Natural Math: 100+ Activities & Resources

math through play, everyday math

image: pixabay.com

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

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

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

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

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

Natural math, according to math expert Maria Droujkova, is about,

people making mathematics their own, by posing their own problems, pursuing their own projects, and remixing other people’s activities in personally meaningful ways. We believe that “learning math” means two things—developing mathematical state of mind and acquiring mathematical skills.

Droujkova goes on to say,

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

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

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

natural math, math through play,

Applied math (images: morguefile.com)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

~Get into statistics.

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

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

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

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

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

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

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

~Compete. 

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

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

~Use logic. Apply critical thinking to current events.

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

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

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

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

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

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

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

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

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

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

~Play with math and critical thinking, together.

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

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

~Read-aloud math stories for children under 8.

~Math Stories for Children 8 and up.

~Math inspiration for older kids.  

Enjoy math-y videos.

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

This post is third in a series on natural math. 

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

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

image adapted from livescience.com

image adapted from livescience.com

Posted in early education, homeschooling, learning, math, play | Tagged , , , , , , , , , , , , | 1 Comment

The Benefits of Natural Math

natural math, exploratory math, hands-on learning,

images: public-domain-image.com

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

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

As parents, we believe we’re providing a more direct route to success when we begin math (and other academic) instruction at a young age. Typically we do this with structured enrichment programs, educational iPad games, academic preschools, and other forms of adult-directed early education. Unfortunately we’re overlooking how children actually learn. Real learning has to do with curiosity, exploration, and body-based activities. Recent studies with four-year-olds found, “Direct instruction really can limit young children’s learning.” Direct instruction also limits a child’s creativity, problem solving, and openness to ideas beyond the situation at hand. 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 discuss, and yes, argue among themselves as they try to find the best way to construct a fort, set up a Rube Goldberg machine, keep score in a made-up game, divvy out equal portions of pizza, choreograph a comedy skit, map out a scavenger hunt, decide whose turn it is to walk the dog, or any number of other playful possibilities. These math-y experiences provide instant feedback. For example, it’s obvious cardboard tubes intended to make a racing chute for toy cars don’t fit together unless cut at corresponding angles. Think again, try again, and voila, it works!

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

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

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

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

2. Homeschooling and unschooling families around the world devote much less time to formal mathematics instruction. Studies indicate their children grow up to succeed in college, careers, and life with greater self-reliance and focus than their schooled peers. Interestingly, two different surveys of grown unschoolers showed that a much higher number of them work in STEM careers than schooled adults. The samples were small but intriguing.

3. Democratic schools where children are free to spend their time as they choose without required classes, grades, or tests. As teacher Daniel Greenberg wrote in a chapter titled “And ‘Rithmetic” in his book Free at Last, a group of students at the Sudbury Valley School approached him saying they wanted to learn arithmetic. He tried to dissuade them, explaining that they’d need to meet twice a week for hour and a half each session, plus do homework. The students agreed. In the school library, Greenberg found a math book written in 1898 that was perfect in its simplicity. Memorization, exercises, and quizzes were not ordinarily part of the school day for these students, but they arrived on time, did their homework, and took part eagerly. Greenberg reflects, “In twenty weeks, after twenty contact hours, they had covered it all. Six year’s worth. Every one of them knew the material cold.” A week later he described what he regarded as a miracle to a friend, Alan White, who had worked as a math specialist in public schools. White wasn’t surprised. He said, “…everyone knows that the subject matter itself isn’t that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step. The only way we have a ghost of a chance is to hammer away at the stuff bit by bit every day for years. Even then it does not work. Most of the sixth graders are mathematical illiterates. Give me a kid who wants to learn the stuff—well, twenty hours or so makes sense.”

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

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

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

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

So what is the origin of those false beliefs? It’s hardly a mystery. People form that misconception because of their experience at school. In school mathematics, students are only exposed to problems that (a) are well defined, (b) have a unique correct answer, and (c) whose answer can be obtained with a few lines of calculation.

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

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

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

And Keith Devlin explains in The Math Gene that we’re schooled to express math in formal terms, but that’s not necessary for most of us—no matter what careers we choose. People who rely on mental math in their everyday lives are shown to have an accuracy rate around 98 percent, yet when they’re challenged to do the same math symbolically their performance is closer to 37 percent.

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

There are many ways to advance mathematical understanding. That includes, but isn’t limited to, traditional curricula. As this high school math teacher explains in his TED talk, we need to focus less on higher math and more on analytical and critical thinking skills.

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

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

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

Portions of this article are excerpted from Free Range Learning.

Posted in homeschooling, learning, math, school, success, testing | Tagged , , , , , , , , , , , , | 12 Comments

Math Instruction versus Natural Math: Benezet’s Example

Louis Benezet, natural math,

1930’s classroom (forestpark4.wikidot.com)

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

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

I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year-old child for knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.

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

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

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

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

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

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

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

Yet another difference was apparent. It was something that Benezet had anticipated. He explained, “For some years I had noted that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child’s reasoning faculties.” At the end of that first year, he went from classroom to classroom and asked children the same mathematical story problem. The traditionally taught students grabbed at numbers but came up with few correct results, while the experimental students reasoned out correct answers eagerly, despite having minimal exposure to formal math.

Based on these successes, the experiment expanded. By 1932, half of the third- to fifth-grade classes in the city operated under the experimental program. Due to pressure from some school principals, children in the experimental classrooms were back to learning from a math book in the second half of sixth grade. All sixth-grade children were tested. By spring of that year all the classes tested equally. When the final tests were given at the end of the school year, one of the experimental groups led the city. In other words, those children exposed to traditional math curricula for only part of the sixth-grade year had mastered the same skills as those who had spent years on drills, times tables, and exams.

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

As we know, schools went in the opposite direction.

Louis Paul Bénézet

Louis Paul Bénézet

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

Posted in history, homeschooling, learning, math, reading, school, testing | Tagged , , , , , , , | 18 Comments

Are You An Anthropocentrist?

 

animal intelligence, anthropocentrism,

Paradise, by Gillis d’Hondecoeter circa 1575

When I was growing up we were taught humans were at the top of every chart, far superior to all other living beings. Our textbooks, illustrated with stereotypical images of “cave men,” proved the assertion with a long list of what our species could do that others could not. The list was so smug that I was a bit embarrassed on behalf of my fellow homo sapiens. A skeptic even then, I thought the list was somewhat prejudicial. Worse, it didn’t acknowledge what feels obvious to young children, that we are all things and all things are us.

I don’t for a moment dismiss our many human accomplishments—among them language, science, the arts, and shared rules meant to advance mutual compassion. I simply mean to point out that we’re not better, we’re different.

Besides, what I was taught as a kid doesn’t really hold up. Here are some reasons why.

Tool use was a biggie on that list. It’s true, animals haven’t developed the smart phone (thus are spared walking into traffic while texting) but they naturally incorporate tool use when it makes sense for them.

  1. Crows make tools like hooks and rakes out of twigs, leaves, even their own feathers to obtain items just out of reach and can use three tools in sequence.  They also will drop pebbles into a container in order to raise the level of water, understanding cause and effect as well as a seven to 10-year-old child. Other examples of tool use by crows? They’re known to drop nuts on a roadway so cars will crack the shells, then wait for a break in traffic to retrieve the nutmeat. Interestingly, they’re more proficient when they grow up watching adult crows fashion tools. (Crows might wonder why we segregate human kids away from the interesting work-a-day world of adults.)
  2. Naked mole rats dig with their teeth, but to keep from inhaling dust and dirt they’re known to position wood shavings in their mouths as rudimentary face masks.
  3. The octopus is more closely related to clams than to people, yet these invertebrates plan ahead, tool-wise. For example they’ve been seen carrying coconut shell halves they can hide under later in order to grab unsuspecting prey as it passes.
  4. Orangutans fold leaves into a usable “musical instrument” that modifies their calls, making them sound lower and therefore more threatening to large predators.

Math was another obvious difference. We were taught that numerical sense is evidence of higher order thinking. Yet the animal kingdom uses math when necessary.

  1. Bears can count. Although they don’t benefit from the intelligence-boosting effect of living in social groups, research shows bears can estimate quantities just as well as primates. One particular study taught bears to discriminate between dots on a touchscreen computer, a situation about as far removed from relevant bear smarts as possible. Their abilities in natural habitat are likely to be far more impressive.
  2. Elephants have substantial numerical skills, outperforming primates and even human children when tested for their ability to find the difference between two quantities. A study found elephants can discriminate between one and two as well as between larger numbers.
  3. Baby chicks can not only count, they can even can add up numbers based on groups of objects they can’t see at the moment.  And that’s when they’re a few days old!  By two weeks of age, chickens can take into account the sun’s height and position to navigate. Plus they’re able to draw inferences and plan ahead, for example choosing to delay gratification in order to reap a greater reward. And who’d have guessed, but chickens prefer to count from left to right.
  4. Pigeons are able to learn abstract rules about numbers and order pairs. Aside from humans, only rhesus monkeys have been able to perform at this level.
  5. Insects also use math. Honeybees can distinguish between and remember quantities up to four. They can also match patterns. Ants operate with a collective form of intelligence, able to use complex problem-solving strategies to optimize time and energy spent feeding the colony.

People, we were told, communicate in complex ways while animals are, well, just animals. Again, not true.

  1. Elephants communicate sophisticated ideas in a variety of ways including low-frequency sounds from 1 and 20 Hz that can travel over miles. So far, researchers have identified nearly 200 expressions and gestures, along with nearly 100 vocalizations. Elephants can recognize at least 100 other unseen elephants by voice alone.  Their remarkable ability to understand communication isn’t limited to their own species. African elephants can differentiate between languages, gender, and age of human speakers.
  2. Dolphins remember one another, without contact, for at least 20 years. In fact, researchers have found that dolphins call each other by name (in this case, distinctive signature whistles).
  3. Koko, a western lowland gorilla, has been taught American Sign Language and, according to her trainer, understands about 1,000 signs along with nearly 2,000 words of spoken English. Sometimes, when there’s not a relevant sign, Koko invents her own signs. For example, she “compounded the sign for scratch with the sign for comb to mean, “brush” (scratch-comb).”
  4. Alex, an African gray parrot, learned well over 100 words that he used appropriately in unique contexts, demonstrating the intelligence of a five year old human child. He died suddenly in 2007. The last thing he said to his trainer upon going to his cage for the night was, “You be good, see you tomorrow. I love you.”

Which brings us to emotion and personality. Yup, non-human types are brimming with it.

  1. Chimps not only create social traditions, they’re interested in what’s trendy. Researchers are just now catching on (academic types are not known for fashion forwardness) to the latest thing, chimps wearing grass in their ears.
  2. Stressed-out honeybees show an increased expectation of bad outcomes. In other words, they become pessimists. The bees also showed altered levels of neurochemicals associated with depression. Other invertebrates, such as crayfish, can exhibit anxiety and respond well to medications that relieve anxiety in humans.
  3. Dogs traumatized by military service or abuse exhibit signs of canine Post Traumatic Stress Disorder 
  4. Rats feel regret after making poor choices.
  5. Crows will eat nearly anything, but prefer French fries from a McDonalds bag to the same fries in a plain brown sack. They not only hold grudges against specific humans who have done them wrong, but will teach other crows to react badly upon seeing them as well.
  6. And play? There’s plenty of it. Crows like to ski down icy rooftops and snow-covered slopes holding sticks or boards in their talons. River otters, elephants, and whales are known for playful behavior.

 

Let me push it one step farther, to compassion and even spirituality. We’ve been told that only humans have evolved beyond survival-based selfishness to establish ethics and morality. We’ve been taught we’re the only species to perform rituals as we mourn the passing of our departed, the only ones to meditate in silence, the only ones to experience a sense of awe akin to reverence. Apparently not true either.

  1. Altruism? There’s plenty of evidence. A dolphin saving a beached whale and its calf. Gorillas working together to dismantle dangerous poachers’ traps. A pod of sperm whales adopting a disabled dolphin. Rats gnawing through cages to help other imprisoned rats. A bear assisting an injured crowLions chasing away an Ethiopian child’s kidnappers and guarding her until human help arrived.
  2. How about awe? Chimps are known to ritualistically dance at the advent of thunderstorms and dance at waterfalls. They’ve also been observed dancing (rather than fleeing instinctively) in the face of wild grass fires.
  3. Meditation? Baboons have been observed performing a sangha, sitting in silence for over a half hour gazing at a stream of water, even the juveniles remaining quiet.
  4. Love? Probably yes according to research with cats and dogs who seem to be tapping into fields beyond our conscious awareness to know when their owners are coming home.
  5. Funerals, those too. Elephants weep in sorrow and grieve their dead. They’ve also been known to sense the death of humans important to them, even from great distances, as two tribes of African elephants did when they walked for hours to mourn at the home of a conservationist who’d once rescued them. Ritualized behavior to mourn death is common in animals including foxes,  magpieswolves, dolphins, and gorillas.
  6. Maybe even religion. Cetologist Hal Whitehead‘s research indicates that sperm whales not only transmit culture to their young, they may have have evolved a form of religion to make sense of their purpose.

 

Even these terribly incomplete examples have probably taxed your patience although there are thousands of other fascinating proofs out there. Let’s remember, all these observations are human-centric, further evidence that we judge animals against one species—-us.

We wouldn’t have particularly good scores if tested according to the abilities of our fellow creatures. It’s not as if we can age in reverse as a jellyfish named Turritopsis dohrnii does, possess a snake’s infrared vision able to assess the difference in temperature between moving prey and surrounding area on the scale of milliKelvins, emit a protein that neutralizes nearly every poison as an opossum does, regrow limbs and organs as the salamander can, or are able to hear as well as the wax moth Galleria mellonella which is capable of detecting frequencies of up to 300kHz, (we humans at best hear to about 20kHz).

According to evolutionary biologists, we humans aren’t better than animals, just different. Researchers in fields like comparative psychology and language study, say there’s an “emerging consensus among scientists that animals share functional parallels with humans’ conscious metacognition — that is, our ability to reflect on our own mental processes and guide and optimize them.”

As naturalist Henry Beston wrote in The Outermost House,

We need another and a wiser and perhaps a more mystical concept of animals. Remote from universal nature and living by complicated artifice, man in civilization surveys the creature through the glass of his knowledge and sees thereby a feather magnified and the whole image in distortion. We patronize them for their incompleteness, for their tragic fate for having taken form so far below ourselves. And therein do we err. For the animal shall not be measured by man. In a world older and more complete than ours, they move finished and complete, gifted with the extension of the senses we have lost or never attained, living by voices we shall never hear. They are not brethren, they are not underlings: they are other nations, caught with ourselves in the net of life and time, fellow prisoners of the splendour and travail of the earth.

 

animal capacities, Eden,

“Paradijs met dieren” by Jan Brueghel the Elder, 1620

Posted in deep ecology, meaning, nature, perspective, science, stereotypes, truth | Tagged , , , , , , , , , , , , , , , , , | 9 Comments

Understanding Children Through Imitation

follow your child's example, what it feels like to be a child, child's experience,

Mirror a child’s movements. (morguefile)

So much of a child’s experience, from infancy on, is constantly being shaped by adults. Their behavior, posture, movement, and sound are restricted by structured activities, confining seats, and grown-up expectations . If we allow ourselves, we can drop into a child’s world for few moments by replicating his or her movements. It’s a form of listening at the bodily level that can be instructive as well as enlightening.

I’ve admitted to trying this the very first time as a new mother, imitating my newborn’s movements in an experience so profound it felt like a ceremony.

I didn’t try it again until I was the mother of three kids under six. I’d dashed over to a friend’s house to drop something off, feeling rushed to get back to my nursing baby. My friend’s children were asleep. I stood in her quiet kitchen telling her how much I wanted to sit down and chat, but couldn’t spare the time. She answered my complaint with mock outrage, “Don’t you dare relax! What were you thinking?”

In my best imitation toddler voice I said, “WANT TO!”

She wagged her finger. “That’s enough out of you. Do what you’re told right this minute.”

Then I dropped to the floor in a full-on act of defiance; lying on my back, kicking my legs, and squalling, “You can’t maaaaaake me!”

By this time our hilarity was well out of proportion to this brief moment of improv. When I got up I felt different—wonderfully de-stressed and energized.

I insisted my friend give it a try. She resisted, until I admonished her with the same phrases I’d heard her use on her kids. I even flung out her full name accompanied by finger wagging. That did it. She twirled around whining “Noooooo. No no no!” till she was out of breath, with hair in her mouth and a smile on her face.

We both agreed we felt incredible.

I don’t for a minute suggest you do this, ever, in front of any child. Our children should never be ridiculed for expressing themselves. But if they’re not home, give it a try. What this did, for me as well as my friend, was let us fully express strong emotions through our bodies as our children do, as we used to do when we were children. We may have been well-educated, reasonably sophisticated women but the need to indulge in some primal venting hadn’t left us. A little method acting gave us both new insight into what our children experience.

After that, I looked for ways to learn from my children through imitation. We adults do this all the time when we play with our kids. We chase and let them chase us. When they pretend to be an animal or make-believe character we join in. We’re the big bad wolf blowing down a child’s fort made of cushions. We’re the sotto-voiced doll talking to another doll or the train engine struggling up an imaginary hill. Playing is a window into a child’s experience, and remarkably restorative for us as well.

But what truly let me honor my children’s world was letting them choreograph my movements. Sometimes we’d play what we called “mirror”— standard actor training done face to face. The child is the leader, the parent the “mirror.” As the child makes gestures, facial expressions, and hand movements the parent tries to duplicate the movements exactly. Then we ‘d switch so the child got a turn being the mirror. I always ended up laughing first.

Sometimes we played a variant of this, making each other into emotion mirrors. One would call out a feeling like “surprised” or “angry” or “wild” and the other would try to convey the word through facial expression. (This is also a great way to advance emotional intelligence.)

My favorite imitation was through dance. We’d turn on some lively music and I’d try to copy my child’s dance moves. This is much more difficult than it sounds. It’s nearly impossible to keep up with a child’s energy level for long!

My kids are past the stage where they want me to imitate their dance moves. But I haven’t forgotten how much letting my kids choreograph my movements taught me. Even now, they’ll catch my eye across a crowded room for a brief moment of mirroring. It’s funny, warm, and lets us both feel understood.

Don’t miss this wonderfully expressive choreography by Zaya, imitated by real dancers.

Posted in attention, early education, family, listening, mindfulness, mothering, music, parenting, perspective, play, selfhood | Tagged , , , , , , , , , , , , , , | Leave a comment

“It is good to love many things” ~Vincent Van Gogh

Sunflowers, by Vincent van Gogh

“It is good to love many things, for therein lies the true strength, and whosoever loves much performs much, and can accomplish much, and what is done in love, is well done.”

L’Arlésienne, L’Arlésienne, by Vincent van Gogh

L’Arlésienne, L’Arlésienne, by Vincent van Gogh

“Poetry surrounds us everywhere, but putting it on paper is, alas, not so easy as looking at it.”

child 2

Memory of the Garden at Etten (Ladies of Arles), by Vincent van Gogh

“Even as a boy I would often look up with infinite sympathy, indeed with respect, at a woman’s face past its prime, inscribed as it were with the words: here life and reality have left their mark.”

Self-portrait in front of easel, by Vincent van Gogh

“If you hear a voice within you say you cannot paint, then by all means paint and that voice will be silenced.”

The Sower, by Vincent van Gogh

The Sower, by Vincent van Gogh

“Great things are done by a series of small things brought together.”

child 2

“Normality is a paved road: It’s comfortable to walk, but no flowers grow on it.”

The Church in Auvers-sur-Oise, View from the Chevet by Vincent van Gogh

“That God of the clergymen, He is for me as dead as a doornail. But am I an atheist for all that? The clergymen consider me as such — be it so; but I love, and how could I feel love if I did not live, and if others did not live, and then, if we live, there is something mysterious in that. Now call that God, or human nature, or whatever you like, but there is something which I cannot define systematically though it is very much alive and very real, and see, that is God, or as good as God. To believe in God for me is to feel that there is a God, not a dead one, or a stuffed one, but a living one, who with irresistible force urges us toward aimer encore; that is my opinion.”

“I cannot help thinking that the best way of knowing God is to love many things. Love this friend, this person, this thing, whatever you like, and you will be on the right road to understanding Him better, that is what I keep telling myself. But you must love with a sublime, genuine, profound sympathy, with devotion, with intelligence, and you must try all the time to understand Him more, better and yet more.”

Bedroom in Arles, by Vincent van Gogh

“I dream my painting, and then I paint my dream.”

“I am always doing what I cannot do yet, in order to learn how to do it.”

Portrait of Père Tanguy, by Vincent van Gogh

“It is with the reading of books the same as with looking at pictures; one must, without doubt, without hesitations, with assurance, admire what is beautiful.”

“If only we try to live sincerely, it will go well with us, even though we are certain to experience real sorrow, and great disappointments, and also will probably commit great faults and do wrong things, but it certainly is true, that it is better to be high-spirited, even though one makes more mistakes, than to be narrow-minded and all too prudent.”

Sorrowing Old Man (At Eternity’s Gate), by Vincent van Gogh

“Well, right now it seems that things are going very badly for me, have been doing so for some considerable time, and may continue to do so well into the future. But it is possible that everything will get better after it has all seemed to go wrong. I am not counting on it, it may never happen, but if there should be a change for the better I should regard that as a gain, I should rejoice, I should say, at last! So there was something after all! “

Prisoners’ Round, by Vincent van Gogh

“People are often unable to do anything, imprisoned as they are in I don’t know what kind of terrible, terrible, oh such terrible cage.I do know that there is a release, the belated release. A justly or unjustly ruined reputation, poverty, disastrous circumstances, misfortune, they all turn you into a prisoner. You cannot always tell what keeps you confined, what immures you, what seems to bury you, and yet you can feel those elusive bars, railings, walls.

Is all this illusion, imagination? I don’t think so. And then one asks: My God! will it be for long, will it be for ever, will it be for eternity? 

Do you know what makes the prison disappear? Every deep, genuine affection. Being friends, being brothers, loving, that is what opens the prison, with supreme power, by some magic force. Without these one stays dead. But whenever affection is revived, there life revives.” 

A Pair of Shoes, by Vincent van Gogh

A Pair of Shoes, by Vincent van Gogh

“What preys on my mind is simply this one question: what am I good for, could I not be of service or use in some way, how can I become more knowledgeable and study some subject or other in depth?

The Potato Eaters, by Vincent van Gogh

The Potato Eaters, by Vincent van Gogh

“One must not be afraid of going wrong, one must not be afraid of making mistakes now and then. Many people think that they will become good just by doing no harm — but that’s a lie, and you yourself used to call it that. That way lies stagnation, mediocrity.”

Starry night over the Rhone, by Vincent van Gogh

“Be clearly aware of the stars and infinity on high. Then life seems almost enchanted after all.”

Starry Night, by Vincent van Gogh

Starry Night, by Vincent van Gogh

“When I have a terrible need of — shall I say the word — religion. Then I go out and paint the stars.”

First Steps, by Vincent van Gogh

First Steps, by Vincent van Gogh

“Love is eternal — the aspect may change, but not the essence. There is the same difference in a person before and after he is in love as there is in an unlighted lamp and one that is burning. The lamp was there and was a good lamp, but now it is shedding light too, and that is its real function.”

The Man is at Sea (after Demont-Breton), by Vincent van Gogh

“The more I think it over, the more I feel that there is nothing more truly artistic than to love people.”

Self-portrait with straw hat, by Vincent van Gogh

Self-portrait with straw hat, by Vincent van Gogh

Posted in art, imagination, perspective | Tagged , , , | 9 Comments

Gifting a Week of Meals

giving meals, cooking for others, meal sharing,

Yum. (CC by 2.0 thebittenword.com on flickr)

Soon after my second baby was born, I was informed that I’d be receiving a week of meals delivered by my friends. The next seven nights our doorbell rang and there stood someone dear to me holding warm dishes filled with delights.

A break from planning and making dinner was a blessed relief. It also exposed my family to a wider array of foods. More importantly, each night we sat down to eat a relaxed dinner lovingly made for us.

We were given so much food that we tucked lots of it in the freezer, spreading the bounty of kindness into the following weeks. One friend came laden with two different kinds of lasagna, one with garlicky white sauce and spinach, another layered with black beans and lots of veggies. Years later I still make both of her recipes.

A week of meals for families with new babies became a tradition in my circle of friends and my Le Leche League chapter. Here’s what worked for us.

1. Someone particularly close to the new mom and her family usually broached the idea to their mutual friends. We never designated a person in charge of planning. But your group of friends, or church, or neighborhood may decide that putting one person in charge of noting who will make a meal which night makes it easier.

2. We contacted the new mom with some basic questions such as best days and times to drop off food, food preferences, and if she wanted food brought ready to eat at dinner time or in advance to heat up later that day. Some moms preferred to have meal deliveries every other day.

3. Then we verified the plans with all potential participants. It worked best to accommodate a variety of needs among people contributing meals. Some preferred to drop off bags of Mid-Eastern salads or trays of sushi they picked up on the way home from work. Some didn’t have time to deliver a meal during the week but happily provided brunch on the weekend. It helped to jot down what people were planning to make so the family didn’t end up with three enchilada entrees on three consecutive nights.

4. We sent out a full schedule to everyone participating. It functioned as a reminder, listed who was bringing what, and offered suggestions such as labeling pans and including recipes. A shared Google doc can uncomplicate things. Or use one of these online meal scheduling sites to make this easier:

Meal Baby

Take Them a Meal

Meal Train

Care Calendar

Lotsa Helping Hands

Caring Meals

Of course, a new baby isn’t the only reason to provide a series of meals. It’s a great way to welcome someone home when they return from service project or military assignment. It’s a godsend when people are dealing with illness or injury. And it’s remarkably helpful during the time a family is undergoing a major home renovation. Mix it up. Rather than arranging a week of steady meals, you might offer a meal every Wednesday or set up a regular potluck date to eat together.

There may be no more basic gesture of kindness than feeding people. Food sharing is a tradition found in every culture, stretching back to our earliest history. It’s a stomach-filling, community-building kindness like no other. It can also swing back around remarkably. By the time my fourth child was born I was gifted with a full three weeks of meals, nearly all made by people I’d once cooked for. It was an embarrassment of riches but oh how those delicious foods warmed our hearts.

Other ways to build community:

Bring Kids Back to the Commons

Engage the Window Box Effect

It Really Does Take a Village

We Don’t Need No Age Segregation 

Welcome Kids Into the Workplace More Than Once a Year

Odd Second Saturday Suppers

Better Together: Restoring the American Community

The Abundant Community: Awakening the Power of Families and Neighborhoods

All That We Share

This is a repost from our farm site

Posted in community, food, service, subversive cooking, volunteering | Tagged , , , , , , , , , , | 4 Comments