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 fives or tens 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.

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!

Estimation, an important skill for young children to have, can be touched on in many different activities. Benita Johnson was an Early Math Education Project participant in 2007-2008. As a prekindergarten teacher at Christopher House, she played a game after reading Mouse Count by Ellen Stoll Walsh. She 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 a participant of the program in 2008-2009 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.

The data display the children created was then analyzed. Which amount has the most “votes?” Why did children pick the amounts they did? Finally, a child is selected to count the manipulatives, and the class learned whether they each selected the right estimate. This fun exercise tackled number and operations, number sense, and data analysis in one fell swoop.

“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.

The Early Math Education Project is doing something it has never done before: it’s going back to college. With funding from the Robert R. McCormick Foundation, planning is underway to spread the project’s brand of foundational mathematics to community colleges throughout the Chicago area. The new model: train the trainers, which involves using materials and learning already developed during the Early Math Project’s five years of in-service professional development with CPS.

During the 2011-12 year, data have been collected from community college instructors and students through interviews, focus groups, syllabus reviews, and surveys. This is informing the planning for a pilot implementation program in 2012-2013. Early childhood education faculty from community colleges will complete a Summer Institute — which includes a day of learning at Erikson and a month of learning online, all designed to encourage them to think deeply about foundational mathematics and how to encourage their students to deepen their understanding.

“As we continue to grow,” says instructional coordinator Lisa Ginet, “we realize that a train the trainer model makes a lot of sense. We can’t be everywhere at once. Providing ideas and materials to instructors at community colleges allows us to influence early math teaching on a larger scale and earlier in a teacher’s learning process. Plus, we’ve found that many colleges don’t spend nearly enough time focused on math content.”

Faculty will be asked to develop a plan to incorporate materials from four content modules into their math methods courses in the upcoming academic year.

Thanks to the Motorola Solutions Foundation, this year the coaches and teachers of the Early Math Project’s Innovations program have benefited from new mobile technology and an innovative way to use it. With tablet computers creating easier ways to view and document experiences, it was natural to leverage that technology to bolster the coaching cycle that the project has been doing for years.

With ten Motorola Xoom tablet computers and a new website where videos and coaching data can be posted and shared with teachers and their coaches, this year moves the program to the “cloud.” Coaches document their coaching cycle by inputting online forms rather than paper forms. Rather than burning video observations onto DVDs, videos are posted on the Early Math Project’s website. All of this remains password-protected and secured so that only teacher and coach have access to their videos and coaching documentation.

With the Innovations program lasting until 2015, the end result of the website will be a multimedia portfolio of each teacher’s coaching experience over years of learning, full of videos, reflections, and ongoing goals.

“The Xooms and website really help us be more efficient,” points out Donna Johnson, an instructor, coach, and coaching supervisor.. “If we are more efficient, then that means there is more opportunity to reflect on teacher practice and the overall coaching process.”

The website, Early Math Online, was built in the Fall of 2011 to incorporate many pieces of what the Early Math Project does into one secure online resource. While participating teachers are logged in, they can not only view their own observation videos, but they can also access videos and view Powerpoint presentations that were shown in previous learning labs, download documentation and forms, and view announcements about upcoming programming.

These services are made possible by a generous grant from the Motorola Solutions Foundation’s Innovation Generation program.

The coaching experience is an integral part of programming for the Early Mathematics Education Project. “Coaching” means many things, but what sets the project apart from others is its intentional and focused approach. No one knows this more than the coaches who perform this coaching model, and particularly, when they reflect on the first one or two coaching cycles of the year.

“I think that now that our first coaching cycles are complete,” says Veronica Castro, a coach with the Early Math Project’s Innovations program, “and teachers have a feel for what our coaching cycle is like (as opposed to other coaching they may have received in the past), they feel more comfortable sharing their thoughts and ideas about the lessons and what they want their students to learn.”

She adds, “I feel like most of them are not waiting for me to give them the answers so much anymore but instead are throwing their ideas at me and looking for feedback. For me, that is important. I want my teachers to feel like I am a partner in their classroom, not an inspector, evaluator, etc. I don’t feel like that happens enough in the teaching profession.”

The coaching cycle is a three step process. It begins with a planning conversation in which the teacher sets goals for the students’ learning and for the teacher’s practice. The second part of the cycle is the classroom observation of a lesson, usually videotaped, while the coach takes notes that will allow the teacher to see progress toward the goals set in the planning conversation. The final component of the cycle is the reflection. During this conversation, the coach and teacher reflect on the video of the lesson as well as the notes that were taken.

Jill Sapoznick, another Innovations coach, comments on both creating those moments of learning and merely being an advocate for learning. “From my point of view, the coaching cycle allows for some intimate discussions regarding teaching and allows for each teacher to think through his or her process,” she says. “The most interesting and useful part is being the sounding board and catalyst for growth, all at the same time. Each component of the coaching cycle provides an opportunity to examine one thought, one moment in time, in depth. This is a rare privilege that all educators should get.”

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 the game 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.

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.

In early November, Erikson Early Math Team members made several presentations at NAEYC’s annual conference. In a three-hour session, instructors Rebeca Itzkowich and Mary Hynes-Berry had participants engaged in exploring Big Ideas about measurement as they used both direct and indirect measurements in order to figure out how long a piece of leather would need to be to make a lovely tree “bracelet.” They later turned their adding machine tape lengths into rulers with arbitrary units so they could determine not just which tree girth was bigger, but how much bigger it was.

As further proof that math is all around us, we were delighted to see that in December examples of “yarn bombing” or knitting cozies for trees showed up in Evanston! We’re sure there was all kinds of measuring involved.

There are any number of wonderful books that give children very concrete images to show how big creatures are and to make comparisons to their own size. Steve Jenkins’ Actual Size and Prehistoric Actual Size have been favorite choices in our past workshops on measurement. Find the author’s website here: http://www.stevejenkinsbooks.com

In last year’s Idea Book, Laura Miller reported how she and and her preschoolers from Prieto Academy explored linear measurement after studying polar animals. They wondered, “Are we taller or shorter than an Emperor Penguin?” They used the book Penguins! by Anne Schreiber to find out that the average height of an emperor penguin was about 44 inches tall.

They used measuring tape and large butcher paper to mark this height and, with the help of an overhead projector and a toy penguin, they created a shadow of a penguin. Students worked together to make the penguin shadow match the mark they made on the butcher paper. Children compared the height of the shadow to the mark on the butcher paper. Comments such as “We have to make it bigger,” “Not that big!” and “Move it over!” were made until the height of the shadow and the mark on the butcher paper were the same. The class traced the shadow of the penguin and then they were ready to compare their heights to that of the penguin. Comparison statements that the children used included, “I am the same size as the penguin,” and “I am a little bit taller.”

After this experience with direct comparison, Ms. Miller asked the students, “How else can we measure this penguin?” The first response from a student in the morning class was, “Books!” Discussion arose surrounding how they could use books to measure. There were a number of questions to consider:

• Should they lay the books flat or stand them up?
• Would they fall down?
• Are all books the same size?

Ms. Miller shares the conversation that ensued:

One student, Laura, suggested we could lay the books flat so they would balance better than if we stood them up. After realizing that not all books are the same size and it would take a lot of books to complete the task, Jonathan suggested we use blocks! “We can use blocks; they’re the same!” Jonathan said in reference to blocks being the same size. Children shared their ideas of what they thought would be the best blocks to use. It was decided that triangles would be too tricky and rectangular ones would work best. After stacking the blocks and counting how many long rectangle blocks (8) and how many square rectangle blocks (16) would equal the size of a penguin, one of our reading buddies pointed out that the number was doubled because it takes two square rectangles to make a long rectangle.

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