This is a joint blog post by Kara Shane Colley, the author of My Hundred Friends that is soon to be published at Natural Math, and Dr. Maria Droujkova, the founder of Natural Math. We’ll be posting periodic updates about the book on the blog.

Kids are natural storytellers because they are still open to the world of possibilities and imagination. How powerful it could be to encourage students to use their imagination when doing math, or even better, when playing with math.

Maria: I’d like to start a conversation about timely and untimely math stories so we can get better at mathematical storytelling as parents, teachers, and organizers. But first, I’d like to invite you to join the crowdfunding campaign for My Hundred Friends book before June 27th and tell your friends and colleagues about it!

This could have been such a good story!

We understand, remember, share, and guide our experiences through stories. That’s how we comprehend the world as more than a pile of data points and collaborate as more than disjointed individuals. That’s why storytelling is a human universal in every culture ever.

Comic by Loonar Baboon

Kara: Kids are natural storytellers because they are still open to the world of possibilities and imagination. I love the comic above by Loonar Baboon. The adult sees a messy room. The child sees a room full of lions, robots, and mountains. How can we use math storytelling to leverage a child’s imagination? And how can we use a child’s imagination to leverage math skill-building?

Maria told me about the book, The Children’s Machine by Seymour Papert. He says, “Fantasy has always been encouraged in good creative writing and art classes. Excluding it from science is a foolish neglect of an opportunity to develop bonding between children and science.” The same goes for math! We encourage students to use their imagination when writing stories or making art. How powerful it could be to encourage students to use their imagination when doing math, or even better, when playing with math. 

Maria: Like everything else we make, stories can be good but also ugly or bad, from an anecdote that elicits only a confused “Huh?” to hostile propaganda. It all depends on how, why, and when the story is told.

Mathematics is a culture and storytelling is a part of every culture. And so as math people, we tell math stories. One of the most popular math stories, Hotel Infinity, starts like this: Somewhere, there’s that hotel with rooms numbered 1, 2, 3,… and the rooms never end, just like the numbers. Every room is full, but a guest comes to the door. The manager messages everyone to move one room up, so the person from #1 goes to #2, from #2 to #3, and so on. Since there’s always the next number, every single person still has a room, including the new guest, who moves to #1! Yet they never have to build rooms. Adding one to infinity makes… the same infinity. ∞ + 1 = ∞

When would you share a story like this with people learning about infinities? I like to use it as an introduction to spark curiosity and make people giggle. That’s a great mood for math, and moreover, the story boosts mathematical reasoning. The hotel gives structure to the sheer weirdness, complexity, and abstraction of the infinite, turning it into something concrete and relatable: rooms. Children see rooms and move from room to room daily, so rooms are easy to imagine or pretend-play.

And that’s not all! Whenever our characters intrinsically enact our math, we activate the problem-solving technique called Smart Little People. The original technique imagines physical and chemical phenomena enacted by microscopic crowds to solve engineering problems. For example, if two parts must stick together and come apart as needed, imagine friction as people holding hands, then releasing, then holding again. That’s how Velcro could have been invented. In the hotel story, numbers are people. What is ∞ + 2? We imagine 2 as two guests, and if they come to the door, the manager can ask everyone to…

Prime factorization of 56; “My Hundred Friends” is entirely made of Smart Little People

…move two rooms up, so #1 goes to #3, #2 goes to #4, and the first two rooms welcome the new guests.

∞ + 2 = ∞

From the Arabian Nights to Star Trek, people like to keep telling stories set in the same imaginary world. When would you go back to the infinite hotel and tell more of the story? I like to do that to help students level up within a topic, pose a tough problem, or tackle a mind-bending idea. Can we multiply infinities, for example? If another infinite hotel in the next galaxy over goes out of business, can the manager move guests to make enough room for all? (Spoiler: yes, and ∞ × 2 = ∞.) In time, we can even use the story to navigate abstract proofs like, “The set of all rational numbers is countable.”

When would you tell a different and related math story, such as “Life on the Infinite Farm?” I like to do that separately, after a break, giving each story its own space and time, even if we do that on the same day like some sort of intense Barbenheimer experience.

Ask yourself: when would you NOT tell a math story? Here’s what I’ve tried, and tried, and it never worked. Even a beloved story or a reliable anecdote falls flat if I first tell it during exercises for fluency. Won’t a good story in the middle make a worksheet fun? Won’t students have more stamina if some new characters introduce a tedious computation? Nope!

When students have just explored a new topic? When they are first applying their fresh understanding to some exercises and problems? It’s both too late and too early to introduce new story-worlds. Too late because the imagery of the story might clash with new math images just formed in each student’s mind and derail everything. Too early because applying a fresh new math idea to a new context won’t work without fluency in the old context. That’s why children often feel that “story problems” are both boring and hard.

Kara: I know from 20 years of teaching math that story problems are nearly universally disliked and feared by students. Story problems are often contrived, confusing, and overly complicated.

I heard math educator Dan Meyer talking about breaking math problems into a grid. One parameter is real world vs. fake world, and the other parameter is real work vs. fake work. Too often, story problems fall into the “real world – fake work” box. The problems are realistic, but there’s not a lot of real math going on. Often, the students just need to plug the given numbers into a formula, and they’re done.

Maria: By the time your story has built up enough context and depth for real work to happen, you’ve probably used too much storytelling technique for the world to remain entirely real.

Diagram by Dan Meyer

Meyer encourages math teachers to pose problems in the “fake world – real work” box. His example below is “Pick any number you want. Multiply it by 40 and add 200. What questions could we ask about all our numbers?”

Okay, I pick 1:

40(1)+200 = 240

Next, I pick 10:

40(10)+200 = 600

What questions could we ask? 

Hmm, will the result always be even? Will the results always be positive? Will the results always be a whole number? Do the results lie on a certain line? 

Maria: I read world-building books like My Hundred Friends to unlock powerful new ideas like factors and multiples for children who’ve never done that math before. I also read My Hundred Friends as a complementary and separate experience with children who study or have studied multiplication, to reach new depth and build new connections. But I try not to be like, “What, factoring 60 makes you cry? Here, now numbers have faces, and 2 is a nice shade of pink, go go go!” 

Kara: Ever since I’ve heard this analysis, I felt more liberated to move towards the “fake world – real work” box. I built My Hundred Friends in that box: it’s a cozy world that takes place in a math scrapbook where cheeky little number-characters romp around. There’s a lot of math going on, but the math results naturally from the context of the imaginary world in the story. The number-characters and I happily play with factors, multiples, and primes, and the story invites readers to do the same.

This is a joint blog post by Kara Shane Colley, the author of My Hundred Friends that is soon to be published at Natural Math, and Dr. Maria Droujkova, the founder of Natural Math. We’ll be posting periodic updates about the book on the blog.

I like how it’s funny and learnful.
Elizabeth, 4th grade reader

Maria: In My Hundred Friends by Kara Shane Colley, the math of factors and multiples is friendship. It’s a cozy little world, inviting readers to play and explore. From April 27 to June 27, head to My Hundred Friends crowdfunding campaign to preorder hardcover copies for your family or group, reserve your complimentary PDF, and support mathematical storytelling!

Kara: While editing the book, I spent several afternoons reading early drafts of My Hundred Friends with a Montessori class of 4th, 5th, and 6th graders. The teacher gave me a corner of the room and set me up with pairs of students.

I greeted the children and explained, “I’m writing a book and I’d like your feedback on it. I’d like you to read it and tell me what you like, what you don’t like, what’s confusing. I’ve illustrated the book so far, but I am not going to be the actual illustrator.” I handed them my sketchbook, and they turned to page 1.

In the first few pages, the children noticed right away how excited the numbers are to see themselves in the other numbers. On page 4 in the bottom right corner was 4’s prime factorization, 2 ✕ 2, with Two exclaiming, “Look! It’s me and me again.” On page 8 in the bottom right corner was 8’s prime factorization, 2 ✕ 2 ✕ 2, with Two doing excited cartwheels and exclaiming, “Wow, it’s me! And me! And me!”

Quoting Two’s line, Beck, a 4th grade reader, said, “‘It’s me!’ I’ve never seen a book like that.”

The numbers being numbers have inherent, repetitive structure just bursting to come out. The children turned the page from page 8 to page 9. In the bottom right corner, they saw 9’s prime factorization, 3 ✕ 3. The children chuckled to themselves when they saw Three exclaim, “Look! It’s me and me again.”

And it wasn’t very long before the children reached page 16, where they saw 16’s prime factorization, 2 ✕ 2 ✕ 2 ✕ 2. They smiled and laughed again. Helen, a 6th grade reader, told me, “I like 2. He’s always there!” Theo, another 6th grade reader, remarked, “I like the patterns in the book. I feel rewarded.”

Maria: Humans are social creatures, and big areas of our brains are devoted to people and relationships. Our imaginations spark playful bonds with characters and toys. What if we could use all that social reasoning to boost our mathematical power? Yet merely adding a face and limbs to a number won’t give it enough life for a good story or game. My favorite authors create their number-characters and storylines with deep care for each number’s mathematical behavior. In beloved books like The Cat in Numberland and shows like Numberblocks, numbers are persons, and everyday adventures spring from their mathematical personalities.

Kara: The children also liked the connections to daily life. When they reached page 13 with its connections to bagels, 4th grade readers Beck and Elizabeth became animated, telling me that the bagel should be “tastier-looking” and that I should “toast it and color it.” Based on this feedback, the bagel did become tastier as I made edits to the book. (I can’t wait to see illustrator Coley Nielsen’s bagel! Yum!)

The children interacted with the drawings in ways that I didn’t expect, reaching out to grab the bagel or tracing the path of a jumping number-character. They touched the pages, brushing their hands over the number-characters, almost like they were petting them. Theo said, “Your drawings are so cute, so sweet!”

Maria: Why do we care so deeply about children touching their mathematics? Because that physical play develops excellent habits of mind. Children learn to order numbers in space to hunt for complex patterns. They use visual structures like graphs and tables to solve problems, or read data from clever diagrams such as factor trees in My Hundred Friends. Our society is full of complex number patterns and data. Early, happy familiarity with organizing numbers in space helps every child.

Kara: I usually spent about 15 minutes with one pair of children, and then the next pair of children came to my corner. Many pairs just read the book straight through, starting at page 1 and going forward. One pair of children, Theo and Pascale, reached page 24 and noticed that the number-character Twenty-Four had one strip of brown at the bottom. Pascale, a 4th grade reader, asked, “What’s brown?” She paused and then wondered, “ Is it 12?” as they flipped back to page 12. Pascale guessed, “6?” as they flipped back to page 6. “No, it’s 3!” she exclaimed as they landed on page 3.

They were on to something! They had picked up some sort of pattern with the colors. From that point forward, Theo and Pascale started reading with an eye towards what was going on with the colors. I didn’t explain the role of colors in the book; they were having fun trying to figure it out. I held myself back, asking a few open-ended questions, like “What do you notice?” or “What do you see?” 

This was such an exciting moment for me. I had created a world where kids were exploring and asking themselves math questions. They were playing with the number-characters, playing with math. At the end of our session together, Elizabeth remarked, “I like how it’s funny and learnful.” That felt like high praise! 

If you have a child, a classroom, or a fondness for cozy math, I invite you to come explore My Hundred Friends. What number connections might your child make? How might your student pretend-play with their favorite number-characters? 

This is a joint blog post by Kara Shane Colley, the author of My Hundred Friends that is soon to be published at Natural Math, and Dr. Maria Droujkova, the founder of Natural Math. We’ll be posting periodic updates about the book on the blog.

“The chalkboard appeared when Two wanted to draw on it.”

Maria: In My Hundred Friends by Kara Shane Colley, the math of factors and multiples is friendship. It’s a cozy little world, inviting readers to play and explore. From April 27 to June 27, head to My Hundred Friends crowdfunding campaign to preorder hardcover copies for your family or group, reserve your complimentary PDF, and support math joy for all!

Evolution of the cover

Kara: In making drafts of the cover, you can see that some aspects of the cover stayed the same. On every draft, there are number-characters hanging from the title, sitting on the title, running up a path, and cheeky number Seven trying to get our attention!

It was my publisher Maria’s idea that the number-characters should be doing math on the cover. In the illustrator Coley Nielsen’s cover, you can see Two building a cubic model with blocks, Fifty-Five playing tag with their factors, and Seventeen proudly displaying their factor rainbow.

Maria: It is said that sci-fi isn’t about the future and fantasy isn’t about magic: it’s all about our life, here and now. Where does math come from? We make mathematics as we live our lives, create, and discover together. Kara and the number-characters built such a friendly village with their mathematics! Imagine that. Let’s go!

Evolution of page 9

Kara: Collaborating with Coley has been exhilarating. The process of working with her helped me develop some ideas that weren’t totally fleshed out in my head. As you can see in the early draft, I originally thought of the pages of the story as loose leaf paper with a horizontal line at the top of the page and holes punched down the left side. As you can see on the second draft, the horizontal line at the top remained but the hole punches were gone. As we prepared to send over the second draft for Coley to draw her version, I realized that the whole book was like a math journal; it was a scrapbook that I made of my hundred friends. Because it was a scrapbook, I asked Coley to draw graph paper and make it look like it was taped onto the page.

Maria: Our numbers, our math – is it out there, hiding in plain sight all around us, to be discovered? Or does mathematics come alive by our creativity, the same way pretend-play toys or fiction characters do? The readers can see Kara’s and Coley’s art either way.

Evolution of page 12

Kara: Another evolution of the world occurred as Coley began to draw drafts of page 12. In my mind, I was the one writing “12 is one dozen” and I was the one who drew the egg carton. Coley drew a couple of versions of the egg carton for us and that made me realize that I didn’t want a totally realistic egg carton. I wanted a rough “sketch-like” version of the egg carton.

On the other hand, the chalkboard at the bottom was part of the imaginary number-character world. I didn’t draw the chalkboard. The number-characters came to play in the math scrapbook, and the chalkboard appeared when Two wanted to draw on it.

Come check out My Hundred Friends. The book is a kind, welcoming math gathering. We invite you to come, hang out, and meet a few new friends.