The Growing Story and Three Feet Small are two wonderful picture books that address a “math all around us” question that young children find compelling. Both of these stories are built around a question: Am I getting bigger? They also offer rich possibilities for exploring two important Big Ideas of measurement:

• Many different attributes can be measured, even when measuring a single object, and
• Isolating the attribute to be measured is an important precursor to measuring.

The Growing Story by Ruth Krauss, illustrated by Helen Oxenbury, was first published in 1947 but the story is timeless. From early spring through fall, a little boy easily can see how baby chicks, puppies, and plants in the garden grow. But his own growth isn’t nearly as easy to measure—until he finds he has outgrown the warm clothes his mother put away in the spring.

Helen Oxenbury’s warm pastel illustrations will help young readers track precisely how the chicks change in appearance and size; how the rolly-polly puppy becomes a dog; and how the apple tree goes from barren, to bud, to apples ripe for picking. The story links beautifully with an inquiry unit looking at farm animals; it invites rich discussions of how different creatures grow at different rates—a baby chick’s childhood is a matter of months while a human’s lasts for years.

Michael Rosen’s Three Feet Small, illustrated by Valeri Gorbachev, takes a different tack, but one that is equally important to little ones. The concern about getting bigger is not just a question of physical growth—it has just as much to do with how getting both older and bigger means we can do more things. The little bear who is the center of this book is just not sure that he is growing—especially when he sees all the things his older sister and his adult relatives can do that he is still too little for. However by the end of the book, he also finds that he needs new clothes and that he is beginning to do some of those things that seemed out of his reach earlier.

These books could be used alone or together to open up a wonderful exploration of the benchmarks children can identify concerning what it means to “get bigger”—either in terms of size or abilities. Rosen’s book includes at least one picture showing a measuring chart—if you don’t already have one in the classroom to mark the growth over several months, you could start one now. However, be sure to emphasize that height isn’t the only attribute that shows growth. Include discussion of other physical attributes such as weight. The most emphasis should be put on how we can measure how much we are growing our brains as well as increasing physical abilities. What are things the children know now and can do now that they couldn’t when they were one year old? How about things they couldn’t do at the beginning of the school year?

Estimation, an important skill for young children to have, can be touched on in many different activities. Benita Johnson, a teacher trained by the Collaborative, was a prekindergarten teacher at Christopher House. She played a game after reading Mouse Count by Ellen Stoll Walsh to bring out the math that was inherent in the story. Games are often best brought out with a good book.

Ms. Johnson used a large jar and “mice” she had created out of three sizes of Styrofoam balls. She first asked each child to estimate how many large mice would fit in the jar and recorded the answers; then the children put the mice in one by one to compare their estimates with the results. She repeated the process with the smaller mice; all along she kept the children thinking actively by discussing what was happening. Using small, medium-sized, and large mice resulted in different questions to the children and different answers from them.

“I could see that it was really important to have concrete materials to do this activity,” Ms. Johnson said. “I also found that it was a good idea to keep asking children questions after placing each mouse in the jar, because it allowed the children to modify their thinking. I could see that the children did begin to understand the idea that larger items take up more space than smaller ones.”

And all along, estimating helped the children think about the reasonableness of their answers.

Tracey Fisher was another participant of the program and tried out a similar activity. With her prekindergarten class at Peirce Elementary, she combined an estimation exercise with a data display to help children order and compare amounts.

For the activity, Ms. Fisher provided a clear jar with a small set of manipulatives inside, in this case bunnies. She put up a chart with three possible amounts, only one of which was correct. Children placed their name card in the column with the amount they think was in the jar. By constraining the number choices, Ms. Fisher helped children focus on reasonable estimates and encourage strategic comparison and logical thinking. If it seemed like more than 4, it had to be either 6 or 10.

“Every time I play the estimation game,” Ms. Fisher commented, “I always realize what a good reinforcement it is for the concepts of highest, middle, and lowest when comparing how many children guessed each number. It is easy for them to see. I also remind them that estimation is not only in math but also is important for scientific purposes.”

With the right questions to the children, both activities can illustrate estimation in purposeful ways.

All kinds of confusion can result when children are asked to rattle off the numbers from 1 to 10 or 20 or higher without actually counting something. In our learning labs and activities we are working to help teachers find ways to avoid “Naked Numbers” and to help children understand that quantity is an attribute, not a noun.

One of our research lessons on number arrangements gives children opportunities to develop visual number sense of quantity and to build subitizing skills (the ability to rapidly and accurately quantify a set of items without counting). The children find many different ways to arrange three popsicle sticks. As they look at their own and other children’s creations, they develop a sense of “three-ness” and the different ways it can be constructed (composed and decomposed).

In turn, teachers themselves have found many ways to give children the repeated experiences they need to solidify their understanding of number.

Program participant David Newman at Chappell Elementary finds it easy to give his preschool children plenty of experience with comparing the value of numbers by having them play a simplified version of the classic card game “War.” The pack of cards he uses only have two suits, stars and dots. While a few cards use the number symbols, most arrange the dots and stars in configurations similar to those found on dice so that the game reinforces subitizing and visual number sense. Appropriately for prekindergarten, the numbers only go up to 6.

Mr. Newman finds it important to introduce educational math games for kids by modeling. Playing a few rounds with a child while others watch is all that is needed. All cards are distributed face down between the two players. Each player turns up one and the higher number takes both cards. If the same number is turned up, the one marked with a star wins. Otherwise, the winner of the next round takes the cards. Mr. Newman notes that this game is a great way to introduce children to playing a game with rules without adult supervision. He noticed that the game also helps with fine motor skills as the children learn to handle cards and keep them in piles.

My card looks like one, two and if I turn it this way it looks like two, one. So it could go here or there.

Another program participant, Melinda Chum, and her kindergarteners at Bridge School enjoyed creating a Three Museum after doing number arrangements with craft sticks. Later after reading 10 Black Dots by Donald Crews one child suggested, “We could use the dots to make the Three Museum like we did with the sticks.” Ms. Chum put this idea into action.

She invited small groups of children to join her as she brought out a basket of black dots and 3” x 5” index cards. She then modeled how many different ways she could arrange the dots. She labeled one card with a 1-1-1 pattern as she said, “I put one here, one here, and one here.” She went on to demonstrate 1-2 and 2-1 patterns that put two closer together and one further away. The children went on to create their own patterns by sticking their three black dots on the white index cards.

Ms. Chum really understands how important it is for children to think and talk about what they are doing in order to deepen their understanding. So once they completed their patterns, she asked them to place their cards under the labeled number pattern cards and explain why it fit where they put it.

She reports, “It was wonderful to see how they think and decide that their card could belong just about anywhere on the number pattern graph depending on how you turn the card and how you are looking at it. During center time, they continued to think and rearrange their cards, commenting on things like ‘My card looks like one, two and if I turn it this way it looks like two, one. So it could go here or there.”

That rich conversation gave Ms. Chum clear evidence that her children really understood the Big Idea that pictures and objects can be arranged in different ways and that no matter which way they arrange their three-dots-card, there would still be three dots.

Melinda Chum is one of many teachers who have found great ways to do math with Donald Crews’ wonderful picture book 10 Black Dots. Children love going through the pages, exploring how 2 black dots form the eyes on a fox or 4 black dots can be seen as the tires on a vehicle.

Program participant Leandra Gonzalez at Lara Elementary Academy found she could extend the book to do an authentic assessment of how well children understood numerosity. After reading the book to her prekindergarten class, she invited small groups of children to decide how many dots they wanted to use to create their own picture and to dictate something about what they had created. Ms. Gonzalez had each group share their pictures to continue the good conversations.

She says, “When students were given the opportunity to create something and decide on their own how many dots they needed for their creation, I feel I got a true idea of their comprehension of the concept. I had the opportunity to hear them count, use number words, and apply quantity.”

Crews is not the only author to use the simple dot to bring numbers to life. Herve Tullet’s Press Here is an amusing commentary on how many young children have become proficient at pressing on computer screens to bring about some kind of change. The book uses instructions on one page such as “Five quick taps on the yellow dot” and so when the child turns the page, they see a line of 5 yellow dots! It’s the magic of the imagination. Here again, all kinds of learning fun can happen as children count taps, give shakes, and find the patterns. Inviting them to do their own variations on the book will extend the delight and the understanding even further.

Students explore spatial relationships by describing and traversing an obstacle course, then making a map of it.

Ever exaggerate a good story? The little boy Hugh Thomas sure does in the book A Million Fish… More or Less by Patricia C. McKissack, illustrations by Dena Schutzer. In the book Hugh Thomas goes fishing in a swampy bayou. After listening to the tall tales of his elders, he catches on to the art of embellishing a story. He recounts catching a million fish in a half-hour and defending his catch against “one hundred alligators one hundred feet long [who] can move at one hundred yards per second.” Then he tells of battling a pirate raccoon who is after his fish in a contest of skipping rope. Hugh Thomas beats the raccoon by jumping 5,553 times before missing. Eventually he makes his way home with only three fish left, but a good story to tell!

This book is a delightful way to start a discussion about estimation in the early grades. Is it reasonable that Hugh Thomas caught a million fish in 30 minutes? What about 100 fish? What about 10 fish? Building children’s estimation skills supports their number sense and helps them judge whether a number is reasonable or not.

Estimation is the process of evaluating a quantity when the situation calls for a rough or tentative number. An estimate is not merely a “guess.” A reasonable estimate depends upon mathematical understandings of both numerosity and measurement.

For example, many teachers use an “estimation jar” to provide practice with estimation. In this activity, the teacher puts a collection of objects, such as marbles, in a clear plastic jar with a lid. The children estimate how many marbles are in it, recording their estimates and their names on sticky notes to post by the jar. At the end of the week, the class counts the collection and compares the count to the estimates.

When children look at a collection of marbles in a jar, they have to “picture” a number they can visualize such as 5 or 10 (numerosity). Then, they have to estimate how many of those units (5s or 10s) might fit in the jar (measurement). Finally, children count by 5s or 10s that many times (counting). Thus children need to learn the foundational skills of subitizing and counting well and build mental images of “benchmark” collections (10 looks like this; 50 looks like that) to estimate accurately.

An estimate is not merely a “guess.” A reasonable estimate depends upon mathematical understandings of both numerosity and measurement.

One suggestion is to add a “reference jar” next to your estimation jar. The two jars should be the same size and shape. Clearly label a benchmark collection, such as “50 marbles,” in the reference jar to support children’s estimation. Another suggestion is to allow children to revise their estimates based on a partial counting of the collection. For example, count roughly half of the marbles. Invite children to revise their estimates based on this new information. With experience, children can make increasingly accurate estimates… and that’s no exaggeration!