Why Play This Book

Children dream big. They crave exciting and beautiful adventures to pretend-play. Just ask them who they want to be when they grow up. The answers will run a gamut from astronauts to zoologists and from ballerinas to Jedi masters. So how come children don’t dream of becoming mathematicians?

Kids don’t dream of becoming mathematicians because they already are mathematicians. Children have more imagination than it takes to do differential calculus. They are frequently all too literate like logicians and precise like set theorists. They are persistent, fascinated with strange outcomes, and are out to explore the “what-if” scenarios. These are the qualities of good mathematicians!

As for mathematics itself, it’s one of the most adventurous endeavors a young child can experience. Mathematics is exotic, even bizarre. It is surprising and unpredictable. And it can be more exciting, scary, and dangerous than sailing on high seas!

But most of the time math is not presented this way. Instead, children are required to develop their mathematical skills rather than being encouraged to work on something more nebulous, like the mathematical state of mind. Along the way the struggle and danger are de-emphasized, not celebrated – with good intentions, such as safety and security. In order to achieve this, children are introduced to the tame, accessible scraps of math, starting with counting, shapes, and simple patterns. In the process, everything else mathematical gets left behind “for when the kids are ready.” For the vast majority of kids, that readiness never comes. Their math stays simplified, impoverished, and limited. That’s because you can’t get there from here. If you don’t start walking the path of those exotic and dangerous math adventures, you never arrive.

It is as tragic as if parents were to read nothing but the alphabet to children, until they are “ready” for something more complex. Or if kids had to learn “The Itsy-Bitsy Spider” by heart before being allowed to listen to any more involved music. Or if they were not allowed on any slide until, well, learning to slide down in completely safe manner. This would be sad and frustrating, wouldn’t it? Yet that’s exactly what happens with early math. Instead of math adventures – observations, meaningful play, and discovery of complex systems – children get primitive, simplistic math. This is boring not only to children, but to adults as well. And boredom leads to frustration. The excitement of an adventure is replaced by the gnawing anxiety of busy work.

We want to create rich, multi-sensory, deeply mathematical experiences for young children. The activities in this book will help you see that with a bit of know-how every parent and teacher can stage exciting, meaningful and beautiful early math experiences. It takes no fancy equipment or software beyond everyday household or outdoor items, and a bit of imagination – which can be borrowed from other parents in our online community. You will learn how to make rich mathematical properties of everyday objects accessible to young children. Everything around you becomes a learning tool, a prompt full of possibilities for math improvisation, a conversation starter. The everyday world of children turns into a mathematical playground.

Children marvel as snowflakes magically become fractals, inviting explorations of infinity, symmetry, and recursion. Cookies offer gameplay in combinatorics and calculus. Paint chips come in beautiful gradients, and floor tiles form tessellations. Bedtime routines turn into children’s first algorithms. Cooking, then mashing potatoes (and not the other way around!) humorously introduces commutative property. Noticing and exploring math becomes a lot more interesting, even addictive. Unlike simplistic math that quickly becomes boring, these deep experiences remain fresh, because they grow together with children’s and parents’ understanding of mathematics.

Can math be interesting? A lot of it already is! Can your children be strong at advanced math? They are natural geniuses at some aspects of it! Your mission, should you accept it: to join thrilling young math adventures! Ready? Then let’s play!

Questions & Answers

How can I help my child to solve math problems?

Use modeling, especially with young kids. Represent the problem with objects and toys, or roleplay it with people. Use drawings and paper models, too. Different colors can stand for different mathematical features of the problem. Retell the problem as a story with characters, if your child likes fiction. Make simple diagrams out of objects by lining up toys or using sorting grids, and use pictures to make timelines (such as bedtime routine). Capture math processes, such as sorting, as series of photos.

What should I do when my child makes a mistake?

Any story, number, shape is an example of some math. It may not be the example you were originally looking for, but that’s OK. Take the answer and love it and find a problem that would fit it. Another good exercise: for a month or two, don’t ask questions for which you know answers. If you really feel the urge to ask “What is two plus three?” rephrase as, “What would you make two plus three be, and why?” Investigate all answers. You will learn a lot about math and about your kid!

When I ask my child a word problem she always solves it, but if I pose the same problem as an equation, say 2+3=?, she doesn’t know what to do. Why?

Because you broke The Rule of Three! Any idea needs at least three very different examples, more for young kids. In this case, you offered two examples of seeing the problem: stories and numbers. Offer more examples, and make them varied! Try jumps (two jumps and three jumps), lengths (two units and three units), counters (two raisins and three raisins), sounds (two claps and three claps), and so on.

How can I make math engaging? How do I help my kid love math?

Find math you love. There is some out there, for sure. Try artistic math videos, puzzles, apps, and games. Use math to make money, to hack your computer, to bring social justice to your city – any cause you find worthwhile, or at least cool. After you figure out how to be a good math role model, find math in activities your kid already likes, as well.

How do you collect math manipulatives without feeling overwhelmed by “stuff”?

Reuse and recycle! Many regular household items make excellent math manipulatives and game props. For example, cups and spoons provide gradients; stairs can become a number line. Dedicate a box or a basket to math treasures (such as shells of different colors and shapes) and math tools (such as a ruler, a compass, and graph paper). This way, the kid can grab the manipulatives and start working right away.

Does teaching a young child to count or to instantly recognize a group of dots undermine deeper mathematical learning?

Here are the four main ways toward building the basic concept of number:

• subitizing – the ability to instantly recognize quantities without counting
• counting – addition and subtraction are all about sequences: dealing with objects one-by-one
• unitizing – multiplication and division are based on equal groups or units
• exponentiating – self-similar structures, such as fractals

None of these ways toward number is dangerous in itself. The lack of balance is dangerous! The four ways have to live in harmony, as one happy, connected ecosystem. If one way toward number takes over, the ecosystem becomes unstable and some concepts become very difficult for children to learn, for many years to come. For example, many US curricula do not have enough early unitizing and exponentiating. You can observe a huge increase in math failures around the third and fourth grades, when children are faced with problems that depend on groups and units. Don’t let it happen to you and yours.

Which math concepts come easier to young children and which are hard? Does it vary from child to child?

We agree with Bruner: “Any subject can be taught effectively in some intellectually honest form to any child at any stage of development.” It is not the concepts that are easy or difficult, but how you present them. Help the young kids experience the concept through the whole body and all the senses. For example, mimic one another’s poses, cut folded paper, or play with mirrors to explore symmetry. Invite children to answer in actions, rather than words or symbols. For example, a child can prepare enough tires for two toy cars, rather than answer “What is 2x4?” And use technology, because children love it: take photos of kids doing math, record their math stories, search for math pictures and math videos online, use math computer games. You can teach any concept at any age.

How can I stay enthusiastic when my child wants to play the same game again and again, or asks the same question over and over?

Kids get into “vicious cycles”: doing something that used to be meaningful again and again with diminishing returns on learning and joy. Kids can never get enough of what they don’t really want.

Sometimes kids suddenly burst into tears or run away from the game to break the cycle dramatically. Help the kid escape the vicious cycle! Turn it into a spiral leading to new heights. For example, if the kid asks, “Why, why, why?” without paying much attention to answers, make each answer twice as long and detailed as the previous one. And agree to take turns, so you can put in your own, more varied questions. Instead of desperately trying to get some joy out of an old-and-tired game, invite the kid to tweak and redesign its rules. Offer closely related, but different activities.

What is the single best material to have on hand for children to play with math?

We always keep a large stack of graph paper, color pencils or markers, glue, and scissors at the ready. Other than that, we have lots of building toys and construction kits available for both children and adults. Sure, babies may knock your constructions down, but they absorb the ideas anyway. And for sorting, use household or toy objects that are similar, but vary by attributes (color, size, shape) – such as pots and pans, shells, miniature cars, or model animals.

Can young children understand the concept of negative numbers? How should I introduce it?

You can use the opposites to introduce the idea of negative numbers even to a baby. You can put toys on the “underwater elevator” in the bathtub, marking up levels over and under water. Toddlers are happy to play with “objects and anti-objects” that explode, eat, or otherwise cancel one another. For example, three hungry caterpillars can cancel two leaves, but one will stay hungry! In adult talk, -3+2=-1. Pegs (positive numbers) can cancel out holes (negative numbers). If you have three barefoot kids and five socks, how many feet will stay bare? In general, the answer to “Can young children understand the concept of… ?” is always “Yes!”

How old should a child be before I start writing math on paper?

Paper is an excellent cheap medium, and you can use it from birth. At all ages, use pictures and cutouts in addition to symbols. Babies have a grab reflex, so you can put a thick marker into your baby’s hand and hold a notepad near enough to the marker’s tip for the baby to “draw” for a few seconds. Babies think it’s a fun game, and you can always frame the resulting art piece and call it “abstract.” Toddlers often enjoy it when they hold a marker, and you hold their hand in yours and draw shapes with it. This develops hand-eye coordination; the hand talks to the brain, and can teach it a thing or two about shapes and symbols!

You always talk about adventures and explorations. What about adding numbers fluently, or memorizing multiplication facts?

Make more mundane work meaningful and productive, and track progress. The work is meaningful when the child has deep personal reasons to perform “grind” tasks. Children’s personal reasons come down to play, beauty, community, or utility. Children, like grown-ups, will do rather routine tasks as parts of games, or to create art, or because their friends do that, or because they need the task for something else in life.

To track progress, use simple time and task management tools. How can you help children easily see their progress with a task? For example, computer games track progress with levels, achievements, badges. Stay away from rewards, but track and celebrate milestones.

To stay productive, you need good workflow and healthy work habits. Eat before or during math time (math is hungry work), run or jump for a bit before any memorizing (so the brain has enough oxygen), find best time of the day for math (kids know, just ask them), use music or total quiet (depends on the person), and so on.

How can my family get into the habit of doing a bit of interesting math every day?

There are several easy tools you can use. All of them either remind you to do something little, right now – or help you prepare those little math activities for the future.

• Put math art around your house and car, for conversation starters. Change it once in a while.
• Write math graffiti on your walls (or on removable papers) – beautiful words like “fractal” or beautiful symbols like ∞.
• Strew little mathy objects and manipulatives around. Mini-puzzles such as Traffic Jam, small objects for sorting, a card game, construction kits, origami paper – any object that says, “Grab me and do some math!”
• Tie math time to other daily routines. For example, read some math stories before bedtime, or play guessing games while waiting for the bus.
• Once in a month or so, reflect on all the above – get some new math words, math pictures, math objects, and think of your math routines! For ideas, chat with friends who have kids of similar ages and interests.

But seriously, can you teach any concept at any age? Can you teach a toddler something like the transitive property?

Oh, a challenge! Try playing “Rock, paper, scissors.” It illustrates lack of transitivity very nicely. Also, experiment with storytelling: “If I sit on a chair, and the chair stands on the floor, does it mean I sit on the floor?” Use examples when transitivity does work: “Alice is taller than Bob, and Bob is taller than Carol. Does it mean that Alice is taller than Carol?” All order relations – “heavier” and “faster” and “longer” – are transitive. Toddlers love to sort and order things. Take, for example, the dishes: if the saucer fits into a pan, and the pan fits into a pot, then the saucer will fit into the pot as well. And yes, you can and should use the word “transitive” while playing such games, and your toddler will pick it up. Eventually.

How do I make math manipulatives and game props without spending all my spare time on this?

The medium is the message. In this case, the making of manipulatives is the math. If you can’t make it together with the kid, as a part of math explorations, don’t make it at all. Designing and making manipulatives is where math is at! Keep even the littlest baby on your knee when you make math pictures or mobiles for him or her, and you will see precious reactions and interest. DIY – Do It Yourself – strongly supports a child’s love and understanding of math, but only if “yourself” refers to the child.

Pose your questions and help others at our online Q&A hub:

ask.moebiusnoodles.com

Symmetry : Live mirror

Stand in front of each other and mimic each other’s gestures and expressions. That’s it! Sounds too easy? As soon as you try crisscross poses, asymmetric finger shapes, or fast motions, the game provides enough challenge even for an adult! You can easily adjust the difficulty to match each child’s gross and fine motor skills, as well as attentiveness to details.

Bring Variety

Many dances have parts for two people mirroring one another. Put some music on and compose a dance as you play the game. Choreographers break the symmetry to add tension and drama!

For a hardcore challenge, draw two perpendicular lines and play the game with four people. Expect much confusion! This is just like playing Special Snowflake games, where you and your friends are the snowflake.

Play live mirrors when your child’s day is not going right, you can’t stop arguing, or a homework session is not going well. A sympathetic human mirror gives an immediate boost to the child’s thinking abilities and emotional well-being! This happens, in part, because brain’s special “mirror neurons” fire when we see an action performed by another person.

For All Ages

Grown-ups: Find finger positions and interesting movements that challenge children. Help kids who get confused by mimicking them in return, or gently positioning their limbs with your hands. Ponder why some motions are harder to mimic than others.

Babies: Let your baby lead. Follow by mirroring the baby’s gestures and facial expressions. Holding the baby in your lap, mirror someone else’s gestures by moving baby’s hands or feet.

Toddlers and Young Kids: Offer whole body or limb movements, rather than fine gestures. Help with more challenging movements by positioning your child. You can also help by telling the story of your movements using math words: up/down, forward/backward, front/back, perpendicular, across, and so on. Avoid using “left/right” because it confuses in the context of mirrors.

Big Kids: Try more complicated movements – for example, rub your tummy and pat your head. Experiment with friends using two or more “mirrors” (that is, lines of symmetry).

Extend and Connect

Instead of your own bodies, manipulate dolls, plush toys, models made from construction sets, or posable action figures. It’s a math lesson taught by LEGO®, a Barbie®, and a Transformer®!

Take pictures of your games – this will inspire kids to strike more interesting poses! If you take pictures from the side, with the symmetry line in the middle, you can cut them in half and play a matching game with the pieces.

Go on scavenger hunts for characters striking mirror poses or doing copycat movements in games, cartoons, paintings, or dances. Directors, choreographers, and artists use symmetry to express both harmony and conflict. For example, mirrored confrontation shots often appear on posters for action movies.

Add objects to the game – give each player a ball, a hula hoop, a large wooden block, or something you can climb. Objects can help kids notice and discuss the idea that “my right is your left.”

Go on scavenger hunts for characters striking mirror poses or doing copycat movements in games, cartoons, paintings, or dances. Directors, choreographers, and artists use symmetry to express both harmony and conflict. For example, mirrored confrontation shots often appear on posters for action movies.

Symmetry: Double doodle zoo

Fold a piece of paper in two. Make a random doodle in such a way that it overlaps the fold line – otherwise, the result will fall apart. Now comes the fun part: invite your game partner to see something in the doodle! Add features to it and draw details to make the image more apparent. Cut out the result and open it. Turn the paper over, see something else in the opened shape, and draw the details to make it apparent!

Bring Variety

Use the same doodle with different paper folds and predict what it will look like depending on the type of symmetry. Cut out the doodle and check the accuracy of the predictions.

Introduce some randomness into the game – doodle with your eyes closed.

Make it delicious – bake a fairly large and complex cookie (e.g., gingerbread man, butterfly) and try to eat it one bite at a time while preserving the symmetry.

For All Ages

Grown-ups: Observe kids growing their math eyes as they notice more and more key points, angles, symmetries, quantities, and other mathematical features. Pay attention to these features yourself when it is your turn to see something in the doodle, say their names out loud, and otherwise encourage looking for properties.

Babies: Parents can play this game by themselves and give the resulting paper toys to the baby. You can use the cutouts to make a mobile or a garland. A version for babies who can point or use signs: have a lot of pictures of toys, animals, and so on shown at once, and invite the baby to point or put the doodle next to the object it resembles.

Toddlers and Young Kids: Toddlers will be more successful drawing on large sheets of paper. Many toddlers can see animals, houses, cars, or other familiar shapes in clouds, tree bark, or abstract rug designs. Help toddlers add more details to their scribbles until something familiar emerges.

Big Kids: It’s better to play the game quickly until you create many examples. Only through many examples do children develop property-noticing strategies. However, some older kids may want to tinker with a shape they particularly like, turning it into a more polished project, such as the Japanese art of Notan (positive-negative space).

Extend and Connect

Fold the paper into more than two parts. Observe what changes.

Play this game in reverse – give a child a cutout of a shape or a doodle and suggest to fold along its line (or lines) of symmetry, if any.

Use paint instead of pencil or marker. Paint or just drop paint on paper then fold. Resulting color mixing will add beautiful details to the game.