Erikson Institute Early Math Collaborative https://earlymath.erikson.edu Early Math Collaborative Mon, 12 Feb 2024 15:55:28 +0000 en-US hourly 1 https://wordpress.org/?v=6.4.3 https://earlymath.erikson.edu/wp-content/uploads/2018/05/cropped-Favicon-150x150.png Erikson Institute Early Math Collaborative https://earlymath.erikson.edu 32 32 What Do We Mean by Counting Principles and What Do Early Childhood Educators Need to Know? https://earlymath.erikson.edu/what-do-we-mean-by-counting-principles-and-what-do-early-childhood-educators/ https://earlymath.erikson.edu/what-do-we-mean-by-counting-principles-and-what-do-early-childhood-educators/#respond Tue, 23 Jan 2024 19:04:48 +0000 https://earlymath.erikson.edu/?p=20459 Counting is a part of young children’s everyday life. They love to count everything from the stairs they climb to  More →

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Counting is a part of young children’s everyday life. They love to count everything from the stairs they climb to the crackers they eat. But what is counting?

To adults, counting seems very simple, but it is actually quite complex. Counting is a precise, step-by-step procedure governed by rules. In order for counting to work—that is, in order to use it and get the right number—children have to coordinate all of these rules at the same time into one set of actions.

As a teacher, it can be helpful to know about these rules so you can be more specific about what each child is working on and how you can help. There are 5 Counting Principles:

  1. Stable Order
  2. One-to-One Correspondence
  3. Cardinality
  4. Order Irrelevance
  5. Abstraction

This article focuses on the first three, since they are absolutely required for a counting procedure to be accurate.

What’s the stable order principle?

Counting words must be said in the same order every time. As they learn, children often skip numbers, or get them out of order. They may even get the sequence right one day but mix it up the next! This is a normal part of the development of counting skills.

Put this into practice:

One of the best ways to help children clarify the sequence of the number words is with counting rhymes and songs that incorporate using fingers to indicate amounts. Young children love to recite these stories—whether about ducks, sausages, or elephants!—and become invested in remembering which number comes next. Using their fingers to show each number as they say it out loud adds an element of meaningful understanding. If they can show “four,” they must “know” four!

Math at your Fingertips: Songs and Fingerplay for Preschoolers

What’s the one-to-one correspondence principle?

Each object in the set to be counted must be counted once, and only once. Keeping track of which objects you’ve already counted and which ones remain to be counted is an additional task children have to learn to manage. As they learn, they may count some objects twice, skip objects, or even skim their finger across the whole set as they recite counting words.

Put this into practice:

Modeling and supporting precise counting is key. Children need opportunities to see a mature counter slowly and deliberately tracking objects as they count. They can benefit, too, from the suggestion to point at each object or move objects to the side as they are counted. Helping children become accurate in this way supports their success and will inspire confidence.

Visual tools such as the number path seen below support both the stable order principle and one-to-one correspondence. Placing one object in each corresponding box can support children’s ability to keep track of the count.

One-to-One Correspondence

Ask young children, “How does this tool help you keep track of your count?” An empty egg carton is a useful tool as well.

Number paths – a visual model of our counting system

What is the cardinality principle?

The last word in the counting sequence indicates the total amount. It’s not obvious to young children that they should use the last number they say to describe the total amount. They are often so absorbed in getting the sequence of words right and not missing any of the objects to be counted that they arrive at the last number and simply stop. If you ask them, “how many is that all together?” they may even start their counting procedure over from one, as though producing a counting procedure is what you want them to do.

Put this into practice:

Asking “how many is that all together?” after children count a set is the key here. If you find they don’t know, you can model and support their understanding by demonstrating the cardinality principle yourself, “for example, “…three, four, five…that’s five crayons.” If you make a habit of this, they will eventually get it.

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RJEM Project Announces the 2023-24 Teaching Fellowship Cohort https://earlymath.erikson.edu/2023-24-rjem-teaching-fellowship-cohort/ https://earlymath.erikson.edu/2023-24-rjem-teaching-fellowship-cohort/#respond Mon, 14 Aug 2023 13:16:07 +0000 https://earlymath.erikson.edu/?p=20305 In May of 2023, a diverse group of six kindergarten teachers from across the United States were selected from a  More →

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In May of 2023, a diverse group of six kindergarten teachers from across the United States were selected from a national pool of talented and committed kindergarten teachers to participate in the RJEM Teaching Fellowship. The RJEM Teaching Fellowship is a one-year professional development, mentoring, and research opportunity intended to support kindergarten teachers who are committed to racial justice in early math education.

Representing different racialized and gendered groups, geosociopolitical contexts as well as years of experience and levels of expertise, the six RJEM Teaching Fellows have demonstrated strong commitments to racial justice, personal and professional growth, and working collaboratively. Racial justice work requires commitment, courage, patience, and humility. Thus, the RJEM Local Planning Committee intentionally put together a cohort of teachers who have demonstrated these characteristics and who will support each others’ learning.

At the RJEM Teaching Fellowship Summer Orientation held in June of 2023, the Fellows spent two days at the Erikson Institute in Chicago getting to know each other and the many others who will support their experience (mentors, members of the Local Planning Committee, and members of the national RJEM Advisory Committee).

Fellows also learned more about the structure of the Fellowship and engaged with foundational critical theories (e.g., positionality, white supremacy, antiblackness, systemic oppression) and resources (e.g., Applying RJEM to the Classroom, and How to Have Hard Conversations) that will support their learning in the upcoming academic year.

During the fellowship year and through individual mentoring, group seminars, and a self-selected Action Research Project, RJEM Teaching Fellows will be supported, motivated, and challenged to engage in self-reflection, develop a robust understanding of early math education that fosters racial justice, and implement pedagogical practices that honor the humanity of all children along with their families and communities, especially those who have been historically dehumanized in the United States.

We are excited for the amazing work that the RJEM Fellows will develop throughout the course of the fellowship and look forward to sharing our learning and experiences with you in the months ahead. Stay tuned.

racial justice early math fellowship

RJEM Teaching Fellowship

Read more about the Fellowship and learn more about the 2023-24 Fellowship Cohort.

Learn more

 

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The Power of Books to Support Racial Justice-Centered Math Teaching and Learning https://earlymath.erikson.edu/books-to-support-racial-justice-centered-math-teaching-and-learning/ https://earlymath.erikson.edu/books-to-support-racial-justice-centered-math-teaching-and-learning/#respond Thu, 10 Aug 2023 13:36:36 +0000 https://earlymath.erikson.edu/?p=20313 From a racial justice centered perspective, books can be powerful tools to engage children in mathematics learning. The worlds, stories,  More →

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From a racial justice centered perspective, books can be powerful tools to engage children in mathematics learning. The worlds, stories, adventures, characters, and possibilities that are made available through books – explicitly math focused and otherwise – allow teachers and children to wonder, question, make predictions, engage with different ways of knowing, solve problems, organize and communicate complex ideas, and see the world through different lenses. All these are necessary attributes to a robust math learning experience.

Alongside the positive impacts of diverse racial representation in books, a racial-justice centered approach also utilizes books as crucial opportunities to counter deficit-oriented perspectives that position historically marginalized groups as inferior, abnormal, and incapable of complex thinking and knowing. In this sense, books are used to honor, celebrate and learn about people’s history, culture, (math) knowledge, accomplishments, and potentials.

To help incorporate racial-justice centered books in your school and classroom libraries, Donna Johnson – a founding member of RJEM and one of the mentors for the RJEM Teaching Fellowship – has curated a rich list of books that includes math focus and/or sociocultural centered titles. Take advantage of this wonderful resource as you get ready for the upcoming school year.

Beautiful Me by Nabela Noor. Illustrated by Nabi H. Ali

Meet Zubi, a joyful Bangladeshi girl excited about her first day of school. But when Zubi sees her mother frowning in the mirror and talking about being “too big,” she starts to worry about her own body and how she looks. As her day goes on, she hears more and more people being critical of each other’s and their own bodies, until her outburst over dinner leads her family to see what they’ve been doing wrong—and to help Zubi see that we can all make the world a more beautiful place by being beautifully ourselves (From author’s Page). Link »

Count Me In!: A Parade of Mexican Folk Art Numbers in English and Spanish by Cynthia Weill. Illustrated by Aguilar Sister

We have written about this book previously numerous times. Check out more information and resources. Link »

Count On Me by Miguel Tanco

A young girl sees the world differently in this beautiful picture book celebration of math. This book is a gorgeous ode to something vital but rarely celebrated. In the eyes of this little girl, math takes its place alongside painting, drawing and song as a way to ponder the beauty of the world. (From author’s Page) Activity Idea »

Counting Kindness: Ten Ways to Welcome Refugee Children by Hollis Kurman. Illustrated by Barroux

Arriving in a new place is stressful for newcomers, especially when the newcomers are little ones. But this beautiful counting book helps readers see the journey of finding a new home and the joys of being welcomed into a new community. From playing to sleeping, eating to reading, celebrating to learning, Counting Kindness proves we can lift the heaviest hearts when we come together. (From publisher’s page)

Counting to Diwali by S. C. Baheti. Illustrated by Rohan Dahotre

Bring the magic of Diwali home with this incredible book. Learn how to count from one to ten in Hindi, all the while sharing in the excitement of the festival of lights. This book is a great way to reinforce language learning, as well as building an integral part of Indian culture with your family, in a fun and colorful way.

Dr. Zola and Daddy Explores Ear Infection by Darrin “1831” Collins. Illustrated by Doran Starks

On this adventure, Zola is suffering from a common childhood ailment-an ear infection! Her father explains the source of her pain by taking her into his high-tech laboratory. Once inside, Zola learns everything about the inner workings of her ear and her ear infection, through an unforgettable virtual tour. Every moment is memorable as Zola and daddy learn from one another!

Dumpling Day by Meera Sriram. Illustrated by Inés de Antuñano

Readers follow ten diverse families as they cook dumplings inside their homes in preparation for a neighborhood potluck. Dumplings are added to plates one by one, encouraging children to count with each new addition. Authentic recipes for all the dumplings and a map showing their regions of origin are included in the endnotes. (From author’s page)

dumpling day

Feast For 10 by Cathryn Falwell

A counting book that features an African-American family shopping for food, preparing a meal, and sitting down to enjoy a festive dinner together. (From author’s page) Activity Ideas »

How Many? by Christopher Danielson

Written by a math educator, this innovative book encourages critical thinking and sparks memorable mathematical conversations. You and your child decide what to count on each page. You have many choices, and the longer you look, the more possibilities you’ll notice. There are no wrong answers in this book. As long as you’re talking about what you see, think, and wonder, you’re talking math! (From publisher’s page)

Just A Minute: A Trickster Tale and Counting Book by Yuri Morales

This spirited tribute to the rich traditions of Mexican culture is the perfect introduction to counting in both English and Spanish. The vivacious illustrations and universal depiction of a family celebration are sure to be adored by young readers everywhere. (From author’s page)

Lia & Luis: What Has More? by Ana Crespo. Illustrated by Giovana Medeiros

Twins Lia and Luís argue over who has more of their favorite snacks. Can the siblings use math–and a little sharing–to pick the winner? A playful exploration of measurement, counting, and estimation, featuring Brazilian American characters and a glossary of Brazilian Portuguese words. (from Amazon’s page)

One Sun and Countless Stars: A Muslim Book of Number by Hean Khan. Illustrated by Mehrdokht Amini

From one sun to countless stars, this gentle introduction to numbers also celebrates the many diverse traditions of the Muslim world, encouraging readers young and old to reflect upon—and count—their many blessings. (From author’ page)

We All Count: A Book of Cree Number by Julie Fleet

We All Count a Book of Cree Numbers provides insight into contemporary Cree life. It teaches Cree numbers and provides pronunciation. This board book is wonderfully illustrated by Canadian based Cree/Metis artist Julie Flett. This book is important in enhancing learning of Cree numbers and making Cree culture accessible to young readers. (From author’s page)

we all count

She Persisted: 13 American Women Who Changed The World by Chesea Clinton. Illustrated by Alexandra Boiger

She Persisted is for everyone who has ever wanted to speak up but has been told to quiet down, for everyone who has ever tried to reach for the stars but was told to sit down, and for everyone who has ever been made to feel unworthy or unimportant or small. With vivid, compelling art by Alexandra Boiger, this book shows readers that no matter what obstacles may be in their paths, they shouldn’t give up on their dreams. (From author’s page)

Sometimes We Do by Omo Moses. Illustrated by Diego Chaves

Modeling ways for fathers to teach math to their small children in everyday interactions. Johari loves Daddy days, when he and his father make pancakes and play with trains. Together Johari and his father chat about size, number, amount, recipes, and family chores. Playful illustrations allow a glimpse of Johari’s fantasy world and invite further discussion of the book’s early math concepts. (From publisher’s page)

10 Gulab Jumans by Sandhya Achayra. Illustrated by Vanessa Alexandre

Idu (Ee-doo) and Adu (Aa-doo) want to eat the Gulab Jamuns Mama has made, but Mama has asked them to wait for the guests. But how does one stay away from soft, sticky, melt-in-your-mouth Gulab Jamuns? Before long… 1, 2, 3… slurp! How many Gulab Jamuns do you think are left? This sweet story combines STEM elements, sibling love and fun characters. (From author’s page)

Think Big, Little One by Vashti Harrison

Featuring 18 trailblazing women artists and scientists, Think Big, Little One is the irresistible board book adaptation of Vashti Harrison’s second book Little Dreamers: Visionary Women Around the World. (From author’s page)

Up to My Knees by Grace Lin

Mei is so excited about her new garden! Discover the foundations of measurement and comparison with her as she waters, tends, and harvests her carefully-cultivated crop. Yum! Math is delicious. (From author’s page)

Usha And The Big Digger by Amitha Jagannath Knight. Illustrated by Sandhya Prabhat

When sisters Usha and Aarti look up at the stars, they see different things. Aarti sees the Big Dipper, but Usha sees the Big DIGGER. And cousin Gloria sees the Big Kite! Could they all be right? A playful introduction to geometry and spatial relationships, featuring Indian American characters and a note about cultures and constellations. (From author’s page)

We Are One by Susan Hood. Illustrated by Linda Yan

This book offers a mind-expanding look at early math concepts like part/whole relationships, fractions, and addition. Underlying themes of cooperation, peace, and unity make this visually stunning volume one to be enjoyed by anyone at any age. (From author’s page)

Zero by Kathryn Otoshi

Zero is about a big round numero, Zero. When she looks at herself, she just sees a hole right through her center. She admires the other number who can count. She wants to count too, but wonders who can a number worth nothing become something? Thus, begins the story of Zero’s research to find value in herself and in others. (From author’s page)

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Big Idea Talk and Play: Co-Discovering the Power of Open-Ended Math Exploration https://earlymath.erikson.edu/big-idea-talk-and-play-co-discovering-the-power-of-open-ended-math-exploration/ https://earlymath.erikson.edu/big-idea-talk-and-play-co-discovering-the-power-of-open-ended-math-exploration/#respond Tue, 08 Aug 2023 18:26:54 +0000 https://earlymath.erikson.edu/?p=20311 RJEM at Work highlights the everyday efforts of caregivers, community organizations, teachers, researchers, and educational institutions to promote racial justice  More →

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RJEM at Work highlights the everyday efforts of caregivers, community organizations, teachers, researchers, and educational institutions to promote racial justice in the mathematical experiences of children. For the RJEM Local Planning Committee, there are multiple ways to foster racial justice in early mathematics. As such, RJEM at Work aims to showcase a diverse set of practices. By doing so, we hope not only to share interesting work but also to inspire action at different levels and points of entry in the area of early mathematics education.

Dr. Rebecca Anne Swartz, Assistant Professor at Southern Illinois University- Edwardsville, shared with us her experiences at the Let’s Talk Math Project – a project intended to re-envision a math pedagogy course for teacher candidates. Countering mainstream deficit-oriented perspectives that position students as passive learners, Let’s Talk Math positions both teacher candidates and their students, especially those of historically marginalized communities, as producers of knowledge who are curious and fully capable of learning.

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Photo Chat about Quantities Puts the Focus on Units https://earlymath.erikson.edu/photo-chat-quantities-units/ https://earlymath.erikson.edu/photo-chat-quantities-units/#respond Wed, 10 May 2023 18:49:25 +0000 https://earlymath.erikson.edu/?p=20258 In this video, preschoolers share the quantities they see when looking at a photo of a box of doughnuts. There are many ways to answer the question, how many of what?, depending on the unit. 

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Twelve doughnuts or one dozen? How Many of What photo chats invite children to talk about how many of what they see. A quantity is expressed as a number and its unit (e.g., 3 chocolate doughnuts or 1 doughnut with sprinkles).

In this video, preschoolers share the quantities they see when looking at a photo of a box of doughnuts. There are many ways to answer the question, how many of what?, depending on the unit. 

Mathematical thinking is strongly connected to oral language development; therefore, we encourage conversations about not only how many children see but how they see it. Offering children an image with many smaller quantities within a larger total offers all children a point of entry. The quantities in the photo are composed so that children can practice looking for and making use of structure to determine how many of what. 

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What is a Number Path and How Does It Build Number Relationships? https://earlymath.erikson.edu/number-path/ https://earlymath.erikson.edu/number-path/#comments Wed, 22 Feb 2023 18:17:44 +0000 https://earlymath.erikson.edu/?p=20211 Many of children’s favorite board games follow a path to a goal. Exploring a number path with young children also  More →

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Many of children’s favorite board games follow a path to a goal. Exploring a number path with young children also leads to a goal that is critical for early mathematics: understanding number relationships.

What is a Number Path?

A number path is a visual model of our counting system. It simply shows the order of the natural numbers in a linear fashion. Natural numbers are also known as counting numbers because these numbers can be counted on our hands. For example, the numbers that we count on our hands are 1, 2, 3, 4, 5, 6, and so on.

A number path does not include 0 and only shows whole numbers—no fractions, no decimals, no negative numbers—so it’s a great place to start with preschool and kindergarten aged children.

number path

How is a Number Path Different from a Number Line?

A number line is also a visual model of numbers in a straight line, but a key difference is that it is a measurement model. This means that numbers are placed at equal intervals along a given scale. Zero is the origin of the number line and it extends in both directions, representing both positive and negative numbers along with fractions and decimals. The magnitude, or size, of numbers is represented by their distance from zero.

For young children, it can be difficult to understand that a number line requires us to count the intervals of distance, not the tick marks or numerals themselves.

number line

What Are the Advantages of a Number Path for Young Children?

In preschool and kindergarten, children have many opportunities to count blocks or other objects. As they count a set of objects, children don’t automatically think about how one number is related to another. Their goal is matching one number word at a time to one object until they reach the end of the set.

Number paths help children connect this “set-representation” view of numbers to a model that puts numbers in relationship to one another. The number path has great advantages in helping children understand the order and relative position of numbers.

Children need to explore how numbers relate to one another in order to build a robust and flexible number sense. Take the number 6 for example. Six is one more than 5 and two less than 8. A number path can help develop a rich understanding of numbers – their meanings, their relationships to one another, and how we operate with them.

How Can I Use Number Paths with Young Children?

Life-Size Number Paths

It’s easy to make a life-size number path with painter’s tape on the floor or using sidewalk chalk outdoors. There are many relationships that children can explore through movement and visualization such as numbers next to and numbers far away from their position on the path. Give children number riddles they can solve by moving forward and back on the number path. For example, “Stand on the number that is one more than 7.”

number path example

Number Path Board Games

Turn a number path into a simple board game. Children take turns rolling their die and moving a game piece along the path. The goal is to be the player whose game piece reaches the end first. A fun version is Jumping on the Lily Pad from the Young Mathematicians project at EDC. Click here for game materials including game boards to print and video directions.

Lift-the-Flap Number Paths

One of our favorite versions of a number path has flaps that cover each number. This is a playful opportunity to reason about hidden numbers using their relative positions. For younger children, start with numbers 1 to 10. For older students, you can shift the range higher—for example, 35 to 45. You can see this type of number path in the following videos, one from preschool (the first) and one from kindergarten (the second).

Related Ideas

Get on the Right Path: Make Your Own Number Sense Activities

Path games are fantastic ways for families to spend time together and have fun while doing math.

Learn more

Using Number Paths with Child 9

A child counts forwards and backwards using a number path with hidden digits.

Learn more


Photo by Austris Augusts on Unsplash

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Fostering Racial Justice and Deep Mathematical Thinking in Chicago PreK https://earlymath.erikson.edu/racial-justice-deep-mathematical-thinking-chicago-prek/ https://earlymath.erikson.edu/racial-justice-deep-mathematical-thinking-chicago-prek/#respond Mon, 23 Jan 2023 14:29:21 +0000 https://earlymath.erikson.edu/?p=20141 RJEM at Work highlights the everyday efforts of caregivers, community organizations, teachers, researchers, and educational institutions to promote racial justice  More →

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RJEM at Work highlights the everyday efforts of caregivers, community organizations, teachers, researchers, and educational institutions to promote racial justice in the mathematical experiences of children. For the RJEM Local Planning Committee, there are multiple ways to foster racial justice in early mathematics. As such, RJEM at Work aims to showcase a diverse set of practices. By doing so, we hope not only to share interesting work but also to inspire action at different levels and points of entry in the area of early mathematics education.

This video features the work of teachers Kristin Roberts and Anne Kellogg and their students in a Blended PreK Classroom in the Chicago Public Schools.

Inspired by the work of Dr. Margaret Burroughs – a Black Chicago artist, teacher and institution builder – students engaged in detailed observations of the city and their neighborhood and discussed issues of city occupation, labor, and history.

Aligned with the goal of developing children’s deep mathematical thinking, Ms. Roberts and Ms. Kellog emphasized respect to the many ways children of different racial and ethnic backgrounds engage with the city as well as created spaces in which their ways of knowing were accounted for and celebrated.

This project, presented at the 2022 Promising Math Conference, exemplifies a robust understanding of racial justice in early math that goes beyond focusing on content alone. Here, racial justice is centered in lesson planning and classroom structure.

We welcome suggestions of practices to be featured in the RJEM at Work series. Please reach out to us at rjem@erikson.edu to share your work and/or other recommendations.

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4 Playful Counting Books That Bring Numbers to Life https://earlymath.erikson.edu/books-about-numbers/ https://earlymath.erikson.edu/books-about-numbers/#respond Wed, 28 Sep 2022 16:37:00 +0000 https://earlymath.erikson.edu/?p=19495 Counting books that offer children opportunities to participate bring play into math learning. These books engage children physically, cognitively, and emotionally—making them favorites that children will ask to read again and again.

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Counting books that offer children opportunities to participate bring play into math learning. These books about numbers and counting engage children physically, cognitively, and emotionally —- making them favorites that children will ask to read again and again.

Whether at home or in a classroom, children can create their own artwork in the style of these books. Read on for ideas how to play around with numbers!

  1. 1 Smile, 10 Toes by Nelleke Verhoeff

    This mix-and-match counting book is clever and won the Mathical Book Prize in 2022! Children can flip the split pages around in any order to make creatures with different tops and bottoms. The silly creatures have 4 ears, 5 feet, and so on. With so many combinations, children will have ample counting practice in a non-routine way.

    The book practically begs children to draw their own wacky creatures and talk about how many different body parts they drew. One way to do this is to fold a sheet of paper in half. At the fold line, mark where the creature’s torso will meet from the top and the bottom. Have one child draw the top of the creature and a second child (without looking) draw the bottom. Then open the page to reveal the silly creature! Have children label their creature with the number of different body parts like the book does.

1 smile 10 toes

  1. How to Two by David Soman

    Numbers words are used as verbs in this counting book. How do you “two?” In this story, a young child makes “two” by finding a friend on the see-saw at the playground and makes “three” by turning one end of a jump rope with another child holding the other end while a third child jumps. Using number words in this novel way may require some conversation, but isn’t that the whole point anyway?!

books about numbers

    The illustrations are uncluttered and clearly show the growing set of children playing at the playground. Yet, look closer and there are other representations of the given number on each page. As the number of children increases, readers will have to look more closely to find all the children. This brings the opportunity to discuss number composition and the number relationship of “one more.” 

    Ask children to draw their own playground pictures and talk about the number of friends they like to play with. Contextualizing numbers within a child’s social-emotional life provides a natural and meaningful reason to count! 

  1. Two Dogs on a Trike by Gabi Snyder, illustrated by Robin Rosenthal

    What a zany and fun counting book! The narrative is told chiefly through the illustrations, which show a growing group of dogs traveling by wacky modes of transportation. As the number of dogs grows, the jumbled and sometime overlapping arrangement of the dogs might be challenging for young children to count. But, for more proficient counters, finding all the dogs could be part of the fun.

    The text is simple. Each page clearly states the number of dogs. But when it comes to books about numbers and counting, there is much to bring children back for repeated readings. The subplot of a cat’s pursuit of the dogs provides humor and adds to the variety of vehicles in the story. This would be a fun book to physically reenact, building both transportation vocabulary and a sense of numbers in a playful way. 

  1. Dozens of Dachshunds by Stephanie Calmenson, illustrated by Zoe Persico

    More dogs on the move! This charming counting book about numbers shows dachshunds in a town pet parade. The count goes up to 12, building the concept of a “dozen” as a unit. The illustrations of the dogs are distinct and easy to count. The dogs and humans represent a wide diversity, including a girl and a dog using wheeled mobility equipment.

    The text is pleasant to read aloud with a rhyming scheme. Children can sing along! There are extra features too that not just any book about numbers and counting will have. The text mentions the total number of dachshunds at the end of the parade (a good problem to have older children figure out for themselves) and there is robust information about the dachshund breed at the end. 

books about numbers with Dozens of Dachshunds

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Collaborative Math: Creating Sustainable Excellence for Head Start Programs https://earlymath.erikson.edu/creating-sustainable-excellence-math-for-head-start-programs/ https://earlymath.erikson.edu/creating-sustainable-excellence-math-for-head-start-programs/#respond Tue, 26 Jul 2022 19:04:10 +0000 https://earlymath.erikson.edu/?p=19465 Early childhood center leaders play a crucial role in improving and sustaining program quality. New research supports the idea that  More →

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Early childhood center leaders play a crucial role in improving and sustaining program quality. New research supports the idea that involving center leaders in PD initiatives is a way to produce lasting systemic change.

Research on effective professional development (PD) is clear that there must be school-embedded systems to provide ongoing, on-the-job support for teachers to grow their practice.

Collaborative Math takes such a “whole-center” approach to PD by bringing together center leaders and the entire teaching staff around the shared purpose of improving preschool math learning.

With funding from the National Science Foundation, researchers from Erikson Institute, SRI International, and Digital Promise designed Collaborative Math to build on the strengths of and respond to the needs of Head Start teachers to create centers of mathematical excellence.

A Whole-Center Approach to Professional Development

Collaborative Math is a 1-year program designed to foster, celebrate, and sustain high impact, developmentally appropriate mathematics instruction in Head Start centers. It is guided by the premise that teaching improvement is driven by the support and development of teachers’ knowledge, confidence, and practice. Collaborative Math brings together leadership (directors and education coordinators) and all teaching staff (i.e., lead teachers, assistant teachers, and paraprofessionals) from multiple Head Start centers to study early mathematics. Center leaders and teachers return to their sites to implement what they learn with project support.

Teachers and center leaders participated in six cycles of workshops, coaching, and consultation focused on sets and sorting, number sense, and shape. Each 4-week cycle began with a 3.5-hour off-site learning lab for center leaders and teachers, followed by a 4-hour leadership training for center leaders. Then, center leaders participated in up to 2 hours of on-site consultation from an Erikson math coach. The focus of consultation was supporting classroom coaching activities and implementing center wide core components of the Collaborative Math program.

Erikson coaches also provided on-site coaching to classroom teaching teams. Coaching teachers in teams represents an innovative approach, designed to support collaborative work among all staff in a single classroom. Each classroom coaching cycle consisted of three 30-minute sessions – a planning meeting, a classroom observation, and a reflecting meeting.

Center leaders held a dual role:

  1. help sustain coaching cycle meetings, including coordinating coverage as needed; and
  2. participate in one coaching cycle per month alongside their Erikson coach, rotating through all the participating teaching teams at their site.

With support from the Erikson coach, center leaders and teaching staff implemented center-wide math lessons, family math activities, and a center-wide celebration of math learning. Center leaders also assisted in the set up and maintenance of classroom math resource libraries and displays documenting math-related activities to share with others at the site.

Development and Evaluation

Over the course of three years, 31 centers, 57 center leaders, 209 teachers, and 1,508 children participated in the project. During the first year, three centers helped to develop and pilot the intervention. During the second and third years, two evaluation studies examined whether Collaborative Math had positive impacts on teacher, center leaders, and child outcomes.

  • Study 1 (2016-17 school year): Researchers randomly assigned 28 centers to participate in the PD or continue to implement business-as-usual math lessons.
  • Study 2 (2017-18 school year): Researchers worked with the centers who did not participate in the PD during the previous year.
  • Study 3 (2017-18 school year): Researchers followed the treatment centers from the 2016-17 school year to examine sustainability.

Center leader and teacher outcomes included measures of math content knowledge, confidence in math teaching, math instructional practice for teachers, and math leadership practice for center leaders. Child outcomes included two standardized measures of mathematics knowledge and skills.

Impact

Participation in Collaborative Math led to improved confidence in math teaching for teachers and center leaders at the end of each evaluation study (see Figure 1). Children attending centers that participated in the PD during Study 2 outperformed their peers who participated during Study 1 on one of the math measures. During the follow-up year, Study 3, teachers who participated in Collaborative Math the previous year reported continuing to implement program activities such as math games, transition activities, and family engagement. Both center leaders and teachers indicated a desire to continue coaching and use of Collaborative Math approaches.

Implications

The project’s findings have important implications for providing PD in Head Start and other early childhood education settings. Early childhood center leaders play a crucial role in improving and sustaining program quality. Involving center leaders in PD initiatives can produce lasting systemic change. If the organization as a whole changes, teachers will not have to take sole responsibility for continuing program effects but will instead be supported by—and continue to contribute to—a professional learning community focused on mathematics.

Importantly, the outcomes of this project suggest that a one-year PD intervention may not be sufficient to cause long lasting, systemic changes in teacher practice and student learning outcomes in early childhood settings. More robust effects on child outcomes may only emerge after the PD model has been implemented successfully over a longer period of time.

This material is based upon work supported by the National Science Foundation under Grant DRL-1503486. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Photo by Yan Krukov

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Racial Justice in Early Math Resource List https://earlymath.erikson.edu/racial-justice-early-math-resource-list/ https://earlymath.erikson.edu/racial-justice-early-math-resource-list/#respond Wed, 20 Jul 2022 18:10:33 +0000 https://earlymath.erikson.edu/?p=19459 One important component of racial justice work is the commitment to continuous learning and reflection. Regardless of how long you  More →

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One important component of racial justice work is the commitment to continuous learning and reflection. Regardless of how long you have been engaged in racial justice in early math, there is always something interesting, relevant, and new to learn.

Based on the feedback from participants of the Promising Math 2022: Racial Justice in Early Mathematics Conference, we prepared a resource list that speaks to racial justice in early math. We hope to add to it on an ongoing basis.

The list currently includes two journal articles, one lecture video, and one essay. Some of the topics discussed in these resources are directly related to early math education. Others, while broader in scope, are necessary to be engaged with for the advancement of racial justice in early math education. Together, these resources highlight particular systems, dynamics, and beliefs that inform children’s math experiences. Enjoy your learning!

Centering Children in Mathematics Education Classroom Research by Amy Parks (Journal Article, 2019)

Journal Information: American Educational Research Journal, 57(4), 1443-1484.

Details

Over the course of three consecutive years–from pre-kindergarten to first grade, Amy Parks analyzed the experiences of Aisha and Jeremiah–a Black girl and a Black boy respectively. Parks found that teachers’ negative gendered perceptions about Black girls and Black boys impacted Aisha’s and Jeremiah’s mathematics development and mathematics identity. Specifically, Parks shows that perceptions of (i) Black girlhood as disruptive and defiant as well as (ii) ‘proper behavior for girls’ informed the teacher’s constant reinforcement of Aisha’s silence and stillness over the years. This reinforcement inferred with her mathematical engagement by giving positive feedback for being passive, rather than active learner. Similarly, assumptions of Black boys as dangerous and always in need of correction informed how teachers interpreted Jeremiah’s participation in the classroom. In spite of being an active learner, confident, and a leader in mathematics, Jeremiah was often perceived as disruptive and disproportionately subjected to punishment. Often positioned as troublesome, Jeremiah’s mathematics engagement was overlooked leading to his exclusion from the category of good student and positioned as troublesome. Parks suggests that teachers’ perceptions and interactions with students also impacted how they saw themselves as mathematics learners and doers.

Link »

Race, Ideology, and Academic Ability: A Relational Analysis of Racial Narratives in Mathematics by Niral Shah (Journal Article, 2017)

Journal Information: Teachers College Record, 119(7), 1-42.

Details

Moving beyond mainstream comparative analysis between the performance of white and Black students, Niral Shah worked with a racially and ethnically diverse group of students–Black, Asian, Latinx, Polyenisian–to investigate the kinds of stories students tell about their race and academic ability–and math in particular. Shah calls these stories racial stories or racial narratives. Shah’s analysis shows that racial stories about mathematics abilities shared by students are rooted in widespread stereotypical views about different racial groups (e.g., Asians are good in math, Black people are inherently good at sports, Mexicans are fit for manual labor). Shah also demonstrates how such discourses are created by social, political, and historical configurations of US society and always in relation to each other. For example, one Black student shared that after getting an “A” on a test, another Black student alluded to his high grade being that he must “have some Asian on you.” A Polynesian student shared that others may not see him as good at math because “Polynesian are so big they think we don’t know how to do math.” Shah suggests that racial narratives create a hierarchy of mathematics ability in which non-Asian students of color are perceived to be not as good at math when compared to Asian and white students. Shah also discusses the negative impacts racial stories have on students’ perceptions of themselves and of others as well as how such perceptions may impact their engagement in mathematics.

Link »

White Rage: The Unspoken Truth About Our Nation’s Divide by Carol Anderson (Lecture Video, 2018)

Details

Based on her New York Times bestselling book White Rage: The Unspoken Truth About Our Racial Divide, Carol Anderson provides a historical analysis–from the Civil War to 2016–on the systemic ways in which white rage punishes Black resilience and Black resolve. For Anderson, white rage is a systemic response to Black people with ambition, resolve, aspiration, and who demand their rights. Anderson explains that white rage is not necessarily about overt forms of violence, but subtle and cohesive ways that state legislature, congress, judiciary, school boards, and legalities have been used to prevent Black and Brown advancement and full participation in US society. White rage is about creating systems in which Black and Brown people cannot thrive. From Brown vs Board of Education, voting suppression, driving while Black, education (dis)investment to the war on drugs, Anderson discusses specific policies that have resulted in the educational disinfrenchament, mass incarceration, and economic exploitation of Black and Brown people. Speaking to the intricacies of interconnected systems, this engaging lecture is a great introduction to historical and systemic analysis of anti-Black policies and governmental practices in the US as well as to Anderson’s book, White Rage.

Link »

A Talk to Teachers by James Baldwin (Essay, 1963)

Details

In this short and provocative essay, James Baldwin draws from both his personal experiences and sociological analysis to shed light on myths that reinforce the subjugation of Black people and impede white people from fully developing their identity and critical consciousness. Baldwin believes that education can play a crucial role in the advancement of US as long as it (i) challenges the normalization of white superiority, (ii) provides the tools for Black and white children to critique and change the society, and (ii) exposes the complexities of history, politics, and the world. Baldwin provides a powerful note on the education he wishes for Black children–an education that (i) is explicit about the wrongs of society against Black people, (ii) reinforces the brilliance and humanity of Black children, and (iii) prepares Black children to change what is not working for their people. That many of Baldwin’s observations from 1963 still are relevant to the present moment signals that we still have a long way to go when it comes to racial justice in education.

Link »

Note from the editor: A special thank you to Sisa Pon Renie for her collaboration in earlier versions of this resource list. Thank you also to Nora Kane for compiling the resources shared by participants on the PMRJEM Conference.

Photo by Gift Habeshaw on Unsplash

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Noticing Same and Different: The Concept of Comparison with Infants and Toddlers https://earlymath.erikson.edu/same-and-different-babies-toddlers/ https://earlymath.erikson.edu/same-and-different-babies-toddlers/#respond Tue, 19 Jul 2022 18:12:33 +0000 https://earlymath.erikson.edu/?p=19432 “More” is often one of children’s first words. Infants attend to differences, though they don’t yet discriminate between “more” as  More →

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“More” is often one of children’s first words. Infants attend to differences, though they don’t yet discriminate between “more” as quantity or magnitude/size or intensity of sensation. Toddlers are beginning to differentiate between attributes and develop a spectrum of preferences. These are all examples of the origins of the precursor math concept of Comparison, that is noticing sameness and difference.

More specifically, comparison involves identifying how the attributes of one set are the same and different from the attributes of another set. The comparison allows us to create increasingly precise categories. For example, it would take no time to see the unique set of attributes that distinguish your family’s charming Jack Russell terrier from a bunch of other dogs.

Comparison is noticing sameness & difference.

  • Comparison depends on recognizing attributes.
  • Recognizing attributes makes it possible to notice sameness and difference.
  • Noticing sameness and difference allows for matching, sorting, & problem-solving with increasing precision.

In the first three years, infants and toddlers are in what we call the precursor stage—their thinking is still developing. Yet, all children seek to make sense of their world and solve the many math all around us problem situations involving specifically mathematical attributes:

Quantity (Which set has more?) Even before children learn to count, they are very good at figuring out more and less, using all their senses. Comparison is especially important for number learning since the number “three,” for example, is held by a set of three objects. So comparing 3 toy cars with 3 toy trains draws attention to the “three-ness” of the sets—the equivalency is what makes these sets the same.

Magnitude/Size (Which set is bigger? Will it fit?) Little ones, like many adults, often don’t distinguish between number and size. For example, two bigger pieces of watermelon can be “more” than three smaller pieces. Filling, packing, and stacking are fun ways for children to compare size and capacity.

Spatial-Temporal Relationships (Where is it? How long do I have to wait?) Even at birth, infants can compare how close or how far away something or someone is and when they can expect important others to attend to their need for food, toileting, rest, and of course, for play!

Sequence & Regularity (What comes next?) In their second and third years, many children can line things up from biggest to smallest, identify which sounds are the loudest or quietest, or what they expect will happen next in a familiar routine like bath time.

same and different

Take a look at this caregiver playing with a couple of toddlers. She takes advantage of the natural opportunity to compare the sizes of balls and cardboard tubes that the children want to explore.

3 questions to think about:

  • In what ways does the teacher highlight comparisons of size?
  • How does she involve the children in problem solving?
  • How do the teacher and child communicate–even though the child doesn’t say any words?

Starting in infancy, and throughout childhood, understanding of how comparison allows us to solve problems depends on important others who closely attend and intentionally respond by using words, actions, and gestures to make comments and connections, ask and answer questions, read books, and offer many opportunities to play with thinking about sameness and difference.

Conversational Math Expressing COMPARISON:

Examples of Comparative and Superlative Adjectives or Expressions

  • Quantity: more, fewer, less, equal to, as well as ordinal numbers; use of numbers to quantify units of measurement or to precisely indicate movement and direction
  • Magnitude: more, bigger, greater, smaller, including more precise identification of a dimension in various dimensions such as longer/est, taller/est, wider/est, deeper/est
  • Time: sooner, later, next day, week, month, after, before
  • Location of items or others in reference to another item or other uses comparison; adjectives will be in the comparative case such as closer, further, deeper down, on top of, behind
  • Sensory perceptions: hotter/est, colder/est, softer/est, smellier/iest, sweeter/est

Photos by Yuri Shirota and Jelleke Vanooteghem

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Using Math to Learn about Racial Representation in K-2 https://earlymath.erikson.edu/math-racial-representation-kindergarten-2nd-grade/ https://earlymath.erikson.edu/math-racial-representation-kindergarten-2nd-grade/#respond Sun, 17 Jul 2022 18:09:21 +0000 https://earlymath.erikson.edu/?p=19458 RJEM at Work highlights the everyday efforts of caregivers, community organizations, teachers, researchers, and educational institutions to promote racial justice  More →

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RJEM at Work highlights the everyday efforts of caregivers, community organizations, teachers, researchers, and educational institutions to promote racial justice in the mathematical experiences of children. For the RJEM Local Planning Committee, there are multiple ways to foster racial justice in early mathematics. As such, RJEM at Work aims to showcase a diverse set of practices. By doing so, we hope not only to share interesting work but also to inspire action at different levels and points of entry in the area of early mathematics education.

This video features the work of a group of K-2 teachers and one instructional leader – Kristen Burke, Amy Christensen, Dora Cottrol, and Holy Tate – that was highlighted at the 2022 Promising Math Conference. Their insightful presentation highlights how they used mathematical concepts to explore racial representation in their classrooms in one public school in Alexandria, Virginia.

While collecting and interpreting mathematical data from books and classmates, students engaged with questions of fairness, justice, and how to address representational inequalities. Because such work was implemented in different grades, it also exposes some of the particularities and possibilities of engaging in racial justice in early math at each grade level.

We welcome suggestions of practices to be featured in the RJEM at Work series. Please reach out to us at rjem@erikson.edu to share your work and/or other recommendations.

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RJEM in 12: Dr. Danny Bernard Martin on the Growing Amount of Research on Racial Justice https://earlymath.erikson.edu/rjem-danny-bernard-martin/ https://earlymath.erikson.edu/rjem-danny-bernard-martin/#respond Fri, 15 Jul 2022 18:11:12 +0000 https://earlymath.erikson.edu/?p=19453 In this interview, we talked with Dr. Danny Bernard Martin, professor of Mathematics and Education at the University of Illinois  More →

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In this interview, we talked with Dr. Danny Bernard Martin, professor of Mathematics and Education at the University of Illinois Chicago. Dr. Martin’s extensive research has focused primarily on understanding the salience of race and mathematics identity in Black learners’ mathematical experiences. He is the recipient of multiple national awards as well as author and editor of multiple books on the intersection of race, identity, and mathematics education (see list of suggested reading below).

The focus on racial justice in early math education is a relatively new phenomenon. Yet, racial justice-centered research in math education has increased over the last two decades. In fact, when thinking about the changes he has noted in math education in the last 20 years, Dr. Martin highlighted the larger (and growing) number of scholars doing such research, bringing with them different methodological frameworks and a more diverse set of ideas and perspectives to the field.

While the growing interest and diverse contributions are certainly welcomed, Dr. Martin called attention to the fact that having more people addressing racism, antiblackness, and white supremacy in math education signals that these issues are persistent. “These issues haven’t gone anywhere,” he says. “So there is a constant need to address them. When you talk about change, as I said, you also have to talk about what hasn’t changed.” More specifically to the world of early math, Dr. Martin identified the lack of quality justice-centered research on Black children in particular as a concern for him. Along with the absence of research, Dr. Martin also wonders about the kinds of policies (if any) and practices – grassroots, institutional, and others – that are in place to foster the mathematical development and brilliance of Black children.

At the end of the interview, Dr. Martin shared thoughts for those who are interested in doing racial justice in early math. For him, “it starts with the self,” and it is very important that we understand how we engage in white supremacy and antiblackness. Dr. Martin also discussed the importance of analyzing the interconnectedness and self-correcting nature of the oppressive systems, since no meaningful change can really occur when the foundational purpose of these systems is to take away the humanity of Black people. Lastly, Dr. Martin asked those who are committed to racial justice work, and racial justice in early math in particular, to keep up the good work!

Suggested Reading

Refusing systemic violence against Black children: Toward a Black liberatory mathematics education (2019) by Danny Bernard Martin, Paula Groves Price, Roxanne Moore

Equity, inclusion, and antiblackness in mathematics education (2019) by Danny Martin

Mathematics Success and Failure among African American Youth (2000) by Danny Martin

Mathematics Teaching, Learning, and Liberation in the Lives of Black Children (2000) edited by Danny Martin

The Impact of Identity in K–8 Mathematics Learning and Teaching (2013) co-edited by Julia Aguirre, Karen Mayfield-Ingram, Danny Martin

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How to Use Digital Photos to Build a Mathematical Language Routine for Kids https://earlymath.erikson.edu/mathematical-language-routines-photos/ https://earlymath.erikson.edu/mathematical-language-routines-photos/#respond Mon, 06 Jun 2022 17:27:25 +0000 https://earlymath.erikson.edu/?p=19398 It turns out that mathematical thinking is developed in conversation with others; asking students “how do you know that?” will reveal a child’s mathematical reasoning skills. The ability to explain one’s thinking is a cognitive leap that requires protected time for meaningful conversation.

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What does oral language have to do with mathematics? It turns out that mathematical thinking is developed in conversation with others; asking children “how do you know that?” will reveal their mathematical reasoning skills. The ability to explain one’s thinking is a cognitive leap that requires protected time for meaningful conversation, and having a mathematical language routine can make a difference.

Oracy, or oral language development, blossoms when children have many opportunities to talk and be listened to. Young children make sense of the world through language. We use language to reason, express emotions, solve problems, and get things done! Teaching oracy means putting more intention behind how you guide and elevate your students’ math talk.

What is Photo Chat?

At Erikson’s Early Math Collaborative, we have created a mathematical language routine to incorporate all these ideas – we call it Photo Chat. It begins with a compelling photograph as a catalyst for your classroom community to chat! Digital photographs are one of the most accessible materials at our fingertips. Conversations around compelling, culturally relevant photographs build language skills that boost and help formalize mathematical thinking.

Notice & Wonder Photo Chat

Two of the most powerful questions we can ask young children are: What do you notice? and What do you wonder? To begin this routine in your classroom we recommend starting with these two questions. What students notice reveals a window into their thinking.

Prompts such as “I notice…: and “I see…” offer an entry into the chat. Children will notice various attributes and reason about what they observe. This is using math to make sense of the world!

mathematical language routine

Let’s look at a Photo Chat conversation in a Kindergarten classroom.

Notice

Antoine: I notice 2 kids riding a bike.
Dakota: They are stopped. I see a street. Maybe they see a car.
Rashawn: I notice they are riding a bike and maybe they are stopping because 2 cars are coming.
James: I see a fire hydrant.
Zoe: I see a tree. I see grass.
Karsyn: I see a birdhouse.
Genesis: I see a house.
Francois: I think they’re going to their dad’s house. It’s a boy and a girl.
Karter: I see a blue and a pink bike.
Journee: Maybe they’re going home.
Allure: I notice the kids are riding a bike and wearing flower clothes and word clothes. The girl’s shoes are different from the boy’s shoes. (sandals vs. sneakers)
Farrah: They are both wearing a helmet but the boy’s has spots.

Ideally children will pivot from first noticing to then building onto each other’s thinking. Rashawn notices the kids riding their bikes. Francois later thinks they are going to their dad’s house, and Journee thinks they may be headed home.

Once students are aware of what they notice they are primed to ask deeper questions or wonderings. The first wondering will fuel a host of other questions in the classroom. The teacher is not concerned with answering these wonderings. The purpose of wondering is to extend conversation, drive future learning and foster a community of curiosity about the world. Let’s return to the same kindergarten classroom…

Wonder

Allure: Are they going to their mom’s house? Does their dad know they are riding a bike?
Rashawn: I wonder if their dad is across the street.
James: I wonder if water will come out of that hydrant.
Zoe: I wonder if they will cross the street.
Damien: I wonder if they are going to the park.
Dakota: Maybe they are catching up with their dad to go to the pumpkin patch.
Journee: Sky Zone

Role of the Teacher

To begin this mathematical language routine in your classroom, we have created a ‘starter kit’ of Notice & Wonder photos. We encourage teachers to add photos from their community or shared classroom experiences such as field trips to their personal photo chat albums.

We encourage teachers to create a talk routine.

First students talk to a partner, a common classroom routine known as “turn and talk” before sharing with the whole class. This ensures all children get a chance to practice speaking in pairs before speaking to whole class.

Think carefully about how you will support children’s conversation.

Say what you see using descriptive language

Model precise language to describe what you see. This supports children’s receptive understanding of language.

“I notice a robin is holding a tiny, squishy worm in her beak.”

Total Physical Response (TPR) Gestures for specific academic vocabulary

Layering gestures with your words supports children’s receptive language understanding. Model for children and expect them to use the gesture, too.
A high impact strategy, especially for English Learners.

Ivette notices the baby robins have opened their beaks very wide.
Create a beak with your hand – open your hands to support the understanding of wide.

Provoke and Build the conversation

Extend and deepen the thinking or reasoning. Support language building.

“Build on’ gesture

Does anyone want to build on that?

The Value of Routines

Routines are repeated learning activities or procedures. They are powerful tools in a classroom because children are offered many opportunities to engage in an activity over time. Photo chat routines offer a playful, open-ended routine with multiple points of entry. Let’s unpack that idea: open-ended means there is more than one thing to talk about, and multiple points of entry means that students at all levels of expressive language skills will benefit from the discussion. We encourage children to express themselves in their home language and bring all their funds of knowledge to the conversation.

Once the practice of noticing and wondering becomes routine in your classroom, you are ready to ask more focused questions such as, “How are these two things the same AND different?” or “How many of what do you see?” Using a variety of questions builds a range of vocabulary and language structures needed to express comparisons and justify reasoning. Over time, photo chats grow increasingly focused on the attribute of quantity and the demand for precise language grows.

Resources

It’s easy to find online photos to start a mathematical language routine such as this. We’ve also created a starter album of images that are sure to spark conversation: Notice & Wonder Photo Album. But, the very best photos are ones that you or your families take of your local community!

Photo Chat 101

Notice and Wonder Protocol

Gesture Poster for Classroom

Talk Routine Poster (Spanish)

Talk Routine Poster (English)


Photos by Nicola Barts and Rakhee Dodia.

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Benchmarks of 5 and 10 with Child 39 https://earlymath.erikson.edu/benchmarks-of-5-and-10-child-39/ https://earlymath.erikson.edu/benchmarks-of-5-and-10-child-39/#respond Tue, 22 Mar 2022 17:21:46 +0000 https://earlymath.erikson.edu/?p=19347 This child uses the visual arrangement of a ten frame to conceptually subitize quantities into the teen numbers. He recognizes  More →

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This child uses the visual arrangement of a ten frame to conceptually subitize quantities into the teen numbers. He recognizes a full row as a unit of 5 and uses that to derive larger numbers as five-and-some-more. The adult is gently nudging him to unitize the full frame as a ten.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Solving a Problem with Both Parts Unknown with Child 39 https://earlymath.erikson.edu/solving-a-problem-with-both-parts-unknown-child-39/ https://earlymath.erikson.edu/solving-a-problem-with-both-parts-unknown-child-39/#respond Tue, 22 Mar 2022 17:19:48 +0000 https://earlymath.erikson.edu/?p=19346 This child encounters a new type of problem situation: a part-part-whole story where both parts are unknown. He is uncertain  More →

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This child encounters a new type of problem situation: a part-part-whole story where both parts are unknown. He is uncertain how to model the problem at first but has some success with adult support.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Solving a Part-Part-Whole Problem with Child 39 https://earlymath.erikson.edu/solving-a-part-part-whole-problem-child-39/ https://earlymath.erikson.edu/solving-a-part-part-whole-problem-child-39/#respond Tue, 22 Mar 2022 16:00:59 +0000 https://earlymath.erikson.edu/?p=19345 This child uses cubes to model a part-part-whole problem where the whole is unknown. He uses a counting all strategy.  More →

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This child uses cubes to model a part-part-whole problem where the whole is unknown. He uses a counting all strategy. He counts out each part and then counts them altogether, starting from one.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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The CAIR Principle to Tune into Math for Babies and Toddlers https://earlymath.erikson.edu/math-for-babies-and-toddlers/ https://earlymath.erikson.edu/math-for-babies-and-toddlers/#respond Thu, 10 Mar 2022 13:24:43 +0000 https://earlymath.erikson.edu/?p=19327 We all know that babies aren’t born walking, talking, following directions, counting, or measuring size. But they do enter the  More →

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We all know that babies aren’t born walking, talking, following directions, counting, or measuring size. But they do enter the world hard-wired with instincts that will mature and develop into those skills in the preschool and school years. However, that only happens when those mathematical capabilities have been nurtured from the start.

In our Math All Around Me program (MAAM), we associate nurturing with the CAIR Principle. CAIR is an acronym which stands for Closely Attend and Intentionally Respond. This principle guides the way familiar adults interact with babies and toddlers in their care:

  • Closely Attend to a child’s gaze, actions, and non-verbal signals such as gestures, pointing, facial expression, and body language. These all are important indicators of children’s current feelings, capabilities, interests, concerns, and needs in all the domains—social-emotional, physical and cognitive.
  • Intentionally Respond using positive interactive communication that support babies’ and toddlers’ natural curiosity and their developing ability to make sense of the world around them.

Infant and toddler caregivers don’t need to be instructed about how to use the CAIR principle as such. What they get from the MAAM program is a deeper understanding that math for babies and toddlers is not about drilling numbers and shapes. Rather it is helping the little ones explore and make sense of the many mathematical problem-situations that are built into everyday routines and interactions as well as into play.

However, the strategies they use in their intentional responses must be tailored to where each little one is as they move through the three phases of precursor understanding.

  1. Emerging Phase (0-14 months) Babies’ sensory-motor systems are still developing. While they don’t talk, they do rely on loving caregivers’ words, gestures, gaze, touch, and interactions to help them recognize and process the sights, sounds, smells, tastes, and textures they are being bombarded with in their immediate environment.
  2. Developing Phase (12-26 months) Toddlers are developing increasing ability to take actions themselves as well as to use words and other forms of communication to show their thinking. But they still rely on caregivers to step in to take care of their needs as well as to provide opportunities for the toddler to safely explore and make discoveries about quantity and size as well as about issues related to where, when, and what comes next. It is also up to the caregivers to scaffold the toddlers understanding that mathematical thinking is involved in all these issues.
  3. Transforming Phase (22-36 months) Toddlers’ ability to form mental images, draw on memories and have ideas is growing by leaps and bounds. However, they still depend on caregivers closely attending and intentionally responding to keep them safe and scaffold their thinking about increasingly precise solutions to the math that is all around us.

In all the phases, it is equally the caregiver’s responsibility to respect and draw on the linguistic and cultural funds of knowledge that come from the child’s home environment. At the same time, caregivers recognize and respect that each child follows a unique learning path. The age any individual enters one phase or transitions to the next will vary widely. Progress depends on nurturing.

Reading a book is a natural, nurturing way to have conversations about math for babies and toddlers that is all around us. As you watch these two short clips, notice how each caregiver tailors what they closely attend and intentionally respond to the precursor phase the children are at.

Watch Ms. Dee use a reading of Fuzzy Fuzzy by Sandra Boynton to support a child in the Developing Phase. Her focus, like his, is on identifying sensory attributes.

Here are some things to notice:

  • How does Ms. Dee show that she is following the child’s lead as they talk about the book? She pays attention to where he points and to what he is saying. He is at the jargon stage of babble so that while we may not understand his words, Ms. Dee uses the rise and fall of his voice to tell when he is asking a question or making a statement.
  • What do you notice about how her tone of voice and the way that she draws out certain words, as well as the way she gently guides his hand makes this a very positive interaction?

Now notice how Ms. Leticia closely attends and intentionally responds to two children in the Transforming Phase. She uses a counting book to carry on a rich conversation that gets them thinking about age and size.



Here are some things to notice:

  • Did you notice how Ms. Leticia stops reading the text to use the girls’ interest in the baby instead to guide a rich conversation about age and size?
  • What are examples of how she and the children use words as well as gestures and pointing to express themselves?
  • What are examples of Ms. Leticia asking questions or re-voicing the girls’ comments to emphasize there are many cues about appearance as well as size that help them understand the complex relationship between size and age?

When caregivers tune into the math all around and the young ones in their care, then no lesson plan is needed! Children’s natural interest in making sense of their world is all that is needed when you closely attend and intentionally respond in a nurturing way.

Precursor Math Concepts: The Wonder of Mathematical Worlds With Infants and Toddlers

This groundbreaking book looks at the development of mathematical thinking in infants and toddlers, with an emphasis on the earliest stage, from zero to three, when mathematical thinking and problem solving first emerge as natural instincts.

Buy Amazon

 

Photo by William Fortunato from Pexels

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RJEM in 12: Dr. Jennifer McCray on Why Racial Justice in Early Mathematics Matters https://earlymath.erikson.edu/rjem-in-12-mccray-racial-justice/ https://earlymath.erikson.edu/rjem-in-12-mccray-racial-justice/#respond Mon, 28 Feb 2022 16:02:20 +0000 https://earlymath.erikson.edu/?p=19296 We are excited to launch RJEM in 12, a series of short and insightful interviews with researchers, teachers, community members,  More →

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We are excited to launch RJEM in 12, a series of short and insightful interviews with researchers, teachers, community members, and other stakeholders whose work advances racial justice in early mathematics.

Our first guest is Dr. Jennifer McCray, Associate Research Scientist and Principal Investigator of the Collaborative as well as co-chair of the Racial Justice in Early Mathematics Project. Dr. McCray is a former preschool teacher who now teaches adults about early childhood and studies how these adults learn.

In this interview, Dr. McCray shared how she first became interested in racial justice in early mathematics and talked about some of the complexities that emerge from having an overwhelmingly white teacher population teaching a student body that is majority composed of children of color. Dr. McCray also spoke about the role of institutionalized mathematics education in the U.S. and the importance of guaranteeing that children, especially children of color, have opportunities to develop their inherent mathematics abilities early on. For Dr. McCray “no one is invulnerable to injustice” and it is paramount for educators to be invested in promoting justice.

At the end of the interview, Dr. McCray invites teachers, especially those who might not be directly impacted by antiblackness, antibrowness, and other forms of racial discrimination, to

  1. commit to learn about white supremacy, antiblackness, and other forms of systemic oppression;
  2. listen to the voices of those who are made most vulnerable in US society, and
  3. take action to change their practices and contexts they occupy. As Dr. McCray said, “racial injustice is everywhere, including in early mathematics and we should be aware of it”.

Click on the video to watch the full interview. Be sure to share it in your networks and social media!

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Developing a Framework for Racial Justice Centered Research in Early Math https://earlymath.erikson.edu/racial-justice-centered-research-early-math/ https://earlymath.erikson.edu/racial-justice-centered-research-early-math/#respond Sat, 26 Feb 2022 19:23:52 +0000 https://earlymath.erikson.edu/?p=19298 This series from the Racial Justice in Early Math project highlights researchers, program facilitators, educators, and others who are actively  More →

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This series from the Racial Justice in Early Math project highlights researchers, program facilitators, educators, and others who are actively working to advance the discussion around racial justice and the field of early mathematics.

To disrupt racial injustice in mathematics education, a growing number of education scholars are developing research that challenges the ways in which racism, antiblackness, and white supremacy impacts the math learning of Black children. Dr. Danny Bernard Martin (University of Illinois at Chicago) has developed a multilevel framework to help those who are interested in doing racial justice centered research while studying math learning of Black children.

In Learning Mathematics While Black (2012), Dr. Martin invites those who are interested in producing knowledge about Black children’s mathematics learning to move away from deficit-oriented ideas about Black children and their mathematics ability. Instead, mathematics education research should be informed by critical understandings of race and the history of racism in the U.S., honor Black children as whole human beings, and account for the ways in which Black children’s mathematics learning is impacted by

  • anti-Black structures, practices, and policies,
  • the realities of Black children’s lives, and
  • the agency, resilience, and mathematical knowledge and practices of Black children, their families, and their communities.

Utilizing a composite case of examples from the existing literature, Martin shows the limitations of studies that do not account for the different factors (e.g., classroom socialization, educational policies, material constraints caused by the state disinvestment in Black communities) that impact Black children’s mathematics learning. Martin suggests that, by not accounting for these factors, mainstream mathematics education research

  • continues to promote negative perceptions about the mathematics abilities of Black children,
  • fails to identify and critique structural issues that contribute to low performance (e.g., teaching for standardized tests, unprepared teachers), and
  • overlooks Black children’s mathematical knowledge and their mathematical practices–inside and outside schools.

Drawing from decades of research with Black learners and Alan Schoenfeld’s (1992) problem-solving framework, Martin offers a multidimensional and multilevel framework that integrates macro-, meso-, and micro- level concerns while researching Black children’s mathematics learning.

Above all, the author calls on researchers to develop approaches to studying Black children’s mathematical development that are not focused on showing how Black children differ from white children. Instead mathematics education researchers should be informed by the belief that Black children are brilliant and have the intellectual capacity to learn mathematics as well as any other children.

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In Appreciation of Bob Moses: His Legacy Spurs Action Today https://earlymath.erikson.edu/bob-moses-legacy/ https://earlymath.erikson.edu/bob-moses-legacy/#respond Mon, 27 Dec 2021 14:14:46 +0000 https://earlymath.erikson.edu/?p=18260 Although I never met him personally, Bob Moses has shaped my life for the past two decades. Reading his book Radical Equations inspired me to focus my doctoral dissertation on math teaching and learning and to join Erikson’s Early Math Collaborative.

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Bob Moses, who died this past July, was a hero to many. During the civil rights struggles of the 1950s and ‘60s, Moses was a leader in the Student Nonviolent Coordinating Committee in segregated Mississippi, working to empower Black people through voting education and voter registration drives. Later, he founded the Algebra Project in 1982 on the belief that math literacy is a critical part of every child’s education. Moses worked to ensure students were able to graduate high school and go onto study math at the college level. For him, this was another civil rights crusade, this time against the inequalities of the public education system.

Although I never met him personally, Bob Moses has shaped my life for the past two decades. In Radical Equations in 2001, Moses wrote “I believe that the absence of math literacy in urban and rural communities throughout this country is an issue as the lack of registered Black voters in Mississippi was in 1961.” (p.5) Reading his book inspired me to focus my doctoral dissertation on math teaching and learning and to join Erikson’s Early Math Collaborative.

Moses’ story of building the Algebra Project in the 1980s and ’90s clarified three important ideas that guide my work:

  • There’s no magic to math; it is a natural way of thinking that all humans can enjoy and use.
  • Math helps us to understand and describe the real world.
  • When we’re working together to solve systemic social problems, math is a tool that will help us.

Math literacy for all students is something we still strive for. The Early Math Collaborative has convened a working group for racial justice in early mathematics comprised of a diverse group of early education stakeholders including teacher educators, parents, and community activists. Together, we have issued a call to action to raise awareness and spur action against the racism baked into curricula, instruction, assessment, and classroom and school structures.

I believe that the absence of math literacy in urban and rural communities throughout this country is an issue as the lack of registered Black voters in Mississippi was in 1961.

Much as I admire the Algebra Project, I hope that someday it won’t be needed because all children will have such a welcoming and authentic experience of math that algebra will be another tool they pick up when they are ready for it. Of course, they will also need teachers who are comfortable with math and build an effective learning community, as well as family members who invite them to explore the world’s mathy-ness together (and throw out the flashcards).

Omo Moses, Bob’s son and the founder of MathTalk, is bringing this vision to life through community math installations and other tools that inspire curiosity and thinking. I am lucky to have had the chance to meet and work with Omo, and his passion and innovations inspire me in much the same way as his father’s writing.

I believe that we can help teachers and parents (and all the adults who spend their lives with young children) to enjoy doing math together and to understand that real math is different from the school math many of them hated. I believe we can build a new generation of math doers who will change the world. Bob Moses, I will keep the faith!

Sign the Call

We call on the field of early mathematics education to name and dismantle white supremacy and anti-Blackness in all of their manifestations. By signing, we hope you will pledge to seek out, participate in, and create opportunities to answer this Call.

Read more

 

MathTalk Leaders Illustrate How Efforts Can Transform Neighborhoods

Read more about Omo Moses and MathTalk’s efforts to mathematize the neighborhood.

Read more

 

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Groundbreaking Book Explains the Beginning of Math Concepts for Infants and Toddlers https://earlymath.erikson.edu/math-for-infants-and-toddlers/ https://earlymath.erikson.edu/math-for-infants-and-toddlers/#respond Wed, 08 Dec 2021 17:42:56 +0000 https://earlymath.erikson.edu/?p=18250 Released in 2021 by Teachers College Press, this book extends professional knowledge for mathematics teaching to teachers who work with the very youngest learners.

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We sat down with Mary Hynes-Berry, one of the authors of the new book, Precursor Math Concepts: The Wonder of Mathematical Worlds with Infants and Toddlers. Released in 2021 by Teachers College Press, this book extends professional knowledge for mathematics teaching to teachers who work with the very youngest learners. We know that children’s brains develop connections faster in the first five years than at any other time in their lives. This book helps us think about how to support children ages 0-3 to explore the mathematics in their world right from the start.

How did you come up with the idea of precursor math concepts? What was your motivation for identifying those concepts and eventually writing this book?

Ten years ago, Jie-Qi, Barbara, and I were working with the teachers at Educare Chicago. We had just finished a professional learning series with the preschool teachers, talking about the Big Ideas of Early Math. As we reflected on the program, Barbara said to us, “You know, 3 is too late. We have to go back much earlier if we want children and their teachers to understand what math really is.” There was virtually no research on math learning for children ages 0-3 at the time. So, we began to struggle together to identify the groundwork for math learning. What must happen before children reach age three and start to do things that are more explicitly mathematical in preschool? And that’s where we came up with the idea for the Precursor Concepts.

Math Concepts for Infants and Toddlers

What are precursor math concepts and how do they relate to the Big Ideas?

There are four Precursor Concepts: Attribute, Comparison, Change, and Pattern. These concepts develop in stages during the precursor stage of development; the age at which children use their senses to perceive and make sense of the world around them. From about 0-14 months, children are thinking about, “what is this?” They’re in that emerging stage, which is sensory. Then somewhere between about 12-26 months – or even longer because every child is different – they start to think a little more abstractly. They have begun symbolic thought and start to think a bit more about “what can I do with this?” They’re developing agency. Then as they get to be between about 22 and 36 months, they go through an important cognitive shift, and they begin to be able to think in a more abstract way. That is, they can generalize and begin to understand ideas that go beyond their immediate experience. Soon, they are making sense of the Big Ideas.

The title of your book is Precursor Math Concepts: The Wonder of Mathematical Worlds with Infants and Toddlers. What does math teaching and learning look like for infants and toddlers? Is 0-3 too young to be teaching math to kids?

The book is about nurturing mathematical thinking, understanding, and problem solving. When people think about math, they often think of numbers and counting but number is not the most salient aspect of math for infants and toddlers, and counting is not even really a part of math at that age. Our definition is that math is a logical way of thinking that allows for precision and that math is found in everyday routines, interactions, and activities. For children 0-3, we really must find the math that’s right there in their world. This has been a major discovery of our Math All Around Me (MAAM) program, which we’ve done with enormous help as part of the Early Math Collaborative.

Infants are born with amazing cognitive capacity, but they’re still infants and they need nurturing. It’s using our language and actions to heighten these everyday interactions of eating, dramatic play, diapering, etc. and to emphasize the math that is already present in the interactions to support math development. We can emphasize that math by offering them provocations. There is one video that illustrates that really well.

This is a teacher working with a young child of about 15-18 months to experience and think about how different things taste.

There’s a huge amount of math in this video but there are no numbers. There’s comparison, there’s sorting, there is a lot of thinking about what attributes are and different uses. These are all important math concepts for infants and toddlers to be developing – and for their teachers and the adults in their life to be cultivating – and none of it involves numbers.

Can you explain the CAIR principle?

This is our term for something that has been known since the beginning of human time. When we CAIR, we Closely Attend and then we Intentionally Respond to that individual child, keeping in mind that child’s interests, their needs, their capabilities, their wants. We keep in mind, “what are the funds of knowledge this child brings from their home culture and language?” And we do everything we can to nurture. Humans are social creatures who need to be nurtured if we are to survive and certainly if we are to thrive. That’s the idea behind all the enormous evidence we’re finding at places like the Center on the Developing Child at Harvard University, that children need this positive, interactive communication. From the moment we are born until the moment we die, we need nurturing. And we believe that here at Erikson, that learning takes place in relationships.

Who is the audience for this book?

We wanted to write this book for a wide audience but, primarily, we hope this book will be used in community colleges and other teacher education and certification programs.

Precursor Math Concepts: The Wonder of Mathematical Worlds With Infants and Toddlers

This groundbreaking book looks at the development of mathematical thinking in infants and toddlers, with an emphasis on the earliest stage, from zero to three, when mathematical thinking and problem solving first emerge as natural instincts.

Buy

 

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Counting a Collection of 101 – How Did You Count Them? https://earlymath.erikson.edu/counting-a-collection-of-101/ https://earlymath.erikson.edu/counting-a-collection-of-101/#respond Sat, 27 Nov 2021 19:25:49 +0000 https://earlymath.erikson.edu/?p=20404 Counting Collections is an activity rooted in sense-making and joy. It invites students to count a collection of objects, talk  More →

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Counting Collections is an activity rooted in sense-making and joy. It invites students to count a collection of objects, talk to a partner about how they counted and then draw/represent their collection on paper.

In this video, a collection of 101 markers—ten packs of ten markers each and one extra—is intentionally chosen. The teacher poses the question “How many do we have all together?” and then guides two children through three layers of engagement, thinking about the amount in this collection.

The first layer of understanding is concrete, as students are encouraged to manipulate the objects in the collection. The next layer of sense-making is talk, as in “Why do you think we should count by 10’s?” The children share their thinking and come to an agreement that 101 is the total amount. This is where most counting activities end, but in Counting Collections we then ask children to represent their collection on paper. This is the third layer of understanding, a pictorial or symbolic representation of their collections. In the video this sparks a deeper conversation “How many 10’s are in 100?” Students now are engaged in concepts of unitizing and their thinking about this quantity has become a rich learning experience.

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Quantity Cards for Subitizing https://earlymath.erikson.edu/quantity-subitizing-cards/ https://earlymath.erikson.edu/quantity-subitizing-cards/#respond Wed, 17 Nov 2021 13:31:40 +0000 https://earlymath.erikson.edu/?p=15505 Quantity is a particular amount of something, expressed as a number. Quantity cards for young children can have pictures of small sets such as dots, finger patterns, or 5- or 10- frames. The ones we have available on our site use a variety of pictures or "suits," since matching quantities such as fingers and dots can help develop ideas about equivalence.

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Quantity is a particular amount of something, expressed as a number. Our playing cards represent numbers, not with written numerals, but with visual models of the quantities. Children can see how many with these subitizing cards.

Subitizing is the ability to instantly recognize small quantities, without counting. The first stage of subitizing is perceptual subitizing. We see a small set such as 3 fingers simultaneously and “just know” that it is three. This ability is innate to human cognition and does not require learning other mathematical processes such as counting. The second stage of subitizing is conceptual subitizing when we see small quantities within larger quantities and use number relationships to perceive the whole. For example, 7 on a ten frame can be seen as five and two simultaneously. The whole is instantly recognized as a composite of its parts.

Our subitizing cards for young children use the visual models of traditional dots, but also include finger patterns, lines, and 5- or 10- frames. There are a variety of pictures or “suits,” since matching quantities such as fingers and dots can help develop ideas about equivalence. Experience recognizing how amounts that look different can be equivalent helps children abstract, for instance, the “four-ness” of four when playing card games.

quantity cards quantity definition

1-5 in Six Suits

These subtizing cards use dots, 5-frames, lines, and fingers for the quantities 1-5. Use this deck of cards for younger children and to emphasize perceptual subitizing, the ability to tell the number of objects in a set quickly, without counting.

Download

math card games using quantity definition

1-10 in Four Suits

These subitizing cards use different sets for the quantities 1-10, starting with a traditional dot arrangements and then also using lines, ten-frames, and hands. Use this deck of cards for older children and to emphasize conceptual subitizing, when children see small quantities within larger quantities and use number relationships to perceive the whole.

Download

Related Ideas

playing card games early math

A New Focus for Familiar Card Games

Here you can learn much more about Quantity Cards and how they build an understanding of early math concepts such as cardinality and composing and comparing numbers.

Learn more

subitizing cards

How to Adapt Math Card Games to Children’s Skill Level

This article and accompanying downloadable instructions explore common card games that can be played and revisited as well as adapted to children’s own math skills as they develop over time.

Learn more

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Math Activities for Infants and Toddlers are Everyday Explorations https://earlymath.erikson.edu/math-activities-for-infants-and-toddlers/ https://earlymath.erikson.edu/math-activities-for-infants-and-toddlers/#respond Wed, 22 Sep 2021 16:07:12 +0000 https://earlymath.erikson.edu/?p=18161 Long before they can talk or explain, babies are thinking logically and exploring ordinary everyday objects, sensations, and experiences that are new to them. Encouraging and celebrating their explorations and discoveries is a highly effective way to help those little brains grow.

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Long before they can talk or explain, babies are thinking logically and exploring ordinary everyday objects, sensations, and experiences that are new to them. They are trying to find out more about questions such as:

  • “What is this?”
  • “How does it work?” and
  • “How is it the same and different from something I already know?”

This kind of thinking and problem solving when participating in math activities for infants and toddlers goes on all the time — it is so natural and commonplace that we often overlook it or dismiss it as “just play.” However, encouraging and celebrating their explorations and discoveries is a highly effective way to help those little brains grow. Instead of looking for expensive learning toys, look to everyday explorations.

For example, take this one-minute video of an 11-month-old playing with the magnets at the bottom of the family fridge. Watch for observable actions and expressions that indicate any of the following:

  • the child is engaged — that she finds what she is doing interesting (consider actions, facial expression, gaze, etc.)
  • the child’s actions are intentional, not just random
  • the child is problem-solving, that is, puzzled about something and testing out some possibilities


Watch the video again. This time, focus on the way those general approaches to learning might be considered mathematical thinking, related to the Precursor Math Concepts of Attribute, Comparison, Change and Pattern.

What do you notice that suggests the child is

  • Intentionally exploring how the attributes of the front of the magnet compare to those of the back and how the attributes of the board book might compare to the fridge surface?
  • Puzzling about change as she finds a different result when the magnet is put on the book instead of the refrigerator, even though both surfaces have many of the same observable attributes?
  • Expecting a pattern, that is, assuming that when the things called “magnets” are put on a hard, shiny surfaces, they will stick rather than slide off?
  • Encouraged by her father who is taking the video to explore and discover?

The father doesn’t try to tell the child what to do but supports and celebrates her discoveries by asking open-ended questions in an interested tone of voice. Letting young ones figure out everyday real life situations for themselves is how to nurture their mathematical thinking. Math activities for infants and toddlers naturally follow their interest in exploring What is bigger? and Which has more? and How are these the same and different? These inquiries as well as all kinds of questions about spatial relationships, sequence, and regularity help infants and toddlers make sense of the world with math.

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Beloved Eric Carle Books Illustrate the Math in Story Time https://earlymath.erikson.edu/eric-carle-books-math/ https://earlymath.erikson.edu/eric-carle-books-math/#respond Wed, 18 Aug 2021 15:44:10 +0000 https://earlymath.erikson.edu/?p=18128 For over 50 years, millions of children have returned again and again to the books illustrated and written by Eric Carle. Kids can’t get enough of his colorful collages, animal characters, and rhythmic texts.

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For over 50 years, millions of children have returned again and again to the books illustrated and written by Eric Carle. Kids can’t get enough of his colorful collages, animal characters, and rhythmic texts. The beloved author died this spring at age 91, and what a legacy he left behind.

Although enjoyed by many, not all of these children — or their parents and teachers — realize that Eric Carle very intentionally built math and science into most of his books. What a wonderful time to celebrate Eric Carle books and the math that can be found in his amazing and timeless work.

Pattern Found in the Works of Eric Carle

Nearly all of his books bring out Big Ideas related to Pattern, which is the foundation for mathematical thinking. Children and all people innately look for and find patterns in the world around them. It is one of the most natural parts of being human. And, searching for pattern and structure is a key practice of mathematics.

For children, reading books is a natural time in which they can discover patterns in the illustrations, the words, and even the page design. Adults can point out the beginning “rules” of patterns while making observations along the way.

After a few readings of a book like Brown Bear Brown Bear, many children see the repeating pattern found in the language that structures this earliest book illustrated by Eric Carle. By the time they are 4- or 5-years-old, many can accurately “picture-read” the book with great expression. But with a little encouragement, they also will be able to contribute to a personal or classroom version of the story. There are endless possibilities for using the same structure of the book with different animals or themes.

Growing patterns abound in Eric Carle books as well. Take for example Today is Monday.

eric carle books, today is monday

The book’s pages list a menu of foods that add one more dish for each day of the week. As the week progresses, so does the length of the list of foods. As the rhythmic text is sung aloud, children experience the +1 growing pattern in a multisensory way. They hear the growing length of the verses. They hold an ever-increasing list of foods in their working memory… and keep them in a fixed and predictable sequence. And finally, they may even run out of breath by the time Sunday comes around and the list of foods totals seven dishes!

Another way to make the growing pattern evident is to create a set of cards showing the different dishes. Cut pictures from grocery store fliers and glue to index cards. Or better yet, ask children to make the illustrations. You can learn about the way Eric Carle paints tissues on his website. The cards can be displayed along with the song to depict the fixed and predictable sequence of the growing pattern.

Number Sense and Counting in His Books

Many of Eric Carle’s best known books explore Big Ideas about number sense and counting also. These math topics are also governed by mathematical patterns, but the focus is on seeing quantity. They support children’s developing understanding that numbers are attributes that name the quantity of a set — they are adjectives that tell us how many of what.

eric carle books, roosters off to see the world

Rooster’s Off to See the World and 1, 2, 3 to the Zoo allow children to practice counting, a growing pattern that follows a very strict rule that each number names a quantity that is one more than the one before it and one less than the one that follows. That makes it possible to predict and calculate exact quantities, even though the “what” may be made up of many different items. The illustrations in both of these books show the cumulative nature of counting by repeating the sets of animals as more are added on. This mathematical idea, known as hierarchical inclusion, helps children understand that numbers are composed of smaller quantities. That is, four includes a set of 3, two sets of 2, and four ones.

The Very Hungry Caterpillar is another well-known Eric Carle book that distinctly uses number sense and counting to show how many of each type of food the caterpillar worms his way through… before transforming into a beautiful butterfly! Children love the way the width of each page expands. This allows them to count the holes as well as the food items he eats.

Classic book The Very Hungry Caterpiller and other Eric Carle books are found in most classrooms and at-home bookshelves. Like many early math resources, educators and parents don’t have to purchase a new book or buy expensive manipulatives to find pattern, counting and number sense learning opportunities. Books such as Eric Carle’s have had them right there all along. And they align with the math concepts that are developing in children’s mathematical thinking at early ages.

eric carle books, very hungry caterpillar

Additional Resources

Introducing Patterns to Preschoolers (pdf)

Brown Bear, Brown Bear, What Do You See? Teaching Plan

Patterning and The Very Hungry Caterpillar (pdf)

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Why Cardinality is the Goal of Counting https://earlymath.erikson.edu/cardinality-set/ https://earlymath.erikson.edu/cardinality-set/#comments Wed, 19 May 2021 15:27:29 +0000 https://earlymath.erikson.edu/?p=18007 What is cardinality? And how is it related to counting? The common definition of cardinality states that it’s the understanding that the last number word said when counting tells how many in all.

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What is cardinality? And how is it related to counting?

The common definition of cardinality states that it’s the understanding that the last number word said when counting tells how many in all. That is, we count a set by matching number words to objects — “1, 2, 3, 4, 5.” This procedure enumerates each object in order but does not reveal how many until we recognize, “1, 2, 3, 4, 5. That’s 5 blocks.” Now the number 5 has both an ordinal meaning (the number that comes after 4) and a cardinal meaning (the number that represents the quantity of the entire set).

Watch a kindergartner counting blocks. She is confident saying the number words in order. She matches each number to a block as she points to them. But does she understand how many blocks are on her mat?



This kindergartner clearly states the total number of blocks each time after she counts. According to the common definition of cardinality, then, we might say that she understands cardinality.

But the fact that each time a block or two is added, she goes back to counting all the blocks one by one, is significant. It is evidence that she hasn’t yet developed a full understanding of cardinality. Cardinality is more than the act of repeating the final count number. Rather it is understanding that the purpose of counting is to answer the question, “How many?” Cardinality as a concept connects the final count number to its quantity, the amount of the set.

At the same time, it is likely she also hasn’t really grasped that the number sequence is not random; it is a pattern that follows a +1 rule. Each counting number identifies a quantity that is one more than the number before it. If she did understand these concepts, then we would see her counting on from the number of blocks she knew she had. Her thinking might go something like this, “I know I had 6 blocks and you gave me 2 more so now I have 7, 8.”

See an example of counting on in this video of a kindergartner from the same class, counting the same blocks. The contrast is striking.



Both of these children are within the normal range of development in kindergarten. What’s important is that all children gain extensive experience counting in contexts where they need to know “how many.” Whether it’s at school or home, when we ask children to count, we need to give them concrete objects to count and consistently ask the question, “How many in all?” at the end of the count. In this way, we emphasize the cardinality of the set, not just the act of counting it.

Cardinality is Critical Preschool Concept with Barbara Sarnecka

After years of studying 3- and 4-year old children of diverse linguistic and cultural backgrounds, Barbara Sarnecka has zeroed in on the importance of cardinality.

Learn more

 

How Children Learn about Numbers: A Conversation with Kelly Mix

In her presentation “Cognition and Early Childhood Numeracy: How Number Concepts are Built and Why Input Matters,” Kelly Mix bridged research and practice in her discussion of math language and learning.

Learn more

 

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Early Childhood Data Collection Activities Are for the Birds https://earlymath.erikson.edu/data-collection-activities-birds/ https://earlymath.erikson.edu/data-collection-activities-birds/#respond Wed, 05 May 2021 18:07:34 +0000 https://earlymath.erikson.edu/?p=17991 Children are naturally interested in birds, so springtime is the perfect opportunity for early childhood data collection activities. Here are some favorites, along with book recommendations.

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Any time of year is a great time to get out and see birds, but migration time is when the variety of colorful birds is at its peak. World Migratory Bird Day is officially celebrated on the second Saturday of May in the US and Canada, but many outdoor events and citizen science projects take place all month long. Children are naturally interested in birds, so springtime is the perfect opportunity for early childhood data collection activities.

You don’t need to go far. Birds can be seen on city streets as well as in parks, yards, and nature preserves. You may want to install a window feeder or feeders near your classroom to observe birds up close. Make a list of target birds to look for and keep count. Remind children that birds are hard to spot. Practice staying still and looking for the movements of birds. To develop children’s powers of observation, make DIY cardboard binoculars to help them focus on birds they see.

Let children’s own questions drive your data activities. Whether they are interested in the colors of birds, comparing types of birds or size of flocks, or investigating what kind of birdseed draws the most birds to the feeder, children will explore the Big Ideas of Data Analysis as they collect bird data to answer their questions.

These mathy books for kids will spark curiosity and inspire bird-related investigations:

  1. Birds by Kevin Henkes

    Birds come in all sizes, shapes, and colors. The simple text and vibrant illustrations of this book invite preschool or kindergarten children to notice the many attributes of birds. What do children wonder about the birds they see?

    Get (or make!) a bird guide. Bird guides can be fun to review before going outdoors as well as once you spy birds. Figure out how your bird guide is arranged. Is it according to the different families of birds or by their colors? You may even want to make your own bird guide with children’s own illustrations and descriptions of when and where you can find the species you see.

data collection activities, Birds by Kevin Henkes
  1. Bird Count by Susan Edwards Richmond

    In this Mathical-award winning book, we read about a young girl who participates in an annual census of birds in her community. Her excitement for finding the various species she sees is contagious. As her count grows with each turn of the page, we see her handwritten tallies, and children can notice how she makes groups of five for efficient counting.

    Turn birding into a scavenger hunt with a goal: To see as many birds as possible. For younger children, counting up to a target number—5 different kinds of birds, for example, or 20 birds in one outing—will be enough to stay focused and have fun. For first- and second-graders, data collection activities can be more formal with tally charts and adding counts over several days, including birds they see with their families over a period of time.

data collection activities, Bird Count by Susan Edwards Richmond
  1. Pigeon Math by Asia Citro

    Birds on a wire may sound like a cliché number story, but Asia Citro surprises us in her book, Pigeon Math. Another Mathical-award winner, this funny story about pigeons plays around with a familiar urban sight as birds (and a cat!) leave and return to a telephone wire. The constantly changing quantities provide many opportunities to add and subtract.

    Have first graders discuss the number combinations within 10 that occur within the story. Then have them write their own pigeon math stories to create a class book! Second graders can do the same with a flock of 20 pigeons or another common local species such as seagulls, geese, or sparrows.

data collection activities, Pigeon Math by Asia Citro

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]]> https://earlymath.erikson.edu/data-collection-activities-birds/feed/ 0 DIY Rekenrek Ideas for Home and School https://earlymath.erikson.edu/diy-rekenrek-home-school/ https://earlymath.erikson.edu/diy-rekenrek-home-school/#respond Wed, 14 Apr 2021 15:29:22 +0000 https://earlymath.erikson.edu/?p=17927 To encourage teachers and other adults to use rekenreks, it's important to not only go over ways to use a rekenrek, but also we thought we would make sure everyone has opportunities to have them, whatever the budget.

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One thing we love is the rekenrek. There are many tools that can be used at home and in classrooms to reinforce number sense and subitizing skills, and the rekenrek is one of our favorites. This “arithmetic rack” features two rows of 10 beads, with each row broken into two sets of five, usually in red and white colors. It is a simple tool that can be quite powerful. To encourage teachers and other adults to use them, it’s important to not only go over ways to use a rekenrek (links below), but also we thought we would make sure everyone has opportunities to have them, whatever the budget. So let’s make some DIY rekenreks!

Small DIY Rekenrek Ideas for Home and Classrooms

Here are two common, easy-to-use and cheap ways to create your own rekenreks. They are great for home use, or maybe you want a budget-friendly way to make ones for an entire classroom of kids.

  1. Acquire (a) colored heavy or card-stock paper, (b) black pipe cleaners, and (c) red and white beads. Punch holes at each end of the heavy paper and pull the pipe cleaners through, and then string the beads, 5 reds and 5 whites together. You may want to start with just a single row for younger children, concentrating on different ways of working with 10 beads. When children are ready, begin to use two rows of beads. Make sure you create rows that are wide enough to leave a large space between the beads on the left and the beds remaining on the right, apart from the set to be counted. Another tip: draw a smiley face to direct children where to look to find the set of beads to be counted (see photo examples below).
  2. diy rekenreks
    diy rekenreks
  3. Here is a variation of the same DIY rekenreks we found some teachers making. Teachers or craftsy people often have use for these plastic mesh canvas sheets. So make a rekenrek out of one! You might find them more durable and easier to manage than card-stock paper with holes in them.
  4. diy rekenrek

Rekenrek Attendance Activity for School

Smaller, individual rekenreks are great for children to be hands-on, but you may want something larger for demonstrating to the entire class. Use common materials and these guidelines to create a rekenrek chart with the same mathematical structure as the ones with beads. For more detailed instructions, click the download button below.

  1. Purchase a) a large poster board, b) library pockets, and c) popsicle sticks or something else to represent the children. Create rows of ten using 5 red and 5 white pockets. Depending on your class size you may want to use two, three, or four rows. Be sure that there are more pockets than children so that there’s room for grouping and rearranging quantities.
  2. diy rekenreks
    diy rekenreks
  3. Here is a variation of the same DIY rekenrek chart, only using a laminated board and velcro. Something like this might better survive the rigors of classroom life, and switching out laminated representations of the children means that they can be seasonal or can represent the children in different and unique ways.
  4. diy attendance chart

Use the rekenrek chart to collect attendance data. Using this tool daily (see a demonstration here) provides repeated opportunities for children to build number relationships. The rekenrek helps them see small quantities inside larger numbers and learn about the 5- and 10-structure of our number system.

Another idea to try with the rekenrek chart is classroom voting. See an example of a class using the rekenrek chart to organize votes to determine which book to read at story time (below). If you want to purchase rekenreks like shown in the images, check out Math Rack.

Related Ideas

rekenrek activity

Rekenrek Chart for Voting in the Classroom

Looking for new rekenrek activity ideas? Try voting with a rekenrek chart. In this video, we see a rekenrek chart used to solve a dilemma common to early childhood classrooms: choosing between two favorite books at story time.

Learn more

rekenrek

How to Use a Rekenrek

New to using rekenreks? Its deceivingly simple design — two rows of ten beads, five red and five white — leaves some wondering how to use it. Here is a detailed video demonstrating how to use one with children.

Learn more

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Rekenrek Chart for Voting in the Classroom https://earlymath.erikson.edu/rekenrek-chart-rekenrek-activity-voting/ https://earlymath.erikson.edu/rekenrek-chart-rekenrek-activity-voting/#respond Tue, 13 Apr 2021 15:31:00 +0000 https://earlymath.erikson.edu/?p=17941 Looking for new rekenrek activity ideas? Try voting with a rekenrek chart. In this video, we see a rekenrek chart used to solve a dilemma common to early childhood classrooms: choosing between two favorite books at story time.

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Looking for new rekenrek activity ideas? Try voting with a rekenrek chart. In this video, we see a rekenrek chart used to solve a dilemma common to early childhood classrooms: choosing between two favorite books at story time. When a quick vote is called for, the rekenrek chart is a handy tool. The structure of this mathematical pocket chart is like that of a rekenrek, an arithmetic rack with red and white beads. Here, children recognize the same rows of ten, composed of 5 red and 5 white pockets in each row. This 5- and 10-structure helps children see the smaller quantities within a larger number—perfect for voting and making comparisons.

Watch as the children place a popsicle stick into red pockets on the left or the white pockets on the right of the rekenrek chart to indicate the book they would most like to hear. Efficient and fair, this voting activity settles the question of what to read while building number sense. Notice how quickly children can compare the quantities of votes when they are visually organized and structured!

Want to make your own rekrenrek chart? Or just need more information about how to use one? See additional rekenrek information below. Read about how to use your chart for a mathematically-powerful attendance routine here along with other rekenrek activity ideas.

Related Ideas

rekenrek activity

DIY Rekenrek Ideas for Home and School

One thing we love is the rekenrek. To encourage teachers and other adults to use them, it’s important to not only go over ways to use a rekenrek, but also we thought we would make sure everyone has opportunities to have them, whatever the budget.

Learn more

rekenrek

How to Use a Rekenrek

New to using rekenreks? Its deceivingly simple design — two rows of ten beads, five red and five white — leaves some wondering how to use it. Here is a detailed video demonstrating how to use one with children.

Learn more

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Making the Number Ten on a Rekenrek https://earlymath.erikson.edu/making-rekenrek-number-rack/ https://earlymath.erikson.edu/making-rekenrek-number-rack/#respond Wed, 17 Mar 2021 16:54:30 +0000 https://earlymath.erikson.edu/?p=17914 A good sense of the number ten is critical for building young children’s reasoning strategies. Here we see kindergarteners making the number ten on a rekenrek, a tool with red and white beads in groups of 5s with 10 beads on each row.

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A good sense of the number ten is critical for building young children’s reasoning strategies. Decomposing 10 into pairs in more than one way is a core activity in kindergarten. Here we see kindergarteners making the number ten on a rekenrek number rack, a tool with red and white beads in groups of 5s with 10 beads on each row. The rekenreks have two rows of beads, so the teacher uses the context of a double-decker bus so that children imagine the beads to represent people riding on the top and bottom levels of the bus.

First, children find many ways to make 10. Then the teacher challenges them to find the number that makes 10 when added to a given number. Knowing how any number from 1 to 9 relates to the benchmarks of 5 and 10 is foundational for building fluency in the early grades.

Want to give this lesson a try? Here are simple instructions to make your own rekenreks like you see in the video. Also visit the Math Learning Center to download their free PDF booklets for introducing and using a rekenrek number rack. Their free Number Rack app is a great digital tool for class demonstrations and online learning.

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Denver Head Start Teachers Sustain Children’s Growth in Math https://earlymath.erikson.edu/denver-head-start-teachers-build-momentum-for-childrens-growth-in-math/ https://earlymath.erikson.edu/denver-head-start-teachers-build-momentum-for-childrens-growth-in-math/#respond Mon, 22 Feb 2021 15:45:26 +0000 https://earlymath.erikson.edu/?p=17884 An ongoing professional development program with Denver Great Kids Head Start (DGKHS) shows the power of a multi-year partnership to build sustainable growth in math achievement for young children.

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An ongoing professional development program with Denver Great Kids Head Start (DGKHS) shows the power of a multi-year partnership to build sustainable growth in math achievement for young children. Since 2014, the Collaborative has worked with the Denver Office of Children’s Affairs to develop Math Champions, a program designed to deepen teachers’ knowledge about the Big Ideas of early math and how to engage all children in joyful math learning.

The partnership began when DGKHS noticed that child performance data were lowest in math and sought professional development opportunities for their teachers. “We knew we wanted to invest in improving our teaching practices around the area of math,” says Liane Martinez, School Readiness Director at DGKHS. “After looking into several options, we selected the Erikson Institute Early Math Collaborative. We invested in training that was not a one-and-done.”

And that investment has paid off. Martinez reports a positive trend in the data. Math is now the area with the largest growth on the Teaching Strategies GOLD assessment. Some years, it’s been as large as a 68% gain!

A Long-Term Focus

Four times each year, two facilitators from the Collaborative fly out to Denver for two full days of math professional development with Head Start teachers and coaches. One day is spent with teachers who are new to the training and the other day is for returning teachers who are deepening their understanding and application of the Big Ideas of early math. Some teachers have been with the program for 3 or 4 years.

Martinez knows the importance of time. “We know that trainings must continue in order for practices to begin, change, or develop in staff. It is easy to forget if never revisited. Initially it was one group, moved to two groups with returning teachers a second year and another new cohort of new staff. We followed the model and then began to develop a peer coaching cohort along with new and returning staff,” explains Martinez.

Training, application of new practice, and continuity is the best way to invest for greater outcomes.
– Liane Martinez, School Readiness Director

Research supports continuity in professional development. A long-term focus provides teachers with adequate time to learn, practice, implement, and reflect upon new strategies that facilitate changes in their practice.

A Team Effort

Over the years, Math Champions has grown to include coaches working in Head Start centers, in addition to the teachers. Coaches are co-learners with teachers in the morning and in the afternoon, with the guidance of Erikson facilitators, they plan activities with the teachers they support. In doing so, coaches create spaces for teachers to share ideas and collaborate in their learning in a job-embedded context.

It’s been great! I appreciate the interactive approach and holding us accountable for it!
– Denver Great Kids Head Start Coach

In sessions specifically designed for them, coaches deepen their expertise about content and evidence-based practices. They focus on how to meet teachers’ individual needs through built-in time for teachers to think about, receive input on, and make changes to their practice by facilitating reflection.

Teacher assistants are also included as essential team members. By working collaboratively, teachers, teacher assistants, and coaches are creating positive and lasting change in math achievement for children in Denver Great Kids Head Start.

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3 Ways to See Mathematical Structure in Everyday Kitchen Math https://earlymath.erikson.edu/mathematical-structures-kitchen-math/ https://earlymath.erikson.edu/mathematical-structures-kitchen-math/#comments Tue, 26 Jan 2021 20:20:35 +0000 https://earlymath.erikson.edu/?p=17855 Cooking with kids is a natural way to do math together. But we're not talking about turning meal preparation into a formal math lesson. Cooking together presents an opportunity that is more about noticing and wondering rather than teaching.

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Cooking with kids is a natural way to do math together. But we’re not talking about turning meal preparation into a formal math lesson. Cooking together presents an opportunity that is more about noticing and wondering rather than teaching. Think of the kitchen as a place to build children’s intuition about measurement, multiplication, division, and fractions.

Look for these common mathematical structures in the kitchen — equal spacing, equal groups, and partitioning — and then make use of them to figure out the task at hand. Research has consistently shown that drawing children’s attention to mathematical structure helps them to recognize and make use of mathematical patterns in new situations.

Structure is so central to mathematics that we have a Common Core Standard for Mathematical Practice that says, “Look for and make use of structure.”

What is “structure?” The term “structure” in mathematics is the way a mathematical pattern is organized. It is expressed in terms of a numerical, spatial, or logical relationship. Structure is so central to mathematics that we have a Common Core Standard for Mathematical Practice that says, “Look for and make use of structure.” The phrase “look for” suggests that awareness of structure is a habit of mind, a way of seeing the world through a mathematical lens. Luckily, you don’t have to look hard to see mathematical structure in the kitchen!

Equal Spacing

Equal spacing is the foundation for understanding units and measurement. Equal spacing is also an essential prerequisite for fractions and data representation. Guess what? It’s important for baking, too. Imagine how you drop cookie dough on a baking sheet or arrange frozen foods on a tray. Visualizing how to evenly space food and fit it within a given space is important mathematical thinking and can add to any kitchen math opportunities.

Here’s how a mom of young boys explored equal spacing in her kitchen one day.

mathematical structures, kitchen math

Boy, I have fussy eaters who don’t like eating the same thing.  I have very little time to cook for them separately. So two kinds of ravioli had to fit on the same tray. I called the boys and asked them how they thought we could do it.

I couldn’t believe how much math talk we had. How many ravioli will fit along the bottom? How many ravioli will fit going down the side? Could we add an extra row? 

The boys had their own questions, too. How come we had more left over of one kind? How will we remember which ravioli are which? Then we got into counting the rows and columns.

kitchen math
mathematical structures

Equal Groups

For young children, counting objects one-by-one with accuracy is a big achievement. However, counting objects individually is gradually superseded by counting equal groups. The structure of equal groups is the foundation of our base-10 number system and the basis for multiplication. The kitchen provides plentiful ways for you to help children recognize and count equal groups. Here are just a couple.

kitchen math
  • Count the rows or columns in a grid array. We can structure our counting by noticing equal rows and columns. Think of muffin tins, egg cartons, and drink containers that come in 6- and 12-packs. We can invite children to count the rows and columns as units rather than count each object. Counting by 2s, 5, and 10s are typically the first structured counting sequences that children use.
  • Counting out equal portions. We can ask children to put 5 strawberries in each bowl and then count the total number by fives. If everyone gets two cookies, how many cookies will we need? Sharing food provides many opportunities to count equal groups.

Partitioning

Partitioning occurs naturally in many kitchen math contexts, including cutting equal-sized pieces, sharing portions of a whole, and fractions. All of the structures discussed here are closely related. You will find that partitioning often requires the construction of equal spacing or equal groups. We’ve all heard the common complaint from a child, “You got more than me!” Use children’s concern about “fairness” to motivate their mathematical thinking and involve them in the partitioning and sharing of food.

Knowledge of fractions is needed to follow a recipe. Young children won’t be familiar with the formal symbols of fractions but can make sense of equal parts of a whole. Cooking is the perfect opportunity to build intuition about what fractions mean.

kitchen math

Here’s a conversation overheard in the kitchen when a second grader helps make pudding for dessert.

I had my son, Xadrian, read the directions on the pudding box to see what measurement cups we might need. Being a typical 2nd grader, he didn’t actually read the box but skimmed the text over to find numbers and quickly responded, “Four, one, two!”

“Huh?!” I read the directions and noticed he skipped over all the other numbers and went straight to the fraction. “Oh! You’re reading this fraction number. That’s actually 4 half cups and it’s talking about how many servings of pudding we’ll have after we make it. To make it though, we just need 2 cups of milk. Can you take out the measuring cups please?” He brings over the set of measuring cups we use most regularly and hands me the half cup saying, “This one has the one-two like it says on the box.”

mathematical structures, kitchen math, pudding directions

“You’re right! That means that when we’re done, we’ll have 4 of those size cups to share with your dad and brothers. That ‘one-two’ is called a half, but to make the pudding, we need 2 cups.” As I say this, I realize that four ½ cups is equal to 2 cups, so I begin to explain that the bottom number of a fraction, the denominator, tells us how many of those parts fit inside the whole (1 cup).“Bring those cups over to the sink and let’s see if that’s true.”

As he begins to investigate, we talk about filling the cups all the way up to the top each time so that all the parts are the same amount (equal). He carefully fills the ½ cup to the top and pours it into the 1 cup, then fills it again and pours it in. The smile on his face is priceless! “Let me try this one,” he beams, as he grabs the 1/3 cup.

mathematical structures, pudding directions

I ask, “So how many of these 1/3 cups do you need to fill the 1 cup?”

“Three,” he shouts as he begins pouring.

“Let’s see if that works. We call that a third.”

“It does!” he says and quickly grabs the ¼ cup. “This will be 4!”

“Let’s see! We call that a fourth, or a quarter.”

“A quarter?”

“Yup! Like the coin. How many quarters do you need to make a dollar?”

His eyes open wider as he thinks about it. “Ooooh! Four! So that’s why it’s called a quarter!”

I smile and nod then redirect our attention to the pudding that we have not even started to prepare yet. I say, “So the box says we need 2 cups of milk and you just saw that 2 half cups makes 1 cup. So how many half cups would it take to fill the 1 cup twice, or 2 cups?”

He thinks momentarily and the smile says it all, “It’s 4!”

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4 Key Findings for Early Childhood Coaching Strategies with Teachers https://earlymath.erikson.edu/early-childhood-coaching-strategies-teachers/ https://earlymath.erikson.edu/early-childhood-coaching-strategies-teachers/#respond Mon, 18 Jan 2021 16:41:54 +0000 https://earlymath.erikson.edu/?p=17837 Instructional coaching is central to the Collaborative's professional development efforts. There is a lot to learn from what works and what doesn't when it comes to early childhood coaching strategies, particularly when focused on the teaching of early mathematics.

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Instructional coaching is central to the Collaborative’s professional development efforts. There is a lot to learn from what works and what doesn’t when it comes to early childhood coaching strategies, particularly when focused on the teaching of early mathematics. Lauren Solarski’s recent dissertation, “Clarifying Coaching: A Mixed Methods Analysis of a Math Content-focused Model and Its Impact on Teachers’ Practice, Content Knowledge, and Dispositions” sheds light on the topic.

Lauren Solarski first came to know the Early Math Collaborative through our partnership with Big Shoulders Fund. As a kindergarten teacher, she was inspired by the Big Ideas and motivated to come study at Erikson Institute. While earning her doctorate, Solarski also worked as an early math coach with the Collaborative. This informed her work dissecting what early math coaching entails, what we have learned from our coaching model, and what the research says.

Coaching is complex, and it looks different in different settings. “Research is still at the point of defining what coaching is,” says Solarski. Yet, there are key findings to consider when developing early childhood coaching strategies for teachers.

Lauren Solarski, early childhood coaching strategies

1. The Importance of Planning

Coaches and teachers planning together seems to be an important component of coaching. “Overall,” Solarski says, “published studies are not always clear about the components of coaching.” But most instructional coaching tends to involve an observation of the teacher, and most involves a reflection of that observation afterwards. One thing it doesn’t always include is a coach’s involvement with the planning of the activity or lesson. Solarski thinks that is a missed opportunity.

“If something doesn’t work with students,” she says, “the coach and teacher both have ownership over that and therefore the problem solving in the reflecting conversation becomes truly collaborative. Planning together also allows for some of the logistics such as materials, number of students, and location in the classroom to get taken care of up-front so that the reflecting conversation can really get to what teachers and students said and did.”

Being a part of the planning means that coaches are more involved with the learning process of the teacher. This can be just as important as seeing the learning of the children take place in the classroom.

Planning becomes especially important in content-focused coaching models to ensure that teachers understand the new knowledge they are acquiring and how to integrate it with children. My study found that teachers were still processing what they learned in the workshops with their coaches during planning since the new information is complex.

2. Role-Playing is a Useful Tool

Another key component Solarski found in her study: role-playing. It often isn’t enough to just say what you are going to do. It is bound to be a bit different when teachers are in the classroom with children.

“Actually embodying the words and actions,” she says, “allows teachers to further internalize new knowledge about math teaching. In my study, participants attributed their increased confidence in math teaching to role-play.”

Acting out how the lesson will go has all sorts of benefits, including creating a safe space for educators to think through the lesson without the curve balls that the children might throw during the lesson or activity. The teachers practice how to respond to questions, confusions, and students’ own ideas.

“Teachers rehearse how they will introduce the activity with children,” she describes. “They scaffold children’s thinking, and then conclude the lesson. It’s also an opportunity to explore the lesson materials such as game pieces or collections for sorting. One teacher pretends to be the lead while the other teacher(s) and coach act as students. This allows them to anticipate how children might engage with the Big Ideas or math concepts that are the lesson’s focus.”

3. Teaching Assistants Can Play a Pivotal Role

Teacher assistants are sometimes overlooked as instructional resources. For example, in classrooms with monolingual teachers, it is often the teacher assistant who shares the children’s home language and who provides language support during math. Yet, they aren’t always part of coaching nearly as much as they should be.

“Most of the literature from K-12 assumes coaching is a one-on-one relationship,” Solarski says. “But in early childhood settings that have multiple adults working together in the same classroom, my study’s findings highlight the importance of a group coaching format that is inclusive of lead and assistant teachers and promotes collaboration.”

There are plenty of pitfalls when teachers and teacher assistants don’t have the same benefits of the coaching process and knowledge that is shared during it.

It is important – particularly in early math – for teachers to have a unified understanding. For example, one goal of the intervention that is based on research is to get children to deepen their conceptual understanding of numbers 1-5 and the relationships between them, rather than the common approach to have children memorize and recite the number sequence to 100. If an assistant teacher is denied the opportunity to participate in coaching there may be a disconnect, and they might even think their co-teacher isn’t challenging the children enough. There could be tension and ultimately confusion for the children.

4. Confidence and Attitudes of the Teachers Matter

Addressing the mathematical mindset of teachers is critical, and the Collaborative is clear about the importance of teachers’ attitudes and beliefs when it comes to teaching math. Solarski’s study confirms this emphasis.

“Confidence and attitudes are important in the context of early childhood mathematics,” Solarski says, “because early childhood teachers report high levels of math anxiety and negative experiences around math in their own schooling histories. These can negatively impact their work with children.”

Shifting mindsets can also be seen as the part of coaching that creates the most opportunity. It can be difficult to increase comfort with mathematics in a workshop or online session with many educators hearing the same thing. A coaching process that delves into diverse early childhood coaching strategies gives each individual teacher a voice with a knowledgeable other. It gives them an ability to be themselves, with whatever educational history and backgrounds they may have.

Just as education has trended towards thinking about the “whole child,” a coaching process can help schools think more about the whole teacher.

“I think it’s important that interventions consider teachers as whole persons and not just vessels to deliver student learning.”

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New Use of Quantity Cards for Remote Preschool Math Games https://earlymath.erikson.edu/preschool-math-games-new-quantity-cards-remote/ https://earlymath.erikson.edu/preschool-math-games-new-quantity-cards-remote/#respond Tue, 29 Dec 2020 21:12:22 +0000 https://earlymath.erikson.edu/?p=17807 Whether teaching in-person with social distancing or online, the number one question we hear from teachers is how to adapt preschool math games for remote learning. Here are some game ideas that work equally well in person or online.

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The Collaborative sends a shout-out to all the hard-working early childhood teachers who have found ways to continue playing games during the Covid-19 pandemic. Whether teaching in-person with social distancing or online, the number one question we hear from teachers is how to adapt preschool math games for remote learning.

Elaine Bennett, an early years teacher in the UK, used our quantity cards to create an interactive subitizing spinner that generates random visual representations of quantities 1-5. We’ve happily shared the spinner (and another one she created with dice) with teachers here in Chicago and have seen how useful it is for preschool math games.

Here are some preschool game ideas that work equally well in person or online.

Three in a Row

Have each player create their own 3 x 3 array of 1-5 quantity cards, face up. Spin the spinner. Players can turn over a card if it shows the same representation of the quantity on the spinner (an exact match such as 3 fingers and 3 fingers). The first player to turn over three cards in a row wins the game.

Stepping Stones

Create a stepping stone path with sidewalk chalk (outside) or painters’ tape (inside). Children at home can create “stepping stones” with common household items such as paper plates, washcloths, or small pillows. Spin the spinner and have player jump that number of stepping stones. The player who completes the path first wins the game. When partners can’t play together, children can race the teacher or play against a stuffed animal at home.

preschool math games, activity jumping games for kids using pillows

Credit: AndNextComesL

Cover All BINGO

Have each player create their own 3 x 3 array of 1-5 quantity cards, face up. Spin the spinner. Players turn over all the cards that represent the same quantity as the shown on the spinner. For example, if the spinner shows 3 fingers, players can turn over cards with 3 fingers, 3 dots, 3 on the ten-frame, 3 tallies, etc. The first player to turn over all their cards wins the game.

Fill the Frame

Use a ten frame and two colors of game pieces such as red and black checkers. Two players take turns spinning and adding that number of pieces to the ten frame. The player who adds the tenth piece to fill the frame wins the game. When partners can’t play together, then children can race to fill their own frame first.

Keep on playing with math! It’s not easy, but we know the importance of math games, especially during remote learning. For other game ideas, see our collection of game-related posts.

Related Ideas

playing card games early math

A New Focus for Familiar Card Games

Here you can learn much more about Quantity Cards and how they build an understanding of early math concepts such as cardinality and composing and comparing numbers.

Learn more

playing math card games early math

Quantity Cards

These Quantity Card downloads are available and cater to different skill levels and ages. They include a variety of pictures or “suits” since matching quantities such as fingers and dots can help develop ideas about numbers and equivalence.

Learn more

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Early Childhood Teachers Look Back on a Year Like No Other https://earlymath.erikson.edu/early-childhood-teachers-look-back-on-a-year-like-no-other-covid-pandemic/ https://earlymath.erikson.edu/early-childhood-teachers-look-back-on-a-year-like-no-other-covid-pandemic/#respond Sun, 27 Dec 2020 21:14:56 +0000 https://earlymath.erikson.edu/?p=17810 The Collaborative interviewed three Chicago-area early childhood teachers who spoke honestly about the challenges and rewards of teaching during the pandemic.

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The year 2020 has been a year like no other! As the year comes to a close, we want to acknowledge the tremendous work by early childhood teachers to quickly shift their programs and educational practices—all while caring for their own families and loved ones. As truly essential workers, these preschool teachers responded by learning new technology and teaching virtually for the very first time as well as responding to new health guidelines for in-person programming.

The Collaborative interviewed three Chicago-area early childhood teachers who spoke honestly about the challenges and rewards of teaching during the pandemic. At the start of the pandemic in March, all three preschools closed in-person services.

Donely, who teaches in the Edgewater neighborhood of Chicago, quickly learned how to connect with Zoom despite never having used this technology before. However, he soon realized this wasn’t meeting the needs of families at his center. “In my class, the parents are working from home and don’t want to be distracted. I know parents were starting to hire babysitters while they were working.” Donely really missed his students and felt frustrated he couldn’t do more to support their families and engage his young learners. “I didn’t know what to do [about the 10 kids who didn’t join]. I tried my best to get the families to participate.”

I want people to understand that our job is not easy. It’s not just babysitting. If the kids are learning and the parents are seeing this, that tells me I’m doing a good job. – Donely

In the suburb of Evanston, Gail had another challenge when she tried virtual story-time, songs, and dance. “Parents just really didn’t understand the preschool perspective and what kids are learning from play. Some families wanted more paper-and-pencil type things like their other kids were doing.”

The third teacher we interviewed, Beth, teaches at a private school in the Near North neighborhood. She noticed a “split between two parent working families and the families that have a parent or other adult that stays home with the child or children.” These two types of families had very different needs. Parents who were busy with their own work schedule had a hard time joining scheduled calls while other families were looking for a more structured schedule that followed school routines. “So, that’s where we tried to find a balance of live video times and recorded things. That way if a child couldn’t make a meeting, they were still getting some of the content and information.”

As summer faded and the school year began, these three preschools re-opened and welcomed children back to their schools. The adults fretted over all the new safety protocols, but the children have been resilient.

early childhood teachers, covid-19, remote learning

Credit: Chicago Tribune

Gail in Evanston shared with us, “You’d be surprised. Kids seem to get used to almost anything. We adults have had to change our routines.” While her students are very delighted to be back in school even with these restrictions, she has noticed that adults’ anxiety can transfer to children. “Maybe when the children are challenging with us, it’s because they feel this stress. The children pick it up from the adults around them. But otherwise, the children adapt and just accept these new rules and routines.”

Beth noticed similar trends in her Near North preschool, “Things have changed on a month-to-month basis and so really it kind of just teaches you to go with the flow. One thing that we’ve said all along is that the kids have taken really well to what school is like this year.” Her school limits the number of people in the building and had to implement a new morning arrival. Beth asked herself, “‘How is a three-year-old going to handle a car drop off?’ Well, all of them just walk happily into the door.” As we know, young children often use play to help make sense of the world around them. Beth has seen this happening in her classroom. “In our dramatic play area, we have baby dolls. We have diapers and little hats for them. And the kids asked, ‘Where are their masks?’ and I thought, ‘Oh I should make some of those for them!’”

Despite all the changes, math learning is still happening. We know that math is all around us, and that hasn’t changed. In fact, these teachers have discovered new opportunities.

Beth incorporates math into her morning meeting each day. “The children help us solve a math problem. I love when the parents come along [and join us for virtual lessons] because they get to see how we’re talking about math. I hope they are listening to the language that we’re using and replicating that at home too.”

Donely also turned a challenge into an opportunity. His classroom can no longer play on the neighborhood playground, so they take more walks around the park and down the street. This allows lots of time for fitting math in along the way. “My kids are counting how many steps they are taking. And when we go down the street, they started recognizing numbers on the houses. They might also say, ‘Look that tree is bigger than this tree.’ They are visually measuring.”

When we’re virtually learning, it makes parents feel nervous like they’re missing out on things we could do in the classroom. There’s so much in everyday activities that your child can learn from. – Beth

Beth couldn’t agree more about the joy of informal learning. She told us about a recent Zoom meeting when the class made playdough cookies together. While they were making their cookies and adding pretend sprinkles, she would say “How many sprinkles did you put on your cookie?” and “Who has the same number of sprinkles?” Beth describes her goal, “It’s letting them play and pulling out those concepts as they come out naturally. It doesn’t have to be that every lesson is a formal math lesson. It is really giving kids different materials and talking with them about what they’re doing. So much of their play IS math.”

There are other silver linings that make teaching rewarding for these educators. Gail recognizes new ways to partner with families. “I’m finding communication with the parents has improved because I am really thinking about what the children are doing all week.” Beth has learned to cherish times when she is together with her students in person. “Every week, it kind of felt like we didn’t know how many weeks we were going to be together. So, knowing that we might not see them, it makes you enjoy every moment a little bit more.”

A special thank you to Beth, Donely, and Gail for their time and willingness to share their experiences. Hearing directly from these teachers gives us hope for the future, no matter what 2021 brings.

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Best Books for Toddlers to Spark More Talk about More and Less https://earlymath.erikson.edu/best-books-for-toddlers-to-spark-more-talk-about-more-and-less/ https://earlymath.erikson.edu/best-books-for-toddlers-to-spark-more-talk-about-more-and-less/#respond Wed, 11 Nov 2020 20:39:49 +0000 https://earlymath.erikson.edu/?p=17774 It’s easy to turn reading books with toddlers into conversations about the math that is all around us. The idea of more and less comes into all kinds of books and finger plays toddlers love—including counting books.

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It’s easy to turn reading books with toddlers into conversations about the math that is all around us. The idea of more and less comes into all kinds of books and finger plays toddlers love—including counting books. What we need to emphasize as we read and chat about these favorites is all the examples of more and less of something that are at the heart of these books.

For example, there’s More, More, More Said the Baby by Vera Williams, a great book where toddlers are quick to join in on the “more, more, more” refrain used in the rhyming text. They love the colorful illustrations and will be quick to point to or chat about the many kinds of “more” we all want and need — be it more as a growing number of kisses and hugs, more as bigger in size, longer in time, faster in movement, or cozier in terms of texture or in settling down together.

We use our Precursor Concepts of Attribute, Comparison, Pattern, and Change as a framework for discussing math thinking during the ages of 0-3. Caregivers have a special role in mathematizing the Precursor Concepts. Infants and toddlers instinctively notice the math all around, but it is the caregiver who supports their receptive understanding by voicing and showing the words to use.

While reading to babies involves “proto-conversations” (see our list of best books for babies), when children are toddlers they begin to interact with the books more and can start a more robust “conversation” while a book is being read to them. And, this back-and-forth about what you see on the page can definitely become math talk!

Counting books invite more talk about more and less — and exact numbers too. While we shouldn’t expect toddlers to count on their own, there is a great deal of research that shows that the more adults model counting to find out how many, the more children connect number words to what they represent. Cardinality means understanding that the last number counted represents the total amount of the set. Research shows that later mathematical proficiency correlates with how often infants and toddlers hear cardinality expressed (Levine, Wang). For example, you might say to a toddler, “How many toes do you have?” then wiggle each toe and say “1, 2, 3, 4, 5″… or you can take it one step further and do the same thing but say, “You’ve got 5 little toes! 1, 2, 3, 4, 5!”

What are the best books for toddlers? Look for books that are sturdy to stand up to lots of use and handling by young ones. Books with flaps, slides, and other interactive elements help create interest. Colorful illustrations that aren’t too busy will invite toddlers to look and see, eventually pointing at things they recognize and can name.

Start the “conversation” about more, less, and the number of things in these best books for toddlers.

  1. More, More, More Said the Baby by Vera Williams

    This book found in board, paper, or hardcover is a Caldecott-winning treat for toddlers that takes advantage of something they all love, which is rhyming and song. Young ones Little Guy, Little Pumpkin, and Little Bird are absolutely adored by the adults around them. Multi-ethnic children fill the colorful pages, while toddlers are sure to like all the repetition (Patterns) of things they like so much, such as kisses, giggles, and hugs.

  1. Un Elefante by Patty Rodriguez and Ariana Stein

    This bi-lingual board book is colorful and full of appealing illustrations. The text is limited to number symbols/words as well as elephants/elefantes. This helps highlight each additional elephant as shown at the end of a line with others. These kind of small things in books for toddlers make a difference. This book gives many opportunities not just to sing the song but to explore and reinforce vocabulary related to more and less.

  2. 10, 9, 8 by Molly Bang

    This board or paper book features a young child getting ready for bed. (Author website here) This classic picture book is an example of a shrinking Pattern (less) while using wonderful and relatable Attributes.

10 9 8 Molly Bang, Best Books for Toddlers
  1. Feast for 10 by Cathryn Falwell

    This popular book features a family shopping and preparing a meal. It comes as a paper or board book, as well as in English or Spanish. The limited text offers many opportunities for children to converse about Attributes, Comparison, Pattern and Change while appreciating the rich illustrations.

  2. Rah, Rah, Radishes!: A Vegetable Chant by April Pulley Sayre

    A photo essay from a farmer’s market provides many ways to notice more and less. Are there more yellow potatoes or red ones? How many green tomatoes hanging on the stem? This book is sure to give you lots to talk about with your toddlers. Also see the companion book, Go, Go Grapes!: A Fruit Chant.

  3. Soup Day by Melissa Iwai

    This multicultural board book is another great book that involves soup-making. It features food shopping and preparation in a way that is inviting and mathematical for young children. The illustrations draw attention to small quantities (1 to 8) as well as shapes and Attributes. Pizza Day is a fun alternative by the same author.

Soup Day, Melissa Iwai, best books toddlers

Best Books for Babies to Find Math in Their World

Books are great for babies too! Find our list of books that are great for the Emerging Stage (0-15 months).

Learn more

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Best Books for Babies to Find Math in Their World https://earlymath.erikson.edu/best-books-for-babies-to-find-math-in-their-world/ https://earlymath.erikson.edu/best-books-for-babies-to-find-math-in-their-world/#respond Tue, 03 Nov 2020 18:14:39 +0000 https://earlymath.erikson.edu/?p=17733 Books are great for babies. Research clearly supports this. What are the best books for babies? We use our Precursor Concepts as a framework for discussing math thinking during the ages of 0-3.

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Books are great for babies. Research clearly supports this. What are the best books for babies? On a broad level, books that are easily handled are best – small size, made of board or cloth. Books that have large uncluttered illustrations are good – an infant’s ability to visually focus is developing through 6 months. Photos of faces are especially appealing. But can math be a part of the books babies interact with? Yes. Well, the concepts that lead to math.

We use our Precursor Concepts as a framework for discussing math thinking during the ages of 0-3. We refer to the baby stage as the Emerging Stage (0-15 months). Symbolic thought has not yet developed in infants, so they are most receptive to very generalized concepts that can be introduced when they are this age. This can of course take the form of reading!

Reading to Babies

Caregivers will be holding the baby up until the baby is about one year of age (lovingly, of course). They will need to hold the book also when reading but baby might reach, touch, and mouth the book. Caregivers initiate the reading, which can include pointing, asking questions, and making connections to baby. These are all done as “proto-conversations” with caregivers using “parentese” and encouraging the infant’s coos, gaze, and noises by waiting for a response.

Beginning about 7-10 months the infant will take increasing agency in book handling, pointing, and vocalizing. They may begin to respond with pleasure to the book. In later infancy, as baby moves through the babble stages, the interactions become true conversations.

Talking About the Math in Books for Babies

Reading through the book from beginning to end is not important at this stage – the book is an occasion for a warm conversational interaction. The best books for babies are often concept books and books that are based on songs, especially lullabies.

In regards to math, in the Emerging Stage infants are focused on sensory perceptions. These are connected to the Precursor Concepts: Attribute, Comparison, Pattern, and Change. Caregivers have a special role in mathematizing the Precursor Concepts. Infants and toddlers instinctively notice the math all around, but it is the caregiver who supports their receptive understanding by voicing and showing the words to use. That is, they explain how many, what size, or where something is located. For example, just the cover of Cleaning Up, Up Up (below) shows the father uses actions and words to show where the book belongs.

It’s also important to note that many books for babies make efforts to include multi-racial characters, diverse ethnicities, and rich multicultural experiences. It’s never too early to emphasize and celebrate such things.

The List

  1. Twinkle Twinkle Little Star, Illustrated by Jerry Pinkney

    This board book is a great version of the classic song. It is good for pointing to, gesturing towards, and talking about Attributes. The book loosely suggests a story so that it can also be used with toddlers at the developing and transforming stages, with many opportunities to talk about the Precursor Concepts.

  2. Small Talk Books by Ellen Mayer

    This is a series of books from Star Bright Books specially developed to encourage book conversation experiences with infants and toddlers. Commissioned by Heising-Simons, they are great books for babies that encourage all kinds of early conceptual understanding.

clean up up up ellen meyer, Best Books for Babies
  1. My Heart Fills With Happiness by Monique Gray Smith

    This board or paperback book is illustrated by Julie Flett and is infused with a beautiful question: what fills you with happiness? The Native American characters and wonderful illustrations are inviting and inclusive.

  2. I Love All of Me by Lorie Ann Grover

    This board book encourages children to love all the little things about themselves. Illustrated by Carolina Buzio, it also allows caregivers to point out all the body parts babies have while they read, touching and talking about all the little Attributes.

  3. What Happens Next by Cheryl Christian

    This board book written in both English and Chinese encourages lots of talk about Pattern and Change.

  1. Storytelling Math Series

    This book series by Grace Lin and others is from Charlesbridge Press and includes 11 board books that encourage lots of math thinking. They have titles such as Up to My Knees, What Will Fit, and The Last Marshmallow. Also helped by Heising-Simons, they all include a brief note from a TERC scholar giving tips for finding math all around us.

  2. Peek-a-Boo by Roberta Intrater

    This Baby Faces Board Book is part of a series published by Scholastic. This one is available in Spanish, English and other languages. The one-word text and emotions conveyed in the children’s faces encourage concepts of Attribute and Comparison.

Storytelling Math Series, The Last Marshmallow, Grace Lin, Best Books for Babies

Best Books for Toddlers to Spark More Talk about More and Less

Books are great for toddlers too! Find our list of books that are great for the not-quite-a-baby stage.

Learn more

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Chores for Kids Can Help Them Find Math Around the House https://earlymath.erikson.edu/chores-for-kids-can-help-them-find-math-around-the-house/ https://earlymath.erikson.edu/chores-for-kids-can-help-them-find-math-around-the-house/#comments Wed, 14 Oct 2020 13:36:40 +0000 https://earlymath.erikson.edu/?p=17689 With young children at home, there’s always cleaning to be done. So why not include them in completing the chores? Chores can engage everyone in the household in a little “math all around us” problem-solving.

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With young children at home, there’s always cleaning to be done. So why not include them in completing the chores? If we give children purposeful work that helps them feel responsible, involved, and important, they often are eager to pitch in.

There are different ways we can do this depending on the age of our children. And there are so many ways to support their mathematical thinking while we’re at it. Chores for kids can engage everyone in the household in a little “math all around us” problem-solving.

Chores for Toddlers

Even if you have to follow up after them to get surfaces really clean, toddlers can love tasks like dusting or wiping down tables and counters. Use a spray bottle with water and a cloth. Then comment as you work together to “say what you see” and “say what you do.” This conversational math is an effective way to build their understanding of mathematical ideas in a familiar context.

Spatial relationships are a good focus for the little ones. Describe the wiping motions your child makes. For example, up, down, around, in circles. Use relational word pairs, such as:

  • here/there,
  • above/below,
  • under/on top of,
  • behind/in front of, or
  • next to/farther way from.

For example, in this video, a 20-month-old child can be encouraged to find the math in what he is doing with the language used by adults.

Hey Judah, great job in getting the dust on top of the long shelf; next to the door! And now you are heading across the room to get at the one on this side!

There are natural opportunities to count, too. You can count the number of sprays of water and the number of wipes. For example, “I’m going to do 2 sprays. Here I go: one spray, two sprays. Can you wipe it clean?”

Chores for Kids 4-6 Years of Age

Older children are ready to take on simple chores such as helping set or clear the table or clean up their toys and belongings. Deepen their understanding by asking questions that invite them use spatial language themselves, such as:

  • Which drawer has the silverware in it?
  • Where do we put containers that can be recycled?
  • Can you get the window-spray from the cubboard under the sink?

When children help you map out the places in the room where things go, they build their sense of responsibility as well as increase their awareness of space and location. Ask children describe where things are in relation to each other.

Not interested in making maps? No problem! Just make a list. Depending on what room you’re cleaning, have children look around and think about what tasks need to be done. Then you can talk about the order of jobs and problem-solve a fair way to divide them up.

chores for kids, cleaning, early math, preschool math

If we have 6 things to do, who will do what? Do you want to divide things up equally? If so, how many tasks does that mean we will each have to complete? But are some jobs bigger than others? Could more than one person work on the same chore?

If you have sticky notes (or use small squares of paper and tape), children can help write each chore on a separate note and post them up on a blank wall or door. As more notes go up, the amount of work that needs to be done can be seen. As the discussion about who does what begins, move the notes around accordingly and the finished product will be a clear list of tasks for each person. Then, as the work is completed, notes can be removed, and everyone can watch with satisfaction as the work disappears one task at a time.

Find math in everyday chores for kids and find yourself with a cleaner house at the same time. Win-win!

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Homemade Playdough Recipe for Shape Fun https://earlymath.erikson.edu/homemade-playdough-recipe-for-shape-fun/ https://earlymath.erikson.edu/homemade-playdough-recipe-for-shape-fun/#respond Fri, 11 Sep 2020 14:00:19 +0000 https://earlymath.erikson.edu/?p=17509 Stamping or imprinting with a homemade playdough recipe using everyday household items is an active way for children to explore the big ideas of shape.

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Use a simple homemade playdough recipe to cook up some shape fun! Stamping or imprinting playdough with everyday household items is an active way for children to explore the big ideas of shape, including the discovery that 3-dimensional objects are made up of 2-dimensional shapes. For example, a spaghetti box is made up of six rectangles of three different sizes. A can of beans has two circular bases of the same size. As children flip, turn, and slide these shapes during play, they develop and deepen their understanding of shapes properties such as flat or curved surfaces and pointy or square corners.

Starting in infancy, children interact with the world largely through object play such as rolling a ball, building a tower, or climbing inside a large cardboard box. You can bring out the math when you add language to describe what and how they are interacting with the different objects in their world. Young children do not yet possess the formal words to express what they are noticing about shapes and their properties, but we can use math language and gestures to guide them.

Playdough is a rich material for children to engage in play; there is no limitation to what you can create. To begin, follow this homemade playdough recipe to make the dough and collect common household objects of a variety of shapes for printing. You can find these items in your cabinets and your recycling bin, such as empty toilet paper or paper towel rolls, cans, caps, and small boxes such as a toothpaste box or small jewelry boxes. Use items that will fit in your child’s hand and create an imprint in the playdough.

Once you’ve made the playdough and gathered all the objects you plan to use, ask your child to roll out and flatten the playdough. How can they do that using one of the objects? Which objects roll and which ones don’t?

Homemade Playdough Recipe

Homemade Playdough Recipe

Your role is to layer the experience with what you notice the children discovering. “When you pressed the lid into the playdough, it made a circle!” or, “Look at four lines and four corners you made with the toothpaste box.” For a fun challenge, ask your child to compose a larger shape or picture made up of smaller shapes. Shape composition helps build a child’s spatial reasoning skills.

This purposeful play activity with objects from around the house will continue to engage your children while deepening their mathematical understanding. See how much fun through exploration and conversation you can have!

Do both sides make the same shape? ​

Homemade Playdough Recipe

Homemade Playdough Recipe

An eyeball was created using multiple cylinders.​

Homemade Playdough Recipe

(Adapted from TheBestIdeasForKids.com)

Ingredients

  • 1 cup water
  • 1 cup flour
  • 2 teaspoons cream of tartar
  • 1/2 cup salt
  • 1 tablespoons of olive oil, cooking oil, or vegetable oil
  • food coloring

Instructions

  1. In a large bowl, combine all dry ingredients – flour, cream of tartar, salt – and mix it well.
  2. Mix the food coloring with the water. And then add vegetable oil and the water and food coloring into a big pot. Then mix those together.
  3. Add all dry ingredients to the pot and mix those well.
  4. Cook all of it over low-to-medium heat so that the dough starts to form and becomes a dry dough.
  5. Once it begins to form a ball and looks cooked fully, take it off the heat. Allow for the dough to cool before touching.
  6. Once it is cool, knead the dough for around five minutes so that the dough is soft.

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Families are Pivotal in Designing Math Picture Books for Children https://earlymath.erikson.edu/families-are-pivotal-in-designing-math-picture-books-for-children/ https://earlymath.erikson.edu/families-are-pivotal-in-designing-math-picture-books-for-children/#respond Tue, 25 Aug 2020 13:58:30 +0000 https://earlymath.erikson.edu/?p=17409 How do you design a better picture book for children? How does consulting with families along with research help us to understand what makes educational children's books effective and engaging?

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How do you design a better picture book for children? What does better mean? How does consulting with families along with research help us to understand what makes educational children’s books effective and engaging? At our biennial conference Promising Math we discussed these questions.

Keynote speaker David Purpura worked with colleagues at Purdue University and families in nearby communities to design children’s books that foster mathematical conversations at home (presentation PDF). His research has demonstrated the effectiveness of the books—children who used them at home learned more math-related language and improved their math achievement more than those who did not—but his presentation focused on the book development process itself. He commented, “If we go into this whole research process and find that something is really effective, but we haven’t considered the families and their ability to use it, and we can’t scale it… it’s kind of useless. It’s really important to consider families in the development process.”

One of the ways the project centered the families’ experience was to focus on a familiar activity, reading books together. Picture books tap into a practice that many families already have as part of their routine, rather than asking parents to do something completely new. David points out that “picture books can be this really great opportunity to… integrate math into homes.” Building on families’ love of reading together to add in talk about math seemed like a wise choice.

If we go into this whole research process and find that something is really effective, but we haven’t considered the families and their ability to use it, and we can’t scale it… it’s kind of useless. It’s really important to consider families in the development process.

Second, the books include not only the text of the story, but example questions parents can ask about what is happening on each page. For example, when parents read a section in which the characters are talking about whether they have enough of each ingredient to make a cake, the extra text might suggest parents ask, “Why don’t they have enough eggs?” In this way, the book reading experience supports families to talk about what “enough” means, prompting math-related conversation at a developmental level just right for preschoolers. Each page includes three different questions, each just a little more complicated than the last, so children and caregivers can return to the book later and have a slightly different conversation. This thoughtful “scaffolding” of the reading experience makes it easy for parents.


designing math picture books for children 1 636x477
designing math picture books for children 2 636x477

Additionally, David’s team made the decision to write the books with both Spanish and English versions in mind from the very beginning. Often, books written in English translate poorly to Spanish, or they don’t account for that fact that different dialects of Spanish translate things differently. In one of the books, the authors changed the subject of the book from “too many marshmallows” to “too many pillows” because “marshmallows” can be translated to Spanish in at least 4 different ways, depending on dialect. David pointed out that there are around 40 million people over the age of 5 who speak Spanish in their homes, but the availability of children’s books in Spanish is low, and their cost tends to be higher than books in English, so the need for additional books in Spanish is real.


designing math picture books for children 3
designing math picture books for children 4

Most importantly, the development team involved families in the design at an early point in the process of the picture books for children. At first, parents came into their lab, read the books, and gave feedback. After a round of revision, parents were given three books and sent home with them for a month. They were asked to read the first book three times in the first week, the second book three times in the second week, the third three times in the third week, and in the fourth week, they read each one once more. The families were interviewed extensively about their experiences using the books. Researchers asked about both ease of use and enjoyment, making sure that the books might be something families would read again on their own. It was as a result of parent feedback that the team made all of the books bilingual—that is, the text and prompts are provided in both Spanish and English in every version.

It’s clear that involving parents as partners in designing the books helped make them effective tools for families to generate mathematically rich conversations. David points out, “…a brief intervention, 12 sessions of parents reading to their children, something they already do, was effective at improving children’s math language and numeracy skills, and the effects on numeracy were maintained over eight weeks after the intervention, and…the process was easy for parents to do.” David and colleagues are working with families to develop additional books that focus on spatial reasoning rather than numeracy, and hope to see similar results!

Family-Centered-Approach-to-Picture-Books

Family-Centered Approach to Picture Books with David Purpura

David Purpura describes his team’s research and intervention designed around picture books.

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Family Math

Families can play a fundamental role in encouraging learning and interest in math. Explore all the resources in our Idea Library for Family Math.

Learn more

Promising Math

Promising Math

Promising Math is a biennial conference that aims to share knowledge about the understanding, teaching, and learning of early math.

Learn more

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Family-Centered Approach to Picture Books with David Purpura https://earlymath.erikson.edu/family-centered-approach-to-picture-books-with-david-purpura/ https://earlymath.erikson.edu/family-centered-approach-to-picture-books-with-david-purpura/#respond Mon, 24 Aug 2020 13:57:42 +0000 https://earlymath.erikson.edu/?p=17427 David Purpura, PhD, of Purdue University describes his team's research and intervention designed around picture books and their effectiveness in homes.

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David Purpura, PhD, of Purdue University describes his team’s research and intervention designed around picture books and their effectiveness in homes. This family-centered approach brought parents into the book development process, which meant thinking about the stories in both Spanish and English from the start.

We made sure that the words that we used would be understandable across different dialects.

As as a result of parent feedback, all the books are bilingual rather than different versions for Spanish and English. This is one example of how involving families strengthened the outcomes of the project.

This was part of a keynote speech given at the 2019 Promising Math conference, focused on family and community contexts for early math learning.

Family Centered Approach to Picture Books

Families are Pivotal in Designing Math Picture Books for Children

Read more about David Purpura and his team’s research and intervention.

Learn more

Family Math

Families can play a fundamental role in encouraging learning and interest in math. Explore all the resources in our Idea Library for Family Math.

Learn more

Promising Math

Promising Math

Promising Math is a biennial conference that aims to share knowledge about the understanding, teaching, and learning of early math.

Learn more

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3 Children’s Books Where Small Numbers Matter https://earlymath.erikson.edu/three-books-where-small-numbers-matter-number-words-printable-downloads/ https://earlymath.erikson.edu/three-books-where-small-numbers-matter-number-words-printable-downloads/#respond Tue, 11 Aug 2020 12:15:55 +0000 https://earlymath.erikson.edu/?p=16658 As children start using number words, they don’t always have a sense of what those words really mean. Early on, we guide children to develop a meaningful number sense by focusing on small numbers.

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As children start using number words, they don’t always have a sense of what those words really mean. We guide children to develop a meaningful number sense by focusing on small numbers in preschool and helping children connect those number words to specific quantities.

We can think of a quantity as a number plus its unit—that is, the numerosity of set that is being counted. Not just “two” but “two shoes” or “two paintbrushes.” Adults can use language to describe how many of what we are seeing as we share attention to small numbers with a child. And, math-focused books are one the best ways to look at quantities with your child.

When sets of objects are small—one, two, and three—children can recognize the quantity of without counting. Even young children can instantly “see” how many — subitize — when we provide them with pictures, photos, and real-life sets and ask the question, “how many do you see?” The more opportunities children have, the more they develop this math lens for seeing small numbers everywhere!

Here are three books that can spark talk about small sets of one, two, and three things. When it comes to exploring number words, printable downloadable at-home activity cards such as these can help. Written in both English and Spanish, they are a guide to engaging your child. Many of these books are available in digital form on YouTube. Play the video but turn off the sound and read the book yourself. Remember, you can pause the video just like you would while reading the paper book to ask questions and look for the small numbers mentioned in the stories.

number words printable activity cards

More, Fewer, Less by Tana Hoban

This wordless book contains interesting and attractive photos of sets of items that children and adults can discuss and compare.

Download

Uno, Dos, Tres / One, Two, Three by Pat Mora

This book is beautifully illustrated, is bilingual in Spanish and English, and has a great introduction to counting and rhyme.

Download

Two Mice by Sergio Ruzzier

This is an adorably illustrated tale of two mice on a boating trip. The author tells the story using simple text and groups of one, two, or three items. No counting required here. Readers can easily see and name the small sets. With number words printable guides like these can help.

Download

Download All

Here you can download a PDF that includes all of the at-home activity cards mentioned here. English and Spanish versions.

Download

Get downloadsJoin the website

Want free printable activities and card sets? Join our website (or log in if already joined) to unlock much more.

Join or Log in

 

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Putting Groceries Away is Hands-on Shape Activity for Young Children https://earlymath.erikson.edu/putting-groceries-away-is-hands-on-shape-activity-for-young-children/ https://earlymath.erikson.edu/putting-groceries-away-is-hands-on-shape-activity-for-young-children/#respond Wed, 29 Jul 2020 20:29:21 +0000 https://earlymath.erikson.edu/?p=17313 Putting groceries away is necessary work for families. Sharing this work with our children turns an everyday task into a hands-on shape activity.

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Putting groceries away is necessary work for families. Sharing this work with our children turns an everyday task into a hands-on shape activity — if we slow down and let them figure things out for themselves!

Start by including children in the work in a meaningful way. You might say, “This a heavy bag of groceries! Let’s put the food away so we know where to find it when it’s time to cook.”

Unpacking the grocery bags

Children will begin to notice how some groceries are similar or different by categorizing food packages as they set them out on the kitchen counter or floor. Groceries come in a variety of shapes and sizes. Cans of soup or beans and jars of pickles or peanut butter can be grouped together; so can boxes of cereal, pasta, or crackers. Having a conversation about why cans go together and boxes might go together in another group brings math into this household work.

I notice you’re stacking the boxes flat, on top of each other. All four fit.

Hands-on Shape Activity 2

If we push the boxes over, the milk carton can stand up here.

Hands-on Shape Activity 1

Filling up the cabinet

The conversation you and your child have while placing items on the shelf or in the cabinet can be a way to discover shape characteristics. For example, stacking cans on the flat side versus the curved sides. Children can explore how many cans they can stack to fit inside the shelf; certain items may be too long or wide to fit.

Oh no, the cans are falling out!

If it stands up, then it won’t fall over.

Solving problems with spatial reasoning

When we offer children space and opportunity, we can observe their many attempts to fit items into the cabinet only to stop and restart. The work of trial and error is very beneficial to developing their understanding of the characteristics of shapes. We might watch before we support them with language such as “I notice that long can of Pringles doesn’t fit, it is taller than the cabinet space.”

Let me try the peanut butter jar. Nope it doesn’t fit.

This jar fits on top of the peanut butter.

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Taking Math Out for a Spin: Outside Activities for Kids That Have Math https://earlymath.erikson.edu/taking-math-out-for-a-spin-outside-activities-for-kids-that-have-math/ https://earlymath.erikson.edu/taking-math-out-for-a-spin-outside-activities-for-kids-that-have-math/#respond Mon, 13 Jul 2020 16:54:44 +0000 https://earlymath.erikson.edu/?p=17205 Warm weather and more daylight hours mean now is a great time to take advantage of outdoor spaces. There are so many ways to incorporate math into your child’s outside activity.

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Warm weather and more daylight hours mean now is a great time to take advantage of outdoor spaces. There are so many ways to incorporate math into your child’s outside activity. Obstacle Courses and the Bean Bag Toss are two fun outside activities for kids that are already rich with math: spatial awareness, geometry, measurement, number sense, and counting… you can even find ways to develop data analysis!

Obstacle Courses

There is a lot of math to be found in obstacle courses, from planning the course and the actions to describing the movement as you go. With older children, you can incorporate measurement and data analysis.

Highlighting the math in Obstacle Courses:

  • Use household items, such as sheets, brooms, pots, tables, and chairs and/or outdoor items such sticks, rocks, and trees to create the course. Together you can design the shape of the course: will it be circular, straight or something else?
  • Call out what to do when they get to each obstacle using directions that develop spatial relationships; “walk over the broom,” “walk around the tree,” “step on the stick,” “crawl under the sheet,” etc.
  • After designing and creating the course outdoors, older children can use a stopwatch to time how long it takes to complete the course. Record each time they do the course and compare the results to see if they are getting faster and use the data to analyze why or why not.
  • Children can also draw a map of the course, either before as part of the planning process, or after as a way to share their fun with others. Be sure to have them include arrows that show the movements and actions.
  • You can also involve measurement with obstacle courses by comparing the length and heights of the different obstacles. To keep your obstacle course exciting, after you measure the obstacles, you can decide if you want to change them and measure how much taller, shorter, longer, or wider you want to make them.
outside activities for kids obstacle course math
obstacle course math 2

The best part using an obstacle course is that you can change it whenever you want. Children stay interested and use it over and over again.

Bean Bag Toss

Cornhole or Bags are popular games at many summer gatherings. These are family-friendly outside activities for kids that any age can play. While playing, they develop counting skills, spatial reasoning, and measurement. You don’t even need a board with a hole in it to play. You can easily create your target with household items and play in the grass , or draw your target with chalk on the sidewalk.

Highlighting the math in Bean Bag Toss games:

  • With younger or less experienced children, use a hula hoop or draw a circle with chalk on the sidewalk. Use five bags to toss towards the target. No bags? You can even use sponges or balls of socks! After the child tosses the bags, talk about how many bags are in the circle and how many are outside the circle; are the bags near the target, or are they are far away from the target? Talk about how there are still five bags, even though there are some bags inside and some bags outside the circle.
  • Older or more experienced children can draw a multi-leveled target with different point values on the sidewalk and record their points.
  • Play Bean Bag Toss as a carnival game by drawing a row of circles on the sidewalk or placing buckets, pots, or other types of containers in a row. As you create the row of chalk circles or buckets, you can measure to make sure there is equal distance between each one. Additionally, you can measure the length of the entire row. Start with just three spots and add on to make the row longer.
  • If you are playing with chalk on the sidewalk, you can create a “scoreboard” for your child(ren) to fill in.
activities bean bags 1
bean bags 2

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How to Adapt Math Card Games to Children’s Skill Level https://earlymath.erikson.edu/how-to-adapt-math-card-games-to-childrens-skill-level/ https://earlymath.erikson.edu/how-to-adapt-math-card-games-to-childrens-skill-level/#respond Fri, 05 Jun 2020 19:10:08 +0000 https://earlymath.erikson.edu/?p=17052 Card games provide meaningful practice of the basic number combinations. These common card games that children learn in school or at home can be revisited many times and can be adapted to children’s own math skills as they develop over time.

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Card games provide meaningful practice of the basic number combinations. Becoming automatic with basic number combinations is the key to computational fluency. These common card games that children learn in school or at home can be revisited many, many times and can be adapted to children’s own math skills as they develop over time.

Go Fish!

Go Fish is a pair-seeking game. Players each have their own hand of 5 cards. Players ask their opponents for a card that pairs with a card in their own hand. Basic play begins with making matched pairs. For example, if you have a card with five dots you would ask you opponent for another card with five on it. If your opponent has the card, they give it to you, and then you keep asking for cards until your opponent does not have the card you request and says, “Go Fish!” You get to pick from the pile of face-down cards in the “fishpond.” If you make a match you continue. If not, it is the other player’s turn. The game continues until all the cards are paired up. Players count their pairs, and the player with the most pairs, wins the game.

Match ‘Em / Memory

Match ‘Em or Memory are similar pair-seeking games that allow you to adjust the memory challenge. In both games, cards are arranged in an orderly array on the floor or a table. In Match ‘Em, the cards are face up; in Memory, the card are face down. Players take turns finding pairs. The game is over when all the pairs are matched.

Capture

Capture is a comparison game also known by the traditional name of “War.” To play, deal the entire deck of cards among the players. Players show their cards at the same time, and the person with the greatest number wins the round and takes the cards. When the cards run out, the game is done, and players can count their cards to see who has the most cards.

Things to Consider

While all these games can be played with a standard deck of cards, you will want to give some thought to how you create a deck to match your child’s math skills. Some things to consider:

  • Use quantity cards that show a pictorial representation of a number. This helps children develop their visual number sense as well as their computational fluency. Use our quantity card downloads or, when children are ready to explore multiplication, check out this resource from YouCubed.
  • Start with cards 0-5 for preschoolers; 0-10 for kindergarten and first grader; 0-20 for second grade
  • The size of the deck (number of cards in play) should be appropriate for children’s age and experience. Small hands can only hold a small hand of cards!
  • Use small arrays of cards to play Match ‘Em or Memory at the start such as a 3 x 4 array of 12 cards. Arrange cards face-up or face-down to vary the memory challenge based on children’s age, experience playing, and the math skill they are working on. If it is a new concept, even for older children, face-up may be the way to play the first few times.
  • Fewer cards in the deck means faster games that keep children’s attention. When games are quick, they can be played again and again, which eases the disappointment of losing.

Downloads

Adapting math card games chart

Adapting Math Card Games Chart

This download breaks down the variations of these games into a chart so that adapting them to children’s skills and ages can be easy to do.
Math Card Games Chart

Go Fish playing math card games

Go Fish Game

This download provides an easy take-home game so educators can provide parents with opportunities at home.
Go Fish Directions

Make a Match playing card games

Match ‘Em All / Memory Game

This download provides simple instructions for playing cards at school and at home.
Match ‘Em All Game

Capture playing card games

Capture Game

This download provides directions for the Capture card game, which is a variation on an old favorite. Learn how to vary the game to match your child’s math skills.
Capture Directions

Related Ideas

playing card games early math

A New Focus for Familiar Card Games

Here you can learn much more about Quantity Cards and how they build an understanding of early math concepts such as cardinality and composing and comparing numbers.

Learn more

playing math card games early math

Quantity Cards

These Quantity Card downloads are available and cater to different skill levels and ages. They include a variety of pictures or “suits” since matching quantities such as fingers and dots can help develop ideas about numbers and equivalence.

Learn more

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Math at Your Fingertips: Songs and Fingerplays for Preschoolers https://earlymath.erikson.edu/math-at-your-fingertips-songs-and-fingerplays-for-preschoolers/ https://earlymath.erikson.edu/math-at-your-fingertips-songs-and-fingerplays-for-preschoolers/#comments Wed, 27 May 2020 15:04:06 +0000 https://earlymath.erikson.edu/?p=17006 Do you remember enjoying fingerplay songs like 5 Little Monkeys or Un Elefante Se Balanceaba when you were young? You may not have realized that sharing these songs is a great way to bring math to life.

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Do you remember enjoying fingerplay songs like 5 Little Monkeys or Un Elefante Se Balanceaba when you were young? You may not have realized that sharing these songs and fingerplays for preschoolers is a great way to bring math to life. When we use fingerplays, we help our children’s mathematical understanding by:

  • Connecting a quantity to its number name. For adults it seems like second nature that the word “three” signifies the quantity of 3 objects, but this is a complex and abstract concept! By using our fingers to show a quantity (i.e. holding up three fingers when we say “three squirrels”) we are helping children connect the number name to the quantity it represents.
  • Adding a visual for number relationships. Fingerplay songs for preschoolers can tell the story of squirrels coming to a branch one by one or birdies leaving a nest one by one. These bring out number relationships (i.e. 2 is one less than 3, or 2 is one more than 1), which are essential for building a strong number sense. By using our fingers, children also have a visual that is helping them see and feel the difference between 4 and 3 or 3 and 2.
  • Highlighting the rules of counting. Children can see important principles of counting in action, like only counting each item once.
  • Emphasizing small quantities. A strong foundation with small quantities is essential for young children before they move on to larger numbers.

Good Times for Finger Play Rhymes

Imagine you are sitting looking out the window with your child or taking a walk; they point out a bird or a squirrel running across a tree branch, or ducks in the pond. This is the perfect time to introduce one of many fingerplays for preschoolers in the form of a song like 5 Little Ducks or 5 Little Squirrels. If you had peas for lunch, follow up with 5 Peas in a Pod. The important thing is that you and your child are finding math in a shared interest.

But fingerplay songs are equally fun anytime your child is restless or needs a little purposeful movement break. It’s amazing how you starting to sing one of these songs brings even toddlers over to join in the fun. (Or make it a game.) Children learn naturally through imitation and you will find that as you bring joy and playfulness to these fingerplay songs with your own voice, children will naturally want to join in. Most importantly, you will all be having fun together and connecting through math!

Five little squirrels sat up in a tree; (Hold up five fingers)
The first squirrel said, “Well, what do I see?” (Point to thumb)
The second squirrel said, “I see a dog!”
The third squirrel said, “I see a frog!”
The fourth squirrel said, “Let’s run into the shade!”
The fifth squirrel said, “I’m not afraid!”
Then RUFF went the dog, and away the squirrels ran,
One, two, three, four, five!

Tips & Tricks

  • In order to make the math in these rhymes meaningful for young children we need to take an extra step and connect that math language to gesture. This will engage their bodies, eyes, voices, and minds in the learning, which makes it even more powerful!
  • Don’t worry if your child just watches and listens at first. Little by little, you will find that they will enthusiastically make the big movements such as shaking their finger to scold the monkeys for jumping on the bed in 5 Little Monkeys. It may take longer to manage folding up or down their fingers, so you need to model the change in the number of fingers you have up.
  • Below you will find an example of rhymes and fingerplays for preschoolers. Using rhymes from your own childhood is a wonderful way to share language and culture with your children.

Beehive

This is a beehive (hand cupped)
Where are the bees?
Hidden away where nobody sees…
Now they come creeping out of the hive
One, two, three, four, five (extend fingers one by one)
Five bees!
Bzzzzzzzzz!

Five Little Birdies

Five little birdies went out one day over the hills and far away
The mother bird called “Tweet, tweet, tweet, tweet”
But only four little birdies came to see
(continue decreasing number of birdies)

Las Ardillas

Una ardilla se balanceaba sobre una rama delgadita,
Como veía que resistía, fue a llamar a otra ardilla.
Dos ardillas se balanceaban sobre una rama delgadita,
Como veían que resistía, fueron a llamar a otra ardilla …
Dos ardillas se balanceaban sobre una rama delgadita,
Como veían que resistía, fueron a llamar a otra ardilla.
… (sigue agregando una ardilla)

Ten Little Raindrops

One little, two little, three little raindrops,
Four little, five little, six little raindrops,
Seven little, eight little, nine little raindrops,
Ten little raindrops falling down!

Falling, falling, falling raindrops,
Falling, falling, falling raindrops,
Falling, falling, falling raindrops,
Pitter-patter, pitter-patter, splash!

Get More Songs and Rhymes

Download a PDF version of these and more fingerplay songs for preschoolers.

Download

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Get on the Right Path: Make Your Own Number Sense Activities https://earlymath.erikson.edu/get-on-the-right-path-make-your-own-number-sense-activities/ https://earlymath.erikson.edu/get-on-the-right-path-make-your-own-number-sense-activities/#respond Tue, 26 May 2020 15:08:50 +0000 https://earlymath.erikson.edu/?p=16992 Path games are fantastic ways for families to spend time together and have fun while doing math. Path games develop number sense, counting skills and, depending on children’s ages and the tools you use, computational fluency.

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Path games are fantastic ways for families to spend time together and have fun while doing math. Path games develop number sense, counting skills and, depending on children’s ages and the tools you use, computational fluency. These opportunities come about as players take turns rolling dice, counting spaces to move, and figuring out what they need to roll to win the game.

As we talk with children while playing a path game using a number path, we can highlight their ability to see small quantities without counting (subitizing) and build number relationships, such as: spatial sense, one/two more than and one/two less than a given number. Asking questions such as, “What do you hope to roll to win the game?” or “How many spaces do you need to get to the finish line?” also helps develop their mathematical skills. Kindergartners and primary grade children can develop addition and subtraction fluency by playing with two or more dice and finding the sums or differences to determine the amount of spaces to move. Children will want to play these games often because they have fun doing it, while games and number sense activities can provide a chance to connect with family and friends.

Best of all, there is no need to run out and buy these games! There are simple and creative ways to make path games and other number sense activities at home using materials from around the house. Making a path game at home allows you to tailor the game to the experience and age of your children. Also, it allows for the children to participate in making the game. The game-creation process adds to the math learning by incorporating measurement and geometry skills. Some, but not all of the elements that need to be considered are: How many spaces will the game be? Will you make your own dice? Will the game have a theme? Will you use different shapes to form the path? You can make as many different games as you like. Enjoy the journey.

Materials needed:

  • Paper grocery bag
  • Scissors
  • Markers/Pen
  • Dice (game dice or homemade) or other number generators such as a spinner or playing cards
  • Playing pieces

Directions:

  1. Cut paper bag
  2. The bag should open up fully (see photo below) after you cut it
  3. Cut off the extra piece from the bottom, so that the bag looks like the photo below
  4. Draw out a path game board on the paper bag
    Note: The path of the game can be modified to fit the needs of the child, varying the number sense activities like this require. You can make a short path game (5-10 spaces) or a longer path game (10-20 spaces) or even longer if you like.

Playing the Game

  • Each player needs their own playing piece.
    • Options, pieces from other games, different coins or any other ideas you have.
  • You can use a dice to determine the number of spaces to move,
    • One die for beginning players
    • Two or more dice for more advanced players or boards with longer paths
    • Cover up the faces of the dice with tape and write smaller numerals or quantities on the tape
number sense activities path games diy making
  • If you do not have dice, you can make a spinner from a piece of cardboard. Cut a circle. Divide the circle into 4 or 6 equal pieces and label with dots (and/or numerals). Use a pencil and paper clip to make the pointer of the spinner (see top photo)
  • You can also use a deck of cards, to pull one or more cards to represent how many spaces the child will move on each turn.
    • Pull out only some of the cards to use. For example, you can pull out the cards with quantities 1-3 or 1-5 and only use those as part of the game.
    • If using playing cards, you can have children add the amounts on the cards to compare them to the number of spaces traveled on the path. They should be the same.
number sense activities path games diy

Modifications

  • Develop the number relationship of +1/+2 or –1/–2 by covering the faces of one of the dice with 3 green sides and 3 red sides for adding or subtracting 1 or 2
  • If a player rolls an even amount on the dice, they move one less space but if they roll an odd amount on the dice, they move one more space than on the dice.
  • Create a path with spaces that include spaces marked +1/+2 or –1/–2.
  • Make your own dice or a spinner to change the dot designs or number of dots on the faces.

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Games to Play at Home Using Attributes for Math Thinking https://earlymath.erikson.edu/games-to-play-at-home-using-attributes-for-math-thinking/ https://earlymath.erikson.edu/games-to-play-at-home-using-attributes-for-math-thinking/#respond Mon, 25 May 2020 15:09:38 +0000 https://earlymath.erikson.edu/?p=16997 Language and math have a lot in common! For example, every time you describe something—as red, tall, sticky, or loud, for example—you are helping to define and categorize things. And defining and categorizing is huge in mathematics!

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Language and math have a lot in common! For example, every time you describe something—as red, tall, sticky, or loud, for example—you are helping to define and categorize things. And defining and categorizing is huge in mathematics! You can’t count all the apples until you are sure you know which things are apples and which are oranges. And a square is only a square because of the characteristics that define it: closed figure, four sides of equal length. When children use attribute language to describe and organize objects when playing games at home, they are preparing their brains for math thinking!

There are tons of opportunities to use attributes when talking with your kids at home, whether as a fun game or as part of regular cleaning and organizing activities. Below are some ideas that don’t require any special materials.

I Spy

One way to help kids think about attribute language is the classic game “I Spy.” We have played this for years as a waiting game in restaurants and doctors’ offices. It works just as well as a game to play at home!

To play, someone finds an object that everyone in the room can see, and describes it using one characteristic, as in “I spy something yellow,” and other players guess— “is it a banana?”

The whole family can take turns and get in on the fun. If you want to, you can make it even more math-y by using attributes that encourage mathematical thinking, such as numbers, shapes, measurement, and location words. 

The ways to use features to describe objects is endless… Think outside the box and have fun! 

*Note: younger children may need a reminder that for the game to work, the item they are “spying” needs to be one that everyone can see at that moment. When our kids were younger, our son would often choose things that he saw in the past while in the car or out and about.


  • I spy something round 
  • I spy something that has three parts 
  • I spy something bigger than my hand
  • I spy something heavier than the dog 
  • I spy something taller than daddy 
  • I spy something that is a pair 
  • I spy something that is under the window 
  • I spy something wider than your arms 
  • I spy something half the size of you 

Crazy Train

Another game idea is building a Crazy Train with just about anything you have around the house. Pencils, apples, keys, pillows, containers, toys, and books are all good choices for Crazy Train games to play at home. Gather them together on a table or the floor, and have the kids pick an object to be the locomotive, or the first object in the Crazy Train. In our game, the toy truck is the locomotive. Then look at your other objects: which one has something in common with the truck? As long as your child can justify their choice, they can add that item to the train! We chose a green die because it shares the same color as the truck. Next is an orange die because the dice have the same shape.

The goal of the game is to connect all the items in a long chain with each pair of objects connected by a unique attribute. The hard part comes as the game nears its end, when the remaining, passed-over objects have to be included. That’s when the attributes might get really crazy, like pointing out that both an eraser and a flashlight are “taller than an ant” or “don’t speak French” — as long as the statements are true, it’s a train!

Once all your objects are “train-ed” comes the fun part: Can you go back to the locomotive and name all the relationships that link everything, all the way to the caboose?

attribute train Games to Play at Home playing games at home

Sorting your Shoes

Is there a place in your house where people take their shoes off when they come in? If so, that’s a natural opportunity for talking about attributes. It can be a household cleaning project or a game—either way, it all starts with a question: how should we organize these shoes? When the sorting begins, attribute language is unavoidable!

Maybe it makes sense to organize the shoes by owner – all of Nana’s shoes over here and all of Toni’s over there. Maybe it makes sense to organize them according to purpose: sneakers in one place, sandals next to them, and all the rainboots lined up against the wall.

shoes attributes Games to Play at Home

If it’s more of a game than a housekeeping task, you can really get creative, and let the kids take the lead. “Let’s put all the shoes with Velcro closures together.” “Here are two pairs of shoes with red stripes.” “I wonder if we have more shoes with laces or without?”

If you and your family need some inspiration, below are links – in English and Spanish — to a great Pete the Cat video about using attributes to describe shoes.

shoes attributes Games to Play at Home 2

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Dishing Up the Math: Tasks at Home Can Become Math Time https://earlymath.erikson.edu/dishing-up-the-math-tasks-and-chores-can-become-math-time/ https://earlymath.erikson.edu/dishing-up-the-math-tasks-and-chores-can-become-math-time/#respond Tue, 28 Apr 2020 18:50:13 +0000 https://earlymath.erikson.edu/?p=16807 Every day dishes need to get cleaned, and every day dishes need to get put away. By involving children in tasks like doing the dishes, you can help them see mathematics in this work.

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Every day dishes need to get cleaned, and every day dishes need to get put away. It might not seem like an important task… until there are no spoons for your morning cereal! By involving children in chores like doing the dishes, you can help them see mathematics in the work and turn an everyday task into a math task. And even better if you get some help in everyday chores around the house!

Enough is Enough – Measurement

There are many different size comparisons happening throughout your dishwashing experience. That means there are good opportunities for you to provide measurement-related language while children are already engaged.

Essentially, any time you talk about amount – how much, whether more or less would be better, which is smaller—you are talking about math relationships. Just noticing these opportunities and using language to point them out really mathematizes the dishes!

Mother and daughter compare sizes of “baby spoons” and “daddy spoons” in their home language, Gujarati.

Here are some examples of how measurement can be found in your dishes routine, making it a math task:

  • Organizing which dishes will be washed first: “We’ll start with the little things, like utensils, and save the big things for last.”
  • Explaining how you decide how much water to put in your sink or basin: “Let’s fill it right up to that line—that’ll be enough for the pans to get soaked.”
  • Deciding how much dish soap to use: “If we use too much, the bubbles will get all over and we won’t have enough left for tomorrow.”
  • Putting dishes away: “The short spoons go in this tray, and the long spoons go in that one.”

Give your Dishes a Spin! – Spatial Relationships

It’s also true that most of the dishwashing task is consumed with moving objects around. To do that effectively, we all make judgements about whether there is “enough room” and we rotate and flip objects to clean and organize them. This kind of spatial thinking not only connects to geometric thinking but has been shown to connect to many different kinds of math tasks.

Singing makes chores a lot more fun! Use a familiar tune and sing as you work. Here’s a song in Gujarati.

Here are some of the opportunities you might see:

  • Bringing all the dishes from the table to the counter next to the sink: “Let’s stack the plates on top of each other here, right next to the bowl.”
  • Loading the dishwasher: “Put the forks with the tines up, like this, so they have a better chance to get clean.”
  • Hand-washing the cups: “It’s good for you to get the cloth all the way down inside the cup to the bottom to get all the milk out.”
  • Putting dishes away: “The pot lids go in the drawer under the microwave.”

Setting up for success

Just a reminder to be patient the first time you try involving your kids in cleaning up the kitchen. Start small and give little ones ample time. It might take you 10 minutes to complete the task on your own and 20 minutes or longer with your children but trust that process will get easier. Plan on sticking around to share the work – children will need your support and your supervision. When you layer in math language about size and spatial relationships, then your dish-washing does double duty. Clean dishes AND math learning: a win-win situation!

When you layer in math language about size and spatial relationships, then your dish-washing does double duty. Clean dishes AND math learning: a win-win situation!

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We’re Going on a Math Walk, Going to Find Some Math Talk https://earlymath.erikson.edu/were-going-on-a-math-walk-were-going-to-find-some-math-talk/ https://earlymath.erikson.edu/were-going-on-a-math-walk-were-going-to-find-some-math-talk/#respond Sun, 26 Apr 2020 18:49:42 +0000 https://earlymath.erikson.edu/?p=16831 Going for walks is an excellent time to talk about math with your child. You’ll be surprised how much math talk you can have when you look for the math in your very own neighborhood.

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Going for walks is an excellent time to talk about math with your child. You’ll be surprised how much math talk you can have when you look for the math in your very own neighborhood.

Spark your child’s curiosity by noticing numbers, shapes, and sizes of things when you’re out and about. On one day you can focus on counting—everything from the number of stairs to go down, how long to wait for the green light, or the dogs out for a walk. Another time, look at the lines on the buildings and talk about straight lines and curved lines. On another day you can use your fingers to keep track of how many circles or squares you see.

This is not a test for children. A math walk should be fun and enjoyable for you and your child. The moment it stops being fun, it’s time to stop. But, you’re likely to find your child will be pleased with your attention and will start making suggestions to you of what to count and talk about next.

Every neighborhood has interesting math all around. To give you some ideas, we’ve included some pictures from a neighborhood in Chicago.

Counting

  • “How many stairs are there to get from the sidewalk up to our front door? 1-2-3-4. We went up 4 steps. Let’s go back down now: 4-3-2-1.”
  • How many cars will we see as we walk around the block?”
  • Yesterday we counted red cars. Let’s count emergency vehicles today….Today we saw cars and bicycles, but no emergency vehicles.”
  • “Let’s use our fingers to keep track of how many tall dogs and how many short dogs we see.” (you can change what you count, depending on your child’s interests)
  • “I spy a ring around the tree. Let’s step on each stone and count them as we go around.”
math talk walk

Number Sense

  • For a preschooler, you may want to stop counting stairs when there are only 2 more stairs left and ask, “How many more do we need to reach the top?”
  • Would there be the same number of stairs going down? How could you know?”
  • “Do you think they have the same number of stairs at the white house? the red brick house?”

Shape

  • “I spy bricks that have a rectangle shape. They all have straight lines and they have 4 sides. Can you find other rectangles?”
  • “Besides windows and doors, can you find other things that have 4 sides?” (bricks, gate, sidewalk sections, shutters)
  • “You found many shapes with straight lines. Can we find any curved lines?” (decorative scrolling in hand railing, awnings, lamppost)
  • A child might say that the roof of the house that is brick with white trim is a triangle, but you would say “It looks like a triangle, but it’s missing one side. Triangles have 3 sides.”
math talk walk bricks

Spatial Relationships

  • “The snow is on the grass, on top of the bush, but not on the stairs.”
  • “Look at that squirrel jump between the trees! Where is it going next?”
  • “Let’s step over this small puddle, but go around that large puddle.”
  • “Let’s walk backwards. What other silly ways can we walk?”

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Let’s Wash that Math Right Into Laundry Day https://earlymath.erikson.edu/math-time-lets-wash-math-right-laundry-day/ https://earlymath.erikson.edu/math-time-lets-wash-math-right-laundry-day/#respond Sat, 25 Apr 2020 19:55:03 +0000 https://earlymath.erikson.edu/?p=16774 Whether you have a machine at home or visit a laundromat, there is all kinds of math involved in doing laundry. And there are all kinds of ways that children of all ages can join in the thinking and doing.

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Whether you have a machine at home or visit a laundromat, there is all kinds of math involved in doing laundry. And there are all kinds of ways that children of all ages can join in the thinking and doing. The washing and folding will take a little longer when you involve your “little helper” but those extra minutes of math time together are worth it!

Sort It Out

Everyone has their own rules about sorting clothes before and after washing. They all call for logical thinking that even toddlers can understand. Here’s a few ideas about turning often what feels like a chore into math conversations and games.

Clothes Toss: Put out one basket for light colors and a second one for dark colors. Let everyone take turns tossing an item from the pile of dirty clothes into the correct basket. With toddlers, cheer and give a shout out that names the rule: YAY you got your light blue shirt into the Lights basket! You might do this as a kind of bean bag game—you get 2 points for tossing an item into the correct basket; only one point if it doesn’t quite make into the basket.

math time laundry time sorting colors

Matching

Two-year-old children will enjoy finding and creating a pile of their own clothes. By the age of 3 many children are able to match socks — for themselves and others. Chat about the math, noticing and asking them about how they make a match. Older children can sort in more complex ways such as pajamas and school clothes or even type of clothing and family member who wears it.

Fold and Pile

Children ages 3 and up often enjoy folding flat items such as towels — and can get quite good at it. Lining up the corners lets children explore symmetry. As children get older, the conversations can get more elaborate. This video shows a great math conversation about shapes, colors, and fractions.

Predicting

Before sorting, encourage children to make and explain predictions about which basket is likely to end with the most items. Then talk about the results, making it math time without them even noticing.

Making Comparisons

As children get older, there are plenty of ways to keep score as tallies or graphs. This kind of keeping track allows you to chat about how your laundry stacks up.

  • Tally how many items get tossed into each basket or keep a running count of how many are socks, shirts, etc. or who has the most clothes that need washing (or folding). There are all kinds of interesting measurement comparisons possible: Which type of clothing has the most? Which piles folded laundry are biggest (tallest)? Why might the pile of jeans be bigger/taller than the pile of shirts even though the number of shirts is greater than the number of jeans?
  • Keep the record and use it to predict and compare to what happens the next laundry day. First-and second-graders might want to turn some of the comparisons into graphs.

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DIY Puzzles Develop Spatial Thinking https://earlymath.erikson.edu/diy-puzzles-made-from-cardboard-can-develop-spatial-thinking/ https://earlymath.erikson.edu/diy-puzzles-made-from-cardboard-can-develop-spatial-thinking/#respond Fri, 24 Apr 2020 18:47:06 +0000 https://earlymath.erikson.edu/?p=16791 Jigsaw puzzles are a great way for children to develop their spatial thinking and problem-solving skills. Children enjoy doing all kinds of puzzles and making ones from materials found around the house keeps it fresh and new.

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Jigsaw puzzles are a great way for children to develop their spatial thinking and problem-solving skills. Children enjoy doing all kinds of puzzles and making ones from materials found around the house keeps it fresh and new. Do It Yourself puzzles (or DIY puzzles) allow you to adapt and create them to address your child’s personal skill level and age. For a child new to doing puzzles or a young child, you might make a puzzle with just 5 or 6 pieces, but for a more experienced child, you might make a puzzle with 12 pieces or more.

You can make many different puzzles by using any type of cardboard box you have around the house and your own picture (a photo, poster, magazine page, or your child’s drawing). You can even use the design of the box itself such as puzzles made from the cardboard of your favorite cereal box. Here is what you’ll need and the step by step directions.

Materials needed:

  • Cereal box (or other cardboard boxes you have around the house)
  • Pencil
  • Scissors

Optional material:

  • Photo, poster, magazine page, or drawing
  • Glue
  • Dark marker

Directions:

  1. Choose either the image on the box or to use your own picture.
    If you are using the image on the box itself, skip to Step 2.
    If you are using your own picture, glue the image onto your cardboard and let it dry completely, at least two hours.
  2. Cut along the edges of your picture to make the border of your puzzle.
  3. Flip the cardboard over and use a pencil to mark the lines of your puzzle pieces on the box. If necessary, trace the lines with a black or dark marker.
  4. With scissors, cut along the lines.
  5. Now you have your puzzle! Put the pieces in a plastic bag to make sure they stay together.

There are many ways to build conversation related to spatial relationships and shape into doing DIY puzzles. You can cue your child to do spatial thinking and look at the shape of the puzzle piece to think about where it might belong. Give hints that still let children figure it out for themselves, such as:

  • “The shape of that piece makes me think it goes next to/below the one at the top.”
  • “Let’s turn/flip that piece around to see if it fits near the bottom.”

Some questions you might ask include:

  • “By looking at that piece, where do you think it might fit?”
  • “Which way do you have to turn/flip that piece so it will fit?”
  • “What makes you think that?”

As children get more experience, they can make their own puzzle or a puzzle for you. Have fun!

diy puzzles spatial thinking cutting

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At-Home Activity Cards: Counting https://earlymath.erikson.edu/take-home-activity-cards-counting/ https://earlymath.erikson.edu/take-home-activity-cards-counting/#respond Tue, 24 Mar 2020 13:16:02 +0000 https://earlymath.erikson.edu/?p=16144 Activities related to counting books can spark a lot of conversation and creativity. These English and Spanish-language activity cards are great for printing and using at home and school.

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Count

This book by Denise Fleming illustrates numbers one through ten and then counts on by tens. Each number is illustrated with an uncommon animal, from gnus to kangaroos. Creative activities at home related to it can spark a lot of conversation and fun. This activity card is great for printing (English and Spanish versions) to demonstrate how it can be used by families at home.

Download

Count and See

In this book by Tana Hoban, there are things to count all around you if you look, think, and see. Exciting photographs — reinforced by the number as word, as numeral, and as model set — show you how to count what you see, and make what you see count. This printable activity card creates opportunities for rich discussion around the question, “How Many?”

Download

Ten Black Dots

This activity card paired with the book by Donald Crews is in both English and Spanish. On each page the artist adds one more dot and thinks of new designs to make with the growing number. It’s an effective springboard for mathematical art projects for parents, libraries, and other home uses.

Download

Anno’s Counting Book

This book by Mitsumasa Anno takes the reader on a journey through the four seasons with numbers one to twelve. This activity counting card can accompany the journey at home to help bring out important counting and cardinality concepts.

Download

Count Me In / Cuenta Conmigo

By Cynthia Weill, this bilingual book combines number and Oaxacan clay art to provide creative activities at home. This downloadable activity card can help bring math learning and its cultural connections to families at home.

Download

Counting Cards creative activities at home

Mouse Count

This book by Ellen Stoll Walsh tells the story of a very hungry snake. This downloadable activity counting card provides a great jumping off spot for discussions at home. You can find a classroom example of an activity on our website: Mouse Collections.

Download

 

Fish Eyes

This storybook by Lois Ehlert follows a fish on a journey as it meets new groups of fish. This at-home activity card builds on math concept of “one more” that is embedded in the story.

Download

creative activities at home for counting cards

1, 2, 3 to the Zoo: A Counting Book

By Eric Carle, this book follows a tiny mouse’s train ride to the zoo. This activity card provides opportunities to take the learning that can happen with the book to at-home settings.

Download

Download All

Here you can download a PDF that includes all of the at-home activity COUNTING cards.

Download

Get downloadsJoin the website

Want free printable activities and card sets? Join our website (or log in if already joined) to unlock much more.

Join or Log in

 

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At-Home Activity Cards: Shapes https://earlymath.erikson.edu/take-home-activity-cards-shapes-preschool-shape-shapes-preschool-activities/ https://earlymath.erikson.edu/take-home-activity-cards-shapes-preschool-shape-shapes-preschool-activities/#comments Mon, 23 Mar 2020 14:46:50 +0000 https://earlymath.erikson.edu/?p=16156 Books can illustrate kindergarten and preschool shape concepts while introducing foundational Big Ideas of math. These activity cards can help take the learning to the home environment where parents and caregivers can spark such discussion.

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Books can illustrate kindergarten and preschool shape concepts while introducing foundational Big Ideas of math. These activity cards can help take the learning to the home environment where parents and caregivers can spark such discussion.

preschool shapes, preschool activities, Take Home Activity Cards Shapes

Round as a Tortilla

This book by Roseanne Thong beautifully illustrates various aspects of Latino culture while introducing some mathematical and geometrical thinking. This activity card can help parents and caregivers find shapes in their own contexts.

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preschool shape, shapes preschool activities, shapes for preschool

Shapes Shapes Shapes

This photo book by Tana Hoban compels readers to engage with the photos, to ask questions and discuss attributes of shapes found in everyday life. This at-home activity card can help initiate such kindergarten and preschool shape talk.

Download

Color Farm activities, Take Home Activity Cards

Color Farm

This book by Lois Ehlert introduces the reader to farm animals with a focus on shapes such as octagon, hexagon and square. This activity card in both English and Spanish can be a great jumping off spot for parents and caregivers to explore shape composition–that is, how smaller shapes can form larger shapes.

Download

preschool shapes, preschool activities, shapes for preschool

Color Zoo

This other book by Lois Ehlert uses zoo animals, rather than farm animals, and uses the turning of the pages to introduce new shapes (and animals) in a fun and engaging way. This activity card can help parents and others to explore kindergarten and preschool shape concepts in fun ways at home.

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Cubes Cones Cylinders Spheres Take Home Activity Cards

Cubes, Cones, Cylinders, & Spheres

Another one from Tana Hoban, this book offers photographs of shapes in a real-world environment. This activity card provides an opening to delve into finding 3-dimensional shapes in a home math environment.

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Here you can download a PDF that includes all of the SHAPES activity cards.

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At-Home Activity Cards: Sets and Sorting https://earlymath.erikson.edu/take-home-activity-cards-sets-sorting/ https://earlymath.erikson.edu/take-home-activity-cards-sets-sorting/#respond Sun, 22 Mar 2020 13:42:35 +0000 https://earlymath.erikson.edu/?p=16146 These printable activity cards in both English and Spanish invite home explorations with sets and sorting using the books and activities related to them. The books can encourage conversations about attributes while also inspiring questions about what is happening from page to page.

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sets and sorting fun activities at home

Five Creatures

This book by Emily Jenkins includes three humans and two cats in one house. That allows a lot of opportunities for sorting! This printable activity card in both English and Spanish invites home explorations with sets and sorting using the book. You can watch and read further about an example activity here in our Ideas at Work series: All Sorts of Families: Connecting School and Home with Math.

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sets and sorting book pair of socks fun activities at home

A Pair of Socks

By Stuart J. Murphy, this book uses illustrations and language to highlight qualities a sock could have. This downloadable activity card provides fun activities at home while developing children’s recognition of attributes as well as deepen their matching abilities when dealing with sets and sorting of objects.

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sets and sorting fun activities at home

Exactly the Opposite

This book by Tana Hoban contains photos that inspire conversation, showing two items that are opposites in some interesting way. This printable Spanish and English activity card can accompany parents for opportunities to engage children at home.

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sets and sorting Is it Red

Is it Red? Is it Yellow? Is it Blue?

This wordless book encourages conversations about an engaging attribute, color, while also inspiring questions about what is happening from page to page. This printable activity card can take the discussion to the at-home setting.

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fun activities at home sets and sorting books opposites

Opuestos/Opposites

By Cynthia Weill, this book is in Spanish and English and is great for discussions around sets and sorting. Explorations of different cultures intermixed with important early math concepts can still provide fun activities at home. This activity card can go home with parents to further the math thinking in more settings beyond the school or library.

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Here you can download a PDF that includes all of the SETS activity cards.

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Creatures at Home Activity https://earlymath.erikson.edu/creatures-at-home-activity-childrens-books-in-spanish-and-english/ https://earlymath.erikson.edu/creatures-at-home-activity-childrens-books-in-spanish-and-english/#respond Tue, 17 Mar 2020 13:55:03 +0000 https://earlymath.erikson.edu/?p=15576 In both English and Spanish, the directions here outline how to do an activity at home that can allow parents to involve themselves with the learning taking place in classrooms, all tied to the book Five Creatures.

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children's books in spanish english

Creatures at Home Directions

In both English and Spanish, the directions here outline how to do an activity at home that can allow families to share in the learning about sets and sorting taking place in classrooms, all tied to the book Five Creatures and aligned with our Big Ideas of math. Children’s books in Spanish or English can be creative springboards for mathematical discovery in the home setting.

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children's books in spanish english

Creatures Who Live in Our Homes Class Book

In both English and Spanish, these two book page templates go together to guide parents as they create sorts based on similarities and differences of the members of the family.

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children's books in spanish english

Creatures at Home Video

In this example video, using children’s books in Spanish or English like the book Five Creatures can act as a springboard for discussing sets and sorting at home.

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How Children Learn about Numbers: A Conversation with Kelly Mix https://earlymath.erikson.edu/how-children-learn-about-numbers-a-conversation-with-kelly-mix/ https://earlymath.erikson.edu/how-children-learn-about-numbers-a-conversation-with-kelly-mix/#respond Wed, 19 Feb 2020 17:10:03 +0000 https://earlymath.erikson.edu/?p=16333 In her presentation “Cognition and Early Childhood Numeracy: How Number Concepts are Built and Why Input Matters,” Kelly Mix bridged research and practice in her discussion of math language and learning.

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Early Mathematics Learning in Family and Community Contexts was the 2019 topic of the Promising Math biennial conference. Framing the event were three plenary presentations that brought cognitive, contextual, and collaborative lenses to the topic. In her presentation “Cognition and Early Childhood Numeracy: How Number Concepts are Built and Why Input Matters” (presentation PDF), Kelly Mix bridged research and practice in her discussion of math language and learning.

We talked with Mix in order to understand better the connections between her research and how children learn mathematics.

Kelly, in your talk you identified four key cognitive processes that help young children learn about math: statistical learning, structure mapping, language, and spatial cognition. I have a question about statistical learning.

The way I understand it, statistical learning is about an ability we all have to detect similarities. That is, in the case of number learning, we eventually come to associate the word “three” with “three-ness” because the word and the cardinal amount are frequently paired. When they are paired again and again, we notice the pairing, and distinguish it from random noise. What do you think this suggests about the kinds of family interactions that will help develop numeracy?

Good question! Yes, you have it right – frequency is a big part of this learning mechanism. Children need lots of experiences that help them pair quantities and number words. In statistical learning terminology, we like to talk about dense co-occurrences and regularity. Dense co-occurrences means how often and how close together in time experiences are paired. So, if a parent labels a set of 3 items such as three tea cups repeatedly in the same play session, that would be an example of dense co-occurrences. If they label a set of 3 on Tuesday and then do it again on Friday, that would provide co-occurrences, but they would not be dense. Regularity means that the words and experiences are structured in a clear, predictable way. For example, if parents count and label sets every time they comment on number (e.g. “1, 2, 3…3 tea cups”), that language structure becomes regular and predictable, and children can readily see how it applies to lots of different situations.

There are many ways parents and caregivers can help by talking about quantities when there is some way for kids to simultaneously see or experience those words and quantities. For example, adults can mention that we need four forks to set the table and involve kids in getting four forks out of the drawer, or saying “just three more pushes on the swing” and then counting out “one, two, three” – these are both the kinds of everyday interactions that activate statistical learning and help children discover the meaning of the number words. Because of how the statistical learning process works, finding these opportunities in your daily interactions with children and presenting them with frequency and regularity can make a big difference.


Kelly Mix Promising_Math 2019 2
Kelly Mix Promising_Math 2019 1

Another of the processes you discussed, structure mapping, suggests that when we compare things, we tend to see the similarities and differences between them in greater detail, and the similarities become generalized. For example, when we compare a car and a motorcycle, we might think about four wheels versus two wheels, but we might also think about the idea of transportation. Can you expand a bit on how this might be relevant to the way a child learns number outside of school?

The thing about number is it’s fairly difficult to “see.” Think about trying to explain to a visitor from space what we mean by “two.” You might point to two mittens, two cookies, and two trees, saying “these are all two.” This is a good approach, but there is so much detail and information in each of these kinds of objects, that it’s hard to focus on the quantity. Partly that’s because the “two-ness” is held by the sets of things, rather than by the things themselves; each mitten by itself is not “two.”

Because of this difficulty, comparison processes may be particularly important for number learning. For example, children can get a sense of “two-ness” by comparing object sets. A good way to “see” number by comparing is to demonstrate one-to-one correspondence. For example, when children see three dogs and three dog dishes, they can match the objects one-to-one and see that the sets are equivalent. Another way is to hear the sets labeled with a number word, and then look to see what attribute the label refers to. If multiple sets are called, “two,” they must have something in common—what is it? One of the things we know about structure mapping is that it works best when the two structures are alignable. A set of dogs and dog dishes is easier to align than a set of airplanes and a set of whales in the ocean, or a set of claps and a set of glasses on the table. So, adults can support structure mapping by pointing out numerical equivalence in sets that are easily aligned, positioning the items in space to show the correspondence (like, lining up the dogs and dog dishes), and using gestures to point back and forth between the items in each set.

Just like statistical learning, this process of structure mapping highlights the importance of adult input for children’s learning. In this case, it’s because adults can use words or actions to prompt comparisons, and these comparisons help children abstract the relevant numerical structures: the more transparent and alignable the structures we present, the more likely children are to make these comparisons and figure out what the number words mean!

Number names, counting, and written numerals are all different symbols for quantities that children must link to each other and to the underlying meaning. Both statistical learning and structure mapping support this grounding process by helping children notice that words and written symbols go together, or that the same quantities are called “three,” are counted, “1-2-3,” and can be labeled by the same written numeral.

In your talk, you suggested it is helpful to think of developing early numeracy as a “symbol grounding” task. Can you clarify what this means and how it connects to children’s interactions with adults?

The central idea of symbol grounding as it applies to early numeracy is that each child must make connections between the symbols we use to talk about quantity and the underlying meaning of these symbols. That is, the word “five” is itself a symbol, and children must learn to map it to examples of five in the world or to the abstract idea of “fiveness”. An easy way to think about symbol grounding is to think about how children learn other words, like words for objects. The word, “tree,” for example, is an arbitrary symbol that has no meaning until it’s linked to examples of trees in the world or to the abstract concept of trees. Symbol grounding is what happens when people make these connections between symbol and meaning. To learn to think mathematically, then, the child is exposed to lots of math-related symbols, and must connect each one to its meaning. When a symbol is “grounded” to its meaning in the thinking of a child, rather than floating freely in an ocean of symbols children have yet to understand, it can be used meaningfully by the child.

It’s for this reason that adult input is so important. Number names, counting, and written numerals are all different symbols for quantities that children must link to each other and to the underlying meaning. Both statistical learning and structure mapping support this grounding process by helping children notice that words and written symbols go together, or that the same quantities are called “three,” are counted, “1-2-3,” and can be labeled by the same written numeral. In the very early years, we can almost think of “symbol grounding” as “language grounding” instead, since language is the child’s most powerful symbol system, both for learning and for being understood. Children are busily grounding words they hear in their environment all day long.

In terms of advice for parents and caregivers, we can boil this down to: talk with your child about small numbers! Whether it’s conversation about the number of fingers on each hand, the number of school days before the weekend, or the number that is one more than four, give your child abundant opportunities to connect the symbols for quantity to their meanings.

Two Ways That Language Scaffolds Understanding of Cardinality

At Promising Math 2019, Kelly Mix gave a talk about how language promotes the development of numerical cognition in young children. Dr. Mix is the Chair of the Human Development and Quantitative Methodologies (HDQM) department at the University of Maryland.

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Two Ways That Language Scaffolds Understanding of Cardinality https://earlymath.erikson.edu/two-ways-that-language-scaffolds-understanding-of-cardinality/ https://earlymath.erikson.edu/two-ways-that-language-scaffolds-understanding-of-cardinality/#respond Tue, 18 Feb 2020 17:17:53 +0000 https://earlymath.erikson.edu/?p=16271 At Promising Math 2019, Kelly Mix gave a talk about how language promotes the development of numerical cognition in young children. Dr. Mix is the Chair of the Human Development and Quantitative Methodologies department at the University of Maryland.

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At Promising Math 2019, Kelly Mix gave a talk about how language promotes the development of numerical cognition in young children. Dr. Mix is the Chair of the Human Development and Quantitative Methodologies (HDQM) department at the University of Maryland.

Mix suggests that adults use language to label sets (e.g., “two shoes”), count sets (“1, 2 shoes”), and match sets with one-to-one correspondence (“Look, 2 shoes and 2 socks”). Parents and caregivers help children focus on the meaning of the number words when language like this is used during everyday experiences along with familiar objects that children can see and touch.

How Children Learn about Numbers: A Conversation with Kelly Mix

We talked with Dr. Mix in order to understand better the connections between her research and how children learn mathematics.

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Launch Problem Solving with Understanding Using Three Reads https://earlymath.erikson.edu/problem-solving-three-reads/ https://earlymath.erikson.edu/problem-solving-three-reads/#respond Tue, 28 Jan 2020 15:03:21 +0000 https://earlymath.erikson.edu/?p=16240 In this lesson launch, we see a third-grade teacher using reading comprehension strategies to help children understand a math story problem using a Three Reads strategy.

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In this lesson launch, we see a third-grade teacher using reading comprehension strategies to help children understand a math story problem. Using a Three Reads strategy helps children understand what is happening in the story and reason about the numerical relationships before making any calculations. By doing so, these children are developing the Common Core Standard for Mathematical Practice, making sense of a problem and persevering to solve it.

Learn more on our website at Exploring the Three Reads Math Protocol for Word Problems. Or check out another resource at SFUSD Mathematics Department, The 3-Read Protocol.

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MathTalk Leaders Illustrate How Efforts Can Transform Neighborhoods https://earlymath.erikson.edu/mathtalk-leaders-illustrate-how-efforts-can-transform-neighborhoods/ https://earlymath.erikson.edu/mathtalk-leaders-illustrate-how-efforts-can-transform-neighborhoods/#respond Wed, 22 Jan 2020 17:27:30 +0000 https://earlymath.erikson.edu/?p=16178 Early Mathematics Learning in Family and Community Contexts was the 2019 topic of the Promising Math biennial conference. Framing the  More →

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Early Mathematics Learning in Family and Community Contexts was the 2019 topic of the Promising Math biennial conference. Framing the event were three plenary presentations that brought cognitive, contextual, and collaborative lenses to the topic. In the first presentation “Aha, See Math Everywhere,” Omo Moses and Keith Griffin of Cambridge, Massachusetts shared their grass-roots efforts, called MathTalk, to engage young children in mathematics in their own neighborhood.

Both Cambridge natives, Moses is the founder of MathTalk, a community-based educational technology lab, where Griffin works as a Family Engagement Leader focusing on product development. Griffin has raised six children in the neighborhood. Community connections are a big part of their story.

Moses started with some personal context, sharing that “growing up, math wasn’t my favorite thing to do, and math time with my dad wasn’t fun time.” It was after becoming a dad himself that he became interested in how he could do math with his young son. “I knew I wanted our relationship to be about learning, doing the things that we enjoy and we love doing. And so for me, math time happened when we were making breakfast…Math happened in the playground and math happened when we rode trains.” MathTalk grew out of his wish to help other families have fun with math in their everyday lives.


I knew I wanted our relationship to be about learning, doing the things that we enjoy and we love doing.

[youtube https://www.youtube.com/watch?v=5g2j46fVm-8&showinfo=0&rel=0]

About MathTalk

Griffin became involved with MathTalk when he brought his young daughter to a weekend basketball program that Moses had created and was running. MathTalk flyers led to a conversation, where he began to see what the vision was. He exclaimed, “Man, I just got to be a part of this.”

As a long-time Cambridge resident and involved parent, Griffin was in a good position to help MathTalk reach out to families in the community. “Essentially what we did was think about…what are the locations, who are the people in the neighborhood, and how can we engage them in thinking about activating those spaces for early math learning?… So, we’re going to the bodega or the Puerto Rican restaurant and we’re having a conversation, saying ‘We want to begin to use your space to help kids learn math’.”

Both men emphasized how much of the process was truly collaborative – that is, community members were involved from the very beginning of the work. Moses reached out to folks at the local school district and got middle and high school students involved in the co-design process. People from housing and neighborhood organizations were also involved, and MathTalk managed to secure space at MIT to hold a “hack-a-thon.”

[youtube https://www.youtube.com/watch?v=CA21sqYPqM0&showinfo=0&rel=0]


I grew up in close proximity to all this innovative stuff that’s happening that has global impact… but kids either don’t see it or don’t feel like they can access it.

“So what you see (in the video) are different images or stations… ideas of what we might want to do in a barbershop, what we might want to do at an eatery, things that we can install in a playground.” Essentially, the hack-a-thon participants took “ideas that got generated at the community level” and built on and extended them.

For Griffin, working on the ideas at MIT with other community members was “a wonderful experience.” He said it was powerful to be involved in the development of an idea, such as a children’s menu for a restaurant that has math ideas built in, and then have it later be actually installed and used in a public space. Part of the excitement of the opportunity came from finally being able to access MIT, an amazing local resource that had not previously been available to him. Moses commented that he “grew up in close proximity to all this innovative stuff that’s happening that has global impact… but kids either don’t see it or don’t feel like they can access it.” He hoped this kind of work would “help kids and parents at an early age have the types of experiences that will inspire them to access all these opportunities that they’re literally surrounded by.”

Last June, MathTalk “went live” in Cambridge, meaning multiple physical installations were put up throughout the neighborhood. A foot pattern was installed in the sidewalk on a corner at one of the housing projects, an Abacus was built on a playground, and a number line was set up in a city park. An app was created that pairs with all those locations and provides prompts, way-finding experiences, and even augmented reality experiences that give people ways to interact with the math. View it in action.

Griffin is clear that the work has made a difference. “When you see the kids smiling, that’s my gift. Seeing the people actually engaging and learning from our product.”

Promising Math 2019 welcomed over 100 attendees from across the United States. Participants included community organizers, math educators, early childhood specialists, researchers, parent leaders, policy makers, and librarians, to name a few. They offered presentations on a wide range of topics, including practices for engaging families, research on how contextual factors influence the early math learning of English learners, discussions of what “public math” is, and demonstrations of games and activities families can use to do math together in fun ways. The conference is funded by Heising-Simons Foundation.

Co-Design Process for Math in the Neighborhood with Omo Moses

Hear Omo Moses and Keith Griffin further discuss MathTalk and the co-design process of their neighborhood installations.

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Co-Design Process for Math in the Neighborhood with Omo Moses https://earlymath.erikson.edu/co-design-process-for-math-in-the-neighborhood-with-omo-moses/ https://earlymath.erikson.edu/co-design-process-for-math-in-the-neighborhood-with-omo-moses/#respond Tue, 21 Jan 2020 17:26:48 +0000 https://earlymath.erikson.edu/?p=16217 Omo Moses, founder of MathTalk, and his colleague, Keith Griffin discuss how they connected over basketball and went on to find local, culturally-relevant ways to engage families in math learning in their own neighborhood.

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Omo Moses, founder of MathTalk, and his colleague Keith Griffin discuss how they connected over basketball and went on to find local, culturally-relevant ways to engage families in math learning in their own neighborhood.

It was easy for me to transition to math from basketball because math is in basketball as well.

Their co-design process involves diverse stakeholders from the community, resulting in public installations that reflect the experiences and interests of families who live nearby.

MathTalk Leaders Illustrate How Efforts Can Transform Neighborhoods

Read more about Omo Moses, Keith Griffin, and MathTalk’s efforts to mathematize the neighborhood.

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8 Counting Books For School and Home That Tackle Big Ideas https://earlymath.erikson.edu/8-counting-books-for-school-and-home-that-tackle-big-ideas/ https://earlymath.erikson.edu/8-counting-books-for-school-and-home-that-tackle-big-ideas/#respond Wed, 08 Jan 2020 17:14:17 +0000 https://earlymath.erikson.edu/?p=16163 To encourage and develop children’s counting knowledge, here are eight books (with an accompanying download) that are common to preschool classrooms.

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Answering the question, “How many?” is central in children’s everyday lives. Seeking to make sense of the world, children count anything and everything as they make order out of chaos. Counting has rules that children learn through many, many experiences when they engage in counting. To count accurately, children say the number names in order, touch objects as they count, and name the set by its total quantity. Books are effective springboards that encourage kids to want to count.

To encourage and develop children’s counting knowledge, here are eight books that are common to preschool classrooms. You can encourage a home-to-school connection by utilizing math books for kids that you possibly already have in your collection. Create a check out system for these books and insert these easily downloadable tasks written in both English and Spanish. Using the books and tasks both at school and at home will create multiple opportunities for the children to engage in the Big Ideas of counting.

  1. Count! by Denise Fleming

    This wonderful book illustrates numbers one through 10 and then counts on by tens. Each number is illustrated with an uncommon animal from gnus to kangaroos. It can spark a lot of conversation and creativity!

  2. Anno’s Counting Book by Mitsumasa Anno

    This favorite book of ours takes the reader on a journey through the four seasons of the year with numbers 1 to 12. Numbers are represented by numerals, cubes and many sets of objects in the beautiful scenes depicted in this wordless counting book.

  3. Ten Black Dots by Donald Crews

    In both English and Spanish, this book has simple pictures and text. On each page the artist adds one more dot and thinks of new designs to make with the growing number. It’s an effective springboard for many hours of creativity, combining art and math.

  1. Count and See by Tana Hoban

    This book connects numerals with dot representations and real life photos for each number one to fifteen. After that it jumps to twenty and continues by tens until one hundred. As a wordless book with everyday photographs, it creates opportunities for rich discussion around the question, “How Many?”

counting books tana hoban count and see
  1. Count Me In / ¡Cuenta Conmigo! by Cynthia Weill

    This is a page turner that combines number and Oaxacan clay art to create a bilingual counting book. Counting books like this can be read and discussed simply with just the number words or you can dive in deeper by using the brief text that accompanies each collection.

  2. Mouse Count by Ellen Stoll Walsh

    This common classroom book tells the story of a very hungry snake. He is on the hunt. He finds some mice to eat but something surprising happens. Children love to listen to this story and retell it many times over. Watch one example in action: Mouse Collections.

  3. Fish Eyes by Lois Ehlert

    This book follows one fish on a journey as it meets new groups of fish. It builds the number relationship of one more as our fish narrator finds the total of each new group of fish and then adds itself to the count. For example, “2 jumping fish plus me makes 3.”

  1. 1, 2, 3 to the Zoo: A Counting Book by Eric Carle

    This book by favorite author Eric Carle follows one tiny mouse’s train ride to the zoo. As the journey continues, a new car is added with one more animal on each car, then the one before it. The entire story is told with the colorful pictures that Carle is well-known for. We see each car’s collection, as well as the entire train each time we turn the page.

counting books, 123 to the zoo

Home Activity Cards: Counting

Educators and parents can download a set of activity cards (available in English and Spanish) to go along with these counting books. A great way to support family math opportunities!

Learn more

 

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5 Kids’ Books that Involve Sets and Sorting https://earlymath.erikson.edu/5-of-the-best-kids-books-that-involve-sets-and-sorting/ https://earlymath.erikson.edu/5-of-the-best-kids-books-that-involve-sets-and-sorting/#respond Wed, 11 Dec 2019 15:24:50 +0000 https://earlymath.erikson.edu/?p=16078 Children look for ways to organize and make sense of their world through play at school and home, as well as at clean-up time. Sorting items into groups by specific attributes gives children the opportunity to define sets according to a rule of what does and does not belong together.

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Children look for ways to organize and make sense of their world through play at school and home, as well as at clean-up time. Sorting items into groups by specific attributes gives children the opportunity to define sets according to a rule of what does and does not belong together. Sometimes objects will change classification from day to day. A ball one day may be in the set of things that roll while the next day it might be in the set of toys that goes outside. When cleaning up, children can think about which collection has the most and which has the fewest items. This information becomes useful when choosing how to organize the cleanup and builds the foundation for reasoning about number, shape, and space.

To encourage and develop children’s thinking about sets and sorting, here are five books that are common to preschool and kindergarten classrooms. You can encourage a home-to-school connection by using math books for kids that you possibly already have in your library. Create a check out system for these books and insert these easily downloadable activity cards written in both English and Spanish for families to enjoy together.

Using the books and activity cards both at school and at home will create multiple opportunities for children to engage in the Big Ideas of sets and sorting.

  1. Exactly the Opposite by Tana Hoban

    This photo book contains conversation-inspiring pictures showing two items that are different from each other in some way. Opportunities to observe, compare, and discuss are abundant.

  2. A Pair of Socks by Stuart J. Murphy

    The descriptive language in this book is used to highlight numerous qualities a sock could have. It is a great way to launch activities to develop children’s matching ability and their recognition of different attributes. Explore other ideas involving this book with Matching and Concepts of Same and Different… with Socks!

Exactly the Opposite by Tana Hoban best kids books sets sorting
  1. Opuestos/Opposites by Cynthia Weill

    This Spanish and English book is a springboard for creating sets by using the vocabulary highlighted in the book. The book’s photographs of unique figurines made in Oaxaca, Mexico give the math a rich cultural context and will spark curiosity and wonder.

  2. Five Creatures  by Emily Jenkins

    This book has two cats and three humans in one home. As the story develops, we see how the creatures are alike in some ways and how they are different. Each turn of the page shows us new groupings as they highlight what makes the creatures in this family who they are. Dive deeper by reading about our People Sort activity idea that stems from the book.

  3. Is it Red? Is it Yellow? Is it Blue? by Tana Hoban

    A wordless book with photographs, this book inspires conversation and questions about color as an attribute. It also inspires discussion about what is happening on each page.

Home Activity Cards: Sets and Sorting

Educators and parents can download a set of activity cards (available in English and Spanish) to go along with these books. A great way to support family math opportunities!

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Egg Carton Story with Child 15 https://earlymath.erikson.edu/egg-carton-story-change-unknown-word-problem/ https://earlymath.erikson.edu/egg-carton-story-change-unknown-word-problem/#respond Thu, 05 Dec 2019 13:25:24 +0000 https://earlymath.erikson.edu/?p=16071 The math story this second grader solves is a change unknown story. There was a full carton of a dozen eggs but some were eaten, leaving only 3 eggs. After making sense of the problem, the child uses a counting up strategy to figure it out.

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The math story this second grader solves is a change unknown story. There was a full carton of a dozen eggs but when the interviewer goes to make her breakfast there are only three eggs left. She wants to know how many were eaten before she got to the eggs. First he answers one egg. She retells him the story with more details to connect to prior knowledge about egg cartons. He seems to understand the story better. The next time he counts up from the number he knows using his fingers as tools to arrive at the correct answer.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Teddy Bear Picnic Math Stories with Child 15 https://earlymath.erikson.edu/teddy-bear-picnic-math-stories-with-child-15/ https://earlymath.erikson.edu/teddy-bear-picnic-math-stories-with-child-15/#respond Wed, 04 Dec 2019 14:21:32 +0000 https://earlymath.erikson.edu/?p=16056 A second-grade student solves a variety of addition and subtraction math stories. Using his fingers as math tools, he is able to solve the stories, even as what is known or unknown shifts.

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A second-grade student solves a variety of addition and subtraction math stories. Using his fingers as math tools, he is able to solve the stories, even as what is known or unknown shifts. Solving a range of problem types encourages children to make sense of the problem and develop flexible thinking. Here, we see the child strategically select a method for solving each story based on the given numbers.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Capture It with Child 15 https://earlymath.erikson.edu/capture-it-addition-game/ https://earlymath.erikson.edu/capture-it-addition-game/#respond Tue, 03 Dec 2019 20:21:21 +0000 https://earlymath.erikson.edu/?p=16068 A second-grader works on basic number combinations as we see him playing a comparison card game, Capture It. He is accurate when comparing sums but is still developing his efficiency and flexibility, key indications of fluency.

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A second-grader works on basic number combinations as we see him playing a comparison card game, Capture It. Comparing a number card to his opponent’s card, he can quickly and accurately share who has the higher number. Then the interviewer and child play with two cards each and compare sums. We see that this child is also accurate when comparing sums but is still developing his efficiency and flexibility, key indications of fluency.

For example, at one point both players have a five card. He observes that her cards make eight and his cards add to only five. He does not notice the fives “cancel each other out.” Although in the video we can observe how much he knows about counting, he is still developing the ability to reason about number relationships.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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All Sorts of Families: Connecting School and Home with Math https://earlymath.erikson.edu/all-sorts-of-families-connecting-school-and-home-with-math/ https://earlymath.erikson.edu/all-sorts-of-families-connecting-school-and-home-with-math/#respond Wed, 30 Oct 2019 14:45:56 +0000 https://earlymath.erikson.edu/?p=14463 All members of a household have things in common and things that set them apart. Exploring the important ways that  More →

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All members of a household have things in common and things that set them apart. Exploring the important ways that loved ones go together and what makes them unique involves both mathematical thinking and a lot of fun.

Kick off your getting-to-know-you or family studies with the book Five Creatures by Emily Jenkins. In this story, a little girl describes the five creatures in her house—3 humans and 2 cats—by putting them together in various groups by their physical attributes, likes and dislikes, and talents.

Three with orange hair, and two with gray.

Three who like to drink milk, one who’s allergic, and one who has it only in coffee.

The myriad attributes of family members provide a meaningful context to explore the Big Ideas of Sets, namely:

  • Attributes can be used to sort collections into sets.
  • The same collection can be sorted in different ways.
  • Sets can be compared and ordered.

Teachers can use the book to explore sets and sorting with our classroom activity People Sort.

But why stop there? After exploring in the classroom, continue the learning at home. Ask families to contribute to a Creatures in our Classroom book. This helps build a bridge between math at school and math at home, while welcoming families to share what makes them unique. Finding different sets of people (and animals) is a way to celebrate what makes us special and what makes us similar.

Resources to involve families include this video to send home with how-to instructions to make pages for the class book about their own family. Here are templates for the pages of the book so that each family can illustrate two ways to sort the “creatures” in their home. Both video and print resources are available in English and Spanish.

children's books in spanish english

Creatures at Home Downloads

Find downloadable and printable directions and templates on the Download page Creatures at Home Activity.

Learn more

 

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Why One-to-One Correspondence Matters https://earlymath.erikson.edu/why-one-to-one-correspondence-math-matters/ https://earlymath.erikson.edu/why-one-to-one-correspondence-math-matters/#comments Tue, 29 Oct 2019 15:30:30 +0000 https://earlymath.erikson.edu/?p=15936 Children love to count. Counting helps them make sense of the world and to find out how many of something. With time and practice, children develop an understanding of the “rules” or principles of counting.

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Children love to count. They count everything from the steps they take to get from their bedroom to the kitchen, to how many friends are in school each day. Counting helps them make sense of the world and to find out how many of something. With time and practice, children develop an understanding of the “rules” or principles of counting.

One such principle is known as one-to-one correspondence. It’s the idea that numbers correspond to specific quantities. For example, in playing a game, a child counts 1, 2, 3, 4, 5 dots on the die and jumps 1, 2, 3, 4, 5 spaces on the board because 5 dots correspond in quantity to 5 jumps. The number “five” always corresponds to that precise quantity, no matter what it is you are counting.

A hallmark of accurate counting, then, is when preschoolers begin to assign one number, and only one number, to each object as they count. We see this achievement when a child touches or tags each object in time with saying the counting words. And, this is no small achievement as it requires coordinating motor movement and speech with exact synchrony.

In this video, we can observe a young mathematician who is just beginning to coordinate his oral counting with tagging each block as he counts. After a few attempts, he shows some confusion between what he sees and what he says. The timing isn’t working out yet.


For full Focus on the Child video, click here.

But even when children tag every object one-to-one with a counting word, they may not yet have full understanding of one-to-one correspondence. Understanding the correspondence between a quantity and its number name (and numeral) is more than the action of tagging or keeping track while counting.

Children often first develop a sense of one-to-one correspondence by playing with toys that require matching one object to one space, such as putting plastic eggs in an egg carton or fitting shapes into a shape puzzle. Eventually children can put objects into one-to-one correspondence themselves, such as setting the table with one plate and one napkin for every seat. But children can do this without fully understanding that the corresponding number of plates, napkins, and seats is the same.

It’s important to discuss correspondences that occur naturally, and meaningfully, in the life of young children. When putting on winter gloves, does every finger find an opening? Are there enough glue sticks for everyone at the table? How many garages do we need to park all the toy trucks?

This student has a completely different question to answer. He needs to figure out how to fairly share cookies between two friends.


For a demonstration of this video being used by teachers, click here.

We watch as this student counts out all the cookies, dealing them one at a time, back and forth, between two plates. Though he counts the 10 cookies accurately, the action of dealing cookies is little more than a matching activity. While he says the two groups are the same, he does not fully understand that the equal groups correspond to the same number. So he counts the 5 cookies on one plate, and then needs to count the cookies on the other plate too, rather than knowing that it also has 5.

Deepen young children’s number knowledge through many experiences and conversations about how one-to-one matching creates equal groups—if you know the number in one of the groups, then you know the number in the other. It will take time. Understanding one-to-one correspondence will deepen children’s number sense and serve them well for counting and beyond.

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Cardinality is Critical Preschool Concept with Barbara Sarnecka https://earlymath.erikson.edu/cardinality-critical-preschool-concept-with-barbara-sarnecka/ https://earlymath.erikson.edu/cardinality-critical-preschool-concept-with-barbara-sarnecka/#respond Mon, 21 Oct 2019 17:39:08 +0000 https://earlymath.erikson.edu/?p=14536 After years of studying 3- and 4-year old children of diverse linguistic and cultural backgrounds, Barbara Sarnecka has zeroed in on the importance of cardinality.

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After years of studying 3- and 4-year old children of diverse linguistic and cultural backgrounds, Barbara Sarnecka, a professor of Cognitive Sciences at the University of California-Irvine, has zeroed in on the importance of cardinality as a critical preschool concept. This key understanding is important for children’s learning and seems to be a prerequisite for accurate assessment of their number knowledge.

Once they understand cardinality, we see big improvements on a whole range of conceptual number tasks that don’t even have to do with counting.

Sarnecka recommends a focus on spoken number words along with many opportunities to practice counting and practice subitizing in order for young children to learn about cardinality in preschool.

The Early Math Collaborative, supported by the CME Group Foundation and the Heising-Simons Foundation, held their first biennial conference in 2017 called Promising Math. A conference linking research and practice, the event brought experts from around the United States to discuss the understanding, teaching, and learning of mathematics for children between birth and eight years. The focus for 2017: Fostering access to mathematics for children who are English Language Learners (ELLs).

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Counting Collections Builds Strong Counting Knowledge https://earlymath.erikson.edu/counting-collections-builds-strong-counting-knowledge/ https://earlymath.erikson.edu/counting-collections-builds-strong-counting-knowledge/#respond Wed, 16 Oct 2019 13:43:37 +0000 https://earlymath.erikson.edu/?p=15872 There are all kinds of things to count in pre-k to second grade classrooms. Counting Collections is an activity that develops the Big Ideas of number sense and counting, such as cardinality, one-to-one correspondence, and unitizing.

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There are all kinds of things to count in pre-k to second grade classrooms, including at several Chicago Archdiocese schools where we work regularly with teachers. Students have many opportunities to count collections of real objects—stickers, bottle caps, glue sticks, glass beads, acorns, LEGOs… you name it. After students determine how many in the collection, they work to record their counts with pictures and symbols.

Counting Collections develops the Big Ideas of number sense and counting, such as cardinality, one-to-one correspondence, and unitizing. This routine activity is based on research from UCLA and the resulting book by Megan Franke, Elham Kazemi, and Angela Chan Turrou, Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom. These teachers endeavor in this work through our partnership with Big Shoulders Fund, which supports many Chicago-area Catholic schools through professional development.

Counting Collections has an easy to follow yet effective structure which includes the following four components: prepare, launch, work, and wrap up.

Prepare

In preparing the collections, teachers gather sets of objects in quantities that reflect students’ number sense and counting knowledge. This will vary by grade level and student experience. Keeping collections organized is important. Collections should be bagged and labeled, and teachers keep a log of each bag and its contents. This log helps teachers stay organized and is a handy reference to confirm students’ findings once they count the collections. It may look like this.

Bag Item Amount
A cubes 40
B crayons 12
C pom poms 22

Launch

To launch, use interactive modeling to prepare students to work with their own collections. This means teachers and students model the steps involved in counting collections, while other students notice and describe those steps. Over time teachers highlight different aspects of the procedure, always making the math goal or social goal explicit so that students are ready to go to work on their own.

Work

During work time, students might work in pairs or individually depending on the teacher’s goals for that day, which might be assessment or it might be for students to support each other in developing more sophisticated counting strategies. Students choose tools to help them keep track as they count, such as cups and bowls or number paths and ten-frames. As students work, the teacher checks in with students by commenting or asking questions. Students record the total of their collection and represent how they came to their total by drawing and/or writing.

counting collections

Wrap up

To wrap up, teachers bring students together to discuss some of the work they did. Possible closure topics may focus on new strategies for counting or effective recording representations.

Pre-k to 2nd grade classroom teachers are excited to see their students deepening and strengthening their number sense and counting understandings through this work. And, students are so excited to count that they bring their own collections from home to school.

Additional Resources:
https://tedd.org/counting-collections
https://www.responsiveclassroom.org

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New Book Tells Where to Find Math Fun in Books, Games, and Routines https://earlymath.erikson.edu/new-book-tells-where-to-find-math-fun-in-books-games-and-routines/ https://earlymath.erikson.edu/new-book-tells-where-to-find-math-fun-in-books-games-and-routines/#respond Wed, 11 Sep 2019 12:53:02 +0000 https://earlymath.erikson.edu/?p=15645 Books, games, and routines are a natural entry point for math fun in the early years. A new book tells how to tap into children's curiosity to explore the math that is found in everyday life and in play.

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Books, games, and routines are a natural entry point for math fun in the early years. These ways of engaging children in math build on their interests, discoveries, and questions. A new book by Mary Hynes-Berry, PhD, and Laura Grandau, PhD, tells how to tap into children’s curiosity to explore the math that is found in everyday life and in play.

The book, called Where’s the Math?: Books, Games, and Routines to Spark Children’s Thinking, evolved from a professional development project in partnership with the Overdeck Family Foundation that took place in 2017-18. Called “Families Finding Math in Storybooks,” Hynes-Berry and Grandau conducted learning labs and consulted with librarians at the Evanston Public Library in order to develop better knowledge and practices for their librarians and the families that come through their doors.

We discuss with co-author Dr. Mary Hynes-Berry her experiences with developing such a book.

Framing math learning by using questions children ask as a jumping off spot is a unique concept. Why did you decide to start there for this book?

Children’s interest and understanding of the math all around us is grounded in questions. In fact, even as adults, we are always trying to come up with answers to questions like “How big is it?” or “How many do we have? need? want?” The hope is that using these questions will shift the attitudes and beliefs of teachers who are used to worksheets instead of math investigations.

You use children’s books often as a recommended tool for expanding children’s math thinking. Is there a recent book you’ve discovered that you find rich in math opportunities?

I’m finding new books every time I turn around! In fact, it’s pretty hard to think of a really good children’s book that doesn’t have some math in it. But one I recently discovered is Alma and How She Got Her Name. It is a beautiful celebration of names and lends itself well to all the math activities we can do around them, such as name towers.

Is there a classic book that most teachers and parents probably have on their bookshelves that they maybe don’t know is very mathematical?

I would guess that most people who are very familiar with Goldilocks and the Three Bears would never think of it as very mathematical. But we have been using in the Collaborative virtually from the beginning.

What advice do you have for teachers and parents when it comes to tailoring activity ideas to their situation and their children?

Effective tailoring of an activity comes down to two things:

  1. Have a clear understanding of the mathematical Big Idea and a specific understanding you want to build.
  2. Have an equally clear sense of the interests, needs, and disposition of the specific children for whom you are tailoring the lesson.

What is an everyday activity or routine that teachers or parents may be surprised to know can be very mathematical?

Everybody knows that songs with movement are wonderful ways to transition or to begin group time. But many teachers and parents are surprised that a song that has no numbers in it such as The Wheels on the Bus or The Hokey Pokey are full of math, especially math related to spatial reasoning and patterns.

Is there something new you learned or was reinforced through the process of writing this book?

I ended up with a digging-deeper understanding of how books, games, and routines have so much mathematical thinking and problem-solving embedded in them. And that playing intentionally with that math is an incredibly powerful, engaging way to make sense of how math truly is all around us. I love to see the enthusiasm of teachers and children both when that message comes across.

Where’s the Math?: Books, Games, and Routines to Spark Children’s Thinking

Make math learning both meaningful and fun by building on children’s natural curiosity to help them grow into confident problem solvers and investigators of math concepts. Using five math-related questions children wonder about as a framework, this book helps you go deeper into everyday math with children.

Buy

 

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5 Books and Songs for Exploring Growing Patterns https://earlymath.erikson.edu/5-books-songs-and-activities-for-teaching-growing-patterns/ https://earlymath.erikson.edu/5-books-songs-and-activities-for-teaching-growing-patterns/#respond Wed, 28 Aug 2019 13:46:17 +0000 https://earlymath.erikson.edu/?p=15562 Cumulative tales and rhymes illustrate growing patterns, typically an increase or decrease by one on each page. As the growing pattern is revealed through the story, children get excited because they can figure out "what comes next."

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Cumulative tales and rhymes such as The Napping House illustrate growing patterns, typically an increase or decrease by one on each page. As the growing pattern is revealed through the story, children get excited because they can figure out “what comes next.” Suddenly a nice pattern book for kindergarten or preschool becomes an adventure. As we point out in our book Big Ideas of Early Mathematics, children are naturally “tuned in” to patterns, which can deepen their understanding of them.

Like all patterns, growing patterns feature repetition and regularity. In this case, it’s not a sequence of elements that repeats such as in the AB pattern of a checkerboard, but a quantitative change that repeats. Our counting system is a basic growing pattern because each number is one more that the number that precedes it. The plus-one rule is what brings predictability to our count sequence.

Because children will be so engaged by these stories and songs, you can guide them to notice the growing pattern, using geometric models such as stacking blocks or pictographs to show how each repetition has a constant change in quantity. For example, if you stack block towers in order of size, children will see the resulting stair-step structure of a plus-one growing pattern. Acting outing out the stories and songs is another way to make the pattern structure come to life!

Here are a few of our favorite mathy books and songs that illustrate all the math going on in growing patterns.

  1. We Had a Picnic This Sunday Past by Jacqueline Woodson

    This tale vibrantly captures an annual family picnic. As guests arrive with food—and Teeka and Grandma dread the arrival of Cousin Martha’s dried out apple pie—a growing story emerges! There are two sets: people at the party and food on the table.

    With preschool children, a reader can pause and comment on how there are more people and more food! They may also be interested in sorting the different types of food into binary sorts by attributes such as “food I like” vs. “food I don’t like” or “dessert” vs. “not dessert.”

    Kindergarten and first grade children may wish to keep track of the quantity of each set and discover that the change does not always follow a growing pattern – especially when Cousin Trevor comes empty handed, and Teeka complains, “you can’t eat air!” The delicious surprises in this book are a result of disrupting the pattern and are what make it such a rewarding read. After enjoying this fun read-aloud many times, children can write their own growing stories about their families’ traditional gatherings.

  2. The Waterhole by Graeme Base

    growing pattern, The Waterhole Graeme Base, how to teach patterns to preschoolersIn this gorgeously illustrated tale of animals convening to drink from a watering hole, author and illustrator Graeme Base takes a counting book and does something magical with it. As each animal comes to drink from the watering hole one by one, the water in it diminishes each time. It is a demonstration of a growing pattern as well as an example of cause and effect, a decreasing or shrinking pattern paired with the growing pattern, and an exploration of volume all in one unique story. What a great way to spark a conversation with children.

  3. Two of Everything by Lilly Hong

    Not all growing patterns use the same plus-one rule for growing. For children ages 6-8, introducing more advanced growing patterns through books can help provide a context for understanding multiplicative change. Two of Everything is a great Chinese folktale that introduces a doubling pattern with wit and wisdom. You can pair this book with activities that involve input and output such as function machines or in/out boxes. Whatever goes in comes out double!

  4. One Grain of Rice by Demi

    Another doubling pattern can be found in this Indian folktale, in which a clever girl, Rani, outsmarts a greedy ruler who is hoarding the people’s rice harvest for himself. Despite her hunger, Rani returns some spilled rice to the raja instead of keeping it and is rewarded for her honesty. What payment does she ask for? One grain of rice, but with the condition that the grains of rice will double each day for 30 days. Thinking this a meager amount, the raja agrees and is surprised to see that the reward doubles to more than a billion grains of rice in 30 days! The beautiful illustrations capture the growing pattern in a powerful way as the volume of rice given to Rani takes up more and more of the page, eventually taking a fold-out, 4-page spread to show it all.

    A chart at the back of the book shows the number of grains of rice given to Rani on each of the 30 days. First and second graders learning about the units of our base-10 system can practice reading these large numbers. Make a photocopy of the chart, cut it apart, and have children put the 30 days back in order according to the doubling pattern.

  5. Songs for Growing or Decreasing Patterns

    The classic song Over in the Meadow uses fun animal sounds and a catchy tune to make its growing pattern. It creates a great opportunity to allow children to illustrate what they hear happening in the song and the change that happens with each new verse.

    Ten in the Bed is another song that has a decreasing pattern that children love to sing and act out. Imagine an activity that gets kids moving while visually demonstrating how there is one less in the bed while also creating an increasing set of people on the floor along the way. It’s a familiar and fun song that can spark great dialogue around the concepts of constant change.

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Different Types of Shapes and Nonexamples to Spark Geometric Learning https://earlymath.erikson.edu/using-different-types-of-shapes-and-nonexamples-to-spark-geometric-learning/ https://earlymath.erikson.edu/using-different-types-of-shapes-and-nonexamples-to-spark-geometric-learning/#comments Mon, 29 Jul 2019 13:40:39 +0000 https://earlymath.erikson.edu/?p=15502 It’s up to us to find, share, and talk about a variety of shapes with children in ways that expand their understanding and build connections between the shapes drawn on paper and the concrete objects in our world.

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Children explore the everyday shapes in their environment from birth. They informally learn about the properties of 3D shapes as they mouth, touch, and play with blocks, balls, and other objects in daily life. Long before children have language to name or describe these different types of shapes, they develop intuition about which shapes roll, how a flat surface feels, and that “corners” are pointy.

Though we live in a 3D world, common language and materials used to teach children about shapes focus on 2D shapes. In fact, the majority of emphasis gets put on just four shapes: circles, triangles, rectangles, and squares. And these shapes are usually presented in only one typical way so that children often develop rigid and fixed notions of what shapes must look like.

So it’s up to us to find, share, and talk about a variety of shapes with children in ways that expand their understanding and build connections between shapes (think, how is a square a special case of a rectangle?) and connections between the shapes drawn on paper and the concrete objects in our world.

Take triangles for example. Most children’s books present triangles as equilateral (3 equal sides) and oriented to sit on a horizontal base. As a result, children may not recognize isosceles and scalene triangles as triangles because they look “stretched out.” Children may say that a triangle with a vertex pointing down is “upside down.” Watch a small group discussion about “upside down” triangles during a preschool activity called Feel for Shapes.

types of shapes, different types of shapesTalking about diverse shape examples along with nonexamples can go a long way to help children understand what defines a shape—what makes a triangle a triangle, for example. Nonexamples are highly visually similar to a type of shape but lack at least one defining attribute.

Luckily, it’s easy to find nonexamples to discuss in everyday life. How often might we point to a piece of pizza and call it a “triangle?” As adults, it’s not our main concern to correct children’s language, but to listen to what shape knowledge children reveal by what they say and to build on it. So, if a child says a pizza slice is a triangle, you might ask them why they think that. Likely, the child will have noticed that the slice has three sides. This is a great opportunity to focus on the defining attributes of a triangle. For example, you might say, “Yes, I see three sides, but the crust side is curved so it’s not exactly a triangle. Triangles have three straight sides.”

Going on a “Shape Hunt” in real life or through picture books such as Shapes, Shapes, Shapes by Tana Hoban is bound to turn up many common 3D objects that get labeled with 2D shape names. This is a great opportunity to help children see the relationship between different types of shapes, particularly 2D and 3D shapes. For example, many children will say that a ball is a “circle” because it is round. You can acknowledge what the child knows while pushing for greater precision—“Yes, the ball is round like a circle, but it’s not flat. A ball is round in every direction so we call its shape a sphere.” The book Now What? A Math Tale by Robie H. Harris tells the story of Puppy putting together blocks to build a bed that’s just the right size for a nap. As Puppy solves problems, readers get to explore how 2D shapes are the flat faces of 3D prisms and share in the delight of discovering “a rectangle block—that’s a square block, too!”

Most of the world around us is not made of regular 2D shapes the way they are drawn in simple shape books and posters. And that’s okay. What a wonderful opportunity to start a conversation with children about different types of shapes and the attributes of shapes!

Surprise!
Every side is a rectangle!
So, it’s a rectangle block —
that’s a square block, too!

types of shapes, Now What? A Math Tale, Robie H. Harris

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School Event Engages Families in Early Math Learning https://earlymath.erikson.edu/engaging-families-in-early-math-learning/ https://earlymath.erikson.edu/engaging-families-in-early-math-learning/#respond Wed, 19 Jun 2019 12:48:46 +0000 https://earlymath.erikson.edu/?p=15448 Families play a fundamental role in shaping children's interest and skills in math. Schools can help connect the math that exists both in and out of school and nurture families' positive relationship to math.

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Families play a fundamental role in shaping children’s interest and skills in math. Schools can help connect the math that exists both in and out of school and nurture families’ positive relationship to math. Teachers can share a variety of fun and engaging activities such as games that children learn at school. Teachers can also learn from families about how they use math in their everyday lives at home. Families are perfectly situated to talk about quantities, shapes, and sizes in meaningful contexts. When parents ask children about these ideas and support their thinking through the use of open-ended questions, math learning can happen anywhere families are–in the kitchen, at the park, or in the grocery store.

Here, a team of preschool teachers and the principal at a Chicago public school discuss a family math event designed to engage parents with the math their children are learning. Parents are invited to stay at school after drop-off. First, parents learn about developmentally appropriate math content and ways to promote positive math interactions at home. This is important since the teaching of mathematics looks different from the instruction that most adults experienced when they were in school. For example, teachers share information about the importance of finger counting and that it is something to encourage. Next, parents join their children in games and activities that the children know from school. Families leave with a goody bag containing engaging math materials such as cards, dice, and counting songs to take home. Teachers also hope that parents take home the message that math is everywhere, and that families have all the tools they need to play, explore, and talk about math with their child, in whatever language they may use at home.

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Build Tangram Shapes with Do-It-Yourself Puzzles https://earlymath.erikson.edu/build-tangram-shapes-with-do-it-yourself-puzzles/ https://earlymath.erikson.edu/build-tangram-shapes-with-do-it-yourself-puzzles/#respond Sat, 08 Jun 2019 15:53:11 +0000 https://earlymath.erikson.edu/?p=17076 A tangram puzzle consists of seven pieces — five triangles and two quadrilaterals made from one square. Tangrams originated in  More →

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A tangram puzzle consists of seven pieces — five triangles and two quadrilaterals made from one square. Tangrams originated in China hundreds of years ago. They traveled to Europe in the 1800’s, where they gained global fame. These seven pieces are used to create thousands of unique figures. A tangram puzzle shows only an outline or silhouette of a figure, and the puzzle solver must create it, using all seven pieces. This requires spatial reasoning as one flips, slides, and rotates the smaller shapes to compose the larger tangram shape.

You can make your own set of paper tangrams by following this easy, step-by-step video. All you need is a piece of paper and a pair of scissors to make these do-it-yourself puzzles. Then, get creative!

Ms. Chen, a second-grade teacher at St. Therese Chinese Catholic School in Chicago, integrates art with math when she uses tangrams with her students. Her math focus is spatial reasoning and shape composition.

Ms. Chen begins the exploration by reading Grandfather Tang’s Story by Ann Tompert, and you can listen to the story here. Have children explore the tangram shapes by trying to create the large square from which they came. Next, try to make the animal figures from the story. Then, invite children to create their own figures, following the rules of tangram puzzles. The rules state that you must use all pieces, each piece must touch another piece, and none of the pieces can overlap. Look at some lovely examples from the 7- and 8-year-old children in Ms. Chen’s class!

tangrams shapes puzzles
tangrams shapes puzzles

Why use tangrams?

Tangram puzzles help learners develop an understanding of the big ideas about shape and spatial relationships. Spatial reasoning is the capacity to visualize relationships between shapes and manipulate shapes through turns (rotations) and flips (reflections), both physically and mentally. Developing children’s spatial reasoning deepens their understanding of geometry as well as develops flexibility for problem solving. Working with do-it-yourself tangram puzzles—at home or at school—can really help your child’s creative, mathematical thinking take shape!

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Video Analysis of Children’s Mathematical Thinking https://earlymath.erikson.edu/video-analysis-of-childrens-mathematical-thinking/ https://earlymath.erikson.edu/video-analysis-of-childrens-mathematical-thinking/#respond Thu, 30 May 2019 11:54:52 +0000 https://earlymath.erikson.edu/?p=15206 Looking at a video from our Focus on the Child series, Collaborative member Donna Johnson leads a discussion about the  More →

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Looking at a video from our Focus on the Child series, Collaborative member Donna Johnson leads a discussion about the mathematical thinking of a child during a comparison task. This is an example of how we use video analysis to hone teachers’ observational skills. Notice how Donna directs the participants to take notes about the evidence of student thinking in the video clip. As they share what they notice, Donna asks them to consider what the evidence indicates about the child’s mathematical thinking and understanding. In this way, video analysis can build teacher’s capacity for formative assessment.

The full clip that is used can be found here: Comparing Quantity with Child 4.

Adult learning videos are portions of professional development sessions designed to engage adults in a meaningful way. They are not activities meant for recreating with children. Learn more about our Whole Teacher Approach and Learning Labs.

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Video Analysis of Preschool Number Sense Activity https://earlymath.erikson.edu/video-analysis-of-preschool-number-sense-activity/ https://earlymath.erikson.edu/video-analysis-of-preschool-number-sense-activity/#respond Wed, 29 May 2019 12:52:51 +0000 https://earlymath.erikson.edu/?p=15262 Collaborative member Donna Johnson leads teachers in a discussion about an activity designed to build number sense in a video  More →

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Collaborative member Donna Johnson leads teachers in a discussion about an activity designed to build number sense in a video from our Focus on the Lesson series. This is an example of how we use video analysis to engage adults in discussions around classroom practice and instructional choices. Examining authentic video vignettes allows participants to apply new understanding of what math learning and teaching looks and sounds like.

Here, Donna uses questions to help teachers think about how an activity changes over time as the children’s understanding about numbers and number sense develops. She asks the them to consider how the teacher in the video responds to what the children say and why she chooses certain materials over others. Donna asks open-ended questions and builds on what the participants notice in the video. In this way, video analysis helps develop the ability to recognize and describe teaching practices that promote children’s mathematical development.

The full clip that is used can be found here: Number Arrangements

Adult learning videos are portions of professional development sessions designed to engage adults in a meaningful way. They are not activities meant for recreating with children. Learn more about our Whole Teacher Approach and Learning Labs.

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A New Focus for Familiar Card Games https://earlymath.erikson.edu/playing-card-games/ https://earlymath.erikson.edu/playing-card-games/#comments Mon, 29 Apr 2019 17:40:33 +0000 https://earlymath.erikson.edu/?p=15357 Here you can download cards and simple-to-learn game ideas to help young children build their understanding of early math concepts such as cardinality, composing and comparing numbers, and allowing them to subitize.

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Familiar, simple-to-learn card games help young children build their understanding of early math concepts such as cardinality and composing and comparing numbers. Using quantity cards with pictures when playing card games, rather than numerals, increases children’s attention to numerosity and adds novelty to old favorites like Go Fish or Memory.

What’s so special about quantity cards?

Quantity cards have pictures of small sets of dots, finger patterns, and 5- and 10- frames showing numbers from 1 to 10. Quantity cards require children to look at a picture and say how many they “see.” This attention to numerosity builds children ability to subitize. That is, children will instantly see a small quantity and label it with a number word. Most preschoolers will recognize three fingers without counting, “one, two, three.”

Matching quantities such as dots and fingers develops ideas about equivalence. Though these two cards look quite different, they are the same in one important way—their numerosity. Experience recognizing how amounts that look different can actually be equivalent numbers helps children abstract the “three-ness” of three when playing card games.

Most importantly, quantity cards helps children build number relationships. Numeral cards present each number separately as a monolithic entity. Quantity cards depict numbers in relationship to other numbers. For example, the picture of 4 fingers also shows the thumb tucked away, allowing for children to think and talk about how 4 is one less than 5. The dot patterns have clusters of dots, helping children to see that six can be thought of as 3 and 3, for example.

Who can play these games?

These games are for all ages. While the numbers only go through 10, the different representations gets children and adults alike thinking and talking. Watch this video of a mom and son playing the game Capture together. Beyond the laughter, you’ll hear lots of math talk that is natural and playful.



How do I get started?

Print our quantity cards on sturdy paper and cut them to make a beginner deck of 20 cards; see link below. That is a perfect place to start with your preschooler or kindergartner.

Or make your own dot cards with index cards and dot stickers or dot stampers. You can experiment with different dot arrangements — dice pattern, shapes, and lines. See our dot pattern chart for ideas. Using dots of two colors helps children to see subgroups within a larger quantity such as two blue and two red dots that compose a set of four dots. Having a variety of configurations strengthens children’s ability to visualize numbers in their minds.

Have fun playing your favorite game such as Go Fish or Memory with quantity cards. Try one of our games, such as Capture. (Or learn a few math tricks with kids to get them having fun while learning.) The important thing is to watch and listen as children play games. You will find that there is a lot to talk about when the focus of early math games is on recognizing, composing, and comparing quantities.

Download

Dot Pattern Chart for playing card games

Dot Pattern Chart

To make your own quantity cards, use this chart about dot patterns for ideas about various dot arrangements to further children’s thinking. There is a lot to think about with just dots!
Dot Pattern Chart

 

Related Ideas

playing math card games early math

Quantity Cards

These Quantity Card downloads are available and cater to different skill levels and ages. They include a variety of pictures or “suits” since matching quantities such as fingers and dots can help develop ideas about equivalence.

Learn more

Math Card Games

How to Adapt Math Card Games to Children’s Skill Level

This article and accompanying downloadable instructions explore common card games that can be played and revisited as well as adapted to children’s own math skills as they develop over time.

Learn more

Get downloadsJoin the website

Want free printable activities and card sets? Join our website (or log in if already joined) to unlock much more.

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Teachers as Knowledgeable Others in the Spotlight at Math Conference https://earlymath.erikson.edu/teachers-as-knowledgeable-others-spotlighted-at-math-conference/ https://earlymath.erikson.edu/teachers-as-knowledgeable-others-spotlighted-at-math-conference/#respond Tue, 23 Apr 2019 15:32:05 +0000 https://earlymath.erikson.edu/?p=15346 Everyone knows it’s better to teach someone how to fish than to hand out fish. In terms of professional development, this philosophy means empowering classroom teachers to grow their own practice by facilitating the learning of other teachers.

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Everyone knows it’s better to teach someone how to fish than to hand out fish. In terms of professional development, this philosophy means empowering classroom teachers to grow their own practice by facilitating the learning of other teachers. This peer-to-peer professional growth is the idea behind the 2019 National Council of Supervisors of Mathematics annual math conference session entitled “Teachers Teaching Teachers How to Fish.”

The session was led by a unique mix of the Collaborative’s own Prof. Mary Hynes-Berry along with early childhood teachers who are a part of a Big Shoulders Fund project aimed at developing teacher Professional Development Leaders who support early mathematics teaching and learning across a network of Catholic schools in Chicago. Karen Griffin, Tia Hanna, Faith Jones and Stephanie Tominov are full-time Catholic schoolteachers who have been working with adult learners for several years. They shared their professional journey with nearly 100 math educators and specialists from all over the country.

I can enhance my professional development as a learner in the process as well as be a knowledgeable other to teach my colleagues. It is a phenomenal feeling!

The four educators talked about how working with Erikson Institute has given them a deeper understanding of math and effective instructional strategies that they can share with their peers.

“The model that we use at Erikson,” Griffin said, “is unique in that the professional development leaders (PDLs) are teachers who are in the classroom everyday. Most professional development I have attended is presented by a person who is not in the classroom anymore and hasn’t been for many years. We as the PDLs are the knowledgeable others but we are open to learning from our peers. I have left every learning lab with a new idea or understanding.”

These teacher leaders lead a four-part series of professional development sessions each year involving about 25 PreK-2nd grade teachers. Precisely because they are speaking teacher-to-teacher rather than as an academic “sage on the stage,” they have an authenticity. They can share how working with Erikson has transformed their own practice They can testify that the best way to engage young children’s minds and hearts is to function as a community of learners who are making discoveries and problem solving.

Ms. Jones adds, “I can enhance my professional development as a learner in the process as well as be a knowledgeable other to teach my colleagues. It is a phenomenal feeling!”

Perhaps the best evidence of how participating in the project has transformed classroom teachers into effective professional development leaders for their peers is the fact that immediately after the presentation, the four educators initiated a thoughtful reflection on the session. While they took pride in how well it went, they were able to identify areas they could see as needing improvement the next time they present at a professional conference.

The impact of creating such professional learning opportunities in early childhood education is not trivial. Far too many early childhood teachers hold negative attitudes about math. There is a huge need for professional development that will meet teachers where they are and help them to change their perspective along with their practice. Learning from a fellow teacher is an effective way to do this.

We are deeply grateful to our partners, Big Shoulders Fund, and to the CME Group Foundation and the Robert R. McCormick Foundation for funding this important work.

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3 Books That Encourage Simple Graph Explorations with Young Ones https://earlymath.erikson.edu/3-books-encourage-simple-graph-explorations-graphs-for-kids/ https://earlymath.erikson.edu/3-books-encourage-simple-graph-explorations-graphs-for-kids/#respond Sun, 24 Mar 2019 15:19:48 +0000 https://earlymath.erikson.edu/?p=15006 At the heart of it, graphing in the early years is about quantifying information in order to answer a question. That requires children to organize data in some visible way so that comparisons and generalizations are possible.

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At the heart of it, graphing in the early years is about quantifying information in order to answer a question. That requires children to organize data in some visible way so that comparisons and generalizations are possible. Here are a few good kids books with good questions to spark simple graph explorations with children.

  1. Anno’s Flea Market by Mitsumasa Anno

    Anno’s Counting Book is a favorite of ours, so we would be remiss if we didn’t mention another title from Mitsumasa Anno, Anno’s Flea Market. The illustrations depict the variety of wares found at a Saturday morning flea market. How many stalls have books? What kind of games and toys are on display? The intricate illustrations prompt many different questions that can be discussed. Then, you can invite children to bring in their trash and treasures for your own indoor flea market. Everyone will be curious to answer the question, “What kinds of things do we have at our flea market?” Sorting the items into categories for an object graph allows children to count, compare, and draw conclusions.

  2. Which Would You Rather Be? by William Steig

    This book of questions by celebrated author/artist William Steig gets everyone talking! Would you rather be a stick or a stone? Rain or snow? A grown-up or a child? Children enjoy taking surveys of friends and family, and this book gives them a simple model for a forced-choice question. A T-chart with tallies or other simple marks can make a visual comparison of preferences easy to see and share. Children are motivated to find answers when it’s their own questions they are asking.

  3. Whose Shoes? by Stephen R. Swinburne

    Discussing and organizing shoes is a natural way to connect math to everyday life. One reason is because footwear is something we all have in common. In Whose Shoes?: A Shoe for Every Job, we have many examples of different types of shoes. Oftentimes shoes are associated with the type of work one does, while other times what we wear on our feet is dictated by the weather outside. The variety of styles makes for good conversation. For example, if we discover that there is a significant number of people wearing boots, we can draw a conclusion about what the weather outside might be. We also might make a generalization about winter footwear versus summer footwear. You can watch an example of an activity in our Focus on the Lesson series called Shoe Graph, along with some planning and reflecting conversations here. Or take a time out from shoe sorting to play a game of heads or tails while collecting data about the fun you’re having.

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Ordering Similar Objects by Measurable Attributes https://earlymath.erikson.edu/ordering-similar-objects-by-measurable-attributes/ https://earlymath.erikson.edu/ordering-similar-objects-by-measurable-attributes/#respond Wed, 27 Feb 2019 15:51:54 +0000 https://earlymath.erikson.edu/?p=15163 Early childhood teacher candidates engage in an adult learning activity that has them order and reorder objects by size according to different attributes.

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Early childhood teacher candidates engage in an adult learning activity that has them order and reorder objects by size according to different attributes. This activity explores the idea that measurement requires us to identify a measurable attribute of an object–“What kind of “big” is it?–and make a comparison to find its size relative to something else. This is one example of how we engage pre- and in-service teachers in the Big Ideas so that they deepen their own mathematical content knowledge.

Adult learning videos are portions of professional development sessions designed to engage adults in a meaningful way. They are not activities meant for recreating with children. Learn more about our Whole Teacher Approach and Learning Labs.

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Exploring Unit Block Relationships https://earlymath.erikson.edu/exploring-unit-block-relationships/ https://earlymath.erikson.edu/exploring-unit-block-relationships/#respond Mon, 25 Feb 2019 16:01:40 +0000 https://earlymath.erikson.edu/?p=15152 Examining sets of unit blocks challenges educators to think about relationships of the blocks. A deeper understanding of the features of unit blocks empowers teachers to support children in ways that promote joyful math learning.

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Examining sets of unit blocks challenges educators to think about relationships since all the blocks are fractional units or multiples of one another. A deeper understanding of the features of unit blocks empowers teachers to support children in ways that promote joyful math learning.

Adult learning videos are portions of professional development sessions designed to engage adults in a meaningful way. They are not activities meant for recreating with children. Learn more about our Whole Teacher Approach and Learning Labs.

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Different Uses of Numbers with Anno’s Counting Book https://earlymath.erikson.edu/different-uses-numbers-annos-counting-book/ https://earlymath.erikson.edu/different-uses-numbers-annos-counting-book/#respond Mon, 25 Feb 2019 15:48:48 +0000 https://earlymath.erikson.edu/?p=11347 Books are a powerful way to launch math investigations with children, but also with adult learners. Our Learning Labs often use picture books as a doorway into mathematics. Here, teachers explore multiple representations of the number five in Anno's Counting Book by Mitsumasa Anno.

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Books are a powerful way to launch math investigations with children, but also with adult learners. Our Learning Labs often use picture books as a doorway into mathematics. Here, teachers explore multiple representations of the number five in Anno’s Counting Book by Mitsumasa Anno. It is critical for teachers of young children to appreciate the complexity of language about numbers and their various meanings.

Adult learning videos are portions of professional development sessions designed to engage adults in a meaningful way. They are not activities meant for recreating with children. Learn more about our Whole Teacher Approach and Learning Labs.

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Math-Related Greeting Builds Safe Learning Community https://earlymath.erikson.edu/math-related-greeting-builds-safe-learning-community/ https://earlymath.erikson.edu/math-related-greeting-builds-safe-learning-community/#respond Sun, 24 Feb 2019 15:58:00 +0000 https://earlymath.erikson.edu/?p=11348 Our Learning Labs open with greetings that serve the dual purpose of building a safe learning community while introducing a math topic in a fun, accessible way. Here, adults directly compare the lengths of paper strips as they mingle around the classroom to find an exact match.

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Our Learning Labs open with greetings that serve the dual purpose of building a safe learning community while introducing a math topic in a fun, accessible way. Here, adults directly compare the lengths of paper strips as they mingle around the classroom to find an exact match. This greeting gets everyone thinking about how measurement involves comparison and how often we use measurement thinking in our daily lives. Mixing up the participants and meeting new people helps ensure that everyone feels included and all voices are heard.

Adult learning videos are portions of Learning Labs designed to engage adults in a meaningful way. They are not activities designed for young children. Learn more about our Whole Teacher Approach and Learning Labs.

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Many Ways to See How Many https://earlymath.erikson.edu/many-ways-to-see-how-many/ https://earlymath.erikson.edu/many-ways-to-see-how-many/#respond Tue, 29 Jan 2019 13:35:47 +0000 https://earlymath.erikson.edu/?p=15087 Regardless of how high a preschooler can rote count, a child’s sense of what those numbers actually mean develops gradually. We call this understanding number sense, and it requires relating numbers to real quantities.

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How many times have you heard young children (or their parents) proudly exclaim how high they can count? Children love to master the sequence of counting words—reciting the number names in the same order every time—just as they love to sing their ABCs.

Regardless of how high a preschooler can rote count, a child’s sense of what those numbers actually mean develops gradually. We call this understanding number sense, and it requires relating numbers to real quantities. Most three-year olds have a solid number sense for the quantities of 1, 2, and 3. That is, they can think about small numbers and their relationships such as 2 cookies is 1 for me and 1 for you or 3 toy cars and 2 tracks to race means 1 car will have to wait.

By the time children leave preschool, most have developed number sense through 5, and many up through 10 and beyond. Two videos featuring the same preschooler vividly demonstrate how children can visualize small quantities in a variety of ways and use those numbers to estimate larger quantities.

What a wonderful demonstration of how important it is that young children get quantities into their eyes, ears, and fingers – that’s how they get them fixed in their brains!

In the first video, this lively preschooler gets a quick look at a dot card, and then uses a small set of counters to produce a set of the same quantity.



For the most part, she has no trouble with this task. She instantly perceives the quantities without counting, an ability called subitizing. She only uses counting to confirm she put out the right amount. However, an unfamiliar configuration of 5 dots in a circle slows her down. She tries a few time before she decides how many counters feels like enough. As she works, she makes a little hum for each of the dots as she places them. What a wonderful demonstration of how important it is that young children get quantities into their eyes, ears, and fingers – that’s how they get them fixed in their brains!

This second task is more intriguing.



This time, the interviewer points to small collection of 7 cubes on one mat and then dumps a larger number of the same cubes on a second mat. The girl quickly says what she knows is a large number—1000! But when the interviewer suggests that it might be 100, the girl cheerfully agrees—she just knows it’s a lot. But, when asked if there are a lot of 7s, she just as quickly rejects that idea. She looks over at the mat with 7 and then back at the larger quantity, hums a bit again, and then puts up 3 fingers to show how many 7s there are.

Both the use of gesture and her estimation of 3 groups of 7 are significant. Her estimate is reasonably close to the precise number of cubes (27). Her 3 fingers show she is still working on getting numbers into words. In fact, when asked how many it would be if there were 7 and 7 and 7, she names the next word in the count sequence—8. The point is that her sense of how many is clearly linked to what she sees, that is to the real quantity. Because she has a sense of 7, she can intuitively use the strategy of visually clustering items into groups of 7 to estimate a reasonable number of 7s within the whole set. With time and opportunity, she will learn to connect those large number words that she can say with the real quantities that she can see.

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4 Fun Books that Explore Math, Patterns, and Sequences https://earlymath.erikson.edu/4-fun-books-that-explore-math-patterns-and-sequences/ https://earlymath.erikson.edu/4-fun-books-that-explore-math-patterns-and-sequences/#comments Fri, 11 Jan 2019 13:42:57 +0000 https://earlymath.erikson.edu/?p=14973 From an early age, children notice and appreciate patterns in the world around them. Patterns and sequences of different kinds begin to pop up all over the place, especially in the books that children love.

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From an early age, children notice and appreciate patterns in the world around them. Patterns and sequences of different kinds begin to pop up all over the place, especially in the books that children love. Pattern books for kindergarten and preschool ages can mean a lot of things, since pattern itself is a complex topic.

Many children’s books contain patterns because that supports literacy development. Children use language and picture patterns to “read” predictable books. Patterns in stories allow children to predict what will happen next. Here are some of our favorite read alouds that invite children to make use of patterns to “read” along with you.

  1. Pattern Bugs by Trudy Harris

    This colorful and fun picture book is full of visual and verbal patterns. Not only do you see worms crawling with every other one arching its back, creating a pattern, you will also find that same pattern in the words of the poem, such as”flutter-float, flutter-float,” and in the decorative border on the page. Have children help to find all the illustrations that share the same pattern structure (AB, ABBA, etc.) on any given page. A pattern hunt like this highlights the Big Idea that the same pattern structure can be found in different forms.

  2. Brown Bear Brown Bear What Do You See? by Bill Martin Jr.

    This classic book illustrated by Eric Carle is on virtually any list of great books to have in the classroom. And it has a clear pattern that young children can quickly notice, and also repeat. Teachers and parents can ask children to identify the words that repeat to help them see the regularity and rhythm in the story.

  3. I Went Walking by Sue Williams

    This vibrant storybook is similar to Brown Bear Brown Bear What Do You See? in some ways, but it adds some interesting complexity to a pattern story. As the red-haired boy comes across different animals on his stroll, the menagerie of creatures trail him on his walking journey. The illustrations reveal a growing pattern. Each page adds one more animal to the long line of followers.

  4. The Napping House by Don and Audrey Wood

    Here is one of the pattern books for kindergarten and preschool ages that we recommend often. The illustrations in The Napping House are fun and humorous and provide a great opportunity for children to identify a plus-one growing pattern. As the characters pile one by one onto the bed for a nap, each page clearly shows the growing, linear sequence and foreshadows what comes next. You can find an example of two teachers exploring this book in our Focus on the Lesson video Who is Napping?

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Transforming Three-Dimensional Shapes with Child 36 https://earlymath.erikson.edu/transforming-three-dimensional-shapes-with-child-36/ https://earlymath.erikson.edu/transforming-three-dimensional-shapes-with-child-36/#respond Wed, 12 Dec 2018 15:16:01 +0000 https://earlymath.erikson.edu/?p=6986 This child successfully attempts to match geo-solids using triangular prism-shaped blocks. Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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This child flips and rotates triangular prism-shaped blocks to match a given arrangement.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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One-to-One Correspondence with Child 15 https://earlymath.erikson.edu/one-to-one-correspondence-with-child-15-one-to-one-correspondence/ https://earlymath.erikson.edu/one-to-one-correspondence-with-child-15-one-to-one-correspondence/#respond Mon, 10 Dec 2018 19:10:18 +0000 https://earlymath.erikson.edu/?p=14944 A child counts an organized collection of bears. Children often sort collections into groups by color. Here, a preschooler assumes he has equal groups until he counts them.

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A child counts an organized collection of bears.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Counting a Large Collection with Child 14 https://earlymath.erikson.edu/counting-a-large-collection-with-child-14-stages-of-development-math/ https://earlymath.erikson.edu/counting-a-large-collection-with-child-14-stages-of-development-math/#respond Sat, 08 Dec 2018 20:33:57 +0000 https://earlymath.erikson.edu/?p=14931 A second grader has a partial understanding of counting in equal groups, but is still working on rational counting. Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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A second grader has a partial understanding of counting equal groups.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Subtraction as the Inverse of Addition with Child 5 https://earlymath.erikson.edu/subtraction-as-the-inverse-of-addition-with-child-5-adding-and-subtracting/ https://earlymath.erikson.edu/subtraction-as-the-inverse-of-addition-with-child-5-adding-and-subtracting/#respond Sat, 17 Nov 2018 20:04:26 +0000 https://earlymath.erikson.edu/?p=14988 This second grader uses a known number combination (3 + 9) to solve a subtraction problem, showing an understanding of how addition and subtraction are related as inverse operations.

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This second grader uses a known number combination (3 + 9) to solve a subtraction problem, showing an understanding of how addition and subtraction are related as inverse operations.

Focus on the Child videos are taken from one-on-one interviews with individual children. The interviews are designed to elicit evidence of children’s mathematical thinking. They are not teaching episodes or formal assessments.

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Counting on Good Food for the Holidays https://earlymath.erikson.edu/counting-on-good-food-for-the-holidays-math-ideas/ https://earlymath.erikson.edu/counting-on-good-food-for-the-holidays-math-ideas/#respond Wed, 07 Nov 2018 15:27:47 +0000 https://earlymath.erikson.edu/?p=14786 Traditions around food and feasts provide rich opportunities to connect math at school with children’s experiences at home. From Diwali to Thanksgiving, fall is a season of special meals with loved ones.

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Traditions around food and feasts provide rich opportunities to connect math at school with children’s experiences at home. From Diwali to Thanksgiving, fall is a season of special meals with loved ones.

holidays math ideasYou can read Feast for 10 by Cathryn Falwall to launch a discussion about counting foods at the grocery store. In this book, a family counts from one to ten and then back again as they grocery shop, cook, and set the table to prepare for a family dinner. Children will have their own stories of helping family members prepare favorite dishes to share.

A familiar, math-rich context like grocery shopping promotes children to think about key grade-level concepts. In preschool, children can make a list of foods for a celebration and represent the quantities with drawings—connecting number words and numerals with cardinality.

In kindergarten, children need a lot of opportunities to compose and decompose the benchmark number 10 (CCSS.MATH.CONTENT.K.OA.A.4). So, as illustrated in this first photo, you can ask them to put 10 of their families’ favorite foods in a shopping cart.

holiday math ideasIn this second illustration, a child decides that his family will share a pizza. There are 9 slices, and he adds 1 hamburger for his uncle (who doesn’t eat dairy) to make 10.

Sheila Houston, a first grade teacher at St. Margaret of Scotland School in Chicago, had her students make their own books about grocery shopping for a feast. On each page of the book, the students added a different healthy food to the cart in quantities of 10 and calculated the total number of items in the cart. Students were given different numbers of items to start with to build the important skill of finding 10 more and 10 less than any two-digit number (CCSS.MATH.CONTENT.1.NBT.C.5).

Read one of the student’s books [pdf]

This holiday season, give every child the opportunity to engage with important math concepts in the context of their own lives. Using children’s own stories makes learning math more appetizing!

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Measuring Math Understanding of ELLs with Barbara Sarnecka and David Purpura https://earlymath.erikson.edu/measuring-math-understanding-of-ells-with-barbara-sarnecka-and-david-purpura/ https://earlymath.erikson.edu/measuring-math-understanding-of-ells-with-barbara-sarnecka-and-david-purpura/#respond Wed, 24 Oct 2018 14:05:24 +0000 https://earlymath.erikson.edu/?p=11257 A panel of early math experts engage with the audience at Promising Math 2017 to consider the challenges of getting a true picture of what young children understand about numbers.

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A panel of early math experts engage with the audience at Promising Math 2017 to consider the challenges of getting a true picture of what young children understand about numbers, especially when children are learning more than one language. Barbara Sarnecka, professor of Cognitive Sciences at the University of California-Irvine, and David Purpura, associate professor at Purdue University, discuss current practices in developmental psychology and early education research.

The Early Math Collaborative, supported by the CME Group Foundation and the Heising-Simons Foundation, held their first biennial conference in 2017 called Promising Math. A conference linking research and practice, the event brought experts from around the United States to discuss the understanding, teaching, and learning of mathematics for children between birth and eight years.The focus for 2017: Fostering access to mathematics for children who are English Language Learners (ELLs).

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Creating Opportunities for Early Math with Rodrigo Gutiérrez https://earlymath.erikson.edu/creating-opportunities-for-early-math-with-rodrigo-gutierrez/ https://earlymath.erikson.edu/creating-opportunities-for-early-math-with-rodrigo-gutierrez/#respond Tue, 23 Oct 2018 13:55:00 +0000 https://earlymath.erikson.edu/?p=14663 The role of play and importance of shared experiences in early math is discussed with Rodrigo Gutiérrez, Co-Director of the Center for Retention-Recruitment for Math Teachers at University of Arizona.

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The role of play and importance of shared experiences in early math is discussed with Rodrigo Gutiérrez, Co-Director of the Center for Retention-Recruitment for Math Teachers at University of Arizona. Gutiérrez makes a case that early childhood classrooms can offer rich opportunities for math learning when teachers capitalize on classroom routines, school-wide events, children’s literature, and play to create shared experiences.

The role of the teacher to create shared experiences is essential.

With young children, you can’t rely on pop culture or shared experiences out of school to provide contexts for math learning. So early childhood teachers can co-construct meaningful contexts with children in classrooms so that math is accessible and relevant to them.

The Early Math Collaborative, supported by the CME Group Foundation and the Heising-Simons Foundation, held their first biennial conference in 2017 called Promising Math. A conference linking research and practice, the event brought experts from around the United States to discuss the understanding, teaching, and learning of mathematics for children between birth and eight years.The focus for 2017: Fostering access to mathematics for children who are English Language Learners (ELLs).

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A Holiday with Too Many Tamales https://earlymath.erikson.edu/a-holiday-with-too-many-tamales-holiday-math-activities/ https://earlymath.erikson.edu/a-holiday-with-too-many-tamales-holiday-math-activities/#respond Mon, 22 Oct 2018 20:03:11 +0000 https://earlymath.erikson.edu/?p=6733 How is it possible to have too many tamales? Well, Maria finds out in the holiday storybook Too Many Tamales by author Gary Soto.

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How is it possible to have too many tamales? Well, Maria finds out in the book Too Many Tamales by Gary Soto. In this holiday storybook, what starts out as a fun experience helping mama make tamales becomes a frantic search for mama’s wedding ring. But it’s in the search that Maria and her cousins are faced with the math concept of too many.

How much is too many? This question is mathematical, making us think of quantity.  What makes it too many turns our thoughts to the context of the problem. Twenty-four tamales was not too many for the family members that Maria and her parents were having over to celebrate Christmas, but when Maria and her 3 cousins had to eat the tamales to search for mama’s missing wedding ring, suddenly there were too many. Why – because the tummies of 4 children were not big enough to hold all those tamales.

Lots of questions come to mind as you can picture Maria, Delores, Teresa, and Danny eating their way through the mountain of tamales, hoping to find the ring with each bite. Was it a fair share? Did they all eat the same amount, or did someone eat more? What are the possible combinations of cousins and tamales that would get them to the total of 24 tamales? It sounds like a lot of number story possibilities could allow for a variety of solution strategies. Kindergartners might work with a smaller number of tamales, like 12, to focus on quantities connected to “ten and some more.” Older students might use the constraints of the story to help frame the problem. We know that “the first one was good, the second one pretty good, but by the third tamale, they were tired of the taste.

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New Book Connects Early Math Teaching with Developmental Research https://earlymath.erikson.edu/new-book-connects-early-math-teaching-with-developmental-research/ https://earlymath.erikson.edu/new-book-connects-early-math-teaching-with-developmental-research/#respond Wed, 03 Oct 2018 15:23:28 +0000 https://earlymath.erikson.edu/?p=14572 A new book released in August 2018 from the Collaborative examines the connections and questions that arise from discussing math experiences in early childhood settings and the research into children's mathematical learning.

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A new book released in August 2018 from the Collaborative examines the connections and questions that arise from discussing math experiences in early childhood settings and the research into children’s mathematical learning. Growing Mathematical Minds documents important research with leading scholars in developmental psychology along with classroom teachers working in the field today. It creates a discussion between these groups, hoping to create a shared understanding that is often lacking between research and practice.

Edited by Jennifer McCray and Jie-Qi Chen of Erikson Institute and Janet Eisenband Sorkin of the University of Chicago, the book is both unique in its format and ambitious in its goal to spark interdisciplinary conversation.

We discuss with Dr. Jennifer McCray her experiences with developing such a book.

How much does a classroom teacher in, say, kindergarten or 1st grade benefit from knowing the research behind the topic they are teaching? Why isn’t it enough just to be told what best practices are?

The answer to this question goes directly to the nature of teaching and learning: what is it? In our view at Erikson, teaching and learning is an interactive exchange of information and ideas that results in the development of new understandings and knowledge. Just conveying information is not sufficient for the excellent educator. Students (children) need structured opportunities to employ ideas and conduct experiments related to what they are learning if they are to “believe in” the end result. If the learning experience requires this kind of engagement, then it follows that the teacher must be an artist—creative—able to take in information about what learners currently know and understand, and find powerful ways to link that to the ideas they wish to convey. This, in turn, requires that teachers be excellent observers of young children. The very young are not good at telling you exactly what they know about and understand; that kind of meta-cognition is beyond them. So coming to the learning situation with some ideas about what course the content follows allows a teacher to adjust and elaborate on the fly. If you understand more about the process children go through as their thinking develops, you are better equipped to support different stages of their learning.

Part of what the book does is allow everyone (researchers and psychologists included) to hear teachers’ voices directly. What part of the research raises the most challenges for teachers? What important teaching questions should the next round of studies address?

How do researchers and developmental psychologists benefit from a discussion like what is found in the book? What can they learn from classroom teachers about early mathematics?

Researchers and developmental psychologists who are interested in early math learning want the knowledge they generate to be useful to teachers. Part of what the book does is allow everyone (researchers and psychologists included) to hear teachers’ voices directly. What part of the research raises the most challenges for teachers? What important teaching questions should the next round of studies address?

Which chapter or topic did you personally learn the most from that you didn’t already know?

I learned the most about gesture and math learning—I was lucky to work with Mary Hynes-Berry on our team and to interview Susan Goldin-Meadow of the University of Chicago about this topic. This research is so new, and so startling. For example, the fact that when children are on the cusp of being able to “conserve” quantity, they are often able to show their understanding in their gestures prior to being able to do so in language. This finding has lots of important implications for how teachers assess children, but it also suggests that gesture provides an additional “pipeline” to what the mind is capable of. The further finding that how teachers gesture as they teach mathematics makes an impact on what children learn is clearly also fundamental, and probably needs further study.

Would you recommend a teacher read this book as an introduction to early mathematics, should they read this as a companion to something else, or should it be read after learning the fundamentals of early mathematics like is found in Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know?

I think the book is useful for the teacher who wants to push herself, to learn more. Most early childhood teachers have little exposure to this kind of research — it is “mind-expanding” stuff. It could be very useful for teachers who are interested in “action research; ” that is, in thoughtfully planning and trying out different interventions in their classrooms to see what will happen as a result, and documenting and sharing their results. No, it should not be an introduction to early mathematics. It is for the teacher who wants to grow beyond that.

This research is so new, and so startling. For example, the fact that when children are on the cusp of being able to “conserve” quantity, they are often able to show their understanding in their gestures prior to being able to do so in language. This finding has lots of important implications for how teachers assess children, but it also suggests that gesture provides an additional “pipeline” to what the mind is capable of.

Having been both an early childhood educator as well as researcher, what do you think are the biggest misconceptions about both of those roles?

I’m not sure that most researchers understand the passion, dedication, and creativity that make an excellent early childhood teacher. People who dedicate their lives to working directly with young children have to have patience, energy, and stamina, but many of them are also brilliant, intuitive, and extremely thoughtful. Because of this, these teachers can be an important resource for researchers who want their work to improve teaching and learning. By the same token, I don’t believe most teachers understand how often the things that drive researchers to understand how children learn are driven by a real desire to improve teaching and learning for all children. Most researchers desperately hope their work will find useful application, but practice itself is not their strength, so they focus on what they can do well. And really, that’s what we all must do. The important thing is that we keep talking to one another, so that the good work we do can be shared.

Growing Mathematical Minds

Available now

Our latest book Growing Mathematical Minds bridges research and practice. We translate research on early mathematics from developmental psychology into terms that are meaningful to teachers and readily applicable in early childhood classrooms.

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The Match ‘Em All Game https://earlymath.erikson.edu/match-em-all-dice-bingo-game/ https://earlymath.erikson.edu/match-em-all-dice-bingo-game/#respond Wed, 26 Sep 2018 14:58:14 +0000 https://earlymath.erikson.edu/?p=14372 This bingo-like game allows children to think about numbers in different ways. It focuses children on the attribute of quantity of small sets and helps them build a more robust number sense.

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This bingo-like game allows children to think about numbers in different ways. Children roll a dot cube and try to find one or more matches on their board. Though the representations may look different, two dots on the cube can match a picture of two blocks, two fingers on a hand, or the numeral 2 when we think about the meaning of the number 2. The first player to match ’em all, wins!

bingo game for math

Download The Card Set

Want to download the printable card set used in this video? Join our website for free (or log in if already joined) to get this and even more downloads. You can also check out our Subitizing Quantity Cards.

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Printable Match em All Cards

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The Hoop Game https://earlymath.erikson.edu/the-hoop-game-simple-tossing-game/ https://earlymath.erikson.edu/the-hoop-game-simple-tossing-game/#comments Tue, 25 Sep 2018 19:54:20 +0000 https://earlymath.erikson.edu/?p=14368 This game played with a hula hoop and bean bags demonstrates all the math that can be explored with a simple tossing game. Each round gives children practice seeing and naming smaller parts of a total number in a variety of ways.

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This game played with a hula hoop and bean bags demonstrates all the math that can be explored with a simple tossing game. Start with a small set of beanbags, no more than 5 or 6. Each child gets to toss the beanbags one by one into the hoop. How many land inside the hoop? How many land outside of the hoop? How many beanbags in all? Each round gives children practice seeing and naming smaller parts of a total number in a variety of ways.

Hoop Game Directions simple tossing game

The Hoop Game Directions

This download provides an opportunity for teachers to easily set up and play this game, or send it home with parents!

Hoop Game Directions

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3 "Mathical" Books for More Advanced Counting https://earlymath.erikson.edu/3-mathical-books-advanced-counting/ https://earlymath.erikson.edu/3-mathical-books-advanced-counting/#respond Tue, 11 Sep 2018 20:05:22 +0000 https://earlymath.erikson.edu/?p=14098 Along with being "mathical" award winners, these books provide compelling contexts that help primary grade children understand why more advanced counting is useful and that it isn’t always efficient to count by ones.

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The Mathical Book Prize is an annual award organized by the Mathematical Sciences Research Institute (MSRI) with an aim to “inspire a love of mathematics in the everyday world in children of all ages.” We previously highlighted some of their selected titles that introduced children to the principals behind counting.
In another look at books awarded the Mathical Book Prize, we focus on three books that go beyond one-to-one counting. These books provide compelling contexts that help primary grade children understand why more advanced counting is useful and that it isn’t always efficient to count by ones.

  1. Sheep Won’t Sleep by Judy Cox

    Advanced Counting, Books for Primary Kids to Go Beyond Counting by Ones Sheep Wont SleepSheep Won’t Sleep (2018 Winner) brings a colorful and creative twist to the dilemma of what to do when counting by ones doesn’t work. When she tries counting sheep, Clarissa’s eyes pop open to discover 10 wooly sheep crowding into her bedroom. So she tries counting alpacas by 2’s only to have 20 of them join the sheep. When she switches to counting llamas by 5’s, 20 more wooly critters show up; counting yaks by 10’s brings another 50 creatures. When she counts them all back down, rolling all their wooly threads into a ball, Clarissa finally gets to sleep.
    Besides the pictures of ever-increasing groups of wooly animals, this book invites questions about how to select the most efficient “friendly” number for skip counting. Create a large classroom chart (that might be fun to share on a classroom bulletin board) about what collections might be best counted by 2’s, 5’s or 10’s. An easy place to start is to figure out how many body parts in total there are in the classroom. It’s easy to count eyes and ears, hands, and feet by 2’s, for example. In the same way, counting how many fingers or toes calls for counting by 5’s and might also be done by 10’s. It might be fun to have groups of children work on illustrating the chart as well as writing out the counts.

  2. Max’s Math by Kate Banks

    Advanced Counting, Books for Primary Kids to Go Beyond Counting Max Math by Kate BanksThe idea of counting sheep to get to sleep comes up again as a way to end a busy night described in Max’s Math (2016 Winner). Max and his brothers travel though Shapeville and Count Town, passing many sets of things to count such as five clouds, two buses, four triangles, and one square. They find numbers everywhere they look! Have your students look for everyday collections of things that come in equal sets (i.e. pairs of shoes, boxes of crayons, packs of trading cards) to make their own counting stories.

  3. One Big Pair of Underwear by Laura Gehl

    One Big Pair of Underwear (2015 Winner) is bound to appeal to 5- to 8-year olds who think just the word underwear is hilarious. However, they are virtually sure to be willing to continue through the book, playing with the idea that when you have an even number of objects and an odd number of critters who want them, it just isn’t fair! For example, if you have 4 large jars of candy bars and 5 greedy goats in red cars, there’s nothing left for the 1 goat who drives up to find 4 empty jars and no candy bars. It isn’t until 20 pigs decide to go piggybacking and take rides on 10 slides that everyone else decides that sharing might be the way to go. The final pictures show all kinds of sharing and swapping going on that kids can have a grand time putting into number sentences. Once they have done that, many might love to pair up with a classmate and think up some fair share tales of their own that push mathematical thinking into more advanced counting and beyond one-to-one counting.

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Preschool Number Sense Routine Perfect for Transitions https://earlymath.erikson.edu/preschool-number-sense-routine-perfect-for-transitions/ https://earlymath.erikson.edu/preschool-number-sense-routine-perfect-for-transitions/#respond Tue, 11 Sep 2018 20:03:44 +0000 https://earlymath.erikson.edu/?p=14320 Moving from one activity to another just got a lot more mathematical with this simple routine that builds early number sense with preschoolers.

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Moving from one activity to another just got a lot more mathematical with this simple routine that builds early number sense with preschoolers. This dot card transition requires only index cards with 1 to 4 dots in various arrangements. Putting them on a ring makes this a handy and quick routine for those in-between times.

In this video you can see how the teacher asks children to show the same amount of fingers as dots on a given card. Much more powerful than traditional numeral flashcards, dot cards focus children on the cardinal amount of a set of dots, while connecting the number word to its quantity. Having to produce a similar-sized set of fingers requires children to translate from one representation (dots) to another representation (fingers) of the same quantity.

Notice that the dot cards are flashed for just a short time. This routine builds children’s ability to subitize — that is, to see small quantities and instantly know how many in the set without counting. Therefore, while this routine can deepen over time, the dot cards remain small quantities up through 4 (or maybe 5) that children can perceive without needing to count.

Instead, vary the questions you ask over time, such as:

  • Can you show me that number of fingers a different way?
  • Can you show me the same number of dots using two hands?
  • Can you show me one more than the number of dots on my card?
  • Can you show me one less than the number of dots on my card?

This routine is easy to differentiate in the moment by the dot card and/or question asked of different children. And, there are low language barriers for dual language learners.

What an effective way to infuse more math into your day!

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