“All the soarings of my mind begin in my blood.” Rainer Maria Rilke
I was called a “fidget” and a “wigglewump” when I was growing up. I was told to sit still and pay attention. I had no problem paying attention to library books I picked out, happily swinging my legs from a chair as I read for hours. But sitting in school or worship services made my whole body feel like a coiled spring. I behaved, but it took a lot of effort. This made it even harder to understand what the adult at the front of the room was droning on about. Long car rides were even worse. I thought everyone felt headachy and nauseated while traveling, so it didn’t occur to me to report this to a grown-up. All I knew is that I wanted the car to stop moving so I could be the one to move.
We aren’t a still sort of species. Even in utero our bodies receive changing stimuli constantly. Our mother’s movements rock us in a watery world. Her footsteps send thud-like vibrations through us. Her heartbeat, along with our far faster fetal heartbeat, sets up a percussive syncopation. Her breath, speech, and digestion, plus sounds from outside her body add to this ever-changing symphony. Sound pairs with sensation, over and over, throughout prenatal development linking movement with meaning.
Within seconds of being born, a newborn will reflexively grasp a finger and turn her head to find a nipple. Within a few months she will teach herself to grab objects, roll over, and stand with support. These aren’t just motor skills; each movement builds ever more complex neural pathways in her growing brain. As Hungarian pediatrician Emmi Pikler advocated back in 1946, babies do best when given freedom of movement. In an article titled “Exercise Affects Baby Brains,” Janet Lansbury writes,
[Dr. Pikler] “studied the contrasts between the children who had been taught, propped, positioned and restricted in devices like infant seats, walkers and bouncers, and those who were given freedom of movement and allowed to develop at their own rate. Dr. Pikler found that the natural approach not only affected the quality of motor skills, but also influenced ‘all other areas of growth – social, emotional, cognitive – and even character development.’ Pikler babies, as the children in her practice were known, could be easily distinguished at the parks in Budapest, because they were ‘poised and graceful, alert and friendly, and so confidently independent.’”
Babies expend effort comparable to world-class athletes as they master new abilities. Child development expert Karen Adolph describes, in a journal article titled “What Changes in Infant Walking and Why,” what it takes the average baby to teach himself to walk.
“Walking infants practice keeping balance in upright stance and locomotion for more than 6 accumulated hours per day. They average between 500 and 1,500 walking steps per hour so that by the end of each day, they may have taken 9,000 walking steps and traveled the length of 29 football fields…. Albeit intense, infants’ practice regimen is not like an enforced march of massed practice where walking experiences are concentrated into continuous time blocks. If practice were massed, the sheer amounts of daily practice would be even more astounding (the average cadence for a 14-month-old toddler walking over the laboratory floor, for example, is 190 steps per minute). Rather, infants’ walking experience is distributed throughout their waking day, with short periods of walking separated by longer rest periods where infants stand still or play….”
It seems exhausting, yet this is what natural learning looks like. The same extraordinary level of motivation continues as the growing child teaches herself, on her own timetable and in ways best for her, a whole spectrum of abilities through direct, real world experiences. That is the way we humans learn best.
Our brains evolved to help us confront and solve problems. We can’t separate learning from the rest of the body, or from the context of an individual life, yet that’s how we expect education to work. Despite the most caring and dedicated staff, the very structure of most schools is top-down, so lessons address students’ brains (mostly left brain) rather than their whole beings. This approach rewards only those students who can most easily narrow their full-body need to DO. Replacing traditional instruction with technology doesn’t make school any more physically engaging.
Learning sticks when our emotions and senses are active because, as psychologist Louis Cozolino explains, “visual, semantic, sensory, motor, and emotional neural networks all contain their own memory systems.”
For example, studies show we don’t master a foreign language best by studying grammar and memorizing words, nor by speaking it before we feel comfortable. Instead we learn most easily and effectively when we are interested in the message it conveys, like trying to decipher music lyrics or follow an instructional video in a different language. We learn better when we have IRL experiences to pair it with, like ethnic food eaten with native language speakers. And we learn better when our bodies are active, even more easily remembering foreign words when we learn them while making gestures.
Research shows that movement, even as small as hand movement, helps people unfamiliar with difficult subjects like organic chemistry understand and remember complex topics.
We learn best, from basic skills to academic subjects, when mind, body, and emotions are involved. Such experiences help to inform later understanding. Consider an introductory physics lesson aimed only at the brain. A student is presented with a concept, perhaps on the page or by online tutorial or lecture, and then must complete comprehension questions for a grade. Contrast this with learning that’s encoded through movement, as happens in play. That same student may already have discovered the principle herself, perhaps learning about centripetal force and acceleration by whirling a bucket of water in a full circle fast enough to keep the water contained or by discovering how fast a toy car needs to go around an upside down loop without falling. These play experiences make her much more likely to retain and build on what she has learned, and more likely to understand principles when they are more formally presented.
This kind of learning sticks with us. That’s what neurologist Frank Wilson noticed when he asked people at the top of their careers about their early experiences. Musicians, mathematicians, surgeons, engineers, artists, and architects all talked about formative hands-on experiences in childhood that were entirely unrelated to formal instruction. What they gained through play and doing chores became so integral to their later success that they recalled it many decades afterwards.
The problem is, kids are immobile for much of the school day. They sit through the journey to and from school. They sit doing homework. And many times they sit through what free time they have left. The average child spends just 4-7 minutes in outdoor free-play every day. All this sitting doesn’t help to develop the vestibular system. Muscle sensors (proprioceptors) react to input from this system to tell the body where it is in space. When the vestibular system isn’t developing properly a child may seem uncoordinated, resist trying new things, be afraid of crowds, bump into people, seem inattentive, have difficulty controlling impulses, or have trouble with reading and other academics.
To develop a strong vestibular system, kids need plenty of time every day to run, jump, climb, balance on uneven surfaces, and otherwise happily move their bodies in all directions. This is important. A developed vestibular sense supports spatial awareness, focus, self-regulation, and other abilities necessary for learning. And sports practice or gym class a few times a week isn’t enough, nor do adult-run programs offer the full-body freedom necessary for this development.
Pediatric occupational therapist Angela Hanscom writes in her book Balanced and Barefoot,
“As adults, we may feel that we always know what is best for our children. A child’s neurological system begs to differ. Children with healthy neurological systems naturally seek out the sensory input they need on their own. They determine how much, how fast, and how high works for them at any given time. They do this without even thinking about it. If they are spinning in circles, it is because they need to. If they are jumping off a rock over and over, it is because they are craving that sensory input. They are trying to organize their senses through practice and repetition.”
In an article titled “Why Children Fidget,” Ms. Hanscom writes about observing a fifth grade classroom near the end of their school day. She saw kids so desperate for movement that they were tilting their chairs, rocking their bodies, chewing on pencils, lightly smacking their heads. She tested the kids and found most had poor balance and core strength.
“In fact, we tested a few other classrooms and found that when compared to children from the early 1980s, only one out of twelve children had normal strength and balance. Only one! Oh my goodness,I thought to myself. These children need to move!
Children are going to class with bodies that are less prepared to learn than ever before. With sensory systems not quite working right, they are asked to sit and pay attention. Children naturally start fidgeting in order to get the movement their body so desperately needs and is not getting enough of to ‘turn their brain on.’ What happens when the children start fidgeting? We ask them to sit still and pay attention; therefore, their brain goes back to ‘sleep.’”
Research continues to indicate that movement is intrinsically linked to healthy development and learning in powerful ways.
Vigorous movement stimulates the birth of new neurons and is correlated with greater brain volume in the hippocampus, which plays a part in short and long-term memory.
Exercise boosts neurotransmitters necessary for attention, positive mood, and learning. It also produces proteins necessary for higher thought processes.
Studies show the sweet spot for kids is somewhere between 40 to 70 minutes of active movement a day for improved executive function, focus, and cognitive flexibility.
One study showed that after just two hours of playful activities like climbing trees, balancing on uneven surfaces, and navigating obstacles people temporarily increased their working memory capacity by 50 percent.
Overall, 60 years of research shows physical activity has overwhelmingly positive effects on kids’ mental health, cognitive abilities, and school achievement. The most fit kids are most likely to be the highest academic achievers.
Some kids can sit still and pay attention longer than others, but that’s not how we’re wired to learn best. In fact some educators point to research saying that after 20 minutes of inactivity, the neural communication networks in our brains function less effectively. And an analysis of nine studies indicate that the more time kids spend sitting, the more anxiety they are likely to experience.
Research shows kids actually fidget in order to better focus on complicated intellectual tasks. This is more noticeable in kids said to have ADHD, but it’s likely that foot-tapping and chair-scooting actually helps most kids store and process information. That’s why they are more likely to be restless working on math problems but relaxed while watching a movie or playing video games.
And several studies show that high levels of physical energy, a.k.a psychomotor overexcitability, is not only common but can be expected in highly intelligent children.
Some kids may grow out of the “fidget” and “wigglewump” stage, but I never did. I can’t even easily sit through a restaurant meal without stifling the urge to misbehave. My kids have blackmail-worthy stories about this. (Fortunately we can’t afford many restaurant meals.) Writing this essay required lots of breaks to walk the dogs, make a snack, do barn chores, talk with dear ones, and otherwise distract myself. If only today’s students were equally free to get out of their seats and move.
“The world is mud-luscious and puddle-wonderful.” ~e.e. cummings
One street over in the neighborhood where I grew up was a small pond where ducks congregated. The ducks lifted from the water with reluctant quacks when we showed up. Despite summer’s heat, the pond was always cool. Aquatic plants waved their greenish fronds just below the surface and the bottom was lined with a thick layer of muck. My sister, a budding naturalist, speculated that the muck was made up of decayed plant matter. When we waded in, our feet sunk into that thick layer of soft goo, a squishy delight for our toes.
It might occur to you that we were standing in duck poo. You would be right.
It smelled a bit when the water was stirred up, but that didn’t bother us. My sister and I would crouch near the edge watching insects. Water striders scurried on the surface. Each of their steps made a faint impression in the water as if they walked on gel. Beetles, ants, and the creatures my sister called by the fairytale name nymphs scampered through pondside plants. She liked to let insects climb up her arms. I was impressed, but too squeamish to copy her. Most magical of all were the dragonflies, their huge eyes looking back at us as they hovered on iridescent wings. This seemed like a separate world.
Eventually we had to return home. Our mother, a registered nurse who strictly adhered to standards like rigorous hand washing and early bedtimes, didn’t miss what we’d been up to. We came home spattered and stinky. But her only rule was that we strip off our clothes and scrub ourselves. She’d call from somewhere in the house, “be sure to use the nail brush!” She didn’t seem to mind that we’d walked a block away to play in a bacteria-infested pond as long as we scrubbed away all traces afterwards.
My mother was on to something.
In her book Balanced And Barefoot, pediatric occupational therapist Angela Hanscom writes about teachers, parents, and medical professionals who are alarmed by ever-growing numbers of children who can’t sit still or pay attention; who have trouble with coordination, balance, or sensory processing; who are fearful, easily frustrated, or act aggressively. She explains that these problems can be connected to an overly contained childhood, one that has become the norm. Restrictions begin in the earliest months, when babies spend hours each day strapped into strollers, car seats, and baby seats. As they get older their movements are curbed by passive indoor activities. Even outdoors, kids are often limited to low-challenge play areas or to prescribed movement in adult-structured programs.
The push for academics, often starting in preschool, strips even more time from active free play, while elementary schools are increasingly limiting or eliminating recess. This is profoundly counterproductive. Reading readiness is strongly influenced by physical movement. So is grasping and using mathematical concepts.
It helps to understand just how closely movement and sensory input is related to development. In the first year of a baby’s life, her brain doubles in size. It reaches 80 percent of its adult volume by age three. Babies are born with vast numbers of neuronal links in their brains and spines, primed to be shaped by what they encounter. Unused networks are not activated and disappear. This is what neuroscientists call “experience-dependent plasticity.” Early experiences rich in movement (plus the nurturance and emotional warmth that set the foundation for learning) activate a wider range of neural connections. This is nature’s wisdom at work, shaping a child’s brain through experience so they develop what they’ll need for the world they’re born into.
Your baby squirms and cries after a few minutes in the high chair. He can be placated with a new food or a spoon to bang on the tray, but only for a few minutes. He wants to get to work on crawling. Your toddler resists being put in her car seat and sometimes cries until she exhausts herself. She wants to run, climb, and play. They’re both responding to an inborn need to learn through movement.
Gill Connell and Cheryl McCarthy, authors ofA Moving Child is a Learning Child, clarify. They write that neural pathways developed in the first years of life,
“determine how a child thinks and learns, but more importantly, they will shape who she becomes… her passions and pursuits, triumphs and challenges, inner reflections, outer reactions, and outlook on life…all flowing through the neural network built by her earliest physical and sensory experiences.
With breathtaking simplicity, nature has created this move-to-learn process to be both dynamic and self-perpetuating, building the body and brain simultaneously. As such, the more a child moves, the more she stimulates her brain. The more the brain is stimulated, the more movement is required to go get more stimulation. In this way, nature gently coaxes the child to explore beyond her current boundaries toward her own curiosity to acquire new capabilities.”
Overall, today’s kids show decreasing core strength and flexibility compared to averages in the 1980’s.”The more we restrict children’s movement and separate children from nature,” Angela Hanscom explains, “the more sensory disorganization we see.” That’s why she advocates sensory-rich, movement-based outdoor free play. Chasing, rolling down slopes, climbing trees, playing with nature’s play-perfect loose parts like leaves and sticks — these and other experiences build spatial awareness, balance, fine motor skills, and bodily control.
Let’s hone in on one sensory-rich experience; going barefoot. Madeline Avci, an Australian pediatric occupational therapist, explains that walking on grass, stones, and sand develops body awareness, called proprioception. Nerve endings in the feet and toes promote the development of sensory pathways, building functional movement patterns while helping children move with a sense of their body in space. When we wear shoes, the quality of sensory information is diminished. A paper published in Podiatry Management details all sorts of ways shoes, including those with flexible soles, interfere with a child’s gait, development, and posture. Walking barefoot also promotes better biomechanics, a more natural gait, and less pressure on our feet. Bones in the feet are not fully ossified until the late teen years, so the more barefoot time possible, the more naturally the foot’s shape can develop. Of course few of us are raising our families in a beachside hut where walking barefoot makes sense year-round. But Katy Bowman, author of Move Your DNA, suggests that all of us try to walk on natural surfaces like sand, rocks, grass, or wood for 10 to 20 minutes a day whenever possible, and to go barefoot at home.
Neurobiologist Gerald Hüther notes that “the most important learning experiences come to us, essentially, by way of our bodies — which means that learning is always an experience of the whole body. At the same time, every learning experience involves emotions. We are only able to learn when the so-called emotional centers in the brain are activated. These centers release neuroplastic messenger substances enabling what has been learned to become anchored in the brain ….[via] emotional activation. The most enjoyable activation we know of is ‘enthusiasm.'”
It’s ridiculous that we need science to confirm the value of enthusiasm. This is the energy each child brings fresh to the world. What they’re able to explore and experience with the whole of themselves, magnified by the capacity for awe, remains with them.
Dr. Hüther gives an example,
“Children living in the Amazon forests learn 120 different shades of green and can name them all, using 120 different terms. Potential of that kind is either used in practice or is little used. Children here can at best distinguish light green, green, and dark green. How far a potential is actually used depends on how important it is .. in a given culture…The result is that what was once a possibility, this potential, …if not used, will just wither away.”
Enthusiasm goes a long way toward explaining why children and nature go together so well. Children are themselves magic — able to shape shift into a toad or hawk, to feel what it’s like to hop nearly hidden under leaves or to glide on the air’s invisible currents. While imagination is alive everywhere, it can’t help but flourish when surrounded by aliveness. The more natural an area, the more kids have a chance to have meaningful encounters with the life around them. In fact, kids play differently in a park with play structures compared to more natural areas like an overgrown field, a row of trees, or a small creek.
As Richard Louv details inLast Child in the Woods, kids confined to structured play areas have poorer balance and agility than those who play in unpaved areas. The social dynamic changes too. Older and physically larger kids dominate on playgrounds but in more natural areas, it’s the creative kids who act as leaders. In wild places, even an overgrown lot, kids are more likely to incorporate each other’s ideas into expressive make-believe scenarios using their dynamic surroundings—tall grasses become a savannah, tree roots become elf houses, pine cones become treasure. The essence of the child comes alive. Outdoor play in natural areas is more likely to include peers of differing ages and abilities. Regular outdoor experiences not only boost emotional health, memory, and problem solving, they also help children learn how to get along with each other.
Outdoor free play also inspires kids to challenge themselves. They are things to climb on and places to explore. In pursuit of fun, kids ignore minor annoyances like cold fingers, sharp briars, stumbles. Kids face and overcome fears. Such play is linked to greater social skill, resilience, and creativity.
And if you’re interested in academic test results, time outdoors has an impact there too. Here are a few encouraging studies.
Kids exposed to more nature had higher scores of working memory than kids who did not.
Simply going for a walk in a nature area, in any weather, can significantly improve memory and attention spans improved by 20 percent after people spent an hour interacting with nature.
If pre-college test results perk up your interest, the children most connected to nature are also most likely to score well in tests including the SAT.
We also know exposure to bacteria can be a good thing. Certain bacteria found in soil, Mycobacterium vaccae, have been found to boost the release of serotonin, a neurotransmitter linked to positive mood and enhanced learning.
Dirt may improve our health too. For example, children who grow up exposed to a greater range of soil microorganisms have been found, in some studies, to have a lower incidence of asthma. Heck, even common bacteria on our skin have been shown to cut down on rashes and reduce inflammation when we’re cut or bruised. A child’s exposure to dirt is part of the body’s education, microbiologist Mary Ruebush explains inWhy Dirt Is Good, “allowing his immune system to explore his environment.” (She adds a caveat, saying that the soil in some urban areas may be contaminated with heavy metals such as lead. That is indeed a wakeup cry. Soil is the structure we need to feed ourselves. When it’s poisoned, so is life.)
The importance of outdoor free play is getting a lot of attention these days. Playground designers, schools, and daycare programs are far more open to the benefits of outdoor free play with natural materials. It’s no surprise that children do better with natural sensory play, places for solitude, and opportunities for self-chosen challenge. This is a step in the right direction. A few steps farther and we’ll let kids back in nature herself to play in woodlands, fields, and beaches as well as back lots, mud puddles, and all the nature around us. It’s a step in the direction of wonder and delight, maybe even in the muck of a duck pond.
“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 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.
Alison Beth Miller — first American woman to win gold at the International Mathematical Olympiad.
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.
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.
School systems often point to families like mine as examples. We prioritized outdoor play, read aloud daily, took our kids to museums, did chores together, and had a family dinner every night. Still, school didn’t really work for my kids. Our five-year-old could read well but still had to complete endless pre-reading worksheets along with his kindergarten class; our eight-year-old’s teacher kept insisting he be medicated for ADHD symptoms we never saw at home; our eleven-year-old was expected to do grade-level busywork although she tested at high school and college levels; our teen was bored by AP classes and hassled by bullies at school.
I dug in, unwilling to give up. How else, I reasoned, can institutions evolve without people pressing for changes from within? Ever since my first child entered school I’d headed PTA committees, volunteered in classrooms, and participated in fund-raisers hoping to effect some of those changes.
Sure, there were a few parents who weren’t fond of my gentle rabble rousing. I never quite shook the negative impression some people had of me as that mom who changed the yearly ritual of first grade hot dog night to first grade popcorn night, or as the one who turned down free Sea World field trips for my kids because I didn’t want them to learn about marine mammals as captive performers.
But all of us parents grumbled in solidarity; united in misery over so many tests, so much homework, so little play. It wasn’t lost on us that we were railing against the very structures that we also “had” to enforce if our kids were to succeed in school. These were overwhelming constraints indeed, many tied to big money.
Corporate influence was present everywhere. Free nutritional posters sent by candy manufacturers on cafeteria walls; software offered by petrochemical companies for science classes; math materials provided by credit card companies. Channel One beamed commercials along with daily snippets of news wrapped in PR-speak.
Parents felt helpless to stem this tide. So did teachers and administrators, who insisted they couldn’t turn down free resources when budgets were so tight. (They’re right. Overall funding for schools nationwide has dropped from 2008 to today due to state and local austerity measures.)
For several years I volunteered as the parent liaison with the district’s food service contractor, but this private company was so focused on profits that fresh produce meant little more than mealy apples, shredded lettuce, and tasteless baby carrots. All my efforts simply resulted in a wider variety of similarly unappealing offerings. When parents demanded that the school stop allowing the sale of chips, ice cream bars, and candy at lunch time the company threatened to back out of their contract altogether. And our offer to develop a school vegetable garden? Turned down. No extra time in the academic calendar for kids to get involved.
Add to that the effect of big money on local schools:
Over 20 million dollars are spent on lobbying by the world’s four largest education corporations to sway policies toward ever more student assessment, effectively trapping students and teachers in a testing mill that steers several billion taxpayer dollars back into those companies.
Extraordinary profits are made on grading software, data tracking, e-schooling, for-profit charter schools, even GED testing and teacher licensing exams. That’s pretty much how we’ve gotten to our current test-crazed educational system.
Decades of standardizing and testing has demoralized teachers, stressed students and families, and dropped public opinion of schools so low that it’s easy for hedge fund managers, education corporations, and other private interests to steer ever more taxpayer dollars into for-profit charters and cyber schools,
But my kids attended a good elementary school with highly motivated parents and positive changes did happen. For example, parent volunteers instituted and ran an annual Art Day, a glorious new tradition. One full school day each year we parents arranged to put artists in every classroom. They demonstrated their work and gave kids hands-on experience. Teachers took students from room to room to learn from sculptors, potters, cartoonists, printmakers, wood carvers, calligraphers, weavers, painters, and others. The whole building was alive with creative enthusiasm. Even then, with so much educational richness available, some teachers didn’t allow children to participate in this all-school program until they’d finished their homework or written “I will keep my hands to myself” 20 times — these kids left in the hallway were, not coincidentally, often minority students.
Although my optimism had waned, I still held on to a shred of hope that there was value in working from within the system to change it. That ended the day my oldest son was threatened by a gun-carrying student in the school hallway. The next day we became homeschoolers.
I may have been slow to react, but that’s often the way collective intelligence looks on an individual level.
We humans form institutions for the value they offer to society. Collectively these structures function with an intelligence based on what works. Ideally, whatever works persists and whatever doesn’t work fades away. But sometimes institutions become resistant to change or change in ways that make them more rigid and therefore less responsive. When that happens, people who work for or are served by that institution tend to suffer. It usually takes a certain amount of irritation, unfairness, or real misery before people step back and take a look at the institution itself. Suffering has a way of making us more fully aware and more authentically invested in change. So we react. We resist, compete, struggle, debate, discuss, break away, collaborate, and reinvent.
More and more we see people resisting the structure of institutional education in its current form.
Families are homeschooling and unschooling in ever-increasing numbers.
We are choosing to integrate learning into our daily lives and our communities. Our choices show the sort of fluid responsiveness that shifts ingrained beliefs about what education is and what it can be.
This is collective intelligence in action.
Whether we intend to impact the larger civic good or not, the collective intelligence of our culture is continually refined as we seek out more conscious and life-enhancing ways to live. It takes a small percentage of people to change a cultural mindset. Often it seems that this kind of wider awareness can’t come soon enough. But as philosopher Arthur Schopenhauer observed, “All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident.”
To learn to read is to light a fire; every syllable that is spelled out is a spark. – Victor Hugo
Reading readiness and reading advancement has little or nothing to do with educational toys, apps, or enrichment programs. It has much more to do with what kids naturally like to do: move their bodies, enjoy stories, take part in conversation, and play freely.
Why?
Movement helps children develop sufficient brain-body maturation so they can successful decode abstract symbols into meaning. This includes complex neurological pathways as well as sufficient kinesthetic awareness and proprioceptive sense. (Find out what movements are essential in “Reading Readiness Has To Do With the Body.”)
Reading aloud every day, starting in babyhood, helps children associate reading with closeness and pleasure. Even a board book builds vocabulary, demonstrates left to right sequencing, and promotes comprehension. We can fold reading time into daily rituals like story time before naps and again after dinner. We can also show how much we value reading by letting kids see us reading our own books and magazines.
As kids get older it’s important to avoid offering rewards for reading or make reading a precondition for privileges. That’s because rewards, even for something kids already enjoy, significantly diminishes their own intrinsic motivation. Telling kids “20 minutes of reading before you can play games on the tablet”casts reading as an obligation, leading kids to devalue reading while enhancing the appeal of digital entertainment. (No wonder “eat your broccoli before you can have ice cream” makes broccoli the enemy and ice cream even more tempting.)
Stories stretch the mind and imagination. They help us, at any age, develop empathy and give us a larger context for our own lives. That’s not limited to the page. There’s extraordinary power in telling family stories. When we share tales of our doubts, misdeeds, and triumphs we’re not only building family cohesiveness, we’re also (according to science) helping kids grow up with greater confidence and self-control.
Daily conversations, including all those questions kids ask, helps them advance in reasoning and social skills while bringing us closer to each other. Let’s admit, a great deal of parent and child interaction isn’t true conversation so much as directives, complaints, and reminders (because, well, life) so it helps to create openings for conversation. Hold a space for kids to talk about what’s on their minds —- this often seems to happen on a walk, a drive, or at bedtime —- good times to avoid earbuds and screens. Make a practice of showing you’re listening by using eye contact and avoiding interruption. Talk about big issues and dilemmas in your lives, in your community, and in the news. Big topics have a way of stretching young minds.
Free play is an essential part of childhood. It also helps kids develop the skills necessary for reading well. It may look like fun, but in ways deeper and more vital than we can imagine play is a process of learning. We don’t have to engineer their play. Play is, and always has been, a universal language. Give kids as much time for free play as possible. But when you want to play along, here are a few ideas.
Play Cherries & Pits to get conversations started. Very simply, each person takes turns telling the best things (Cherries) about their day and the worst things (Pits) about their day.
Tell round robin stories. One person starts a story with a character and setting (“The elf woke up to find a large bird staring at him.”). The next person adds a few sentences before passing it along to the next person. This works well with as little as two people and nearly always becomes amusingly improbable.
Turn socks into puppets for impromptu plays. Puppeteers can hide behind a couch or sheet-covered table to perform, although socks in my house tend to talk on their way to the laundry.
Make story stones (pictures on stones or tiles) and grab a few to prompt a story idea. Other stones can be added as the story goes on.
Ask off-the-wall questions. “Would you rather be a monkey or a lion?” “What would it be like if people had wings?” “If we could go on an adventure together what would we do?”
Write messages to each other. Scratch a few words in the sand, leave a message in magnetic letters, designate a place (under each other’s bed pillows, perhaps) where secret notes can be left, share a question and answer journal (taking turns asking and answering any and all questions), and leave little love letters for kids to find.
Sing songs with familiar tunes and invented lyrics. Those tend to be somewhat scatological in my family, a favorite faux opera here has to do with encouraging dogs to go out and get their elimination duties over with….
Games
Play impromptu memory games. For example, take turns tapping out a beat, seeing if the next person can repeat it. Or try imitating movements in sequence (first person jumps, the other person jumps and adds clapping, first person jumps and claps and adds a turkey gobble, and so on). Or take turns memorizing a sequence of unrelated words to repeat back in two minutes or ten minutes or the next day. Be prepared to lose to your kids!
Encourage classic games like checkers, mancala, and chess. Games of all kinds typically help kids understand sequencing, grouping, and memory. No need to choose specifically educational games.
Set aside one evening a week as a family board game night or set up a kids’ game club with friends. (There are even great games for kids three and under like Roll & Play, First Orchard, and Feed the Woozle.)
Waiting in line with kids? Find objects that begin with each letter of the alphabet together, from avocados to zeros. Or play the classic Going on a Picnic game. Start by saying, “I’m going on a picnic and I’m bringing an aardvark (or any “A” word). The next person continues with “I’m going on a picnic and I’m bringing an aardvark and a basketball (or any word starting with a B) and so on. The last person to remember and repeat the list is the winner.
Encourage kids to draw maps of places they know well (your kitchen, your house, your street) and maps of imaginary places (alien planets, mythic kingdoms, ninja training camps). Draw a map of where you’ve hidden packed lunches for them to discover or the bedtime chapter book you’ll read.
Encourage children to set up obstacle courses. Indoors this may include three somersaults through the hall, chairs to wriggle under, a rope to hop over, and a bunk bed ladder to climb. Outdoors the course can be more ambitious.
Enjoy regular treasure hunts. First hide a prize or two. Then place clues through the house or yard. These can be simple words or sentences, symbols, or pictures. Each clue leads to the next. The prize doesn’t have to be a toy or candy (it could be a note saying “we’re going to the park!”) the fun is in the hunting. Encourage children to set up their up treasure hunts too.
Letterboxing combines walking, navigation, and solving riddles. Clues help seekers find “letterboxes” hidden outdoors. Seekers mark their logbooks with a rubber stamp found in this box, mark a logbook in the box with their own personal stamp, then leave the box for the next seeker. For more information and links to regional clues, check with organizations such as Letterboxing North America or Atlas Quest. Or use the guidebook, It’s a Treasure Hunt! Geocaching & Letterboxing.
Try orienteering. This sport combines navigation, map reading, and decision-making. Participants walk, run, bike, or ski using a map and compass to choose the best route on or off the trail. Consult Orienteering Made Simple And Gps Technology: An Instructional Handbookby Nancy Kelly.
Take turns playing Line Zombie. Draw a line on paper with a pencil or on the ground with chalk, using arrows to indicate direction. The other person must follow the line either by tracing on the paper with marker or walking on the chalk line. Zombie noises optional.
Learning is a whole experience of mind, body, and self in relation to the world
When you pick up an orange you feel its texture and weight in your hand. You breathe in scent emitted by the brightly colored rind. If you’re hungry, you peel and section it to savor piece by piece. A fresh orange has phytonutrients, fiber, minerals, and vitamins that promote health. And it tastes wonderful.
It’s possible to purchase the separate nutritional components of an orange. You simply buy vitamin C, vitamin A, flavonoids, B-complex vitamins, fiber, potassium, and calcium in pill form. Of course replacing an orange with supplements is ridiculously expensive compared to the cost of consuming the fruit itself. And isolated compounds don’t work as effectively in the body as the whole fruit. Besides, where is the sensation of biting into an orange bursting with juice? Lost. Divided into a fraction of the experience.
Imagine being told in your earliest years that pills were superior to food and should replace it as often as possible. Even if handfuls of supplements were deemed more valuable than food by every adult in your life you’d still clamor to eat what you found appetizing. If meal-substitution pills became mandatory for children once they turned five years old, you’d never relate to food (or its replacement) the same way again. The body, mind, and spirit reject what diminishes wholeness.
Don’t argue. Just take it.
Yet that’s an apt analogy for heavily structured education, where learning is set apart from the threads that connect it to what has meaning and purpose for the learner. Conventional education separates learning into thousands of measurable objectives. It has very little to do with a child’s hunger to master a particular skill or thirst to pursue an area of interest, in fact such appetites tend to interfere with institutional requirements. It’s not designed for the whole child but aimed at one hemisphere of the brain, doled out in pre-determined doses and repeatedly evaluated. The most gifted, caring teachers are stuck within systems that don’t acknowledge or understand natural learning. In fact, most of us believe, however grudgingly, that schooling is necessary for learning without recognizing that damage is done.
For the very youngest children, learning is constant. Their wondrous progress from helpless newborn to sophisticated five-year-old happens without explicit teaching. They explore, challenge themselves, make mistakes, and try again with an insatiable eagerness to learn. Young children seem to recognize that knowledge is an essential shared resource, like air or water. They demand a fair share. They actively espouse the right to gain skills and understanding in a way that’s useful to them at the time.
Although we have the idea that learning flows from instruction, when we interfere with natural learning children show us with stubbornness or disinterest that it has nothing to do with coercion. Children often ignore what they aren’t ready to learn only to return to the same concept later, comprehending it with ease and pleasure. What they do is intrinsically tied to why they do it, because they know learning is purposeful. They are curious, motivated, and always pushing in the direction of mastery.
Learning is a hunger too.
But schooling irrevocably alters the natural process of learning for every single child.
The very structure of school makes children passive recipients of education designed by others. They cannot charge ahead fueled by curiosity, pursuing interests wherever they lead. Although interest-driven learning results in high level mastery, the top priority in school is completing assignments correctly and scoring well on tests. Despite what individual children want to learn, value is given to what can be evaluated.
Segregated by age, children are limited to examples of behavior, reasoning, and ability from those at a similar level of maturity. They have little exposure to essential adult role models and minimal engagement in community life. They’re also deprived of the opportunity to practice the sort of nurturance and self-education that happens when children interact in multi-age settings. Even collaboration is defined as cheating.
A child’s natural inclination to discover and experiment is steered instead toward meeting curricular requirements. Gradually the child’s naturally exploratory approach is supplanted by less meaningful ways of gathering and retaining information.
The mind and body are exquisitely cued to work together. Sensory input floods the brain, locking learning into memory. Movement is essential for learning. The emphasis in school, however, is almost entirely static, and almost entirely focused on left-brain analytical thinking. Many children ache for more active involvement, but their attempts to enliven the day are labeled behavior problems. The mismatch between school-like expectations and normal childhood behavior has resulted in millions of children being diagnosed with ADHD.
Coming up with the correct answer leaves little room for trial and error. Thinking too carefully or deeply may result in the wrong answer. The right answer from a child’s personal perspective may actually be the opposite of the correct answer, but to get a good mark the child cannot be true to his or her experience. The grade becomes more important than reality.
Emphasis on the correct answer squeezes out unconventional thinking. The fear of making mistakes squelches creativity and innovation. After years of being taught to avoid making mistakes, the child has also learned to steer clear of originality.
Readiness is pivotal for learning, particularly in reading. In school, reading is used to instruct in every other subject, so the child who doesn’t read at grade level quickly falls behind. The subject matter in school, even when taught well, isn’t necessarily what the child is ready to learn. The way it is presented tends to be indirect, inactive, and irrelevant to the child. Schoolwork repeatedly emphasizes skill areas that are lacking rather than building on strengths, or goes over skills already mastered with stultifying repetition. Neither approach builds real learning
The desire to produce meaningful work, the urge to make contributions of value, the need to be recognized for oneself, and other developmental necessities are undercut by the overriding obligation to complete assignments.
Conventional education takes the same approach to a six-year-old and an 18-year-old: assignments, grades, tests. Self-reliance and independence doesn’t easily flourish in such a closed container.
Children must hurry to do the required work, then change subjects. The information is stuffed into their short-term memories in order to get good grades and pass tests, even though such tests tend to measure superficial thinking. In fact, higher test scores are unrelated to future accomplishments in such career advancement, positive relationships, or leadership. Students aren’t learning to apply information to real life activities nor are they generating wisdom from it. The very essence of learning is ignored.
Schoolwork clearly separates what is deemed “educational” from the rest of a child’s experience. This indicates to children that learning is confined to specific areas of life. A divide appears where before there was a seamless whole. Absorption and play are on one side in opposition to work and learning on another. This sets the inherent joy and meaning in all these things adrift. The energy that formerly prompted a child to explore, ask questions, and eagerly leap ahead becomes a social liability. Often this transforms into cynicism.
When young people are insufficiently challenged or pushed too hard, they do learn but not necessarily what they’re being taught. What they learn is that the educational process is boring or makes them feel bad about themselves or doesn’t acknowledge their deeper gifts. They see that what they achieve is relentlessly judged. They learn to quell enthusiasm and suppress the value-laden questions that normally bubble up as they seek to grow more wholly into themselves. Gradually, their natural moment-to-moment curiosity is distorted until they resist learning anything but what they have to learn. This is how the life force is drained from education.
We’re so committed to structured, top-down instruction that we impose it on kids beyond the school day. Young people are relentlessly shuttled from the classroom to enrichment activities to organized sports and back home to play with educational toys or apps when there’s very little evidence that all this effort, time, and money results in learning of any real value.
Many of us think that education has always been this way—stuffing information into young people who must regurgitate it back on demand. Based on dropout numbers alone, this approach doesn’t work for at least a quarter of U.S. students. So we advocate copying Finland or Singapore, using the newest electronics, taking away testing, increasing testing, adding uniforms or yoga or chess or prayer. We’ve been reforming schools for a long time without recognizing, as Einstein said, “You cannot solve a problem from the same consciousness that created it.”
Figuring something out is itself a delight.
Structured education is actually very new to the human experience. Worse, it actually undermines the way children are primed to advance their abilities and mature into capable adults. That’s because most of the time humanity has spent on Earth has been as nomadic hunter-gatherers, before the advent of agriculture. This time span comprises approximately 98% of human history. Although our culture and lifestyle have changed considerably, our minds and bodies have not. Like our earliest ancestors we are still tuned to nature’s rhythms, cued to react quickly to danger, desire close interdependence with a cohesive group of people, and need in our earliest years highly responsive nurturing that gradually fosters our abilities.
Studies of isolated groups who continue to live in hunter-gatherer ways have shown us that during this era (and throughout most time periods afterward) babies are breastfed and remain in close contact with their mothers for the first few years. This results in securely attached infants who are more likely to grow up independent, conscientious, and intellectually advanced.
Their children play freely in multi-age groups without overt supervision or direction by adults. Such free play promotes self-regulation (ability to control behavior, resist impulse, and exert self-control) which is critical for maturity. Play fosters learning in realms such as language, social skills, and spatial relations. It teaches a child to adapt, innovate, handle stress, and think independently. Even attention span increases in direct correlation to play.
Playfulness can’t be separated from learning. Children watch and imitate the people around them. The child’s natural desire to build his or her capabilities doesn’t have to be enforced. Instruction happens when the child seeks it. The learning environment is particularly rich when young people are surrounded by adults performing the tasks necessary to maintain their way of life. Children naturally learn as they playfully repeat what they see and begin to take part in these real life tasks. Mastering all the skills for self-reliance isn’t easy. Hunger-gatherer children must recognize thousands of species of plants and animals as well as how to best obtain, use, and store them. They must know how to make necessary items such as nets, baskets, darts, carrying devices, clothing, and shelter. They need to learn the lore of their people and pass along wisdom through story, ritual, and art. And perhaps most importantly, they need to be able to cooperate and share in ways that have allowed humanity to thrive. In such cultures, children learn on their own timetables in ways that best use their abilities.
It’s about curiosity and awe.
We don’t have to live as hunter-gatherers do to restore natural learning to children’s lives. Homeschoolers and unschoolers have been doing this, quite easily, for a very long time. Our children learn as they are ready and in ways that augment strong selfhood. They stay up late to stargaze or make music or design video games, knowing they can sleep late the next morning. They may fill an afternoon reading or actively contribute to the community. They have time to delve into topics of interest to them, often in much greater depth and breadth than any curriculum might demand. They explore, ask questions, volunteer, hang out with friends of all ages, take on household responsibilities, daydream, seek challenges, make mistakes and start over. They’re accustomed to thinking for themselves and pursuing their own interests, so they’re more likely to define success on their own terms. Because homeschooing/unschooling gives them the freedom to be who they already are, it pushes back against a world relentlessly promoting narrow definitions of success.
This kind of natural learning isn’t just an antidote to the soul crushing pressure of test-happy schools. It’s the way young people have learned throughout time.
Let children sleep in. Let them dream. Let them wake to their own possibilities.
Flow is “a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it.” ~ Mihaly Csikszentmihalyi
My daughter spent much of this week with a deer skeleton she found in the woods.
As she searched the site she was thrilled to find most bones intact. My only involvement was providing toothbrushes and bleach to clean them.
Today she’s reassembling the skeleton in the driveway. She shows me how the back legs fit into the hip sockets, giving the deer power to leap and run while the front legs are mostly held on by bone and connective tissue.
She points out that the spine is somewhat similar to a human spine in the lower thoracic and upper lumbar regions, but very different where the large cervical vertebrae come in.
I know so little about this topic that I forget what she’s telling me while she speaks.
Handling the bones carefully, she faithfully reconstructs the skeleton. She’s so deeply engrossed in the project that she hasn’t come in for lunch or bothered to put on a jacket to ward off the chill.
Her interests are far different than mine, but I know what it’s like to be this captivated.
You know the feeling too. You become so absorbed in something that time scurries by without your notice. Your whole being is engrossed by the project. You feel invigorated.
Skiers call it becoming “one with the mountain.” Athletes call it being in the “zone.” Psychologist Mihaly Csikszentmihalyi has termed it the “state of flow.”
In this marvelous state the boundaries between you and your experience seem fluid, as if you are merging with what you’re doing. The more opportunities any of us have to immerse ourselves in activities we love, especially those that stretch us to our full capacities, the more capable and centered we feel in other areas of our lives.
Photo by Claire Weldon
Children, especially the youngest ones, slide into flow effortlessly. While playing they concentrate so fully that they lose sense of themselves, of time, even of discomfort. They’re inherently drawn to full-on engagement. As Csikszentmihalyi explains in Flow: The Psychology of Optimal Experience,
Contrary to what we usually believe, moments like these, the best moments in our lives, are not the passive, receptive, relaxing times—although such experiences can also be enjoyable, if we have worked hard to attain them. The best moments usually occur when a person’s body or mind is stretched to its limits in a voluntary effort to accomplish something difficult and worthwhile. Optimal experience is thus something that we make happen.
For a child, it could be placing with trembling fingers the last block on a tower she has built, higher than any she has built so far; for a swimmer, it could be trying to beat his own record; for a violinist, mastering an intricate musical passage. For each person there are thousands of opportunities, challenges to expand ourselves.”
Kids demonstrate flow when they’re eagerly drawing, building, climbing, pretending, reading, exploring—-however rapt involvement captures them. Their intent focus makes a mockery of what is supposedly a child’s developmental handicap — a short attention span.
Flow truly puts a person in the moment. No wonder it can be hard for our kids when we call them away from what they’re doing to what we deem more important. No wonder they might be more enthusiastic about playing with Legos than taking part in a structured geometry lesson.
Imposing too many of our grown-up preoccupations on kids can teach them to block the experience of flow.
What do we need to remember about this state?
Flow is typically triggered:
when a person’s abilities are stretched nearly to their limits
during a self-chosen pursuit
when they are looking to accomplish something worthwhile to them.
These characteristics are also the way we’re primed to learn from infancy on. It’s been called the Goldilocks Effect. This means we are attracted to what holds just the right amount of challenge for us. Not too big a challenge, not too little, but something that sparks our interest and holds it close to the edge of our abilities, moving us toward greater mastery.
That’s pretty much the way science, art, and other major human endeavors happen too. Flow may indeed be our natural state.
It doesn’t have to be complicated. Here are some ways to allow more flow in your kids’ lives (and yours too!).
Foster a calm, relaxed environment.
Engage in what brings out delighted fascination. If you’re not sure what that is, fool around with something hands-on. Tinker, paint, write, sculpt with clay, take something apart, dance, experiment—-whatever feels enticing.
A drive toward complexity, luring us to increase challenges, broaden our range of abilities, even face anxiety and boredom as we access an ever more profound state of engagement. (As A Playful Path author Bernie DeKoven explains here.)
Dr. Csikszentmihalyi’s work tells us achieving the flow state regularly is a key component of happiness.
That’s vital, even if it means you end up with a deer skeleton in your driveway.
“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.
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.
~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.
~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.
~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.
~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.
~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:
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.
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.
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.
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.
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offers many moving math activities for children.
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it’s a great time to pique their interest using almanacs and atlases.
, a group of students at the Sudbury Valley School approached him saying they wanted to learn arithmetic. He tried to dissuade them, explaining that they’d need to meet twice a week for hour and a half each session, plus do homework. The students agreed. In the school library, Greenberg found a math book written in 1898 that was perfect in its simplicity. Memorization, exercises, and quizzes were not ordinarily part of the school day for these students, but they arrived on time, did their homework, and took part eagerly. Greenberg reflects, “In twenty weeks, after twenty contact hours, they had covered it all. Six year’s worth. Every one of them knew the material cold.” A week later he described what he regarded as a miracle to a friend, Alan White, who had worked as a math specialist in public schools. White wasn’t surprised. He said, “…everyone knows that the subject matter itself isn’t that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step. The only way we have a ghost of a chance is to hammer away at the stuff bit by bit every day for years. Even then it does not work. Most of the sixth graders are mathematical illiterates. Give me a kid who wants to learn the stuff—well, twenty hours or so makes sense.”
that we’re schooled to express math in formal terms, but that’s not necessary for most of us—no matter what careers we choose. People who rely on mental math in their everyday lives are shown to have an accuracy rate around 98 percent, yet when they’re challenged to do the same math symbolically their performance is closer to 37 percent.
Laura Grace Weldon 