“My Hundred Friends” Campaign Update #3: When NOT to tell a math story

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.