Math Storytelling (Playful Math Carnival #182)

Once upon a time … 

What do you feel when you read that? Are you eager to hear a story? Well, once upon a time people shared important knowledge through stories. Can we do that with math? We can. 

I don't have any studies to show that this is a better way to learn math, but I feel sure that it can bring in many of the people who are turned off by math the way it's usually taught. Stories move us, and they're a great way to learn.

In this Playful Math Carnival #182, we explore math through storytelling. As always we start with stories about the carnival number. What is 182’s story? 182 = 2*91 = 2*7*13. That’s all I know. I can’t find anything exciting about this number. But maybe I can write a story about it …

The Buried Box

Ada and Bert were digging in their backyard, and struck something solid. It was a box! They knew that their mom sometimes tricked them by burying cool things in the yard, but this box seemed bigger and fancier somehow than the little trinkets she usually buried. [Continued here!] 

I usually write with slightly older protagonists. This was interesting to write and think about. I think I’m not fully satisfied because I don’t know enough about Ada and Bert to feel that they’re fully real yet. I do know that they were named after famous people who are mathematicians (but perhaps more famous in math-adjacent fields). Do you know who? 

And can you write your own story that uses the number 182 in a way that connects its properties to the storyline? 

Playful Math Carnival
This carnival is for July, August, and September. (We are now moving to one post a season.) You can see all of the past and future carnivals at Let's Play Math.

I was excited to write a post including September because September 25th is National Math Storytelling Day. Doctor Maria Droujkova, founder of Natural Math, created National Math Storytelling Day about a decade ago. She doesn’t like birthdays, and this was a way to transform hers. I enjoy birthdays, and think of it as a way to spread birthday joy without involving material gifts. September 25th is her birthday and mine, and we both love telling math stories, so Math Storytelling Day works for us both. 

Would you like to join us in celebrating it? Here's how:

• Write a math story and post it online. (Google docs is an easy way to do that.)
• Link to it in a comment on this post.
• (It would be super cool if we could get 182 math stories. I’ll list a bunch in this post, but we’ll still need lots more from you all to make it to 182.) 

A Little History
The stars have always been important, for navigation and for knowing the time of year. How did people keep track of them? They made up stories to go with groups of stars, and called those groups constellations. The Big Dipper is the most famous, and of course there are many stories from many cultures about this constellation, with people seeing it as a plough, a bear, a wagon, and more.
 

[If anyone reading this post can share more about math or science stories they like that were created between the ancient star stories and the 1600s, I’d be delighted.]

 
Now we jump way up to 1638, when Galileo published The Discourses and Mathematical Demonstrations Relating to Two New Sciences. (#2 on our list, after The Buried Box.) This book has three characters, Simplicio, Sagredo, and Salviati, who discuss mechanics, material strength, and motion. (The study of motion led to Newton's and Leibniz' development of calculus about 30 years later.) The characters share Galileo’s thoughts and discoveries through their discussions, making this work much more accessible than scientific papers written today. 

There are (at least) two sites attempting to catalogue all of mathematical fiction. Alex Kasman's list came first, and MathFiction gives him credit for much of their catalogue. You can see that the stories go back to ancient times. They include the wonderful stories of Lewis Carroll (late 1800s). A Tangled Tale (#3) might be his most overtly mathematical story, but many of them are full of puzzling logical conundrums. 

The first mathematical story I spent time with was a little book by Donald Knuth titled Surreal Numbers (#4). The math in it is college-level and proof-oriented. (My college professor gave this to me for an independent study. I loved it.) The characters, Alice and Bill, are a bit flimsy, and the story completely revolves around them exploring the mysterious math they’ve found on a rock. It worked for me.


The List
Since then I’ve discovered a number of other lovely math stories. The following list starts with five of my favorites, and continues with dozens of stories I’ve enjoyed: 

5. The Cat in Numberland, by Ivar Ekeland, can be shared with kids as young as 5, and will get folks of any age thinking. The premise involves a hotel with an infinite number of rooms. David Hilbert first described this mythical hotel in 1925, and many writers have played with the idea since. This version has lovely illustrations and a cat who does not understand. Sadly, it's out of print. I hope you can find it at your library.
6. 
   How Hungry Are You?, by Donna Napoli and Richard Tchen follows some animal friends as they plan a picnic with more and more of their friends. Equal sharing leads to some great thinking about division and factoring. The story is all dialogue and would make a great play or read-aloud with multiple voices.
7. The Man Who Counted, by Malba Tahan, is set in Arabian lands. The person telling the story recounts a number of mathematical adventures in which the man who counted stars. These mathematical puzzles are delightful. Malba Tahan is the pseudonym of a Brazilian math professor. This is translated from Portuguese.
8. The Number Devil, by Hans Magnus Enzensberger, follows Robert (who does not like math class) into his dreams, where the number devil shows him some pretty neat ideas. (No algebra needed.)
9. Math Girls, by Hiroshi Yuki, follows the narrator and two of his friends as they explore the mathematical questions posed just for them by their high school math teacher. The math gets quite difficult.
10. 
    Beast Academy, by the folks at Art of Problem Solving, is a whole math curriculum for 2nd through 5th grades. The beasts are great. (Many kids will tell you that Grogg is the best.) There isn’t much storyline beyond the math, but those beasts are enough to really entice some kids.
11. Mathemalchemy, by Hosler & Hosler, is a delightful online graphic novel, a fantasy story centered on math. (They say comic book. I think it’s long enough to count as a graphic novel.)
12. Gravity: A Fairy Tale, by Sarah Allen, is about physics, not math, but physics is quite close to math. The story uses a fantasy setting to really get you thinking about how gravity works. (Sarah Allen’s not-yet-published book, Allora and the Puzzles of Archimedes, which explores forces - gravity and friction, and the tools that help us move things - levers and pulleys, includes a number of math puzzles. I love it.)
13. The Adventures of Alexandria Jones is a series of short stories by Denise Gaskins. They are only available online. If enough of you fall in love with them and let her know, maybe she’ll find the time to publish them. I especially like The Secret of the Pharaoh’s Treasure. And …
14.  … Denise especially likes The Mosaic Tile Mystery, along with its conclusion, The Pythagorean Proof.
15. Poetry in the Park, by Kara Colley, has a surprise twist. (Can you figure it out before it's revealed?)
16. John and Betty’s Journey Into Complex Numbers, by Matt Bower, is a short story that explains how we go from counting numbers, to fractions, to the number line, and beyond that. (I've linked to the online version here, but you can now buy a hardcopy version of it.)
17. 
    Ying and the Magic Turtle and …
    
18.  … Farzanah and the 17 Camels, both by Doctor Sue Looney, are some of my favorites published by Natural Math.
19. Funville Adventures, by Sasha Fradkin and Allison Bishop, another gem from Natural Math, inspired John Golden to write some fanfiction. He created a new character, with a new power. (Check this one out if you'd like to do some math storytelling yourself, but aren't sure how to get started. My first fiction-writing, at maybe 10 or 12, was a sequel to Robert Silverberg's The Lost Race of Mars. That's fanfiction.)
20. Marco the Great and the History of Numberville, by SK Bennett, follows Marco through some scary adventures as he also learns quite a bit about numbers. (This is at a pre-algebra level.) If Marco has you hooked, …
21.  … you’ll also enjoy Marco the Great and the Mystery of Phaseville, about Marco's adventures with algebra.
22. Althea’s Math Mysteries, by Sue VanHattum (that’s me). These are not yet published, but you are welcome to ask me (at altheasmathmysteries@gmail.com) for a copy of my manuscript. In return, I ask that you send me some feedback, letting me know whether you liked the story, and how I might improve it. The first book in the series is Althea and the Mystery of the Imaginary Numbers. And …
23.  … the second book in the series is Althea and the Mysteries of Triangles, Circles, and Pi.
24. If you’re up for reading an incomplete manuscript, and you’re into learning calculus, you can ask for Althea and the Mysteries of Calculus.

There are twenty more lovely math stories at the math books page here on my blog, so that puts us up to 44 math stories.

Mathical.org gives yearly awards for “fiction and nonfiction books that inspire children of all ages to see math in the world around them.” (I would love to be one of their readers!) I think there are over a hundred fictional books at their site, so now we’re up to at least 144 math stories, not counting the thousands at those two math fiction cataloguing sites I mentioned at the beginning.

Can you write a math story for Math Storytelling Day?
I’d love to add 38 more stories to this list, to get us up to 182. If we make it to 182, I will randomly pick one of the stories in the comments, and send the author a printed manuscript copy of whichever (one) of Althea’s Math Mysteries they choose.

Here's a story I hope to add to next year’s list. Al, Logical is a graphic novel written by Xavier and John Golden. It will be published by Natural Math.

Do you want to further the cause of math storytelling?
Take this list to your library, and ask them to buy some of these books.


Visual Stories
Most of what I've included in "the list" is conventional written stories. But there are so many wonderful movies and short visual math stories that I've left out. Vi Hart is one of my favorite math storytellers (and I almost forgot her, because she doesn't do much written word). You can check out this interactive story that she co-produced with Nicky Case. It uses the power of mathematical modeling to explain segregation. (The only math is the idea of the triangles and squares wanting at least 1/3rd of their neighbors to be the same as them. Sounds innocent and simple, but it's way too powerful.) Maria Droujkova shared this on her Math Storytelling Day page.  (Sadly, Vi Hart's wonderful youtube channel is gone. If you search there on her name, you at least get videos that others have reposted. I will edit this post once I figure out how she wants people to find her work.) 

Please share other video goodies in the comments.



Write Your Own Story
If you’d like to try writing a story, I have a few story starters that might help:

• Dan Finkel describes the Square of Differences here (aka Diffy Squares, the name I first heard for it). Is there a story that might make it even more delightful?
• My friend John Golden wrote Quest for the Holy Snail. He teaches elementary math teachers, and he wanted to provide a meaningful context for subtraction. Does this give you ideas for your own math story?
  
  
  
  
• Denise Gaskins helped me find some interesting facts about 182. Do any of these suggest a story to you?
• 182 is the product of two consecutive numbers: 182 = 13 x 14. This also makes it double a triangular number. (“Two triangles get together and …”
• 182 is 222 in base 9 and 77 in base 25. It is 20202 in base 3 and 1212 in base 5.
• 182 is the number of connected bipartite graphs with 8 vertices. 
• 182 = 33 + 33 + 43 + 43 = 142 - 14.

On Fiction Writing
I whipped off The Buried Box in an hour or less, and yet I didn’t think of myself as a fiction writer until just the past few years. What has helped me ‘become’ a fiction writer?

• Retirement, for one! (Ahh, to be able to spend my days exploring math in whatever ways I choose. 
• Also, having a particular story that I was burning to tell. (In Althea and the Mystery of the Imaginary Numbers I retell the soap-opera-like stories of the mathematicians who were involved in the creation of complex numbers way back in the 1500’s. I thought the story was hilarious and fascinating, and really wanted to share it.) 
• If you want to get better at writing dialogue, it helps to sit near kids who are the age you’d like to write about, and just transcribe their conversations. 
• After that, write whatever you’re moved to write. Early readers of Althea’s Math Mysteries wanted to know more about the characters. I’ve been writing background from the point of view of each character for a few months now. I’ve learned things about each of them that I didn’t expect. I feel amazed, and lucky, that my characters are talking to me.

New Course, starting in September: A bit of Geometry, a bit of Trig

I'm excited. I've built a course. It's a bit of geometry, a bit of trig. And it uses Althea and the Mysteries of Triangles, Circles, and Pi and its accompanying activity book as its curriculum.

I’m now ready to begin accepting students in the course. It’s for students who:

§   Like math

§   Like reading stories

§   Have a solid foundation of algebra

§   Want to explore geometry and trigonometry

§   Look forward to connecting with others with similar interests

 

Would you like more information? Start here.

 

 

Next Playful Math Carnival coming soon ...

 ... do you have an offering?

I will be putting together the Playful Math Carnival #182 during the coming month. The theme will be math storytelling. If you have blog posts (or other online offerings) that would fit this theme, send me a link!

 

Here's the current Playful Math Carnival, so you can see an example of what they look like.

https://naturestudyaustralia.com.au/playful-math.../

Infinitesimal Land, Or How Infinitesimals Make Calculus Proofs More Natural

I'm writing the 4th book in the series of Althea's MathMysteries, Althea and the Mysteries of Calculus. (I've written the first two, and they still need more beta readers before we publish them. If you're interested, email me at altheasmathmysteries on gmail. The 3rd isn't written yet.) 

I taught calculus almost every semester for 28 years, and then I retired and started writing more. But writing this book is making me think about how I could have taught calculus better. I had already departed from the textbook in many ways, but I'm seeing today how chained to it I was, at least once in a while.

I taught the conventional proof for the derivatives of sine and cosine. I made a (very dense) four-page handout, walking students through the steps necessary. This involves a bit of review of trig, and some geometry, and some sophisticated use of inequalities. It's a lot. I'll guess now that maybe one student a year could really follow all that.

I knew there was another way, and I briefly showed my students the cool way to see this geometrically, if you allow for infinitesimals. But for some reason, I would stumble when explaining that way. So I never threw out the conventional proof.

This post is all about limit-based proofs versus infinitesimal-based proofs in a beginning course in calculus. It will help to know a little of the history. 

1. In the late 1600s, Newton and Leibniz put together what we now call calculus. Newton spoke of fluxions when he wanted to discuss infinitely small quantities. But what the heck is an infinitely small quantity?! Calculus was a huge boon to science, and there was much anxiety about whether its foundations were logically sound. 
2. It took other mathematicians desperately seeking a solution to this issue 150 years (early 1800s) to develop a logically sound foundation for calculus. That tells you how hard a problem it was. And their solution was so logically complicated that you have to be a lover of logic problems to actually get it. Their solution was limits. Back to that in a moment. First let's move another 150 years forward. 
3. In 1960, Abraham Robinson developed a number system called hyperreals (very carefully, one logial step at a time). And in that number system, infinitesimals are fine. But the textbooks had been written long ago, and no one was going to throw out those limits. (So sad.)

Here's a definition (written almost like a poem, because that helps):

limit (x -> c) f(x)=L (read this as "the limit, as x approaches c, of f(x) is L")

means

for any epsilon > 0

there exists a delta > 0

such that

if | x-c | < delta

then | f(x)-L | < epsilon  

Yeah, right. I'm guessing that bit sounds perfectly alien to 80% of the people who read my blog. (Does anyone still read blogs?) And it would be over 99% if you all hadn't already self-selected as math lovers.

 

It might be hard to understand how that solved the problem of infinitely small quantities. But it does. You want an example? Ok. 

limit (x -> 2) (x^2 - 4) / (x - 2) = 4. We can see that by multiplying out the top and factoring out x-2. If we don't do that, we have 0/0, which some of us like to call indeterminate. We know that 0/0 is undefined, but what we really want to know is what happens when x is infinitely close to 2. And then we don't have 0/0, we actually have x+2. And 2+2=4, so we have an answer. And that answer actually tells the slope of the y=x^2 parabola when x=2. That's one of the things calculus can do, and the limit stuff makes sure that it works.

But the limit stuff involves lots of hard algebra. And it's often hard to see where a step in the reasoning came from. So let's look at how infinitesimals solve this problem. That four-page handout for sine and cosine derivatives becomes a one-page diagram with a bit of explanation. I got the online handout, by Alex Alemi, years ago. [It just took me well over an hour to find it tonight online. Does he not see this as valuable? I found other stuff he's written much more quickly. Hmm.]

If you go look at that handout, it might take you a bit, but the connections between sine and cosine and their derivatives make visual sense once you get it. After writing some dialogue between Althea and her friends, as they try to figure out what's going on, I wrote their product rule adventures the next day, and then today they blasted through the quotient rule. (In the conventional proofs, both of these have non-intuitive algebra steps.) After their adventures with the sine and cosine derivatives, Aiden starts to talk about their time in Infinitesimal Land.

Excerpt from Althea and the Mysteries of Calculus

Perhaps I need to study a good calculus through infinitesimals textbook before finishing Althea's story. Any recommendations?