# Book Review: Zero by Kathryn Otoshi

It seems to me that children’s math books fall somewhere along a continuum. On one side of the continuum the books practically shout, “I am a MATH book! I will teach you something!” On the other side, the math is so subtle that you might not even notice it’s there.

I’m much more partial to the subtler side of the continuum, where the author weaves math into the fabric of the story, rather than fitting a story onto the math. It’s not that I think kids will only swallow the math if we sneak it into their literary diet. There are some fantastic, mathiest-of-math books that are engaging and fun and fantastic for children. But sometimes when the mathematics doesn’t jump out and hit you on the head there can be a lot more room for conversation. You just need to know what to look for.

One of my students recently lent me her copy of Zero, by Kathryn Otoshi. I had neither seen nor heard of this book, but I fell in love with it from the very first page. This book hits my sweet spot—simple, gorgeous illustrations; an engaging, child-friendly story; solid mathematics; and plenty of conversation fodder. It’s fantastic. I’m going to give it to myself for Christmas.

In this book, there are two big mathematical ideas:

1. Zero is a different kind of number.
I could write a whole blog post about this idea. (I probably will.) But here’s the short of it: young children don’t initially think of zero as a number because zero is not “part of the count.” When we count, we typically start with “one”. And we never ask our children to count a set of no objects. They’d probably think we were joking if we pointed at an empty tabletop and said, “Count the M&Ms!”

2. With only ten symbols, we are able to write numbers as big as we want to.
This is an under-recognized beauty of our number system. Many historical number systems (like Roman numerals) can only represent quantities up to a certain size before running out of symbols. But our modern Arabic numerals have infinite possibilities for combining just ten symbols into any number we can imagine.

So what can you do with this book (besides read it)? Talk about the math! Here are just a few ideas to get you started, but let your child lead the way and see where the conversation takes you.

• Ask your child, “Is zero a number?” and see what they say. Probe their thinking. You can use the context of the book (but you don’t have to). Why would zero feel left out? What’s different about zero?
• Have your child write the biggest number they can think of, and the smallest number they can think of. Together, see if you can think of bigger and bigger numbers (and for older children, smaller and smaller). It’s a game, but a game that can help your child think about how writing numbers “works.”
• Talk about how the numbers in the book can join themselves together. What kinds of numbers could two numbers make? Four numbers?
• And you don’t have to stop at math. Talk about friendships, fitting in, being yourself, working together.

# Replicate an Experiment! Kids and Algebra (ages 4 – 6)

## Question: Are young children able to reason algebraically?

This experiment comes from a recent study published by researchers at Johns Hopkins University.

Materials
1. A set of about 20-25 each of four different kinds of small objects (buttons, beads, fish crackers, cereal, pennies, etc.)
2. Two identical, opaque cups (paper cups work well)
3. A stuffed animal with a name

Setup
Separate the objects into piles somewhere that will be accessible to you, but out of your child’s view. (You may want to lay out the objects on your table, then hide them behind a book.) For three of the objects, you will want 12 in the first pile (those 12 will go in the “magic cup”), and anywhere from 5 to 9 in the second pile. For the fourth object (I used chocolate chips), you will want 12 in one pile and 4 in the other.

How to Conduct the Experiment
Tell your child you are going to play a number game with the stuffed animal (ours is Puppy). Tell your child that Puppy (or whatever the animal is named) has a magic cup, and show them one of the cups. Tell them that any time you put the cup on a pile of something, it adds the same number of that object to the pile, no matter what the object is. Say, “Let’s try and see what happens.”

Out of your child’s sight, place 12 of the first object (I used pennies) into the magic cup Place the remaining pennies in front of Puppy. Point at the pennies and say, “See Puppy’s pennies?” Then bring the magic cup out, without showing the child the pennies inside. Tilt it upside down over the pile of 5 pennies, then lift it to reveal the (now larger) pile of pennies. Say, “It worked! Shall we try again?”

Remove the pennies and do the same thing with the next two objects (I used cherries and Cheerios). The key is to move fairly rapidly, without rushing. You want to discourage counting, and you want to make sure that your child never actually sees the objects in the magic cup until they have been added to the pile in front of the animal.

Now that you’ve laid the groundwork, you’re going to see if your child can tell how many objects the magic cup creates. Discretely place your fourth objects (chocolate chips in this case) into the two cups, 12 in one and 4 in the other. Bring both cups out and tell your child that you found two cups and you’re not sure which is Puppy’s magic cup. Turn both cups over onto the table and lift them to reveal the 12 objects and the 4 objects. Ask your child to tell you which is Puppy’s cup. You should realize that, at this point, your child has never seen the contents of the magic cup—only the starting amount and the final amount. Will she now be able to tell which of the two is the amount that was added to each pile?

What is the purpose of this experiment?
The researchers who conducted this experiment already knew that children are born with number sense that does not need to be taught. They wanted to know if that same number sense extends to algebraic reasoning.

The task in this experiment may seem simple, but it’s not. There’s a big difference between being able to think about a situation where the unknown quantity is the result of an operation, such as 5 + 12 = ?, and being able to think about a situation where the unknown quantity is within the operation, such as 5 + ? = 17. Identifying the change amount (5 + ? = 17) rather than the result amount (5 + 12 = ?) is something that older kids often struggle with when they first begin algebra.

In this experiment, the children were too young to solve an algebraic situation symbolically. When given equations like 5 + ? = 17, and asked to choose between the numbers 4 and 12, they just guessed. But when the magic cup story presented children with the exact same situation in a non-symbolic form, the children were suddenly successful.

This experiment reveals that your child’s brain is already wired to do simple algebra! Knowing what our children are capable of can give us confidence as parents who want our children to achieve their potential.

What if my kid failed the experiment? Does it mean my child will struggle with algebra?
Of course not! For one, it’s hard to perform an experiment perfectly, and you only had one go at it. And maybe your child was tired or hungry or distracted. Maybe they would have gotten the right cup 4 out of 5 times (better than chance), and you happened to see the one they got wrong. More important than the outcome of your particular experiment is the opportunity you gave your child to think algebraically just by doing the experiment with her. Now give them more opportunities!

Reference: Kibbe, M. M. and Feigenson, L. (2014). Young children ‘solve for x’ using the Approximate Number System. Developmental Science. doi: 10.1111/desc.12177