-play with my mom, dad, and my imaginary friends, like the red velvet bow that was a fire-butterfly who’d gently land on my hand to play with me. Or the friend called Reflection who could escape its mirror, turning invisible. In another couple of years, there were elaborate handicrafts, hours in the making, while my grandpa was meticulously arranging his stamp collection in hand-crafted albums. He worked at the same table, and my crafts only happened if he started his. There was a very different energy, but some of the same timeless feeling, when me and other rough neighbor kids let go of our constant low-key fighting for living as action heroes in one of the traditional games, also rough, like “Cossacks and robbers.”

Once again, it was a different energy and a very recognizable feeling when I started to spend long hours solving delicious problems before my first Math Olympiad.

I don’t think I can live for long without the magic circle experience. It’s somewhere between water and food on the hierarchy of needs. Yet when I first read Flow: The Psychology of Optimal Experience I felt uneasy about the authors’ claims that there are people of the flow, and communities of the flow, maybe even nations of the flow, while other people and groups are not.

Am I doing enough of immersive, productive, joyful work? Are my communities? I’d had none of these worries between building elaborate snowmelt waterworks and making up fantastic worlds for fire butterflies.

How did you imagine its benefits might be useful in other situations?

For the task in hand, a magic circle gives us immersive, focused, joyful motivation. We feel less tired while accomplishing more. Isn’t that dreamy? And also, we feel balanced and peaceful. That’s dreamy too, especially in troubled times. These feelings, in turn, can increase well-being, productivity, and teamwork.

But these feelings can also run amok or be co-opted. They can make us workaholics, media addicts, or viciously competitive. The weekend I played World of Warcraft so long I lost the sight in one eye for a few scary hours? The magic circle had been real! Gamers’ farewell: “It’s been real.” The weekend when I told my husband I’d be there for breakfast after a few minutes of writing? I resurfaced when the manuscript was finished. It was nearly dark by then, and I’d fasted through the day, though my husband kept bringing me water, which I drank but failed to notice. It’s been real, it’s been magic, I still like what I wrote back then, but if I keep doing something like that for several days in a row I get sick.

Some people of the flow have bad health statistics, such as computer coders with their famously long, immersive work-hours.

What “tools” do you use to build magic circles?

I have been collecting toolsets far and wide. One of my favorites is pretend-play: be a character in a different world. Instant immersion! Characters also provide emotional security: if something goes wrong, it’s not me, it’s some other persona who messed up. Sometimes clothes or accessories help to get into character: dress for the job you want, right? I have a collection of math t-shirts that set the mood when I give a talk or run an event. GMs (game masters) of Dungeons and Dragons have a lot of casual roleplay tools to quickly transport the group into the world of play, from evocative sound effects in their phones to action figurines in their pockets.

What does an orchestra, an improv group, and a math circle have in common? They do a warm-up before the main play. Tune the instruments, do a quick skit, solve a puzzle: these are threshold activities for when a group is liminal between our world and its magic circle.

There are many jokes, not always kind, about the stereotypical scientist, mathematician, and engineer clueless about the bigger picture. In reality, magic circles can break from context-switching. A programmer who says, “Hi, how are you?” to a colleague might remain measurably distracted from coding for the next quarter hour or so. As such, STEM professions have developed big toolsets for keeping focus.

Some of these STEM tools for tuning out the world backfire in funny or sad ways. Archimedes running naked through the city shouting his “Eureka!” is still cute to us. Social threats, such as being reminded that you are an oppressed minority or that your job is in jeopardy (or that your clothes or lack of them matter), tend to break magic circles. Yet the habit of tuning out all social issues causes systematic problems in STEM fields. “Oops, this AI is sexist because we never gave gender a single thought while designing it.” That’s one of the forces behind many cultural troubles exemplified by the Silicon Valley.

I like magic circle tools from professions and serious hobbies. Each profession has its way for creating magic circles, but most professionals don’t realize that’s what they are doing. Like speaking a native language, we don’t think about its grammar. Writers talk about their process a lot, because notice-wonder-describe is a part of their trade. Some managers reflect on how they manage. In general, when I want to explore a new field, I go straight for teacher materials. Doctors, mathematicians, or car mechanics may not reflect on their process, but their teachers surely do.

Most recently, I’ve been exploring storytelling communities. Some of those focus on speech games, such as Toastmasters, One Million Cups, or Pechakucha. They have a common toolset, such as format-based stories. For example, One Million Cups has a 5-7 minute story about a startup that must contain certain elements, such as an elevator speech intro and a particular ask of what others can do for the company. The time-tested format helps newbies learn from similarly structured examples. It also gives a checklist for what’s important in business and in business presentations. Checklists and canvases are magic circle tools.

Pechakucha is a rhythm game: a presentation where 20 slides advance automatically every 20 seconds. That deceptively simple storytelling device makes people mindful of their phrases, because they have to write and practice speeches, because it’s impossible to improvise that precisely! The prep provides many hours of a solo magic circle. When the group event comes, the fast Pechakucha rhythm generates high energy in the room, keeping the group magic circle going.

Fanfiction networks gather storytellers who focus on written stories in particular imaginary worlds. The non-profit Organization for Transformative Works has a good amount of peer-reviewed articles and essays published about fanfiction. One of my favorite fanfic tools is a prompt or gift exchange. Imagine a little creative task, a quest if you will, given to the author by someone who shares the author’s love of a particular imaginary world. That person is eagerly awaiting whatever the author makes, and then cheers the gift, gives thanks, and provides content feedback. And maybe there are other peers who admire the creation as well, and leave comments. In any case, the author is guaranteed at least one eager receiver of that gift. Now imagine a homework exercise designed that way: personally requested, anticipated, and loved by at least one like-minded person. Wouldn’t that be a magic circle experience?

Where did this concept originate?

The term “magic circle” as I use it comes from game design. It is also used in gamification and experience design. In a good game, players suspend their disbelief and quit their daily grind to be immersed in the game’s world. They enter the magic circle to start playing, then leave it behind to return to the regular world.

There used to be social stigma against adult play and adult gamers. Talk about work as a game, and you are immature, or else scheming, even Machiavellian. About ten years ago, the average age of a computer gamer grew over thirty. As gamers grow up, gaming is normalized. Now we see a lot of game design tools in the workplace. Naturally, some are used in scary ways, such as workers being nudged to skip breaks by tracking their relative progress on a public display. Magic circles can mutate into harsh competitive prisons. We have to be aware of that.

Where can magic circles happen?

Magic circles happen in the mind. Our actions can help to focus the mind. The activity that invites a magic circle can be shared, creating the sense of oneness with other participants.

A related question: are there places or situations where magic circles cannot happen? What prevents them, what stops them? Can we have magic circles while tired, sick, or scared? What social or mental conditions hinder us, like having a new baby or having ADHD?

I didn’t use the term back then, but I found my first systematic descriptions of creating magic circles with and for children in Janusz Korczak’s books. I’ve read and re-read them as a kid, teen, and adult. “The Child’s Right to Respect” became a motto. “Playful Pedagogy” is still relevant after eighty years in publication. Doctor by calling, Korczak was the founder of one of the first Democratic schools in the world, in a poor Jewish children’s orphanage. It had a children’s parliament and a child-run newspaper. His magic circle tools are based on helping troubled kids find their voice, grow in agency and autonomy, build equality, and care for one another in kindness.

The orphanage was in Warsaw. It was ended by force in 1942. Yet this story is about light, inspiration, and hope. Korczak and his children kept up their classes, their play, their newspaper and their democratic meetings. In troubled times, they had the power to maintain their magic circles.

korczakusa.com

Do you have any resources you’ve written that people can access on this? 

I write about a specific type of magic circles, called math circles. The company I direct, Natural Math  publishes books for math circle leaders. These circles are designed with the purpose of every participant saying enthusiastic “Yes!” to mathematics.  [Laura’s note: Maria’s Natural Math  site also offers a newsletter, blog, courses, and FB group 1001 Math Circles.]

The term “math circle” in its modern sense originated in the Eastern Europe in 1960s. I loved math circles as a child, and they helped to define my career. A few years ago, I interviewed some Western math circle leaders to edit and expand the Wikipedia article on math circles. Now it has a more inclusive definition and a big list of math circle types, such as project-centered clubs and guided exploration circles.

The latest book I co-authored, called Avoid Hard Work, is about kind, accessible, and deep problem-solving. It has ten chapters for ten problem-solving principles. They are tools for creating the math magic, such as the tongue-in-cheek titular slogan: to replace the mindless grind with attentive search for patterns. The book starts with a sample of what hundreds of parents and teachers say when asked, “When it comes to children and mathematics, what are your dreams?” For example, some people dream of math that is friendly, that invites curiosity, that makes sense. These math dreams can be used for value affirmation, for remembering who you are and what you want. Value affirmation is an emotional tool: it combats anxieties, puts us in the right mood for creativity, and makes our circles more robust. There is also a list of teaching techniques, such as moving beyond snap judgment of right-wrong answers to explore the “Why?” behind student reasoning.

(I am attaching these lists from the book, which is published with the Creative Commons open license, so people are welcome to share with attribution.)

Do you have some very basic suggestions about creating circumstances conducive to magic circles happening naturally, such as in the home and community? 

Love this question! Balance is the key. Here is what I mean.

  1. The first balance: self with adventure.You do you – a step aside from your normal routines.  A circle becomes magical if it takes you away from the daily. Choose to make something you don’t normally make: a paper snowflake, a stacked rock sculpture, or a diamante poem. Look up how others make those, try their way, then do it your way. Add a bit of your style. What if you fold the paper differently before you cut the snowflake? What if you stack rocks three by three instead of one by one? What if you shape your poem like a pear rather than a diamond? Seek more and more of these interesting choices to try. Since your project is outside of your life, making a mess of it will impact nothing serious. You can be a bit braver, a bit stronger, a bit more adventurous than your daily self. Yet it’s still you.  
  2. The second balance: inspiration with ease. Do something special – while staying casual. What if it’s hard to feel brave and strong, to shift your focus away from the daily concerns? Children put on cat ears or superhero capes when they have their special play. Adults can also try play-acting and special clothes. Think of a ritual to set the mood. Strike a gavel, play a favorite theme song, or pass treats around. Add something quirky, spicy, or charming as you prepare. Go beyond the bare necessities. Yet keep your prep light so that your circle is easy to start. The feeling of magic is in trying new actions, not in blockbuster-quality props. It’s okay to use a curtain for your cape and pillows for your fort.
  3. The third balance: invitation with consent.One of the main difference between a circle and the ordinary life is agreement. There is a whole lot of have-to in life. Obligations and needs, rules and laws dictate what must be done and how, whether we like it or not. Do establish basic safety and well-being rules for your circle. And then, make the magic. Aim for enthusiastic “Yes!” to every choice, big and small. You can invite, entice, and advertise. But don’t force, dictate, or coerce, neither other participants nor yourself. It’s lovely to point out that you are making amazing seven-sided snowflakes. It’s okay to invite children to count the sides – but only for pleasure. To keep the magic in, leave your worries about math tests out of your circle. This is not the kind of story where a reluctant hero can’t refuse a call to action. Everything’s optional. Don’t push.

What are your favorite tools for building magic circles? I’d love to hear stories and compare notes.

 

Resources

Maria offers a list of teaching techniques, such as moving beyond snap judgment of right-wrong answers to explore the “Why?” behind student reasoning.  (She offers these lists from the book in a free download here: Avoid Hard Work, which is published with a Creative Commons open license, so people are welcome to share with attribution.)

 

1001 Math Circles: Facebook group to share and discuss math circle activities

Natural Math site

Natural Math books to spark enthusiasm and deeper learning. They include Moebius Noodles, with math explorations for children as young as three, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers for young people eight and older.

Avoid Hard Work    free excerpt

“Natural Math: 100 Activities & Resources” 

“Kids Build Together: Math Readiness in Early Childhood” 

“Playing With Math: How Math Circles Bring Learners Together For Fun”

“The Benefits of Natural Math” 

“Math Instruction vs Natural Math: Benezet’s Example” 

“Math is Child’s Play”  Previous interview with Dr.  Droujkova

National Association of Math Circles

Julia Robinson Mathematics Festival

Interview: Math Is Child’s Play

/ 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

Here’s to more math adventures!

Natural Math: 100+ Activities & Resources

/ Laura Grace Weldon / 21 Comments

math through play, everyday math

image: pixabay.com

“The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”   ~Stan Gudder

Today’s children are much less likely than previous generations to learn through play, exploration, and meaningful work. Concern about the math scores of the nation’s youth should instead turn to concern about the manipulation of childhood itself. We’ve substituted tightly structured environments and managed recreation for the very real, messy, and thought-provoking experiences that are the building blocks for higher level thinking.

Learning math requires children to link language with images as they work through equations. It helps if they can easily picture the problem being solved before they move ahead into representational and abstract math. Normally a child who has spent plenty of time playing with manipulatives (water, sand, building blocks, countable objects) and who uses real world applications of math (cooking, carpentry, budgeting) has a wealth of experience to fall back on. This child can call up mental images that are firmly connected to sensory memory, helping him understand more advanced concepts.  Applied math, especially as it relates it a child’s needs and interests, is the bridge to mathematical success.

Computational readiness varies widely from child to child. Some are eager to do mental math, memorize math tricks, and take on increasingly complex calculations. Others need much more time before they are ready to tackle math this way. When readiness is paired with self-motivation there’s no limit to what a child can accomplish.

Benoit Mandelbrot is the Yale mathematics professor credited with identifying structures of self-similarity that he termed fractal geometry. His work changed the way we see patterns in nature, economies, and other systems. Mandelbrot doesn’t believe students need to struggle with Euclidean mathematics. Instead, he says,”Learning mathematics should begin by learning the geometry of mountains, of humans. In a certain sense, the geometry of . . . well, of Mother Nature, and also of buildings, of great architecture.” In other words, by focusing on inspiration found everywhere around them before turning to formal equations.

Natural math, according to math expert Maria Droujkova, 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.

Droujkova goes on to say,

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.

Let’s fling our limiting concept of math education wide open by eagerly using it in our lives.  Math is everywhere. Equations, patterns and probabilities surround us. Sometimes it takes a larger way of thinking about math to celebrate the beauty and perfection it represents.

natural math, math through play,

Applied math (images: morguefile.com)

Here are some of the starting points suggested in Free Range Learning to spark your own math-fueled journey.

 ~Learn more about yourselves. One family hangs a new chart each week to gather data. One week they might mark off where the dog takes a nap, then figure the percentage at the end of the week (40 percent of the time she sleeps in the window seat, 5 percent of the time under the table, etc), another week they might pick a subject like hours of computer use per person. They are also keeping several year-long graphs. One tracks the weight of trash and recyclables they discard weekly and a second graphs the amount of the produce they harvest from the garden. Yet another tracks money they are saving. They notice that in busy weeks, such as holidays, they fall short of sustainability goals they’ve set for themselves.

~Revel in measurement. Investigate joules, BTUs, calories, watts, gallons, degrees, fathoms, meters, hertz, attoseconds and more. Measure your everyday world. Calculate such things as the energy usage to get to grandma’s house in the car compared to taking the train, what angle a paper plane can be thrown and still fly, how much wood it will take to build a shelf for the baby’s toys, how many footsteps are required to walk to the corner. Figure out how to gather measurements and apply data.

~Enjoy math songs. Play them while traveling and sing them casually as you go about your day; you’ll find your children are memorizing math facts effortlessly. There’s something about a catchy tune that helps the mind retain concepts. There are many sources of math songs including Sing About Science and Math with a database of 2,500 songs.

~Say yes. When kids want to explore off the trail, stomp in puddles, mix up ingredients, play in the water, and otherwise investigate they’re making math and science come alive on their own terms. It’ll probably make a mess. Say yes anyway.

~Use wheels. Plan and build a skateboarding ramp. Time relay races using tricycles (the bigger the kids the greater the fun). Estimate how many revolutions different sized bike wheels make to cover the same distance (then get outside to find the answer). Adjust a wheelbarrow load to carry the greatest amount of weight. Use mass transit to get where you are going after figuring out the route and time schedule.

~Make math a moving experience. Instead of relying on flash cards, remember equations by clapping or stomping to them, rhyming and dancing with them, kicking a ball or tossing a bean bag to them, making number lines on the sidewalk with chalk and running to answer them, or any other method that enlivens learning. Games for Math: Playful Ways to Help Your Child Learn Math, From Kindergarten to Third Grade offers many moving math activities for children.

~Learn to dance. The fox trot or the hokey pokey may be funny names to children, but they also describe specific patterned steps. Mastering simple dances are a way of transforming mathematical instruction into art. Choreographers use dance notation to symbolize exact movements. Over the years different methods of dance notation have been used including: track mapping, numerical systems, graphs, symbols, letter and word notations, even figures to represent moves. Choreograph using your own system of dance notation. Draw chalk footprints on the floor to show where the dancer’s feet move to a waltz. Try dance classes. Music and dance enliven math concepts.

~Think in big numbers. Figure out how many days, minutes and seconds each member of the family has been alive. Estimate the mass of the Earth, then look up the answer. Stretch your mind to include Graham’s Number. Talk about why big numbers are best expressed in scientific notation. Check out the Mega Penny Project. Read stories about big numbers, such as Infinity and Me, How Much Is a Million? Millions to Measure, On Beyond a Million: An Amazing Math Journey, Can You Count to a Googol? , and One Grain Of Rice: A Mathematical Folktale

~Fold your way into geometry. Print out paper designs that fold into clever toys and games from The Toy Maker including thaumatropes and windboats. Check out instruction books such as Paper, Scissors, Sculpt!: Creating Cut-and-Fold Animals or Absolute Beginner’s Origami. Although these may seem to be for amusement sake, they teach important lessons in conceptualizing shapes and making inferences about spatial relations.

~Play games. Nearly every board game and card game incorporates arithmetic. Make time to play the games your children enjoy. Try new ones and make up your own. Many homeschoolers set up game days so their children can share games with their friends, this is a worthy tradition for kids whether they’re schooled or homeschooled. Games make strategizing and calculating effortlessly fun. For the latest information on games, check in with the aficionados at Board Game Geek. For educational game reviews, consult Games for Homeschoolers  and The Board Game Family.

~Learn chess. This game is in a class all its own. Research shows that children who play chess have improved spatial and numerical abilities, increased memory and concentration, enhanced problem-solving skills as well as a greater awareness of these skills in action. Interestingly, chess also promotes improved reading ability and self-esteem.

~Get hands-on experience in geometry. Geometrical principles come alive any time we design and build, whether constructing a fort out of couch pillows or a treehouse out of scrap wood. Make models using clay, poster board, craft sticks, or balsa.

~Find out about the math in meteorology. Learn about weather trends and predictions, measurement of precipitation and temperature conversion. Keep a weather log using instruments to measure wind speed, precipitation, temperature, barometric pressure, and humidity: then graph the results to determine average, mean, and median for your data.

~Play with shapes. Enjoy puzzles, tangrams, and tessellations. Notice the way shapes work together in the world around you both in natural and constructed settings. Keep a scrapbook of appealing shapes and designs. Create a sculpture out of toothpicks and miniature marshmallows. Cut paper snowflakes. Make collages out of pictures and three-dimensional objects. Grout bits of tile or broken dishes into mosaic designs. Make mobiles. Cut food into shapes.

~Pick up a musical instrument. Learning to play an instrument advances math skills as well as sharpens memory and attention.

~Learn to code. It’s not only fun, it’s really a basic skill.

~Estimate, then find out how to determine an accurate answer. Predict how much a tablespoon of popcorn will expand, then measure after it has been popped. Before digging into an order of French fries, estimate how many there are or how far their combined length will reach. See how the guess compares with the actual figure. Guessing, then finding out the answer enlivens many endeavors.

~Get into statistics.

  • Kids go through a phase when they want to find out about the fastest, heaviest, most outrageous. Once they’re duly impressed with the facts in such books as Guinness World Records it’s a great time to pique their interest using almanacs and atlases.
  • Sports offer a fun way to use statistics. Player and team stats are used to calculate odds, make comparisons and determine positioning. Children may want to keep track of their favorite teams or of their own activities. The numbers can help them to see patterns, debate trends and make predictions.
  • Data provided by WorldoMeters makes fascinating reading and may lead to further investigation.
  • Collect and interpret your own statistics. You might develop a survey. Or record measurements, weights, and other information about specific data, then analyze the statistics using a graph, histogram, or other instrument.

~Make calculation part of household rules. If children are permitted a certain amount of screen time per week, let them be responsible for charting that time. If children rotate chores or privileges, assist them to create a workable tracking system.

~Learn to knit. This useful skill also provides hands-on experience in basic math including counting, skip counting, multiplication and division, patterning, following a numerical guide, visualizing shapes, and problem solving.

~Make time for calendars. Check out the history of African, Babylonian, Roman, and Egyptian calendars. Learn how our calendar system came into use. Would it make sense to change to 13 equal months of 28 days each, with one remaining “day out of time” set aside? What are the definitions of “mean solar time,” “sidereal time” and “apparent solar time”? Make a homemade sundial to see how accurately you can tell time.

~Make math edible. Cereal, pretzels, crackers, small pieces of fruit or vegetables, cubes of cheese, nuts and other bite-sized foods are excellent tools to demonstrate addition, subtraction, multiplication, division, fractions, percentages, measurement and more. Using food to make math functions visible is a tasty way to solve equations. Your children can calculate recipe changes such as doubling or halving while they learn other useful meal preparation skills at home.

~Use trial and error. This is a fun process, especially when applied to brain teasers, puzzles, and mazes; try making up your own. Other math-related ways to stretch your mind include optical illusions, magic tricks, and drawing in perspective. These activities go well beyond solving equations to figuring out larger concepts.

~Devise your own codes and use them to send messages to one another. Check out the history of codes and code breakers. Set up treasure hunts by hiding a tiny treat and leaving codes or equations to be solved that lead to the next set of hints.

~Compete. 

~Enjoy the intersection of math and art. Muse over puzzling visual patterns, for example the work of M.C. Escher. Learn about rug making, sculpture, weaving, basketry and many other art forms to discover the calculation, patterning, and measurement used to create objects of beauty.

~Delve into maps. Look at maps of the world together. Find maps of your locality. As well as road maps, your child may be intrigued by topographical and relief maps, economic and political maps, navigational and aeronautical charts, weather maps or land ownership maps. Draw maps of your neighborhood, home, yard, or bedroom—notice what details your child includes. Make imaginary maps, perhaps to accompany a story or to demonstrate what an eight-year-old would consider a perfect place. Consider mapping somewhere you know well, but from different time frames—how might this place have looked 100 years ago, now, in the distant future? Some children who are reluctant to keep diaries or sketchbooks will cheerfully keep records of places they’ve been by drawing maps. Maps and mapping can teach measurement, spatial awareness, and complex geographical concepts.

~Use logic. Apply critical thinking to current events.

~Compare related things like the weight of a puppy to a full-grown dog, or the size of a pitcher compared to the number of glasses it can fill.

~Use math at the store. While shopping, have children help check prices as part of the process of choosing a better deal. Talk about what other factors come into play—durability, ecological impact, value, overall worth. If you need to make a bigger purchase like a refrigerator, have the children compare the special features and cost effectiveness of running the appliance.

~Try travel math. Traveling is a great time to use math. Children can figure out fuel usage, keep track of expenditures, consult maps, estimate time of arrival, and more. Playing math games also provides excellent distraction during a long trip!

~Talk about math as if you are thinking out loud. “I wonder how many bricks it took to make this entire wall?” then look up a formula for figuring that out; or “If we don’t buy ____ for a whole month do you think we’ll have enough money left over for a ____?”

~Enjoy hands-on projects requiring sequential instruction. These hone logic and spatial skills as well as patience. Model-building, quilting, making repairs, knitting, carpentry, origami, beading and Legos® are examples of such projects.

~Learn how alternative languages relate to numbers. Check out Morse code, semaphore, Braille and sign language.

~Play pool. The sport known as billiards has a lot to teach about angles, trajectory, speed and calculation. And it’s fun.

~Expect kids to participate in household chores. All sorts of mathematical concepts are learned when the youngest children put away silverware, stack plastic containers in the cupboard, and sweep the floor. Even more while older kids help make meals, do repairs, and brainstorm solutions to make the household runs more smoothly.

~Make puzzles a family tradition. They can increase concentration as well as promote spatial learning and reasoning.

~Start or join a math circle. Meet regularly with others who enjoy making the subject fun and intriguing. Most are run by math experts and include projects, games, and field trips related to math. Some resources to get you started:

~Play with math and critical thinking, together.

~Check out learning games suggested by math teachers and math bloggers.

~Read literature that incorporates math.  Find lists of specific math concepts in children’s literature through the National Association for the Education of Young Children as well as the math in children’s literature list on Love2Learn2Day.  Here are some age-related suggestions.

~Read-aloud math stories for children under 8.

~Math Stories for Children 8 and up.

~Math inspiration for older kids.  

Enjoy math-y videos.

~Keep math references handy, you’ll find them endlessly useful.

This post is third in a series on natural math. 

The Benefits of Natural Math. Data that turns turn our assumptions about math instruction upside down. If you read only one in this series, read this. 

Math Instruction versus Natural Math: Benezet’s Experiment. What happened when formal math instruction was eliminated? 

image adapted from livescience.com

image adapted from livescience.com

The Benefits of Natural Math

/ Laura Grace Weldon / 14 Comments

natural math, exploratory math, hands-on learning,

images: public-domain-image.com

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, interesting, even fun.

We now know babies as young as five months old show a strong understanding of certain mathematical principles. Their comprehension continues to advance 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.”

As parents, we believe we’re providing a more direct route to success when we begin math (and other academic) instruction at a young age. Typically we do this with structured enrichment programs, educational iPad games, academic preschools, and other forms of adult-directed early education. Unfortunately we’re overlooking how children actually learn.

Real learning has to do with curiosity, exploration, and body-based activities. Recent studies with four-year-olds found, “Direct instruction really can limit young children’s learning.” Direct instruction also limits a child’s creativity, problem solving, and openness to ideas beyond the situation at hand. Studies show kids readily understand math when they develop a “number sense,” the ability to use numbers flexibly. This doesn’t come from memorization but instead from relaxed, enjoyable exploratory work with math concepts. In fact, math experts tell us methods such as flash cards, timed tests, and repetitive worksheets are not only unhelpful, but damaging. Teaching math in ways that are disconnected from a child’s life is like teaching music theory without letting them plunk piano keys, or instructing them in the principles of sketching without supplying paper or crayons. It simply makes no sense.

One study followed children from age three to age 10. The most statistically significant predictors of math achievement 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.

There’s another key difference between kids who excel at math and kids who don’t. It’s not intelligence. Instead it’s related to what researcher Carol Dweck terms a growth-mindset. Dweck says we adopt certain self-perceptions early on. Some of us have a fixed mindset. We believe our intelligence is static. Successes confirm this belief in our inherent ability, mistakes threaten it. People with a fixed mindset may avoid challenges and reject higher goals for fear of disproving their inherent talent or intelligence.  People with a growth mindset, on the other hand, understand that intelligence and ability are built through practice. People with this outlook are more likely to embrace new challenges and recognize that mistakes provide valuable learning experience. (For more on this, read about the inverse power of praise.)

Rather than narrowing math education to equations on the board (or worksheet or computer screen) we can allow mathematics to stay as alive as it is when used in play, in work, in the excitement of exploration we call curiosity. Math happens as kids move, discuss, and yes, argue among themselves as they try to find the best way to construct a fort, set up a Rube Goldberg machine, keep score in a made-up game, divvy out equal portions of pizza, choreograph a comedy skit, map out a scavenger hunt, decide whose turn it is to walk the dog, or any number of other playful possibilities. These math-y experiences provide instant feedback. For example, it’s obvious cardboard tubes intended to make a racing chute for toy cars don’t fit together unless cut at corresponding angles. Think again, try again, and voila, it works!

As kids get more and more experience solving real world challenges, they not only begin to develop greater mathematical mastery, they’re also strengthening the ability to look at things from different angles, work collaboratively, apply logic, learn from mistakes, and think creatively. Hands-on math experience and an understanding of oneself as capable of finding answers— these are the portals to enjoying and understanding computational math.

Unfortunately we don’t have a big data pool of students who learn math without conventional instruction. This fosters circular reasoning. We assume structured math instruction is essential, the earlier the better, and if young people don’t master what’s taught exactly as it’s taught we conclude they need more math instruction. (“Insanity: doing the same thing over and over again and expecting different results.”)

But there are inspiring examples of students who aren’t formally instructed yet master the subject matter easily, naturally, when they’re ready.

1. The experiment done over 85 years ago by Louis Benezet showed how elementary school children can blossom when they’re free of structured math instruction.

2. Homeschooling and unschooling families around the world devote much less time to formal mathematics instruction. Studies indicate their children grow up to succeed in college, careers, and life with greater self-reliance and focus than their schooled peers. Interestingly, two different surveys of grown unschoolers showed that a much higher number of them work in STEM careers than schooled adults. The samples were small but intriguing. More proof? Many of our greatest science, technology, engineering, and mathematics contributors have already emerged from the homeschool community.

3. Democratic schools where children are free to spend their time as they choose without required classes, grades, or tests. As teacher Daniel Greenberg wrote in a chapter titled “And ‘Rithmetic” 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 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.”

We know all too well that students can be educated for the test, yet not understand how to apply that information. They can recite multiplication tables without knowing when and how to use multiplication itself in the real world. Rote learning doesn’t build proficiency let alone nurture the sort of delight that lures students to higher, ever more abstract math.

Conventional math education may also limit our concept of what math can do. As Stanford mathematician Keith 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 (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.

Interestingly, people who rely on mental computation every day demonstrate the sort of adroitness that doesn’t fit into our 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 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 their performance is closer to 37 percent.

We have the idea that memorizing, practicing, and testing is the only way to higher achievement. It’s hard to imagine why we still believe that when studies show that high test scores in school don’t correlate with adult accomplishments (but do line up with interpersonal immaturity).

There are all sorts of ways to advance mathematical understanding. That includes, but isn’t limited to, traditional curricula. It’s time to broaden our approach. Let’s offer the next generation a more intrinsically fascinating, more applied relationship to math. Let’s foster analytical and critical thinking skills across all fields. The future is waiting.

This article is one in a series of three on natural math. 

Math Instruction versus Natural Math: Benezet’s Experiment. What happened when formal math instruction was eliminated? 

Natural Math: 100+ Activities and Resources. Finding and learning from math in daily life. 

Portions of this article are excerpted from Free Range Learning.

Math Instruction versus Natural Math: Benezet’s Example

/ Laura Grace Weldon / 29 Comments

Louis Benezet, natural math,

1930’s classroom (forestpark4.wikidot.com)

Children are intrinsically eager and able to learn. If we step back from our limiting preconceptions about education, we discover learning flourishes when we facilitate it rather than try to advance it through force, intimidation, and coercion.

Over 85 years ago a pioneering educator proved that delaying formal instruction, in this case of mathematics, benefits children in wonderfully unexpected ways. Louis P. Benezet, superintendent of the Manchester, New Hampshire schools, advocated the postponement of systematic instruction in math until after sixth grade. Benezet wrote,

I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year-old child for knowledge of long division? 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.

While developing this rationale, Benezet spoke with eighth-grade students. He noted they had difficulties putting their ideas into English and could not explain simple mathematical reasoning. This was not only in his district; he found the same results with fourteen-year-old students in Indiana and Wisconsin. Benezet didn’t blame the children or teachers, he blamed introducing formal equations too early.  So he began an experiment, abandoning traditional arithmetic instruction below the seventh grade.

In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.

To start, he picked out five classrooms, choosing those districts where most students were from immigrant homes and the parents spoke little English. Benezet knew that in other districts the parents with greater English skills and higher education would have vehemently objected, ending the experiment before it started.

In the experimental classrooms, children were exposed to what we’d call naturally occurring math. They learned how to tell time and keep track of the date on the calendar. The students played with toy money, took part in games using numbers, and when dimension terms such as “half” or “double” or “narrower” or “wider” came up incidentally, they were discussed. Instead of math, the emphasis was on language and composition. As Benezet describes these children,

They reported on books that they had read, on incidents which they had seen, on visits that they had made. They told the stories of movies that they had attended and they made up romances on the spur of the moment. It was refreshing to go into one of these rooms. A happy and joyous spirit pervaded them. The children were no longer under the restraint of learning multiplication tables or struggling with long division.

At the end of the first school year, Benezet reported that the contrast between the experimental and traditionally taught students was remarkable. When he visited classrooms to ask children about what they were reading, he described the traditionally taught students as “hesitant, embarrassed and diffident. In one fourth grade I could not find a single child who would admit that he had committed the sin of reading.” Students in the experimental classrooms were eager to talk about what they’d been reading. In those rooms, an hour’s discussion went by with still more children eager to talk.

Benezet hung a reproduction of a well-known painting in the classrooms and asked children to write down anything the art inspired. Another obvious contrast appeared. When he showed the ten best papers from each room to the city’s seventh-grade teachers, they noted that one set of papers showed much greater maturity and command of the language. They observed that the first set of papers had a total of 40 adjectives such as nice, pretty, blue, green, and cold. The second set of papers had 128 adjectives, including magnificent, awe-inspiring, unique, and majestic. When asked to guess which district the papers came from, each teacher assumed that the students who wrote the better papers were from schools where the parents spoke English in the home. In fact, it was the opposite. Those students who wrote the most masterfully were from his experimental classes.

Yet another difference was apparent. It was something that Benezet had anticipated. He explained, “For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning faculties.” At the end of that first year, he went from classroom to classroom and asked children 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. Due to pressure from some school principals, children in the experimental classrooms were back to learning from a math book in the second half of sixth grade. All sixth-grade children were tested. By 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.

In 1936, the Journal of the National Education Association published the final article by Benezet. His results showed the clear benefits of replacing formal math instruction with naturally occurring math while putting a greater emphasis on reading, writing, and reasoning. The journal called on educators to consider similar changes.

As we know, schools went in the opposite direction.

Louis Paul Bénézet

Louis Paul Bénézet

This article is an excerpt from Free Range Learning. (Next post, the extraordinary benefits of emphasizing natural math over math instruction.)