Using Children’s Literature to Reinforce Counting and Cardinality

posted by Dr. Jeanne White

Young children love it when an adult sits down and reads a book to them, carefully studying the illustrations before the adult can turn the page.  Why not seize these opportunities as a way to introduce or reinforce mathematical concepts?  There are four reasons why I like to use children’s literature as a mathematical resource:

  1. Literature can offer examples of real-life problem solving.

When I read The 3 Little Pigs to a child, we discuss how many pigs there are and how each one has a way to solve the problem of how to prevent the wolf from blowing down their house.  Even though pigs can’t really talk or build a house, the young child begins to understand the idea of a problem and solution as well as the lesson that sometimes we have to go back and try a new solution.

  1. Children can discuss and demonstrate how characters use math.

In the book, Pete the Cat and His Four Groovy Buttons (Litwin, 2012), Pete the Cat sings a song about his four buttons.  But as he loses one button at a time, he alters his song to include three buttons, then two, then one.  Young children can see, and hear, how Pete the Cat uses math in his everyday life by counting the remaining buttons each time he loses one.

  1. The text can provide common language and context for problem solving situations.

When I would read the book, The Doorbell Rang (Hutchins, 1986), to my primary students, we used little chocolate chip “cookies” cut out of brown tagboard and small paper plates to act out the story.  On the first page, Mom makes 12 cookies for Victoria and Sam to share.  This provides an opportunity for children to talk about how to distribute the 12 cookies on the two plates and then how to make sure the same number of cookies is on each plate.  Throughout the story, more children come to the house to share the 12 cookies, which are continuously distributed evenly among the growing number of children.

  1. Children can apply mathematical concepts with literature.

In the 12 years I have been teaching math methods for the pre-service teachers at Elmhurst College, the teacher candidates are always amazed at the number of children’s literature books used to teach mathematical concepts, from PreK through eighth grade.  I have also conducted workshops for preschool and kindergarten teachers, as well as for families of young children, who are also surprised at how easy it is to use literature to teach and reinforce mathematical concepts including counting, patterns, geometry and sorting.

Using Children’s Literature for Counting and Cardinality

red dots in a row

Young children learn to count to ten with meaning—they should not only be able to rote count from 1 to 10 but be able to count up to ten objects.  Adults and older children can become role models for counting and demonstrate the concept of cardinality, the awareness that the last number said is the total amount. When first introducing the concept of counting, use a book with the same objects on each page such as Ten Black Dots (Crews, 1986).  As you read each page and model how to count the number of black dots on each page, the child only has to pay attention to quantity since each object is the same size, color and shape.  Then practice counting up to ten circle counters so the child is only paying attention to the quantity.red dots in groups of 5

 

Once children are able to count objects that are the same size, color and shape, read a book such as Math Fables (Tang, 2004), in which animals are shown in different configurations along with a rhyming fable, from one spider up to ten beavers.  This book can be read to children again when they are ready to break down numbers and group them into more manageable and familiar amounts.

colored frogs

 

An activity that can follow this book can be counting familiar objects such as toys or food.  Allow the child to touch each object while counting, whether counting illustrations in the book or toys on a table.  The child should also be able to repeat the total number of objects.  For example, “One, two, three.  Three frogs.”

yellow dots

 

Once children can count up to ten objects, they can begin to learn the complements of ten (one and nine, two and eight, etc.) with the book Ten Flashing Fireflies (Sturges, 1995).  In this book, two children are outside collecting fireflies in a jar.  First there is one firefly in their jar and nine fireflies in the night sky.  Then they catch another firefly and two can be seen in the jar and eight in the night sky.  After repeated readings of this book, try playing a game in which some “fireflies” are in the jar and some are in the sky.  Show the number of fireflies in the sky and have children figure out how many are in the jar.  They can have ten counters of their own to help them figure out the math problem.

 

I hope you have as much fun as I do, reading children’s literature and creating related math activities to introduce and reinforce these counting and cardinality concepts.

Where’s The Math?

Math is a natural way of thinking and making sense of the world. Mathematical situations arise every day. You have to be ready to notice the math all around us and to engage children in doing and talking about math. Attribute is a mathematical idea that arises very early. Attributes are properties or qualities that allow us to describe & classify the world around us. We perceive attributes of the world around us through our senses. Attributes can be used to group. Attributes can be described with increasing precision.

What might you see or hear if children are thinking mathematically about attribute? They might be matching objects, describing objects or sorting objects. They might be paying attention to color or shape or size or texture.  If you notice children noticing and using attributes, you might ask them: Why do these go together? Why do these not go together?

When adults are comfortable talking about math, children will share ideas without prompting. Here’s an example from a preschool classroom:

Chris & Tracy approach their teacher with excitement: “Look, our shoes are the same so they are a group! There are 4 shoes, 1, 2, 3, 4 … It is a group of shoes with holes”

crocs

Teacher: “I see you have a group of 4 shoes with holes. My shoes have little holes on the strap where I buckle the shoes. Can my shoes be part of your group?”

Chris & Tracy: “No, teacher, you have to have big holes all over to be in our group”

Teacher: “I see … you have made a group of shoes with big holes all over. Does anyone else have shoes that belong to your group?”

In a kindergarten classroom, a child runs up to the teacher and says, “A triangle equals a square!” Some teachers might say, “What do you mean? Triangles have 3 sides, and squares have 4. They’re not the same.” However, this mathematically sensitive teacher says, “What do you mean?” The child answers, “Come see!” The child leads the teacher over to the block area, where there are a lot of unit blocks for the children to play with. (Do blocks give you an idea of what the child might be thinking?) This is what the child showed the teacher:

blocks

Two square blocks can be put together to make the same shape as one longer rectangle block.

Two long triangle blocks can also be put together to make the same size rectangle shape.

Therefore, one of the squares takes up the same space as one of the triangles. In other words, they are worth the same, or, as the child says, they are “equal.”

This child was thinking hard about both the attributes of these shapes and the relationships between them. One might even say that the child was doing algebra, because they were using equivalences:

2S = R   and   2T = R           Therefore, 2S = 2T               Therefore, S = T

In both of these scenarios, the teachers are building the children’s understanding of foundational mathematical ideas and their confidence in using math to make sense of their world.

I hope that you will be open to the joy of finding math any time, anywhere with the children in your life!

What Are They Thinking?

posted by Lisa Ginet

It is hard to know what is going on in anyone’s brain. Even when asked to explain ourselves, we cannot always express our ideas clearly. Young children, who are still developing both their communication and reasoning abilities, have an especially hard time explain their own thinking in words or “showing their work” when they are solving problems. Why does it matter to understand what children are thinking about? It helps us to respond to them in ways that nurture competence and confidence.

So, how do we figure out what’s happening in children’s minds? We need to watch what they are doing and listen to what they are saying for clues, then interact with them intentionally. Here is an example from a preschool classroom:

From nearby the teacher has been watching Jenny and Samantha work with the materials at the sorting station.

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materials

 

 

 

 

 

 

The teacher has noticed the children taking the straight items from the collection without talking and putting them on the black paper. The teacher also noticed that Samantha takes each item and rolls it with her hand before she places it on the paper. After the paper is almost full with items, the teacher decides to approach the children.

Teacher: Tell me about your group.

Jenny: Son Palitos (These are sticks)

Samantha: Uh… no..no

Teacher: Samantha you don’t seem to agree with Jenny. What is your group then?

Samantha: A group that rolls

Teacher: Are you saying that what makes this a group is that all the things in it roll?

Samantha: yes, see (Samantha shows how different items in her group roll)

Teacher: Could this be part of your group? (The teacher offers a circular wooden object)

Samantha: (Samantha takes the wooden objects and tests if it rolls) Well, it kind of rolls.

(Samantha then adds the object to her paper)

Teacher: What should we call the group then?

Jenny: Cosas que ruedan (things that roll)

Teacher: I can see why you are naming your group: “things that roll.” There are sticks, pens, brushes, markers, pencils, straws and round pieces of wood but they all roll.

This teacher is drawing from a repertoire of intentional responses that help to surface children’s thinking:

  • Stop & Look – Take the time to observe what a child is doing. Try to figure out what the child’s ideas or goals are.
  • Say what you see – Use precise, descriptive language to describe what you notice. Provide labels to actions or structures that are mathematical.
  • Re-voice what you hear – Use precise, descriptive language to echo and expand on what a child says.
  • Check with the child – Always ask child to confirm your understanding of their ideas and intentions.
  • Wait – Allow child the time to react or respond to an adult prompt
  • Use comments / questions to invite / provoke children’s thinking

We will never be able to be inside children’s brains, but we can closely attend to their actions and words, and then intentionally respond in ways that draw out their ideas. The more we understand about what children are thinking, the easier it will be to help them love, understand and use math.

 

Math is fun? Really it is.

posted by Lisa Ginet

“Let’s do math!” I often say at the start of a workshop. This may lead to some panicked looks or trips to the bathroom. If I say, “no pencils or calculators involved,” then a few people will laugh, and most will look more willing to try what I suggest …

math-workshop

What are these people doing? The Counting Calisthenics! If you need a break from the computer, you can do it, too! Stand up and count while you move:

Touch toes – “1”

Touch knees – “2”

Touch hips – “3”

Touch shoulders – “4”

Throw hands up – “5!”

Continue counting while repeating movements

(toes, 6; knees, 7; hips, 8; shoulders, 9; hands up, 10 …)

Keep going until you want to stop.

After you sit back down, consider this question: If you kept doing the toes, knees, hips, shoulders, hands up counting calisthenics, what movement would you be doing at “456”?

It would be quite exhausting to actually keep doing the counting calisthenics all the way to 456, but I expect that, if you think about it a little, you will be able to figure out that you would be touching our toes at “456.” You probably noticed that your hands were up in the air every 5th number, and then the cycle started again. So, your hands would be up in the air for “455” and back at your toes at “456.”  You are using what you know about the structure of our base-10 number system and the pattern of the calisthenics movements to arrive at an answer. You are doing math!

We at the Early Math Collaborative want to encourage you, and all adults who spend time with young children, to build new math ideas and new math associations by engaging in fun and meaningful math activities. Why? If adults are going to engage children in doing real math and constructing authentic mathematical understanding, then the adults need to exercise their own “math muscles.” Because many adults have bad memories of math in school, they often avoid doing math with children. It can be hard to engage adults in exploring foundational math if they don’t feel good about math and don’t think they can do math. Just as learning math facts from flashcards is not an effective way for children to become fluent and flexible problem solvers, blindly following activity directions will not help adults understand the math in the activities or respond effectively to children’s comments and questions.

“Counting Calisthenics” is just one of our Adult Learning Activities. While they involve basic math concepts, Adult Learning Activities are not children’s math that we are asking adults to do; nor are they activities to repeat with young children. We have designed these activities so that they:

  • pose a puzzle or problem;
  • are interesting enough to capture and retain adult attention;
  • are easy to implement;
  • may have more than one solution or route to solution;
  • clearly focus thinking on foundational mathematical idea(s).

We hope that having fun doing math will help convince you that the world is not divided into those who are good at math and those who aren’t. We can all be doers of math, and we can build the same confidence and excitement in the children in our lives.

 

 

What is Math?

posted by Lisa Ginet

When you hear or see the word “math,” what do you think of? Your high school algebra class? Balancing your checkbook? A geeky engineer with pocket protectors? When you add “early childhood” to “math,” what do you think of then? A little one learning to say, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10”? A bright poster with a circle, triangle and rectangle neatly labeled? All of these are common ideas about what math is and how math starts, but none of them are what I mean when I say “foundational math.” Before I tell you what I do mean, I want you to try something.

Look at this image:
shapes-pictureConsider this question:

Which of the figures are the same?

Often when I ask this, a person says, “They are all different from each other.” Another says, “They are all the same; they are all shapes.” Both of these answers make sense, but I often ask people to keep looking to see if anyone can come up with another answer. Usually, people then generate these six answers:

  • top two shapes are both orange
  • bottom two shapes are both green
  • left two shapes are both striped
  • right two shapes are both solid
  • top left and bottom right are both circles
  • top right and bottom left are both triangles

In fact, although none of the two shapes are identical to each other, any two of them are “the same” in some way. Figuring this out involves logical thinking about the attributes of the shapes.

This shape activity demonstrates one definition of mathematics – a logical way of thinking that allows for increasing precision. We can use math to make sense of the world. We can use math to solve problems. To use math in these ways, though, we cannot just memorize facts. We must build our own understanding, so that we can think flexibly in different situations. Without a strong foundation, a tall building would not stand for long. Likewise, without a strong foundation in mathematical concepts, children can struggle to understand the more complex mathematical thinking they need later in life.

At the Early Math Collaborative, we have developed a set of 26 “Big Ideas” – key mathematical concepts that lay the foundation for life-long mathematical learning and thinking. While these concepts can be explored at any early age, they are powerful enough that children can and should engage with them for years to come. As you engaged in the shape activity earlier, you were using two of the Big Ideas:

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

Most likely, you were not thinking about these ideas consciously; rather, you were looking at the shapes and thinking about them. You were using math to make sense of the puzzle I posed and to come up with a solution. This type of math may not match your prior notion of math as quickly-recalled facts and properly executed procedures. You may need to set aside some of those notions in order to develop a deep understanding of foundational math that will help you have fun doing math with children.

 

Introducing the Guest Blogger for the Month of December and January – Lisa Ginet

The holidays are nearly upon us and for that reason, our guest blogger’s posts will appear over the course of the next two months as we send out 2016 and welcome in 2017.

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Let’s welcome Lisa Ginet to the Math at Home blogoverse.  Lisa comes to us from the Early Math Collaborative at the Erikson Institute where she has been an integral member for many years.  Her expertise comes from years in the early childhood classroom, and as an adjunct faculty member in Chicago.

Since 2009, Lisa Ginet has been a member of Erikson’s Early Math Collaborative, which is transforming the understanding, teaching and learning of early mathematics from the ground up. Before that, Lisa spent more than a quarter century as an educator in various roles: classroom teacher, child care provider, parent educator, home visitor, teacher trainer, and adjunct instructor. She has worked in diverse settings, from child care centers and elementary and middle schools to community colleges and private universities.

 

During her time working with Erikson’s Early Math Collaborative, Lisa has thought a lot about the essence of foundational math, engaging adults in enjoying and doing math, bringing to life children’s mathematical thinking, and authentic mathematical environments in early childhood classrooms. Lisa will reflect on the lessons she has learned and what they mean for those of us who want to help children love, understand and use math.

10 Math Story Books to Gift Children at the Holidays

Information and ideas presented in story form often stick better than rote memorization. As you plan for the holidays this year, consider adding any one of these charming and engaging children’s math picture books to your family library.

Visualizing large numbers, understanding fractions, having fun with division, or just fine-tuning how you approach problems – all of these books present fun and winning ways to bring math into your children’s lives.

How Much is a Million by Stephen Kellogghow-much-is-a-million

Master children’s book writer Stephen Kellogg’s book has been making large numbers understandable to young readers for generations, and is still the best book out there for visualizing them in an accessible way. No one who has ever read this book will forget that if a billion kids made a human tower it would reach past the moon.

The Pancake Menu by Lucy Ravitch

the-pancake-menu

Lucy Ravitch, who blogs at Kids Math Teacher, has created a winning combination of cooking and eating out in this book aimed at getting kids comfortable with adding costs. It encourages readers to make their own fun pancake creations and figure out how much customers would have to pay if they were running their own at-home restaurant. An imaginative, active good time.

 

Sir Cumference and the First Round Table by Cindy Neuschwander

sir-cumference

Clever all around, this title by Cindy Neuschwander is a rarity – a math picture book with a really great and memorable story. It is a smart take on an inventive side story of the Arthurian saga, about King Arthur’s quest to find a table that can accommodate all of his squabbling knights. If you love it, Sir Cumference and the First Round Table is the first in a series.

 

Which One Doesn’t Belong? by Christopher Danielson

which-one-doesnt-belong

This brilliant new book by math educator Christopher Daniels, of Talking Math With Your Kids fame is less about teaching right answers than about getting children to ask the right questions. On each page, readers are invited to pick the shape that doesn’t belong and talk about why they picked that shape. But there is not correct answer – just an opportunity to talk about how we think. A must-have.

 

What Do You Do With a Problem? by Kobi Yamada

what-do-you-do-with-a-problem

It’s a shame that when we talk about solving math equations we equate it with a “problem.” However, this charming and emboldening book What Do You Do With A Problem? by Kobi Yamada. A dark, splotchy problem follows around the unnamed character and haunts his days until he decides to face it head-on. Along the way, this endearing story shows that maybe the problem wasn’t as scary as he thought after all.

The Best of Times by Greg Tang

the-best-of-times

Greg Tang has a number of math picture books out, but I like this one best. In it, he provides simple instruction for multiplying numbers. Here’s a choice example: “Four is very fast to do, / when you multiply by 2. / Here’s a little good advice – / please just always double twice!”

 

 

What’s New at the Zoo by Suzanne Slade

whats-new-at-the-zoo

It’s a simple premise: A visit to the zoo yields possibilities for learning how to add. Four monkeys are all carrying babies – how many monkeys are there? What’s New at the Zoo is perfect for early math learners because of how simple it’s questions are and how it engages readers in a real-world problem. Add to that perfect rhyming and some cute pictures and you have the perfect addition to your library.

 

Full House: An Invitation to Fractions by Dayle Ann Dodds

full-house

Dayle Ann Dodds makes this list twice. In Full House, she wraps two learning stories in one book with the tale of Miss Bloom, the proprietor of an inn with six rooms. As guests arrive, Miss Bloom calculates how many of the rooms have been filled. Later, they all share a cake cut into six equal pieces. The illustrations are charming, the lesson spot-on, and the characters funny as can be.

 

The Great Divide: By Dayle Ann Dodds

the-great-divide

When 100 people set out on a marathon, some of them don’t finish in this rousing story by Dayle Ann Dodds. More a concept book than a memorization story, The Great Divide follows the marathoners as, on each spread, half of them encounter a challenges taking them out of the race. In the end, there can be only one – but getting there is the fun in this book.

 

That’s a Possibility: A Book about What Might Happen by Bruce Goldstone

thats-a-possibility

Some math concepts have as much to do with learning the meaning of words as anything else. Enter That’s A Possibility, a book that teaches students about the subtle differences in the language of possibilities. In popping, colorful illustrations, it leads young readers through a series of situations, defining the terms possible, probable, impossible, and certain along the way. Possibly a must for Christmas?

 

 

10 Things to be Thankful For – The Early Childhood Teacher Edition

10.  In addition to our 3 square meals, we get to eat snack twice a day.

9.  We don’t have to wear pantyhose, ever. 

8.  Our work includes snuggling.

7.  We don’t have to worry about the Common Core.

6.  We know what “ooblek” is even if our smartest friends don’t.

5.  Our workday includes reveling in the little things with childlike wonder and excitement.

4.  No sitting at a desk unless it is pretend.

3.  Nobody cares if you have paint on your sweater or a sticker on your forehead.

2.  Sometimes we mistakenly get called “Mommy” or “Daddy” and it gives us a warm and fuzzy feeling inside. 

and… the #1 reason Early Childhood Teachers are Thankful This Thanksgiving is:

We get to spend each and every day with the best kind of people- the little kind.

Math Can Be Learned Through the Feet

posted by Emily Grosvenor

malkerosenfeld

Q & A: For dancer, educator and homeschooling mom Malke Rosenfeld, math can be learned through the Feet

Educator Malke Rosenfeld believes she had a typical relationship to math as a child.

 

 

“I went K-12 through public school disenchanted with math, never feeling personally connected,” Rosenfeld said. “I always wanted to know the “why” of math but never getting a satisfactory answer.”

For years, she asked the same question, never finding something that clicked. But one day, at a dance workshop in California, after donning a pair of golden tap shoes to learn a French Canadian waltz clog, she realized her feet could be a percussive instrument. This was just the beginning for the creator of Math in Your Feet©, an interdisciplinary approach to learning math through dance. In her new book, Math on the Move, just out from Heinemann Publishing, Rosenfeld takes the idea that it isn’t just the mind, but the mind and the body, that synthesizes learning. Math and dance, it turns out, are mighty bedfellows.

I spoke with Malke Rosenfeld about the Math in Your Feet© program she brings to classrooms as a visiting artist and what children can learn about math through dance.

Emily: When was the turning point for you for discovering that math and dance can be combined for learning? Can you describe that moment?friday-zone

Malke Rosenfeld: After about five or six years of touring with my Celtic band and teaching in the Carolinas my husband got a great job opportunity in Indiana. During the first year or so after we moved I started substitute teaching in local schools. I got placed in a lot of resource rooms and elementary classrooms and interacted with a LOT of kids who were disenchanted with the whole math thing. I thought back to the Drum with Your Feet program I had created about six years earlier, where kids were engaged and highly creative once they were given personal agency to make something on their own. After one particularly difficult day with kids in the resource room I thought to myself, “I wonder if there is math in what I already do with kids?” The idea stuck in the back of my head for a while and then, one night, I had a dream where the whole concept of Math in Your Feet came to me – I woke up knowing I first needed a math education expert to help me make meaningful connections between the disciplines; I wanted to create a five-day math-and-dance residency; I wanted to eventually create teacher PD to help them experience and learn to use this approach in their own classrooms. Finally, I wanted to create a set of curricular materials to support teachers in the implementation of the math and dance work. My new book is the final piece of the puzzle.

Emily: Can you think of any formative moments that changed your relationship to math?

Malke: I think that my relationship to math changed slowly, and over time, for the better as an adult about five years into my teaching career. I had no math anxiety at all when I first came up with the idea

for Math in Your Feet. I knew from my early meetings with my co-collaborator Jane Cooney that I was on to something and the biggest challenge for me was to figure out how to give the math and the dance the same amount of attention in the classroom. In an interdisciplinary context like this we can’t make up the math to fit the dance, or make up the dance to fit the math. So, it takes a little extra effort to see them both happening at the same time.

Overall, I was mostly just curious if my idea would work and was ready to learn as much as I could. Elementary math turned out to be a wonderful place for me to hang out because it’s got big ideas that are absolutely fascinating to me! The idea of composing and decomposing objects or numbers, and thinking about the relationship between parts and wholes totally connects with what I do as a dancer. Thinking about sameness and difference is a happy place for me and, again, is one of the main ways we make our choreography interesting. Finding, making, and describing patterns and exploring transformations is just fantastic fun.

My relationship to math deepened even further when I needed to homeschool my daughter for first and second grades due to health reasons. I used our days as an opportunity for me to experience elementary math learning up close and personal, and as a chance to (re) develop my number sense from the ground up. My biggest resources were the Natural Math and Let’s Play Math! blogs. As I explored and remixed math activities with my daughter I noticed a change in myself. I had found a kind of math experience that was personally relevant to me as a creative, visual, interesting, and conversational experience. It was similar to what we were already doing in Math in Your Feet, but that is the students’ experience, not mine. This time I was experiencing it from the inside! Playing math with my kid helped me (re) build my number sense along with my kid and it was fun to enjoy playing with numbers. One time she was filling in a hundred chart and finding patterns. She exclaimed “I feel like I’m IN the chart, mama!” We also went on lots and lots of math walks around town and found and talked about the math we discovered. It was a mathematically rich few years.

Emily: Why have you decided to devote yourself to this interdisciplinary pursuit?

Malke: If you could peek into my mind you would see a huge, multi-ringed Venn diagram. I see the big picture first and am always seeing overlaps and connections between disparate ideas. Traditional K-12 curriculum presents subjects in isolation from each other but my experience is that learners benefit greatly when they have a chance to explore those places of overlap. The big picture mathematics is about exploring relationships between one idea or object and another. We do the same when we find a nice overlap between two disciplines and this gives us the opportunity to expand our understanding of the world as a whole, not just its individual parts.

Emily: How would you explain “body knowledge” to someone who has never heard of it?

Malke; “Body knowledge” is a phrase coined by the late Seymour Papert who essentially invented computer programming for kids. His work intentionally harnessed the child’s lived experiences in the world as a way to investigate more formal mathematics via computer programming of a little object he

called “turtle.” A lot of this work is similar to the Math in Your Feet work – children working independently or in teams within a specific system/constraint, investigating and creating units of commands or patterns in a spatial and geometric language and, along the way, fine tuning their intentions and results.

mathinyourfeetWe can use body knowledge (more formally known as embodied cognition) in a couple of ways for math learning. The first is to change the scale of a mathematical activity from static, 2D form on a piece of paper to “body – or moving-scale.” In this case we make the investigation big enough for the whole child to be up and moving. An example of this is a large number line or hundred chart taped on the floor. This new scale has the potential to create new insights and understanding in the learner.

In Math in Your Feet we are building body knowledge. The constraints of the dance system require children to enact specific foot positions, movements, and directions to create their patterns. As they do so there is potential for them to be developing new skills and understanding of the mathematical ideas we are using during the choreography.

Emily: I like this idea that it’s not learning object that is memorable or important in learning, but the context. How does Math in Your Feet provide context for learning?

Malke: The context of learning for Math in Your Feet is that we take this idea that experience is key to learning and encounter math in an utterly new and novel way. And, not only that, we explore the math/dance overlap in multiple modes: we use it to make dance patterns, we talk about it, we use it to compare and contrast one dance idea to another, communicate and makes sense of it through written reflections, maps, word studies, and our own performance. No one representation can express a math idea in full; no one learning mode allows the learner, and especially our young learners, to make sense of an idea and make it their own. Emily: What are the mathematical concepts that children can learn Math in Your Feet?

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Malke: I have a whole chapter in the book titled “How is this Math?” In it, I go into great detail with lots of examples of how the moving body is best positioned to express the verbs of math. So often we think of math as the objects or nouns of math, the things we can point to on the page, that don’t move, that are in the perfect position to be identified and talked about. When we have the opportunity to explore the verbs of math by reasoning and thinking spatially with our bodies, physically exploring composition and decomposition of units and patterns, or enacting one or multiple rotations, transforming an object in real time, children get a chance to almost literally wrestle with highly action oriented math ideas, things we can’t always understand by looking at static images on the page. Mathematical practices are the ultimate mathematical verb and are about what we do and how we think as we endeavor to solve mathematical questions. Most of us left school without ever getting a chance to really get to know the action side of math; what better way for elementary school learners to make friends with and explore this kind of math than with a whole, moving body?

Emily: Math in Your Feet seems to create a powerful opportunity for spatial learning. Why is spatial learning important to math? From your book it seems like it’s the great connector, the thing that makes math meaningful to everyone.

Malke: Spatial reasoning is the foundation of mathematical thought and critical to mathematical achievement. It is also one of the verbs of mathematics that helps us “do” math. We use spatial concepts every day in math class to help us make sense of and learn math. For example, a number line is a spatial construct that helps us visualize amount, magnitude, operations, and even negative numbers. Arrays, calendars, and analog clocks are all spatial objects that help us makes sense of numbers in different by ordering information in structured ways. We are all born with the capacity to think spatially, and our cognition is literally built from birth through our physical explorations of the space we live in. Spatial and math concepts refer back to our experiences moving and living in and through our lived physical spaces. The more we include and develop spatial tasks and skills in our learners, or at least recognize their presence in our math classrooms, the better off our learners will be.

Emily: Can you talk about what you do when you visit a class?

Malke: Basically I introduce myself, and then get them dancing with me as soon as possible. I have learned from long experience that nothing I say at the start of our time together will make any sense at all. Children need to explore and experience something tangible first before the concepts make sense; these kind of experiences give them something tangible to think and reason about and from their conversations can begin.

After we have some dancing under our belt and we have previewed the big picture of the challenges ahead we spend the rest of our time making new things, in this case foot-based percussive dance patterns. In a 5-day residency we can usually get pretty far down the math-and-dance making road. In a one-time session I still look for ways to give them some kind of challenge where they are can have some kind of small making experience.

Emily: How do children respond to this kind of learning?

Malke: Sometimes it takes them a little while to accept that they’re in charge of their bodies and their ideas. Though I am at the front of the classroom sometimes, my role is one of facilitator rather than expert. There is usually a point where they realize I’m only going to consult with them, and that they are the main generators of the work.

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Emily: What do you want them to take away from the experience?

Malke: That they are capable of creating something new. That their questions, and wonders, and ideas and thoughts matter and are interesting to me. That math can be a tool for problem solving both inside and outside math time. That being a creator learning in a community of creators is really satisfying.

Get ideas and join the discussion on the Math on the Move book blog and join the Facebook community Math on the Move to keep the conversation going.

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Engage Creative Children in Math Class with Tessellations

tessalation-coverFor math teachers and at-home educators looking to bring some creativity into the classroom, tessellations offer a lot of fun activities and possibilities. For some students, they might be just the right lesson to get students thinking about the practical applications and design possibilities of math.

I should know. I was one of those students. By the time I was in 4th grade, I had decided I was “bad at math.” Looking bath, I wasn’t really all that terrible at it, it just didn’t come as easily to me as other subjects. I struggled to do the timed multiplication and division worksheets in math class, and when I didn’t immediately get it, when I wasn’t the fastest, or the most confident, I was sure it wasn’t for me.

Here’s something I knew I was good at: Drawing.

So when a fourth grade gifted teacher introduced us to tessellations, I didn’t make a direct connection to math application. I didn’t even associate tessellations with math. All I knew was that it was fun to make interlocking patterns and to imagine them spreading in every direction across a plane.

Later, based on this life-long love of pattern from that class, I would write my children’s picture book, Tessalation!, to offer the world a story to use in the classroom, a story about a little girl named Tessa who hides in the patterns of nature.

In the months since, I’ve watched lots of children make their own tessellations. I’ve seen preschoolers identify tessellations out in the world (cutest thing ever, three-year-old’s saying: tessellation!), and I’ve seen the abundance of creativity that happens when school-age children make their own.  A tessellation is a great way to make math fun for kids.

How to introduce elementary school children to tessellations

I’ve written before about how to start talking about patterns and tessellations with children as young as pre-schoolers. But school-age children are capable of having even more fun with pattern by making them themselves. It’s easy, it’s fun, and it’s a perfect way to harness children’s natural creative drive in a math learning setting.

Here’s how I introduce tessellations during my school visits to school-age children.

  1. Read Tessalation! I’ve had success reading the book to younger children, but it is written for ages 5-8, with a rhyming meter created to mimic the feeling of locking pieces and patterns. I generally read it through once and ask the students to talk about the relationship between what is happening on the left side of the page compared to the right. Here’s an :

 

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2.   Talk about what a tessellation is. Most children can identify what a pattern is, so this step is intended to get them thinking about a specific type of pattern – the interlocking, repeated pattern with no spaces in between. Here are some examples you can point to if you’re looking to illustrate what a tessellation is by showing them objects they have already encountered.

  1. A beehive:hexagon1Or even a chain-link fence:chain-link-fence

    A hounds tooth coat:houndstooth

    1. Experiment. Talk about which shapes tessellate. There are three regular polygons that tessellate, square, triangle, and hexagon. You can look back through the book and see which of the tessellations are based on a square, which are based on a triangle, and which started as a hexagon.
    1. DIY Tessellation. Children between the ages of 6-10 are at the perfect age to begin experimenting with making their own tessellations. You can use triangles, squares, or hexagons, but I find, for the sake of saving time and making it easy for all levels, a square works the best. Here’s how to do it.
    2. Step 1: Cut out a square.tessellationtutorial1
    3. Step 2: Cut a shape into the right side of the square. The shape can look like anything, but don’t cut away too much.tessellationtutorial2
    4. Step 3: Take that shape, and add it to the left side of the square.tessellationtutorial3
    5.  

      Step 4: Cut a shape into the bottom of the square.tessellationtutorial5

    6.  

      Step 5: Take that same shape and place it at the top of the square.tessellationtutorial6

    7.  

      Step 6: You now have the shape you are going to tile, or tessellate. Flip it over and tape it so the pieces stay in place. Now take a blank sheet of paper and trace the shape in the middle of it.tessellationtutorial7

    8.  

      Step 7: Move the shape to the right, left, up or down, and tile the original shape into a tessellation.tessellationtutorial8

      Continue to trace the shape. Once the children have the entire page filled, they can decorate the tessellation as they wish. You can share the tessellations you make at the World Tessellation Day Facebook page.

      Don’t have time to do a full DIY tessellation lesson? If you head to my website, you’ll find  four tessellation coloring pages, both simple and challenging made from the pages of Tessalation!. They are free and downloadable PDF’s.

      Do you have a special way you are bringing creativity into your classroom? Share in the comments, we would love to hear from you!