“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.
Here are some of the starting points suggested in Free Range Learning to spark your own math-fueled journey.
~Learn more about yourselves. One family hangs a new chart each week to gather data. One week they might mark off where the dog takes a nap, then figure the percentage at the end of the week (40 percent of the time she sleeps in the window seat, 5 percent of the time under the table, etc), another week they might pick a subject like hours of computer use per person. They are also keeping several year-long graphs. One tracks the weight of trash and recyclables they discard weekly and a second graphs the amount of the produce they harvest from the garden. Yet another tracks money they are saving. They notice that in busy weeks, such as holidays, they fall short of sustainability goals they’ve set for themselves.
~Revel in measurement. Investigate joules, BTUs, calories, watts, gallons, degrees, fathoms, meters, hertz, attoseconds and more. Measure your everyday world. Calculate such things as the energy usage to get to grandma’s house in the car compared to taking the train, what angle a paper plane can be thrown and still fly, how much wood it will take to build a shelf for the baby’s toys, how many footsteps are required to walk to the corner. Figure out how to gather measurements and apply data.
~Enjoy math songs. Play them while traveling and sing them casually as you go about your day; you’ll find your children are memorizing math facts effortlessly. There’s something about a catchy tune that helps the mind retain concepts. There are many sources of math songs including Sing About Science and Math with a database of 2,500 songs.
~Say yes. When kids want to explore off the trail, stomp in puddles, mix up ingredients, play in the water, and otherwise investigate they’re making math and science come alive on their own terms. It’ll probably make a mess. Say yes anyway.
~Use wheels. Plan and build a skateboarding ramp. Time relay races using tricycles (the bigger the kids the greater the fun). Estimate how many revolutions different sized bike wheels make to cover the same distance (then get outside to find the answer). Adjust a wheelbarrow load to carry the greatest amount of weight. Use mass transit to get where you are going after figuring out the route and time schedule.
~Make math a moving experience. Instead of relying on flash cards, remember equations by clapping or stomping to them, rhyming and dancing with them, kicking a ball or tossing a bean bag to them, making number lines on the sidewalk with chalk and running to answer them, or any other method that enlivens learning. Games for Math: Playful Ways to Help Your Child Learn Math, From Kindergarten to Third Grade offers many moving math activities for children.
~Learn to dance. The fox trot or the hokey pokey may be funny names to children, but they also describe specific patterned steps. Mastering simple dances are a way of transforming mathematical instruction into art. Choreographers use dance notation to symbolize exact movements. Over the years different methods of dance notation have been used including: track mapping, numerical systems, graphs, symbols, letter and word notations, even figures to represent moves. Choreograph using your own system of dance notation. Draw chalk footprints on the floor to show where the dancer’s feet move to a waltz. Try dance classes. Music and dance enliven math concepts.
~Think in big numbers. Figure out how many days, minutes and seconds each member of the family has been alive. Estimate the mass of the Earth, then look up the answer. Stretch your mind to include Graham’s Number. Talk about why big numbers are best expressed in scientific notation. Check out the Mega Penny Project. Read stories about big numbers, such as Infinity and Me, How Much Is a Million? Millions to Measure, On Beyond a Million: An Amazing Math Journey, Can You Count to a Googol? , and One Grain Of Rice: A Mathematical Folktale
~Fold your way into geometry. Print out paper designs that fold into clever toys and games from The Toy Maker including thaumatropes and windboats. Check out instruction books such as Paper, Scissors, Sculpt!: Creating Cut-and-Fold Animals or Absolute Beginner’s Origami. Although these may seem to be for amusement sake, they teach important lessons in conceptualizing shapes and making inferences about spatial relations.
~Play games. Nearly every board game and card game incorporates arithmetic. Make time to play the games your children enjoy. Try new ones and make up your own. Many homeschoolers set up game days so their children can share games with their friends, this is a worthy tradition for kids whether they’re schooled or homeschooled. Games make strategizing and calculating effortlessly fun. For the latest information on games, check in with the aficionados at Board Game Geek. For educational game reviews, consult Games for Homeschoolers and The Board Game Family.
~Learn chess. This game is in a class all its own. Research shows that children who play chess have improved spatial and numerical abilities, increased memory and concentration, enhanced problem-solving skills as well as a greater awareness of these skills in action. Interestingly, chess also promotes improved reading ability and self-esteem.
~Get hands-on experience in geometry. Geometrical principles come alive any time we design and build, whether constructing a fort out of couch pillows or a treehouse out of scrap wood. Make models using clay, poster board, craft sticks, or balsa.
~Find out about the math in meteorology. Learn about weather trends and predictions, measurement of precipitation and temperature conversion. Keep a weather log using instruments to measure wind speed, precipitation, temperature, barometric pressure, and humidity: then graph the results to determine average, mean, and median for your data.
~Play with shapes. Enjoy puzzles, tangrams, and tessellations. Notice the way shapes work together in the world around you both in natural and constructed settings. Keep a scrapbook of appealing shapes and designs. Create a sculpture out of toothpicks and miniature marshmallows. Cut paper snowflakes. Make collages out of pictures and three-dimensional objects. Grout bits of tile or broken dishes into mosaic designs. Make mobiles. Cut food into shapes.
~Pick up a musical instrument. Learning to play an instrument advances math skills as well as sharpens memory and attention.
~Learn to code. It’s not only fun, it’s really a basic skill.
~Estimate, then find out how to determine an accurate answer. Predict how much a tablespoon of popcorn will expand, then measure after it has been popped. Before digging into an order of French fries, estimate how many there are or how far their combined length will reach. See how the guess compares with the actual figure. Guessing, then finding out the answer enlivens many endeavors.
~Get into statistics.
- Kids go through a phase when they want to find out about the fastest, heaviest, most outrageous. Once they’re duly impressed with the facts in such books as Guinness World Records it’s a great time to pique their interest using almanacs and atlases.
- Sports offer a fun way to use statistics. Player and team stats are used to calculate odds, make comparisons and determine positioning. Children may want to keep track of their favorite teams or of their own activities. The numbers can help them to see patterns, debate trends and make predictions.
- Data provided by WorldoMeters makes fascinating reading and may lead to further investigation.
- Collect and interpret your own statistics. You might develop a survey. Or record measurements, weights, and other information about specific data, then analyze the statistics using a graph, histogram, or other instrument.
~Make calculation part of household rules. If children are permitted a certain amount of screen time per week, let them be responsible for charting that time. If children rotate chores or privileges, assist them to create a workable tracking system.
~Learn to knit. This useful skill also provides hands-on experience in basic math including counting, skip counting, multiplication and division, patterning, following a numerical guide, visualizing shapes, and problem solving.
~Make time for calendars. Check out the history of African, Babylonian, Roman, and Egyptian calendars. Learn how our calendar system came into use. Would it make sense to change to 13 equal months of 28 days each, with one remaining “day out of time” set aside? What are the definitions of “mean solar time,” “sidereal time” and “apparent solar time”? Make a homemade sundial to see how accurately you can tell time.
~Make math edible. Cereal, pretzels, crackers, small pieces of fruit or vegetables, cubes of cheese, nuts and other bite-sized foods are excellent tools to demonstrate addition, subtraction, multiplication, division, fractions, percentages, measurement and more. Using food to make math functions visible is a tasty way to solve equations. Your children can calculate recipe changes such as doubling or halving while they learn other useful meal preparation skills at home.
~Use trial and error. This is a fun process, especially when applied to brain teasers, puzzles, and mazes; try making up your own. Other math-related ways to stretch your mind include optical illusions, magic tricks, and drawing in perspective. These activities go well beyond solving equations to figuring out larger concepts.
~Devise your own codes and use them to send messages to one another. Check out the history of codes and code breakers. Set up treasure hunts by hiding a tiny treat and leaving codes or equations to be solved that lead to the next set of hints.
~Compete.
~Enjoy the intersection of math and art. Muse over puzzling visual patterns, for example the work of M.C. Escher. Learn about rug making, sculpture, weaving, basketry and many other art forms to discover the calculation, patterning, and measurement used to create objects of beauty.
~Delve into maps. Look at maps of the world together. Find maps of your locality. As well as road maps, your child may be intrigued by topographical and relief maps, economic and political maps, navigational and aeronautical charts, weather maps or land ownership maps. Draw maps of your neighborhood, home, yard, or bedroom—notice what details your child includes. Make imaginary maps, perhaps to accompany a story or to demonstrate what an eight-year-old would consider a perfect place. Consider mapping somewhere you know well, but from different time frames—how might this place have looked 100 years ago, now, in the distant future? Some children who are reluctant to keep diaries or sketchbooks will cheerfully keep records of places they’ve been by drawing maps. Maps and mapping can teach measurement, spatial awareness, and complex geographical concepts.
~Use logic. Apply critical thinking to current events.
~Compare related things like the weight of a puppy to a full-grown dog, or the size of a pitcher compared to the number of glasses it can fill.
~Use math at the store. While shopping, have children help check prices as part of the process of choosing a better deal. Talk about what other factors come into play—durability, ecological impact, value, overall worth. If you need to make a bigger purchase like a refrigerator, have the children compare the special features and cost effectiveness of running the appliance.
~Try travel math. Traveling is a great time to use math. Children can figure out fuel usage, keep track of expenditures, consult maps, estimate time of arrival, and more. Playing math games also provides excellent distraction during a long trip!
~Talk about math as if you are thinking out loud. “I wonder how many bricks it took to make this entire wall?” then look up a formula for figuring that out; or “If we don’t buy ____ for a whole month do you think we’ll have enough money left over for a ____?”
~Enjoy hands-on projects requiring sequential instruction. These hone logic and spatial skills as well as patience. Model-building, quilting, making repairs, knitting, carpentry, origami, beading and Legos® are examples of such projects.
~Learn how alternative languages relate to numbers. Check out Morse code, semaphore, Braille and sign language.
~Play pool. The sport known as billiards has a lot to teach about angles, trajectory, speed and calculation. And it’s fun.
~Expect kids to participate in household chores. All sorts of mathematical concepts are learned when the youngest children put away silverware, stack plastic containers in the cupboard, and sweep the floor. Even more while older kids help make meals, do repairs, and brainstorm solutions to make the household runs more smoothly.
~Make puzzles a family tradition. They can increase concentration as well as promote spatial learning and reasoning.
~Start or join a math circle. Meet regularly with others who enjoy making the subject fun and intriguing. Most are run by math experts and include projects, games, and field trips related to math. Some resources to get you started:
- National Association of Math Circles
- Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers
- Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers
- Nachos and Numbers: …a Step by Step Guide to Hosting a Successful School Family Math Event
- Math Circles for Elementary School Students
- A Decade of the Berkeley Math Circle
- A Moscow Math Circle: Week-by-Week Problem Sets
~Play with math and critical thinking, together.
- Moebius Noodles: Adventurous Math for the Playground Crowd
- Game That Is Worth 1,000 Worksheets
- Junkyard Wars challenges
- Book of Think: Or How to Solve a Problem Twice Your Size
- Brown Paper School book: Math for Smarty Pants
- Games for Math: Playful Ways to Help Your Child Learn Math, From Kindergarten to Third Grade
- The I Hate Mathematics! Book
- Family Math for Young Children: Comparing
- Family Math (Equals Series)
- Family Math II: Achieving Success in Mathematics
- Family Math : The Middle School Years, Algebraic Reasoning and Number Sense
- Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It
~Check out learning games suggested by math teachers and math bloggers.
- Living Math offers reviews, learning ideas, and articles as well as an ever-interesting Living Math Forum.
- Natural Math offers a newsletter, blog, a course for math circle leaders and as well as an idea sharing forum.
- Sue VanHattum’s site, Math Mama Writes
- Christopher Danielson’s Talking Math with Kids
- Denise Gaskin’s Let’s Play Math
- Malke Rosenfeld’s Math in Your Feet
~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.
- Bedtime Math: This Time It’s Personal
- Bedtime Math: A Fun Excuse to Stay Up Late
- Math Curse
- Full House: An Invitation to Fractions
- Anno’s Mysterious Multiplying Jar
- The Boy Who Loved Math: The Improbable Life of Paul Erdos
- That’s a Possibility!: A Book About What Might Happen
- Multiplying Menace: The Revenge of Rumpelstiltskin
and The Multiplying Menace Divides - Any of Cindy Neuschwander’s books including Mummy Math and the Sir Circumference series.
~Math Stories for Children 8 and up.
- The Man Who Counted: A Collection of Mathematical Adventures
- The Number Devil: A Mathematical Adventure
- The Phantom Tollbooth
- 40 Fabulous Math Mysteries Kids Can’t Resist
- Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians
- What’s Your Angle, Pythagoras?
- Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, Vol. 2
- The series Penrose the Mathematical Cat
~Math inspiration for older kids.
- Algebra Unplugged
- Calculus for Cats
- The Cartoon Guide to Algebra
- The Cartoon Guide to Calculus (Cartoon Guides)
- The Cartoon Introduction to Economics: Volume One: Microeconomics
- The Cartoon Introduction to Economics: Volume Two: Macroeconomics
- Agnesi to Zeno: Over 100 Vignettes from the History of Math
- The Moscow Puzzles: 359 Mathematical Recreations
- Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks
Enjoy math-y videos.
- Vi Hart’s series Doodling in Math Class.
- PBS Kids Cyberchase
- Math Bites with Danica McKeller.
- Numberphile.
- Math Nerds.
- Khan Academy
~Keep math references handy, you’ll find them endlessly useful.
- Ask Dr. Math
- S.O.S. Mathematics
- Math Forum
- Wolfram MathWorld
- Chenier’s Practical Math Dictionary
- Chenier’s Practical Math Application Guide: For Do-it-yourselfers, Trades People, Students, Etc.
- Pocket Ref
- Desk Ref
- The Universal Book of Mathematics: From Abracadabra to Zeno’s Paradoxes
- All the Math You’ll Ever Need: A Self-Teaching Guide
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?