A Math Meetup and A Magical Multiplication Puzzle Book

This summer, I held a physically-distanced math meetup for my son and five neighborhood classmates.   Here are my children setting up for our first meeting.

During each meeting, I incorporated a magical math book. One week we read The Boy Who Dreamed of Infinity and another Counting on Katherine. This post is about a magical puzzle book we used often in our 6 weeks together.

The Book

The Original Area Mazes: 100 Addictive Puzzles to Solve with Simple Math―and Clever Logic! features area mazes puzzles created by Ryoichi Murakami and Japanese puzzle master Naoki Inaba. Puzzle master Naoki Inaba was born in Nagoya, Japan in 1979 is one of the greatest puzzle inventors of all time. Inaba “is remarkable not only for the quantity of puzzle types he has invented but also for their minimalism and elegance.” (Alex Bellos, Introduction)

Ryoichi Murakami is CEO of El Camino, a premier cram school in Tokyo, and uses the area mazes with students often. “Area mazes are different, Murakami explains. “They are designed to develop flexible thinking-youthful thinking, if you will. They cannot be solved by repeating a process. You often need a stroke of inspiration to solve them.” ( The Original Area Mazes, p. 5)

Area Mazes was originally published in Japan by Gakken Publishing and later in the United States by The Experiment Publishing in 2017.

I would recommend this book for 3rd grade to adult. And once you’ve solve the puzzles in this book, there’s a second and third book.

The Math

These puzzles have a sudoko puzzle feel. However, in addition to using logical thinking, Area Maze puzzles require two mathematical ideas over and over. The first idea is the area of a rectangle is length times width. The second idea involves the relationship between factors and multiples (although it’s not necessary to use this mathematical language to solve, of course).

If you’ve forgotten what the terms factor and multiple, consider the situation of breaking up counting numbers using multiplication. For example, we can decompose the number 56 using multiplication in various ways. Here are a few examples of decomposing 56 using multiplication:

56=8×7

56=4×14

56=2×2×2×7.

We say 2, 4, 7, 8 and 14 are factors of 56. (Note: Other factors of 56 include 1, 28 and 56)

And we say 56 is a multiple of 2, 4, 7, 8 and 14. (Note: 56 is also a multiple of 1, 28 and 56)

Here is an area maze puzzle from the book. As you solve it, think about how you use the ideas of area, factors and multiples. Note: You will not need any fractions or decimals to solve the puzzle. All area mazes can be solved using only counting numbers (i.e., 1, 2, 3, ….).

From The Original Area Mazes: 100 Addictive Puzzles to Solve with Simple Math

Here is a video of my 10 year old thinking aloud as he solves Puzzle 7.

10 year old solving Puzzle 7 from Area Mazes book

While the puzzle above is simple, these puzzles get complicated for those up for a challenge.

Puzzle found in Guardian article about Inaba’s Area Mazes by Alex Bellos

The Magic

Fluency with multiplication facts makes life easier in math class. At the end of third grade, my son had memorized or found clever ways of deriving most facts quickly. However, there were 8 or so that he did not have ownership of and these facts had become sticking points during problem solving leading to errors and frustration. His teachers encouraged flashcard practice at home. However, my son hated flashcards and our flashcard experiences were miserable. Don’t get me wrong, flashcards work for many students. Also, there are really great activities and well-designed flashcards out there, see Math Flips from Berkley Everett and this reflective post by Michael Pershan about how he uses flashcards in the classroom. However, flashcards weren’t working for my son, so I searched for alternative forms of practice.

Thanks to this blog post by Sarah Carter, I found Inaba and Murakami’s Area mazes. Area puzzles require the solver to decompose numbers over and over, which provides multiplication fact practice. Although he would moan and groan every time I pulled out flashcards, but my sudoko loving kid happily solved these puzzles. He LOVES them. Never complains when asked to solve a few.

This summer during the math meetups, my son delighted in these puzzles and in explaining his methods to the other participants. He proudly recorded the video above. That’s enough magic for me when it comes to practicing multiplication facts.

Puzzles are a great way to practice math facts. In addition to Area mazes, we recommend you check out the puzzles below for practicing multiplication facts.

Find the Factor Puzzles from Iva Salley

Scrambled Multiplication table Geogebra Interactive from Flexbooks

Area maze puzzles brought the joy when it came to my son learning his multiplication facts. I hope your children/students enjoy them too.

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The Boy Who Dreamed of Infinity

“Stories can be windows, but also mirrors.”Jacqueline Woodson

The book featured below holds a special place in my heart. I’m thrilled to finally share it here. But first, a bit about my personal connection to the story.

Five and a half years ago, I applied to The Laura Crawford Mentorship with a picture book manuscript about Fiona, a fairy longing to swim in the sea, and Mira, a mermaid wishing for wings. You probably see where this is going. Their wishes come true, they switch places, and in the end realize the seaweed/grass isn’t always greener.

Opening the email announcing the winner, my heart hummed with anticipation. The winner of the Laura Crawford Mentorship is Chicagoan….

Amy Alznauer.

My heart sunk. Not me. I read on.

Amy’s winning manuscript is the picture book biography THE BOY WHO KNEW INFINITY which captures in lyrical prose the creative spirit and thirst for knowledge of a little known South Indian life-changing mathematician with whom young readers will instantly connect.

I gaped at my computer screen. The winning manuscript’s about Ramanujan and math?! Wait. What?!   I instantly connected to the story and hadn’t even read a page.  Transfixed, I reread the description of Amy’s story. My heart beat deepened. It was if I was looking into a mirror at my own dream.  I want to do that too.  I want to write math stories. 

And with that dream, this blog was born.  As you can see, this post is BIG for me and I’m certain is my longest post, so be warned. I owe a lot to Amy and her story. Since then, I’ve met Amy, and we’ve become dear friends and writing allies. Last year, I blinked back tears as I read her entire manuscript the first time. It’s MAGNIFICENT!  I’m over-joyed to share this dream-filled, magical story with you all!

The Book

Amy Alzauner has been holding Ramujuan’s story in her heart since childhood. In 1976, a 5 year old Amy traveled with her mathematician father, George Andrews, to England. During that visit, Andrews unearthed Ramanujan’s lost notebook in Wren Library in Cambridge.  Alznauer created this video describing the backstory. 

The Boy Who Dreamed of Infinity

Published in April 2020 by Candlewick, The Boy Who Dreamed of Infinity: A Tale of the Genius Ramanujan tells the story of Indian mathematician Ramanujan as a boy through young adulthood.  I’d recommend the book for ages 6 and up. Here’s the description from inside the jacket cover: 

In 1887 in India, a boy named Ramanujan is born with a passion for numbers. He sees numbers in the squares of light pricking his thatched roof and in the beasts dancing on the temple tower. He writes mathematics with his finger in the sand, across the pages of his notebooks, and with chalk on the temple floor. “What is small?” he wonders. “What is big?” Head in the clouds, Ramanujan struggles in school — but his mother knows that her son and his ideas have a purpose. As he grows up, Ramanujan reinvents much of modern mathematics, but where in the world could he find someone to understand what he has conceived? [From jacket flap of The Boy Who Dreamed of Infinity]

Alznauer’s reverence for Ramanujan and his mathematics shines through in this lyrical picture book biography.  The author artfully weaves rich mathematical questions into her story of Ramanujan, a boy with a curiosity and passion for numbers that glows from deep within. 

Speaking of glowing, let’s talk Daniel Miyares’ illustrations.  Miyares brings the magic with his dreamy, lush, layered watercolor illustrations.  Here’s how Miyares described his connection to this project. 

“Numbers haunted Ramanujan in his dreams. He couldn’t get away from thinking about them and how to find new life within them. That is how I feel about drawing and painting. Once I realized that, then it was a passion project for me. And I am so grateful. I think about visual storying telling differently because of this book.” [Source: Daniel Miyares, from Amyalz website]

Below are two of my favorite illustrations from the book. The book opens with:

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Image: Credit Daniel Mirayes, from Amazon website’s “Look inside” pages

How gorgeous is that!

My favorite spread from this book is Mirayes’ playful image of a joyous Ramanujan dreaming of numbers and mathematics as he tries “to catch the golden thoughts” and write them down. [I apologize for cell phone image. Doesn’t do the artwork justice.]

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From the Boy Who Dreamed of Infinity

Some might recognize the nod to 1729, the famous taxicab number, or what is called a Hardy-Ramanujan number. Here’s the famous story, as told by Hardy, about where taxicab numbers got their name (source: Wikipedia).

I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”

 

With this book, you get it all. Rich story-telling, exquisite artwork and… MATH! 

The Math

Srinivasa Ramanujan (1887-1920) was a fascinating Indian mathematician who, despite a lack of formal training in pure mathematics, made substantial contributions to the areas of mathematical analysis, number theory, infinite series, and continued fractions. 

The Boy Who Dreamed of Infinity poses mathematical questions, problems, and puzzles  sprung from Ramanujan’s interests: prime numbers, magic squares, partitions and sums of numbers, infinity and infinite sums.  Alznauer expertly connects mathematics to concrete experiences for children and reimagines how Ramanujan might have seen the world: 

A mango . . . is just one thing. But if I chop it in two, then chop the half in two, and keep on chopping, I get more and more bits, on and on, endlessly, to an infinity I could never ever reach. (From The Boy Who Dreamed of Infinity)

Additional ways for children to connect to Ramanujan’s mathematics are offered on the author’s website.

This book provides a beautiful image of what it means to do mathematics and embodies the idea introduced by mathematician Francis Su that mathematics is for human flourishing. In addition to introducing different mathematical objects and problems, The Boy Who Dreamed of Infinity is the story of a child and young adult tirelessly persevering to do what he loved, share his ideas, and to be heard and understood. 

The Magic

I read the book first with my son Liam (10.5 years). Noting Ramanujan’s interest in primes, Liam was inspired to explore and make his own list of prime numbers.

Next I read the book to my daughter Siena (8 years). She asked with wonder  “Is it real?” [She asked a similar question about a Katherine Johnson book.]  After reading, I asked,  “Is there anything you wonder after hearing his story?”

“I have the same… I know what he’s talking about. Because I write math that not that many people know, what I do, from the pictures. So maybe the math I write is what he writes too.” (Siena)

Wow. I wasn’t expecting that reaction.  My heart melted with empathy for her and for all children yearning and struggling to share and explain their ideas. I loved that she connected to Ramanujan’s story in this way. 

I read the book one more time during a physically distanced math meeting I hosted in my backyard for my 10 year old son and five other rising 5th graders. After reading the book,  I posed a task and asked the children to describe what they noticed about different numbers displayed visually.   (Task from a lesson in Mindset Mathematics (Grade 4), Boaler, Munson and Williams). Here’s the image of the number six from that task.  

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Image from an activity in Mindset Mathematics (Boaler et al.), Activity inspired by the work of Brent Vorgey

Liam eagerly raised his hand to share. “I see there are two triangles. One, two.”

Before I could ask him to explain his idea another boy in the group shouted out, “That’s wrong! There aren’t two triangles. There aren’t even sides. You need sides for a triangle.”

Liam’s face reddened, he looked down at his paper, tears welling up.  “I know. But IF there were…” His voice trailed off. 

UGH. His discomfort was palpable. My mama heart was about to break. In the moment, wanting to spare him further embarrassment, I didn’t press him to explain (which I later regretted). After the meeting, Liam and I talked. First we talked about his idea. He explained it to me. Here is a picture of his “two triangles”.

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I told him that made a lot of sense and I asked him why he didn’t say that when he was challenged. He shrugged. Then I asked him what he thought about the rest of the meeting. 

“I really liked how you read that book. Because it was like me. No one understood Ramanujan and they didn’t understand my idea today either. It’s the same.”

I swallowed hard and replied, “You’re right. It was like that. You have good ideas Liam. Keep sharing them.”

And so, for our family, the magical ingredient in The Boy Who Dreamed of Infinity was connection.  Both my children saw themselves in Ramanujan and his struggle to be understood.  Here’s one last image from the book. Ramanujan is staring out into the sea, waiting to be heard, waiting to be understood. 

“We are all crying out to be understood.”  ( Simone Weil)

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From The Boy Who Dreamed of Infinity

 

I hope you will read The Boy Who Dreamed of Infinity and find your own magic. For those interested in learning more about this book, the author will be presenting virtually for MoMath on August 14, 2020 (and second event TBD with her father), to find out more go here. Thanks to the generous support of Two Sigma, this program is free to attendees. 

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Let’s Have a Laugh and Count Some Goats

Tired of counting the days you’ve been in stuck at home during this global pandemic? Try counting some goats with your…ahem… kids.

Didn’t crack a smile at that last pun?  Here’s a counting book that is actually funny.

Book

Published in 2010 by Beach Lane Books, Let’s Count Goats was written by author and educator Mem Fox and illustrated by Jan Thomas.

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In Let’s Count Goats, Mem Fox’s humorous and fun rhyming text invites children to identify and count a whole host of goats ranging in size, silliness, and profession. The question, Can we count the____goats? is posed throughout the book. Here is one of my son’s favorite pages as an example.

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From Let’s Count Goats by Mem Fox and Jan Thomas

Illustrator Jan Thomas’s bold and colorful illustrations are sure to inspire giggles amidst counting. Here’s a favorite image from the book.

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From Let’s Count Goats by Mem Fox and Jan Thomas

Let’s Count Goats is ideal for ages 3 to 6.

Math

This book offers the opportunity for readers to count sets of goats ranging from 1 to 10.

Magic

After reading this book, I asked my son what he thought. He’s three and gave it a “I love it.” He calls it the “goat book” and we read it often.  Here is a bit about the two main magical ingredients we found in this book.

Magical ingredient 1: Humor

If I were to list some of my children’s favorite picture books, funny always tops the list. Funny books are magical. The sound of a child’s laughter is sure to bring a smile to your face.   The goats in this book are silly. Both the table and the suitcase eating goat garnered a giggle with every read.  The police goat is my son’s favorite goat, with the firefighter goat a close second. Let’s Count Goats is a great way to mix learning with some laughter.

Magical Ingredient 2: Invitation. [Here’s a post I wrote about this magical ingredient of invitation with a few more magical books that illustrate it wonderfully.]

Love how this book invites participation. When the texts prompts the reader,  “Can we count the pilot goats?”, it doesn’t give the answer on the next page. After each question posed in Let’s Count Goats, my son shares his answer by carefully pointing at each goat (an important process as a child is develops one-to-one correspondence). Some pages he silently points to each goat. Other pages he says number words as he points.  The number words he uses typically begin correctly with one, then two.  Often seven was the next number name in the sequence (counting 1, 2, 7).   Some pages I count along with him as he points (sometimes he asks me to, but not always), but mostly I sit back and observe as he works out the process of counting.

The counting process is complex process to work out as a novice.   Imagine being a novice counter and every time you count something in a book, you turn the page and hear a different answer. As my three-year old is making sense of the counting process, I enjoy reading a mix of counting books. We read some books where the number words (i.e., answers) are given on the following page and some where the counts are left solely to the reader. Counting books like Let’s Count Goats provide an opportunity to listen as our children take the lead and share what they are learning about counting from us and the world.

Let’s Count Goats is a keeper on our list of magical counting books. What fun, silly and/or magical books do you enjoy with your children and students? Please share them in the comments.  We’d love to check them out!

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Counting Book Magic: 1-2-3 Peas

Counting is a fascinating process to watch unfold. Picture books are a great way to share counting with a child.  In the sharing process, the reader/co-reader (whether teacher or parent) is able to observe and reflect on how and whether a child participates in counting. We’ve already posted quite a few counting books on this site here). But there are so many more wonderful counting books out there to sift through. My plan in 2020 is to read at least 100 with Landon and to share any magical ones we find. Our counting book count for 2020 is currently 12.  Here is a post about the first magical one we found.

The Book

1-2-3 Peas is actually our second magical book involving peas (Here is the first shared by educators Lana Pavlova and Meredith Wilkes).  1-2-3 Peas is written and illustrated by Keith Baker and was published in 2012 by Beach Lane Books, an imprint of Simon and Schuster. 

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The repetitive, rhyming pattern of the text on each page, all written in three word lines with different verbs, provides a dynamic read.  

One pea searches–Look, look, look. Two peas fishing–hook, hook, hook. 

Baker’s simple text is balanced perfectly with his lively, layered illustrations.  Check out this spread with the Peatles and Pea-oyncé. 

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I would recommend this book for ages 3-6.

The Math

The text begins with 1 and has a page of each of the numbers 1-10 with the corresponding amount of peas to count. For 11-19, only the numerals are shown. And then moves on to counting by 10s starting at 20.  So the numbers 20, 30, 40,  all the way to 100 show up as spreads.

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Since 11-19 are missing illustrations, this could be an opportunity for interested children to make their own illustration for these numbers. I wasn’t concerned with skipping these numbers at all as I was reading this with a three year old.

Peas are scattered in lively scenes in and around the numerals for each number.  There are different groupings and different costumes on many of the peas.  For the child interested in verifying that there are in fact 10 or 50 or 60 or 100 peas on a page, these differences could be helpful in keeping track of a count.

The Magic

Recently a read a piece by Katherine Patterson (author of books like Bridge to Teribethia and The Great Gilly Hopkins) that I think about a lot when observing Landon’s counting.  Patterson is talking about how she supported her children in the process of learning to use language, however the same could be said for learning anything or in the case,  learning to count.

I read them poems and wonderful books, far before an edcuator, indeed before any sensible person would think they could be read to understand the words. But in the midst of this richness, when one of them would stand before me, the little cords straining in his new, as he sought to express the still inexpressible, I would wait with totally uncharacteristic patience, reasoning if they were to learn to speak freely and comfortable to me, I must be willing to listen. Nor would I correct their mistakes. It is rude, I thought, to correct the grammar of someone who is trying his best to tel you something, no matter how tall the person might happen to be…. They would learn quite soon enough, I reasoned, the difference between singular and plural form of a the verb. All they had to do was listen. [page 8-9, Gates of Excellence, Katherine Patterson]

Picture books provide an excellent opportunity to listen to children’s counting.   Here are  two magical things from our reading 1,2,3 peas.

  • BIG NUMBERS. Baker’s illustrations include HUGE numbers. I didn’t think much of this until I watched Landon tracing the shapes of numerals with his little finger. These giant numerals invited interaction.  On the page with the numeral 100, Landon said “O” as he traced. As he is making sense of letters and numbers, Os are zeros and zeros are Os. This surprised me in the moment (although not surprising I’m sure for those that work with young children). 1,2,3 Peas provided me a window into Landon’s navigation between the two worlds of numbers and letters.  Worlds were the shapes are quite similar. A “B” shape is really close to an “8” shape for example, and of course 0 and O are the same shape.
  • MAGICAL SCENES.  Baker’s whimsical pea illustrations weave in and around the the large numerals in varied ways. It was fun to watch Landon count out peas (on the early pages) and search out and share with me the “silly” peas he found on later pages.  Baker’s illustrations are layered and full of surprises. Here is one of Landon’s favorite spreads. The trench coat wearing critters were a bit hit. 
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In addition to picture books, here are some great resources for supporting magical counting experiences for children (https://www.countingwithkids.comhttps://www.playfulinvitations.com/new-blog, Erikson Early Math (here is a recent counting article on the site) and the book Choral Counting & Counting Collections: Transforming the PreK-5 Math Classroom is a MUST have in my opinion for elementary teachers especially. 

Please share your favorite counting books (and why as well) in the comments!

 

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Math Book Magic Holiday List

It’s been 2 1/2 years of searching book lists and library bookshelves for magical books to share with you on our blog. Here is our entire list. Any of them would make great gifts for the holidays, but below is a subset to make shopping a bit easier. Age ranges are not set in stone of course.

Math Book Magic for Children Ages 3-5

OneloneyOne Lonely Fish from creators Alan Mansfield and Thomas Flintham was our ignagural magical math book. Two years later this board book has held up swimmingly to multiple reads. Landon (now 3) is even “reading” it himself now.  This book is great for developing one–to-one correspondence. The book’s nested pages and the growing sequence of fish for the BIG lonely fish to chomp down appeal to my kids and I suspect others. (Counting book, Amazon link: One Lonely Fish.)

Baby goes to marke

Baby Goes to Market by Nigerian-born Atinuke and illustrated Angela Brooksbank follows Baby and Mama through a Nigerian Market. Baby is so adorable that the banana seller gives Baby six bananas. Baby eats one and secretly puts five bananas in Mama’s basket.  Baby continues to collect items at the market, eating one of each, and placing the rest in Mama’s basket.   The story ends with a secret between Baby and reader that is sure to bring a smile. (Counting book, Amazon link: Baby Goes to Market.)

 

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Stack the Cats is written and illustrated by Susie Ghahremani. Cats of all shapes and sizes play, climb and of course stack, across the pages of this counting book. Ghahremini’s crisply illustrated cats pop against mint-green and orange backgrounds. The question at the end of the book,  How will you stack the cats?, encourages children to pull out their stuffed animals (or whatever) and stack some “cats.”  We purchased a set of these cats. WARNING: These cats are not for young children. A few of our cat legs broke with some rough play and could be a choking hazard. Here is another option, Pete the Cat blocks. Pete the Cat is the main character in a picture book series my youngest loves. (Counting and Addition, Amazon link: Stack the Cats.)

51nD9s61BPL._SX364_BO1,204,203,200_ Elephant & Piggie Like Reading! The Cookie Fiasco by Dan Santat is written in a similar style to Mo Willems’s Elephant & Piggie series. Hippo, Croc, and the Squirrels are faced with a problem: How to share three cookies equally among four sweet-toothed creatures. Will Hippo break the cookies to crumbs before they solve the problem?  This book is a blast to read aloud with its humorous and dramatic speech-bubble dialogue. Santat’s comic-style illustrations expertly capture the fear and frustration that “SOMEONE WILL NOT GET A COOKIE!” (Fractions and Division, Amazon link here: The Cookie Fiasco (Elephant & Piggie Like Reading!)

Math Book Magic for Children Ages 6-8

How Many Bundle2

How many?: A Counting Book was created by Christopher Danielson. The book can be purchased either as the Student Guide or a Teacher’s Guide Bundle. I highly recommend purchasing the bundle. There are so many wonderful things one can learn from this Teacher Guide. I never would have guessed 15 years ago that I would STILL be learning about counting.  But listening to children engage with simple prompts and carefully crafted images like in How Many? shows me there are deeper truths and things to be understood about counting and all K-12 mathematics learning. The magic of this book truly lies in sharing it with your own children/students. (Fractions and Division, Author’s website link) 

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How Many Jelly Beans? A Giant Book of Giant Numbers was written by Andrea Menotti and illustrated by Yancey Labat. Aiden and Emma can’t decide how many jelly beans are enough. Is 10 enough? How about 1000?  Each time a certain number of jelly beans is presented, there is an accompanying illustration to support counting practice. Fun fold-out pages show how big 1 million jelly beans really is. Also, Harry Potter Bertie Botts Every Flavor Beans could be a fun thing to give along with this gift. Just make sure to read exactly what “every” flavor means in the description of the link. (Counting and Large Numbers, Amazon link here: How Many Jelly Beans?)

bean13-cover

Bean Thirteen is another book that is a perfect pairing with Harry Potter Bertie Botts Every Flavor Beans. Bean Thirteen was written and illustrated by Matthew McElligott. I first learned about this book while exploring McElligott’s website for a previous post, The Lion’s Share.  McElligott’s bright bug illustrations are the perfect compliment to this delightful bean-sharing story. The main characters are two bugs Ralph and Flora. Ralph warns Flora about picking unlucky bean thirteen. Flora fails to heed Ralph’s warning and now they are stuck with it. In an effort to make bean thirteen disappear, they invite their friends over to share beans. But no matter how they try, there is always at least one bean left over.  This book leaves room for wonder and invites children to participate in the sharing of the beans and mathematizing the story in a way that makes sense to them. (Division and Prime Numbers, Amazon link here: Bean Thirteen)

More-igami

More-igami was written by Dori Kleber and illustrated by G. Brian Karas. Joey loves things that fold: maps, beds, accordions. His interest in foldables increases after watching his classmate’s mother turn an ordinary piece of paper into a beautiful origami crane. Joey sets out to learn origami. However, not everyone appreciates the amount of practice it takes to become an origami master. Luckily, Joey finds a way to fold to his heart’s content and becomes the origami master he set out to be. Karas’s sweet, warmly lit lustrations are the perfect backdrop for Kleber’s simple, beautiful story about the power of perseverance.  The end pages include a simple origami ladybug project, a perfect first step to becoming an origami master. This book and a stack of origami paper would make a great holiday gift. (Geometry, Amazon link here: More-igami)

Math Book Magic for Children ages  8 and up

51dBo8-d8rL._SX386_BO1,204,203,200_This is Not a Math Book: A Smart Art Activity Book was created by Ann Weltman. Ann Weltman is a math teacher and co-founder of Math Munch, an online resource that provides a digest of mathematics for children, parents and teachers. Full full of art-infused invitations to mathematical thinking, Weltman has masterfully curated a collection of activities that opens doors to profound and curious mathematical questions. Couple this book with a some colored pencils, a ruler and a compass and you’ve got a great gift for the artist in your life. (General, Amazon link here: This Is Not a Math Book)

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One Grain of Rice: A Mathematical Folktale by Demi is based on on the Indian tale Sissa and the Troublesome Trifles. In the story, a young girl Rani, returns some spilled rice to the raja (another name for Indian king). The raja had been hoarding all the rice in his land which led to famine for the villagers.  Obligated to reward Rani for returning the rice, the raja tells Rani to name her price. When Rani asks for a single grain of rice, the raja convinces her to ask for more. In addition to the one grain, Rani asks that the raja double that amount for each of 30 days: 1 grain of rice on day one, 2 grains on day two, 4 on day three, 8 on day four and so on.  The raja agrees, certain Rani has once again asked for too little. Doubling is one of my favorite mathematical concepts to share with children. It is such a powerful, magical concept. (Exponential Patterns, Amazon Link here : One Grain Of Rice: A Mathematical Folktale)

the joy of mathematics cover image

 

The Joy of Mathematics: Discovering Mathematics All Around You was written by Theoni Pappas. The book contains problems, puzzles, historical facts, tidbits, and stories that aim to spark curiosity and invite the reader to experiment with a wide variety of ideas and topics. (General, Amazon Link here: The Joy of Mathematics: Discovering Mathematics All Around You)

 

 

 

OK I better stop now or I will just add ALL the books on the blog (Seriously). It was a joy to read through our old blog posts, the memories and the math shared. Hope you have a magical holiday season sharing and creating your own memories.

Happy Holidays and happy magical math book sharing!

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Counting on Katherine (Part 2: The Magic)

Last week’s post about the picture book Counting on Katherine by Helaine Becker and Tiemdow Phumiruk shared about the book and the math.  This post shares some of its magic.

The Magic

A few days before reading Counting on Katherine, I brought my daughter Siena to a STEM book signing at Anderson’s Bookshops.  Authors at the event included Katey Howes, Alice McGinty, Miranda Paul, Teresa Robeson and Ruth Spiro. Here are some of their books.

I was excited to go to the event and wore my favorite #Mathgals shirt with Katherine Johnson’s name displayed first among other female mathematicians (see cover photo, you can purchase the shirts here). Siena was excited too. She brought a book she wrote and illustrated titled The Butterfly Story (The Courage Story) to share with the author of one of her favorite books Be A Maker.

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Be a Maker author Katey Howes and Siena with her double titled book The Butterfly Story (The Courage Story)

 

A few days after the event, Siena and I sat down to read Counting on Katherine for the first time together. Toward the beginning of the book, Siena turned to me and said, “That’s impossible. No one can know that much math. It’s a book.”

She flipped back to the two pages shown below.

 

Surprised I replied, “Yes. It’s a book. But she’s a real person. It’s a true story about her life.” The look of surprise and wonder on Siena’s face was magical.

Then when we got to the spread below,  I started crying.

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“Why are your crying?” Siena asked in her usual “what the heck?” mom tone. (I’ve been known to cry at random commercials and basically anything meant to pull on your heartstrings.)  I could barely finish reading the page. I got it together and all I could say was, “Because she’s amazing. She is an amazing mathematician.”

I cried because I longed for that image earlier in my life.  Too often, mathematical greatness was communicated to me through the histories and images of male mathematicians. The image of Katherine standing above her male co-workers, chalk in hand, working the math in the way only she could, being valued for her ideas and her work was a powerful image for me. A symbol of greatness. Katherine’s word was the last word. She was the one John Glenn trusted. He trusted her with his life. She was that good. Women are that good. Now after learning about Katherine, her modesty would never claim the word greatness, but on those pages I basked in her brilliance and I cried. I cried for Katherine as I had cried when I heard Maryam Mirzakhani, a highly original mathematician who made striking contributions to geometry and dynamical systems, became the only women to win a Fields medal, one of the highest honors a mathematician can received.   I cried because I was grateful to be sharing Katherine’s story with my daughter.  The story of a great mathematical mind.

After sharing Counting on Katherine with my daughter, I wanted to learn more.  I read the picture book biography Hidden Figures,  the Young Readers novel Hidden Figures and watched the movie Hidden Figures for the first time (I know, how could I have missed when it came out in 2017?!). If you’re like me and haven’t watched this movie, please do. And I recommend reading the novel by Marjorie Lee Shetterly before. 

Perseverance doesn’t even begin to capture what Katherine and her colleagues, Dorothy Vaughn, Mary Jackson, and many others, displayed during their time at NASA. Their calculations helped fuel some of America’s greatest achievements in space. They did their work while facing extreme gender and racial discrimination both inside and outside Langley.  They did the work, but often weren’t given credit or respect.

Women were called ” girls.” Grown women, raising children, caring for husbands (Katherine cared for her first husband James Goble who was suffering from a brain tumor up until he passed) and diligently doing their work, important work, were called “girls.”  Even when John Glenn asked for Katherine to check the calculations before his launch, the woman he trusted with his life, he is quoted as saying,  “Go get the girl.” Below is an image of the control room just after 

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Members of the Kennedy Space Center control room team rise from their consoles to see the liftoff of the Apollo 11 mission 16 July 1969. AFP PHOTO/NASA (Photo by NASA / NASA / AFP) (Photo credit should read NASA/AFP/Getty Images)

But while women had difficulty getting in the room, the racial injustices Katherine and her colleagues faced were reprehensible. For example, Katherine couldn’t use a female restroom in the same building she worked, so she had to trek half a mile across the NASA campus to the colored bathroom. She couldn’t even use the same coffee pot as her white colleagues.

Below is an oral history archived by the National Visionary Leadership Project illuminating Kathrine’s perspective on the racial and gender barriers she faced.

At first she [Johnson] worked in a pool of women performing math calculations. Katherine has referred to the women in the pool as virtual “computers who wore skirts”. Their main job was to read the data from the black boxes of planes and carry out other precise mathematical tasks. Then one day, Katherine (and a colleague) were temporarily assigned to help the all-male flight research team. Katherine’s knowledge of analytic geometry helped make quick allies of male bosses and colleagues to the extent that, “they forgot to return me to the pool”. While the racial and gender barriers were always there, Katherine says she ignored them. Katherine was assertive, asking to be included in editorial meetings (where no women had gone before). She simply told people she had done the work and that she belonged.

The link above to the National Visionary Leadership Project has a series of videos of Katherine telling her story.  What a gift. I highly recommend watching these videos. In them, Katherine shares how her father would tell her, “You are as good as anyone in this town. But you are no better.”  This idea was a touchstone in her life and she has shares her father’s advice with the young people she meets.

A few months ago, I bought the print below by the artist Marina Teraud. Teraud was inspired by the Serbian Proverb, “Be humble for you are made of earth. Be noble for you are made of stars.”  I was inspired to buy the print as a message to my daughter Siena:  Be humble and be noble.  

090-36632-sKatherine Johnson is as humble and noble as they come.  I am grateful to have shared this story with my children.  If I had magic wand,  I would manifest this book on shelves of every classroom.  Stories matter. Images matter. Beliefs create reality. 

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Katherine Johnson, ANNIE LEIBOVITZ/VANITY FAIR/TRUNK ARCHIVE

 

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Counting on Katherine (Part 1: The Book and The Math)

America’s history is full of untold stories.  Fortunately, one of these stories was brought to light in the last few years.* This post features a picture book about the brilliant, ground-breaking mathematician, Katherine Johnson. [Cover photo credit: Annie Leibovitz.]

The Book

Written by Helaine Becker and illustrated by Tiemdow Phumiruk Counting on Katherine: How Katherine Saved Apollo 13 tells the story of Katherine Johnson, the mathematical genius who made sure that Apollo 13 returned home safely.  Counting on Katherine was published in Henry Holt and Co. in 2019 and is recommended for children in kindergarten through 4th.

Coutnign on Katherine

The book follows Katherine  along her path to becoming a NASA mathematician.   Katherine loved to count as a child.

“She counted the steps to the road. The steps up to church. The number of dishes and spoons she washed in the bright white sink.” (from Counting on Katherine)

She followed her passion for mathematics all the way to NASA. Along the way, Katherine faced many racial and gender barriers. Phumiruk beautiful art work provides for private moments of reflection during the reading of this remarkable picture book biography about a resilient, humble, and inspirational woman.

Counting on Katherine has won many awards, including, but not limited to, the ILA Teachers’ Choices Reading List Selection award, the Chicago Public Library Best Book of the Year, an NSTA Outstanding Science Trade Book, Mathical honor book, and a Junior Library Guild Selection award.

In 2015, Barack Obama awarded Katherine the Presidential Medal of Freedom for her pioneering work in STEM fields. [Read more about Katherine Johnson here.]  Katherine and three other women who worked at NASA during the space race (engineers Christine Darden and Mary Jackson, and computer programmer Dorothy Vaughan) were recently awarded congressional medals, the highest civilian award in the US.

This past August, Katherine Johnson turned 101.

The Math

It is amazing what Katherine Johnson could do mathematically in an age before calculators and computers. She was the calculator. She was the computer. In fact, “computer” was actually the name for people that did the work that Katherine did.

And Katherine was precise. Here is a quote from Katherine about mathematics.

“[Math’s] just there. You can’t do anything without it. It’s in everything. I like to work problems. If you do your best, nobody can ask you to do it over again. I never had to repeat what I did.” (Interview, AARP)

The mathematics depicted through Phumiruk’s illustrations involve calculations relevant to the physics and the arithmetic/analytic calculations involved in space study.

However, the purpose of this book isn’t so much to spark conversation around mathematical concepts, but instead a conversation about who is a mathematician.  

Think about the mathematicians that you learned about in school. What did they do? Who where they?

Names like Pythagoras (Geometry), Newton and Leibniz (Calculus) come to mind from my own education (ask your students/children). However, this is only part of the story.  There is a rich landscape full of diverse groups of mathematicians and areas of mathematics beyond what is typically found in school textbooks.  Mathematics educator Annie Perkins has a wonderful project titled Mathematicians are Not Just White Dudes where she shares information about different mathematicians. Here is a short description of the project from her website:

  • We as math teachers tend to only talk about white male mathematicians.
  • Most of my students don’t look like that, and thus, they have few mathematical role models they can identify with.
  • Take 10-15 minutes a week to research (read Wikipedia, that’s all you need) a not-old-dead-white-dude mathematician, and then take 5 minutes in class to tell your students about them.  Include a picture. It’s worth it, I swear.

Perkin’s site includes a searchable spreadsheet to serve as a launch for this research.

The Magic

As I’ve been writing this post up over the past few weeks, I’ve found that there was too much  for one post. I will post about the Magic found while sharing this book with my daughter next week.

*The highly acclaimed film Hidden Figures, released in December 2016, was based on the non-fiction book of the same title by Margot Lee Shetterly, which was published earlier that year. It follows Johnson and other female African-American mathematicians (Mary Jackson and Dorothy Vaughan) who worked at NASA.

 

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Trapping Math Magic in Spider Web Shapes

Recently, I was in the library picking up a stack of counting books when I spotted the book featured below. As soon as I noticed the cover, my heart started pounding. I silently shouted, “That’s my idea!”

Let me explain. I keep a notebook full of math picture book ideas  (actually it’s really like four different notebooks, iPhone notes, video memos and computer files. I really need to organize myself better). About a year ago, I wrote down an idea about a math picture book full of spider webs, special spider webs made up of triangles, hexagons and stars. See my notebook image below.

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Some shape webs from my notebook

Every spider web in the wild became a reminder to return to this web shape idea.   But I never did and the idea stayed unformed; a mathematical situation with no story.   For me, the story is the hard part. I come across math situations that interest me every day, but searching for the right story to pair with them takes me the longest time. And this time… time had run out.

I opened the book and I whispered, Please be good. Please be good. I loved the idea of web shapes so much, they deserved to be in magical math book. Please be magical, please be magical.  It was. The perfect story to showcase spider web shape wonder.

The Book

Written and illustrated by Tim Hopgood, Walter’s Wonderful Web is a story about a spider trying to build a perfect web to withstand the wind.

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Walter forms webs containing sets of similar shapes.

Hopgood’s lyrical lines pair perfectly with his adorable images of Walter and his web-making struggles.

After reading Walter’s Wonderful Web, I can’t wait to read more from Hopgood. His website shows about 15 picture books full of intriguing cover art and titles. I immediately placed five of his books on hold at my local library.

Walter’s Wonderful Web was published in 2015 by Farrar Straus Girouz. The book seems like it would be best for 3-6 year olds.

The Math

This book works nicely as a first book of shapes providing an opportunity to notice triangles, squares, rectangles, circles, and diamonds (Note: Diamonds are not a lazy term for rhombus).

Hopgood’s spider web illustrations also open the door to the geometric concepts of similarity and symmetry.

The Magic

Each of my three children read the book.  I didn’t get very far with my 7 year old when I asked what she thought about the book. “It’s for little kids,” she said. It was true for us that the most magical experience occurred with our “little kid” ( my 3 year old son), but my 9 year old enjoyed the book as well.

The end pages provide a set of questions for children to share ideas about geometric objects. My 9 year old shared with me his answer to a question about whether circles have sides.   “The answer is no because it’s round. And it looks like it has one side, but doesn’t because sides are flat not round.”

I asked him what he liked best about the book.

“The last web.”

“Why the last web?”

“Did you look at it?!!”

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The little fingers are the 3 year olds. He tried to snag the book while I was taking the picture for the blog, “It’s MY book” he shouted [Now that’s a promising book review from a 3 year old IMO]
He went on to explain all the shapes he saw in Walter’s wonderful web (pictured above). He explained how by combining the triangles you can see diamonds as well. There’s a lot to explore in that web.

Picture books are just starting to hold my 3 year old’s attention. I wondered whether he would sit for the entire story. He did and he interacted with this book more than any book he has before.

He began a bit afraid. “No. It has spiders.” He said and was about to walk away. “His name is Walter. He’s nice. Look he’s smiling. ” I said to reassure him. Despite his initial hesitation about Walter, by the middle he was emotionally invested. On this spread he pointed and exclaimed at the images of Walter concerned for the sweet spider’s well being.

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On the end pages, I watched curiously as he identified particular shapes and counted the sides of each shape. For him, circles had one side.

“Did you like that book?” I asked. “I loved it, so much.”  And I loved watching him interact with the story SO MUCH.

One more thing about that trip to the library. After I read the book, I went over to one of the librarians to share my excitement.  She mentioned she had just shared it with a group of children during story time. They loved it and they did a toss the yarn activity which looks like fun (see image and link). Here’s an image of a web made by tossing a ball of string. What shapes do you see in the web? 

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Halloween Spider web game From PreK my style blog style.html)

When searching for more spider web activities, I came across this amazing image in an article about the evolution of spider webs. Look at the wonderful shapes the universe creates.

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Orb weavers, from the grouping Orbiculariae, make the classic, wheel-shaped spiderweb, as well as other intriguing designs. This tree hosts a sampling of Orbiculariae, illustrating the web diversity. Evolutionarily older spiders and their ancestors appear on the ground and trunk; more recent arrivals hang from the highest branches. (Credit: F. Vollrath and P. Selden/AR Ecology, Evolution and Systematics 2007 (Modified from Vollrath 1988))Enter a caption

And here’s a Scholastic lesson on spider webs and a video of some spider web art.

 

Spider Web shapes are indeed magical. This book would be a great addition to your math picture book stack this fall, perhaps around Halloween.  Also don’t forget Math Story Telling Day is September 25th if you can’t wait until Halloween to enjoy Walter’s Wonderful Web!

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Celebrate Spring with Anno’s Magic Seeds

Miss Maple’s Seeds by Eliza Wheeler is a family favorite. Nestled among her gorgeous illustrations, Wheeler shares a sweet story celebrating the potential found in every seed. (Makes a nice teacher gift coupled with plant/flowers/pot decorated by your child, if you need ideas).

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“Even the grandest tree once had to grow up from the smallest of seeds.” (Miss Maple’s Seeds)

 

Spring is the perfect time to witness the magic of seeds by planting some of your own or reading picture books like Miss Maple’s Seeds, A Seed is Sleepy, or Anno’s Magic Seeds, the magical math book shared below.

The Book

Mitsumasa Anno is a Japanese illustrator and writer of children’s books. In 1984, he received the international Hans Christian Andersen Medal for his lasting contribution to children’s literature. Many of his picture books reflect his interest in mathematics (e.g,   Anno’s Magic Seeds, Anno’s Counting Book and Anno’s Mysterious Multiplying Jar). 

Annos magic seeds

In Anno’s Magic Seeds, the main character Jack receives a gift of two magic seeds from a wizard. He is instructed to eat one of the seeds (which will sustain him for an entire year) and plant the second seed. Jack’s magic seed keeps growing and growing and growing. His magical crops grow quickly, first by ones, then by twos and faster and faster. But when a terrible flood threatens his livelihood, he must decide whether he will begin again.

Anno’s sweet water color illustrations are clear and purposeful making it both easy and interesting to count the different arrangements of plants. Here is an example of one his  illustrations.

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From Anno’s Magic Seeds showing Jack, 1 red seed he eats, and the brown seeds he plants

There are different ways of counting the seeds above. A child may count by ones, by twos (adding one extra at the end), they might recognize an array of 6 seeds on the left plus 3 extra seeds, or they might count a different way entirely. Anno poses questions throughout the story: How many seeds did he bury? How many seeds grew that year? The answers are not explicitly given in text, but can be determined by following both story and art.

Here are Anno’s own words describing his motivation for sharing this story.

I call this book The Magic Seeds because, in fact, there is a mysterious power in even one tiny seed that seems quite beyond our understanding. Of course, we could not live for a year on one grain of rice or cereal, as Jack did. But if we should bury one seed in the ground and take care of the plant that grows from it, it would not be long before we had a crop of hundreds of thousands of grains. In our real world of nature there are many such magical events–more than are contained in all the most fantastic picture books. 

This book is great for K-3 and beyond. The patterns and questions get complicated enough to intrigue older elementary students. The book is out of print, so prices of a new copy are quite high (used copies are reasonable). You might check your local library.

The Math

There are many counting opportunities in this book and much more math as well.  Here’s the description of the math from inside the jacket cover:

Though the story can be followed without any math skills beyond simple addition and subtraction, sharp witted-young readers will delight in the increasingly tricky arithmetic puzzles woven into text and illustrations.

Even though the patterns in the book do not follow a strictly increasing exponential pattern as is the case in  One Grain of Rice and Two of Everything, the power of exponential growth underlies the ideas in the book.  Since the pattern is not strictly exponential,  more figuring and keeping track is necessary in order to follow patterns and answer Anno’s questions.

THE MAGIC

I read this book twice to my older children (9 and 6 years old).  First we read together, the three of us.

During this first reading, the magical component of this book was the invitation to count. It was lovely watching the variations on how they both counted the images, when and where they were and were not interested in counting, and how they compared and recounted when their answers didn’t match their sibling’s answer. They often stopped me mid-sentence or page flip to count. They directed each other “Move your hand I’m counting.”  This book inspired them to count. This isn’t always the case. Many picture books we read state an answer BEFORE the page turn. This book doesn’t state all its answer in the text, they are hidden in the pictures and the story.  This creates a need to count for the reader that isn’t the case in many of the picture books we read.

In addition to the counting, they were drawn in by the illustrations and the story.   My daughter noticed in an illustration that the mother was going to have a child and was excited when the baby shows up later. “I knew it!” she exclaimed proudly and flipped back to show me the image below (which I had overlooked the first time around).

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Anno’s Magic Seeds. Illustration of Jack’s Wife

They conjectured that the wizard brought the storm that comes at the end. They wondered, how the seed sequence begin asking, How did the wizard get the seeds? The shared why they enjoyed the story,  ” I like how he’s building an empire.”

It was clear during that first reading, they were connected to story through the counting,  story, and  illustrations. I decided to read the story again a few days later.  I was curious if they would connect a bit more to the math (e.g., do more of the figuring and calculating necessary to determine seed pattern from year to year more explicitly).  They did not. The second reading had none of the wonder and joy of the first reading. And later, after coming across a review of Anno’s books written by Carol Hurst, I realized why. Carol writes:

Anno’s precise and detailed watercolors have the air of Japanese elegant simplicity. Most have a gentle humor waiting to be spotted. Not only that, but Anno’s books almost always have a strong mathematical connection. Don’t work them too hard, however. Leave some things for readers to discover on their own.

Anno’s Magic Seeds provides an excellent invitation to children to seek out meaning for themselves, mathematical or otherwise, if we let them. In our second reading, I was working the mathematical connections too hard and my children sensed this. And just like that, the magic, the wonder and joy,  was gone.

To anyone still reading, thank you for listening to our story about this magical math book. We hope you enjoy the magic of spring in the weeks to come and if you have any magical math books you want to share back with us, please go to the Shared booklist to find out how.

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Play with your Math with Little Pea

I am thrilled to introduce our next guest post. Last year, I came across an image posted by educator Lana Pavlova of some shoe arrays her students constructed after reading the book Amanda Bean’s Amazing Dream by Cindy Neuschwander. I immediately messaged Lana in the hopes she’d share some of her math book magic with us in the future.

And the future is now! The post below was written by Lana and her colleague Meredith Wilkes. Lana is  currently a math coach with K-6 students and teachers, and Meredith is a kindergarten teacher. Both live in Calgary.   Their post is an excellent example of two amazing, creative teachers taking a delightful picture book and examining the situations in the book through a mathematical lens. Thanks to both of them for sharing their math book magic with us.  Watch out for flying peas!

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Where do we expect our children to encounter math? Maybe when they have math lessons at school. Or play games that involve numbers. Or when they solve puzzle, or read magical math books. But there is so much math happening in their lives that is surprising and unexpected, that can be found on the pages of the books not written to teach math. We wanted to share how one of the wonderful non-mathy books helped our students to find math in unexpected places.

The Book

Little Pea is written by Amy Krouse Rosenthal and illustrated by Jen Corace. It’s funny and delightful in a way that will make both children and adults smile. Little Pea lives with his Pea family Papa Pea and Mama Pea. He loves when his Mama tells him about her pea childhood, loves when his Papa flings him high into the air off the spoon, and he HATES candy which he has to eat for dinner every night to grow up big strong pea. And if he doesn’t finish his candy, he cannot have desert. Spinach dessert!

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The Math

This book might not have been written with the math lessons in mind, but just as natural children’s curiosity can lead them to mathematical explorations, this book can provoke some. After reading the book with our kindergarten students a couple of times, we showed them…spoons. And this is where magic really started to happen. There are measurement lessons hidden in this book.

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The Magic

Imagine a classroom of kindergarten children who are conducting an experiment to discover how far they can fling their little peas off their spoons. And then looking for the ways to measure, compare and communicate their discoveries.

We didn’t go straight for mathematics, we talked and wondered about the Little Pea family first. The book created a world in which we could play with math. When the students saw the spoons, they were excited to try and make the Little Pea fly. We didn’t really have many peas in our classroom, so students suggested the pompoms would work as well.

And then we waited for the math to come up during their explorations. “My pea went so far!” This was the perfect time to wonder how far it went. We had to do some measuring. Students selected the tools to use and got to work. We did not direct them to the particular tools or strategies, but students have encountered some measuring challenges that were a great starter for this conversation.

 

 

Some students used a variety of objects to measure the path. But how can it help us to describe how far our Little Pea flew? It is five…. five what? Some students also built a meandering path from their measuring objects. Did our Little Pea fly left and right, and left and right before landing or did it go straight? We had to conduct the experiment again. Students continued to develop important ideas about measurement that came up naturally in this exploration like the need to use the consistent units.

 

 

And then we wanted to document what happened to our Little Pea. How can we use pictures, words and diagrams to show how far Little Pea go and how we measured it? Communicating their mathematical reasoning on paper might seem like a challenge for the students who have not learned how to write yet. But young children can use pictorial representation to communicate with admirable precision.

 

 

The Little Pea book prompted a long mathematical exploration of measurement, where students had the opportunity not only to discover some ways to measure with non-standard units, but also practiced counting, and making strategic decisions, and communicating their ideas. They have learned new words and new skills. They have also learned that math magic can live in our favorite books, we just need help each other to find it.

Lana Pavlova and Meredith Wilkes

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Thanks Lana and Meredith for letting us peak in on your kindergarteners students as the use non-standard units to measure. Using non-standard units is a great way to get students thinking deeply about the process of measurement. For those interested on thinking more about this idea, here is a video from teacherchannel.org of Secondary students using non-standard units.  Something I noticed in watching this video, play in secondary math class is as important as it is in kindergarten.

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