# Rethinking Story Problems to Engage Students’ Mathematical Curiosity

## The Arithmetiquities, an Adventurous Merging of Mathematics and Narrative

--

By Jason Ermer

In 2004, while proctoring the American Invitational Mathematics Examination (AIME), I encountered a problem that started me on a years-long adventure of mathematics and storytelling.

A unicorn is tethered by a 20-foot silver rope to the base of a magician’s cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is (a-(sqrt(b))/c feet, where a, b, and c are positive integers, and c is prime. Find a+b+c. (2004 AIME I, problem 14)

Since my youth I have been a fan of tales like The Hobbit, the Choose Your Own Adventure gamebooks, and fantasy roleplaying games like Dungeons & Dragons. Although the context of the unicorn and the magician’s tower is irrelevant to the mathematics of the AIME problem, my fantasy-loving inner teen was drawn so strongly to the context that I couldn’t rest until I had solved the problem. Something about the story motivated me to explore the mathematics it contained.

As I reflected on the feeling, I wondered whether the same might be possible for my students. Could I engage students’ mathematical curiosity and creativity through narrative?

## Reconsidering the Potential of Story Problems

“Story problems” have a bad reputation with students for a variety of reasons. Story problems frequently require a student to extract the relevant information and decide what to do with it, which makes them more challenging than exercises that are purely computational. Contrary to my students’ opinions, I see the opportunity for mathematical modeling as the true virtue of the story problem, especially in a world where technology can assist us with mechanics and computation.

In other ways, though, I understand my students’ consternation. Story problems draw criticism because they so often embed mathematics in what educators have dubbed “pseudo-context.” A pseudo-contextual problem references real objects or situations, but simultaneously asks students to suspend their real-world problem solving skills about the situation. Consider, for example, the farmer who counts how many heads and how many feet are in their barnyard, rather than simply counting the number of pigs and chickens! But pseudo-context has been part of mathematical problem-posing since the Rhind Papyrus, and the prevalence of such “not entirely reasonable” scenarios serves to highlight the power of puzzles to engage the mind.

Furthermore, if we look beyond the two-sentence textbook story problem, we find many skilled authors who have demonstrated the value in merging mathematical content with narrative. An engaging story can draw the reader in and present mathematical ideas in a way that is more clear, compelling, and entertaining than a page of equations or diagrams.

The narrative context of Flatland: A Romance of Many Dimensions by Edwin A. Abbot personifies geometric figures as they explore multi-dimensional spaces. In The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, mathematical ideas arise as a boy named Robert meets the titular Number Devil in his dreams. The Number Devil guides Robert to unexpected locations where the pair discuss concepts like triangular numbers and infinite series. In both Flatland and The Number Devil, deep mathematical ideas are explored as the characters ask questions and have conversations, similar to what might occur in a classroom.

Lewis Carroll, known best for Alice’s Adventures in Wonderland, published many mathematical puzzles, including a serialized set of humorous short stories called A Tangled Tale, each of which includes a mathematical puzzle. Carroll’s goal was to get readers engaged in mathematical thinking by sneaking the puzzles into the story “like medicine so dexterously, but ineffectually, concealed in the jam of our early childhood.”

## Introducing The Arithmetiquities

My own attempt to merge mathematics and narrative has followed in Lewis Carroll’s footsteps. The Arithmetiquities is a serialized fantasy adventure tale told through thirty-six mathematical story problems. The story is a Lord of the Rings spoof, featuring a quirky team of adventurers who fan out across the land in their quest to find the Arithmetiquities, a collection of ancient mathemagical artifacts, such as the Sieve of Eratosthenes and the Roots of Unity.

For example, in chapter 26 Egga, Floora, and Greeta — the dwarven shieldmaidens of Lumparland — have ventured into the snow-capped mountains around Yälä Peak in search of the famed Golden Rectangle.

The shieldmaidens, along with their mountaineering party, reach the Shrine of the Golden Rectangle only to find that Hämähäkki, matriarch of the ice spiders is already there, along with her minions!

Hämähäkki is surrounded by her royal guardian regiment, twenty fierce ice spiders with a long and valiant history of service to their matriarch. They have been through so many battles, in fact, that the members of this regiment have, on average, only seven legs (instead of the standard number of legs for an ice spider which, you may have guessed, is eight). One of them, the most grizzled guardian, has only 4 legs left!

Given this information, and knowing that every spider has a whole number of legs, what is the maximum number of spiders in the regiment who might still have all eight legs?

I have shared the Arithmetiquities with students and teachers, and together we have seen how an ongoing story can engage learners and draw them into mathematics:

`Student and Teacher Reflections on The ArithmetiquitiesIt is a great way to share, be creative, and make math more fun. (Sixth grader)It makes math seem less boring and the stories help me understand the problem instead of just some equation. (Sixth grader)I’m always looking forward to the next chapter. I’m someone who likes to read, so the fact that a continuous adventure, with many heroic characters, was put into these math problems makes it even more fun! (Sixth grader)I like how it's not just like typical equations, it makes me think about different things and stretch myself. I also like how you can follow the characters through the different chapters. (Seventh grader)They have a fun little twist to them and more detail. It’s not like, “Sally had 3 apples,” but instead there are chapters that all come together and are actually fun for kids. (Seventh grader)Having the problems be in a story form with characters and stories you can follow, is helpful for me and also helps me get more creative with how I solve things. (Seventh grader)My students LOVE Dennis. I’m not sure why this was the character that grabbed my 7th grade class’s attention this year, but every Friday when we tackle a new chapter of The Arithmetiquities, they are eager to see if Dennis is involved in that week’s adventures. Dennis has even made it into other math problems because my students are so invested in him! (Seventh grade teacher)`

## Creating Narrative Settings in a Math Circle

Many chapters in The Arithmetiquities are suitable for discussion at a Mathematics Circle for students or teachers. At a recent session of MTC Austin, we discussed creating narrative settings to engage the particular interests of one’s students. We considered story problems that might take place in popular literary or video-game settings, or draw on popular trends from anime to superheroes. Participants were invited to flex their creative writing muscles and spice up some sample “boring” story problems. One participant chose this original:

I have a cylindrical tin of popcorn that is 18 inches tall and has a radius of 4 inches. I want to use the tin for something else and need to empty the popcorn into a box. The box is 8 inches long, 8 inches wide, and 14 inches tall. Will the popcorn fit in the box? Explain.

From that starting place emerged this epic quest with an ecological flair:

The maiden Zeelan carries the Master Flask, a flask of cylindrical shape that is 18 inches tall and has a radius of 4 inches. The Flask contains the last seeds of the great Normwood Tree. This Tree is the most important tree in the realm and there are only three left, which means the seeds are invaluable. She has been tasked with the job of retrieving the ashes of the most celebrated warrior maiden of the land and has been requested to use the famed Master Flask. She must store the most prized seeds in a safe place and has been offered a treasure chest by her trusted warrior, Zeldon. The chest measures 8 inches long, 8 inches wide, and 14 inches tall. Will all of the seeds be safe? Or will some have to be stored in a different container? The future of the Normwood Tree species depends on this answer!

The Arithmetiquities is available online for teachers to explore and use with students (www.arithmetiquities.org). Plus, every textbook contains a wealth of humdrum story problems ready for your own creative reinterpretation! Consider undertaking a mathematical adventure — either for yourself as a creative exercise, or to share with the students or teachers you work with!