Pattern walk at the Field Museum

posted by Lindsay Maldonado

Today I visited the Field Museum of Natural History. Another one of my favorite Chicago museums. The Field Museum houses thousands of artifacts from dinosaur bones to pottery and clothing from ancient civilizations. Again you may be thinking, math? Isn’t this a natural history museum? With thousands of artifacts on display, math is easy to find. Just a quick walk through the halls brings you upon any number of dioramas with countless animals of all shapes and sizes.

It’s easy to count animals (Big Idea: Counting) or classify animals (Big Idea: Sets) by their varying attributes like size or color – but when you start to delve deeper into the exhibit halls you’ll come across other kinds of artifacts. There are cases upon cases of decorative clothing and art from cultures near and far. In my recent visit I happened upon the Hall of Native North Americans exhibit.

Native North American exhibit

At first you’ll be enamored by the craftsmanship. You’ll wonder how long it must have taken to create something so beautiful and intricate. You’ll wonder why Native North Americans wore such adornments but then you’ll notice something else; you’ll notice the shapes and patterns woven together or threaded with beads that make up each artifact. There are circles, squares, rectangles, diamonds, and triangles intricately designed to create simple and complex patterns. We see color patterns too.

Patterns exist in the world, as we see here, and also in mathematics. Through patterns, we find sequences bound by a rule (e.g., a chess board is made up of black and white squares, with a predictable black-white, black-white or AB, AB pattern) that brings predictability and allows us to generalize. Hence, we can predict, with a good amount of certainty, what comes next. Let’s look at a couple of these objects. What patterns can you find?

 

beaded ornaments

beaded bagThe beaded bag has blue and orange flowers arranged in a simple ABAB pattern. Each row alternates orange flower, blue flower, orange flower, blue flower, etc. It’s easy to predict what comes next. We see a similar ABAB pattern in the beaded ornaments (i.e., yellow blue, yellow blue). One big idea of patterns is just this; the same pattern can come in different forms.

We also see more complex patterns when you look more closely at shapes. Can you see the pattern?

 

decorative artPatterns are found in many places and children are particularly attuned to patterns. As we observed, patterns offer a sense of predictability, which children desire (e.g., we create routines for children to add order and predictability to their lives). When children understand the rule of a pattern they are able to extend that thinking to other situations.

Keep talking about patterns in the classroom! You can search for more activities about patterns here.

Museums and Math: The Perfect Pair

posted by Lindsay Maldonado

The shedd aquariumMuseums are likely the most common setting for informal learning. Unlike formal learning (i.e., traditional classroom learning), informal learning is voluntary, unstructured, and learner-led. These settings provide a variety of learning experiences for a diverse group of learners. Museums offer opportunities to be hands-on with objects and even live animals. Museum visitors can observe objects and animals, engage with exhibits, participate in programs, and listen to chats and presentations. Museums afford visitors with flexibility and choice, offering a more customizable learning experience. This is particularly important when you consider the variability of learning styles within one classroom or one family. The ability to create an experience that suits the needs of many makes museums an ideal learning setting.

But you might be asking, museums and math? You might be thinking; how do I teach children math at a museum? There are science museums, art museums, natural history museums – but, there are no math museums. Well, there is one museum in New York that is dedicated to math but in general, math museums are hard to come by so it’s a good thing that math is all around us — all the time, no matter the setting.

Growing up in Chicago I remember visiting Shedd Aquarium often as a child. I would stand in front of the habitats, gazing up to observe small fish, big fish, colorful fish, dull fish, and everything in between. I was in awe of the diversity; it was kind of like reading Dr. Seuss, “One fish Two fish Red fish Blue fish.” There were so many fish, but there were also fish of every color, size, and shape. At the time I wasn’t thinking about math, but as I reflect back on that experience I know that math really was all around me. This experience is not unique; I see thousands of children visiting Shedd every year. As they gaze into the same habitats I did many years earlier, I can see the sense of wonder and awe in their faces. Knowing what I know now, though, I think about taking that moment of wonder and creating a math moment too. I think about using that awe and excitement as a springboard to a conversation about how many fish, how are the fish different, or how are the fish the same. These teachable moments are all around you when you visit a museum.

As we explore museums and math together in the posts to follow, let’s first consider the big ideas of early mathematics: sets, number sense, counting, number operations, pattern, measurement, data analysis, spatial relationships, and shape. These nine ideas laid out by Erikson Institute’s Early Math Collaborative provide the foundation for exploring mathematical concepts in and out museums. We’ll touch on many of these ideas as we explore some of my favorite museum exhibits. So for a moment, let’s focus our exploration on math in museums. Let’s reflect on the ways in which these big ideas exist in museums. Come join me on a mathematical adventure!

Using Children’s Literature to Reinforce Classification

posted by Dr. Jeanne White

When older children and adults perform chores such as doing laundry or putting away dishes, they may not even realize they are making decisions about how to classify objects into categories—washing the white clothes together and putting the plates together on a shelf.  Young children can begin to see how objects can be sorted into categories with the help of several children’s literature books.

Young children naturally sort objects by color.  Have a variety of books available that introduce colors and show objects of a particular color so children can make associations with the object and the color.  A fun book that can be used to explore colors is The Color Box (Dodds, 1992), in which a monkey named Alexander crawls into a box to discover a world where everything is black, then another where everything is white, then yellow, and so on until he ends up back in his world where there are objects of every different color.  Then provide groups of objects that can only be sorted by color such as Unifix cubes or square tiles.  Once children have sorted these objects by color, they can count the number of cubes or tiles in each group and compare them—which group has the most?  Are there more red cubes or more orange cubes? Then make several types of objects available for children to sort by color such as beads, tiles, cubes and blocks.unifix cubes scatteredunifix cubes and other shapes

After children have had several experiences sorting objects by color, they can begin to explore other attributes.  In the book, The Button Box (Reid, 1990), a boy visits his grandma and finds a box with hundreds of buttons inside.  He sorts them into categories such as buttons covered with cloth, sparkly buttons, metal buttons and buttons from uniforms.  He also lines them up based on size and color.  At the end of the story, the boy and his grandma close their eyes and each choose a button from the box.  Then they look at them and talk about all of the ways the two buttons are alike and the ways they are different.  Follow up the story with a game similar to the one in the story, by comparing two buttons or other toys that have similar attributes such as two dolls or two cars.

Another book that can be used for classification of objects is How Many Snails? A Counting Book (Giganti, 1988).  On each page there are illustrations of one type of object but they vary by size, color or design.  On the first page there are eight clouds for children to count.  Then there are more questions to answer, “How many clouds were big and fluffy?  How many clouds were big and fluffy and gray?”  Children can use their toys to count, sort and answer questions such as, “How many frogs?  How many frogs are yellow?  How many frogs are yellow with a green stripe?”  Incorporate science by introducing various types of insects and asking children to tell you how they can sort them—by putting all of the insects that are the same color together or by putting the insects with wings in a group. four frogs

When children sort and classify objects into groups, they are building a foundation for graphing and data collection.  grasshoppersThey can create unique ways of sorting objects into groups and once they know how to put objects into categories, they can help with all of those chores!

Using Children’s Literature to Reinforce Patterns

posted by Dr. Jeanne White

Young children naturally begin to create patterns with objects such as Unifix cubes or colored tiles, even if they do not realize what they’ve created is called a pattern. A child’s early knowledge of color or shape patterns can lead to later recognition of more complex patterns in large numbers and within the four operations.

A book that can be used to introduce young children to patterns in the environment is the book Math Counts: Pattern (Pluckrose, 1995).  The book contains photographs of patterns found in nature such as on leaves, flowers and insects.  The book also shows patterns found in familiar objects such as on a car tire, the sole of a shoe and wallpaper.  Encourage children to draw or photograph their own pattern discoveries such as on clothing, jewelry or furniture

.bracelets

jewelry boxAnother book that can be used to introduce patterns is Rooster’s Off to See the World (Carle, 1972).  In the story, one rooster decides to travel and meets two cats, three frogs, four turtles and lastly, five fish.  As he meets each set of animals, pictures of the animals are displayed in the upper corner of the right page.  Children can see the growing pattern of animals from one rooster up through five fish.  Eventually, all of the animals disappear, starting with the five fish.  The pictures of the animals appear in the upper corner of the left page and gradually disappear until only a picture of one rooster is there.  Children can see another pattern as the number of animals decreases from five down to one again.

Set up activities following this book such as displaying familiar object to create an AB pattern (using only two different elements in the pattern) for a child to continue.  Start with color patterns and say the colors aloud as you display each one, “Red, blue, red, blue.…”  After several examples of color patterns, use toys and say the name of the objects as you display each one, “plate, spoon, plate, spoon….”

patternspoonsOnce children have practiced recognizing and repeating patterns with cubes, blocks, toys and familiar objects, they can begin to listen for patterns in songs, stories and nursery rhymes.  A book that can be used as an example of a pattern set within a story structure is The Napping House (Wood, 1984).  It’s a rainy day and everyone is napping in the house, including a snoring granny.  But then the granny is joined by a dreaming child, followed by a dozing dog, then a snoozing cat, a slumbering mouse, and a wakeful flea.  Each of these nappers pile on the bed with granny one by one, and are introduced on each page, one by one, adding to the words from the previous page:  “And on that granny there is a child…and on that child there is a dog…and on that dog there is a cat….”

Encourage young children to listen for patterns when you read stories or to look for patterns in photographs and illustrations in books, on posters and other media.  Recognizing patterns sets the foundation for algebraic thinking—analyzing patterns, relationships and change throughout the study of mathematics.

Using Children’s Literature to Reinforce Geometry

posted by Dr. Jeanne White

As young children are formally introduced to the names of shapes, they begin to notice these shapes in their surroundings.  They see their plate as a circle and their napkin as a square when they eat dinner.  They look at the windows and doors in a room and recognize them as rectangles.  Tana Hoban’s book Shapes, Shapes, Shapes (1986) uses photographs of familiar objects such as pots and pans, and scenes such as construction sites, to present various shapes. Children will find more shapes on each page as they look at the photos again and again, and as they learn to name more shapes such as trapezoids and ovals.

An activity that can follow the introduction of this book can be allowing children along with family members to take photos of shapes in their home, their neighborhood or school.  They can display and compare the photos and name the shapes in each other’s photo.a door

a bureau

a lamp

In addition to two-dimensional, flat shapes, young children should be introduced to three-dimensional, fat shapes.  Reading the book, Changes, Changes (Hutchins, 1987), can open a child’s mind to the endless possibilities of how to arrange 3D blocks to build structures.  In this wordless picture book, a wooden couple builds a house but it catches on fire, so they must build a fire engine, then a boat to deal with all of the water, and so on.  Encourage children to find 3D objects in their environment such as food containers that represent cubes, cylinders, and rectangular prisms.  They can build their own structure with these containers and name them as they build.a pic of food boxes

Once children are familiar with the names of shapes, they can expand their vocabulary to include attributes of shapes.  The book, If You Were a Triangle (Aboff, 2010), includes illustrations of triangles that are slices of watermelon, Yield signs, faces of pyramids, designs on wallpaper, and more.  The text repeats the phrase, “If you were a triangle…” and lists attributes such as “three sides,” or “three corners” and introduces the terms polygon and angle.  At the end of the book, specific triangles are shown—equilateral, right and isosceles—along with examples of these triangles put together to form a new, composite, shape such as a rectangle or rhombus.  Children can look for triangles in their environment as well as practice putting the triangle Pattern Blocks together to form new shapes.a pic of 2 shapes red and greana pic of 2 shapes blue and green

Another concept children learn in early geometry is relative position.  Young children are gradually exposed to words used to describe the position of an object or person relative to other objects or people such as above, below, beside, in front of, behind, and next to.  Young children are also starting to distinguish between their right and left and are learning to move, and count, forward and backward.  A book that is fun for children to use to learn these concepts is Bug Dance (Murphy, 2002).  The bugs in this book go to school together and in gym class they learn a dance that teaches them to take steps to the right and to the left, then hop forward and backward.  Young children can perform the dance as the book is being read over and over.

After children have practiced their dance moves they can practice the terms in the book, as well as other position words, to describe the position of Pattern Blocks.  For example, children might say: the square is below the hexagon; the triangle is on the right of the square; the trapezoid is on the left of the square; the triangle is next to the square. a pic of 4 shapes

There are many children’s books that are written to introduce shapes, however many use the word “diamond” instead of rhombus.  I try to avoid these books or let children know a diamond shape is called a rhombus when we are learning math.

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.

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.

 

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

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

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      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

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      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!