Mathematics is full of stories. Stories that delight and intrigue, like Fermat’s note promising a proof too large to fit in the margins of the page, and (apocryphal) stories of insight, like a naked Archimedes leaping from his bath shouting ‘Eureka!’.
Stories bring mathematics to life. They also help us to make meaning of mathematical ideas. In his book ‘Mathematics for Human Flourishing’, Francis Su writes: “For millennia, humankind has used stories to convey history or essential truths. A story creates a narrative from disparate events and connects listeners to itself and to one another. It is no different with mathematics. Connecting ideas is essential for building meaning in mathematics, and those who do it become natural story builders and story tellers.” (pp. 38-39)
The power of mathematical stories is that they make connections. Francis gives examples of different genres of stories. A significance story explains why a concept is important, for example, the invention of zero. An explanatory story helps demonstrate a concept, for example, the ‘proof without words’ below helps bring meaning to the formula for the area of a circle, A = πr2.
An experiential story that has stuck with me, and that I would strongly encourage you to read, is the one shared by Sami Shah. Now, instead of trying to remember the formula for an infinite geometric series such as 1/2 = 1/3 + 1/9 + 1/27 + 1/81 + …, I instead recall the story of the greedy friends who demanded more and more paper from the increasingly-harried paper master. I retell myself the tale, reconstructing the symbolic meaning as I go.
At this week’s Maths Teacher Circles in Australia, we examined another collection of stories, beginning with the story of a party: Seven students met at a party. Each student gave every other student a high-five. How many high-fives were there altogether?
This playful scenario invites us to explore. We can act out an experiential story by pretending to be at a party. We can draw diagrams (of what are also known as complete graphs) to visualise different sized parties, and organise our findings into tables.
We can systematically list out what happens when seven students are at the party, and see why 21 high-fives can also be thought of 6 + 5 + 4 + 3 + 2 + 1. The layout of the list might remind us of triangular numbers.
Coming back to our experiential story, we could ask each student how many high-fives they participated in. We calculate 7 (students) x 6 (high-fives per student) = 42 (high fives). But because two students participate in each high-five, we’ve double-counted. So the total number of high-fives is (7 x 6)/2 = 21.
The idea of double counting inspires me to revisit my systematic list. I can represent each high-five with a black dot, and duplicate the triangle as shown by the orange dots. The number of dots in the rectangle is 7 x 6. But we have double counted so we divide by two: (7 x 6)/2 = 21.
I could have also chosen to duplicate the sum: 6 + 5 + 4 + 3 + 2 + 1 + 0. Arranged like shown below, I see 7 groups, each summing to 6. So the total is 7 x 6. But we have double counted so we divide by two: (7 x 6)/2 = 21.
Perhaps this reminds us of the story of Gauss who apparently stunned his teacher by quickly summing the numbers from 1 to 100, a task that was meant to occupy the class for some time. (A story that is likely more fiction than reality.)
To return to the earlier quote, seemingly disparate concepts in mathematics (triangular numbers, the sum of the first n integers, complete graphs, the Handshake Problem) are all connected through a narrative that helps us build meaning.
As Francis says: “Mathematical ideas […] grow richer in meaning the more you play with them — each understanding brings a slightly different perspective — so that when you look at an idea in just the right way, you feel enlightened.”
With great power comes great responsibility
Because mathematical stories can hold such power, we have a responsibility to think of their potential effect. Without care, the mathematical stories we tell can reinforce bias and stereotypes. The diagram below may remind you of a familiar concept.
This is ‘Yang Hui’s triangle’ which was used more than 400 years before what many call ‘Pascal’s triangle’. In fact, the triangle was also known by Indian and Persian mathematicians before its first appearance in Chinese manuscripts. Many of us have internalised a skewed, Eurocentric view of mathematics that is not historically accurate. Our responsibility is to question the veracity of the stories we tell.
In a similar way, the cast of characters of our stories is often incomplete. How many women mathematicians can you name? A 2014 survey found that a quarter of people in Europe were unable to name a single famous female scientist, either living or dead. The #MathGals project is the brainchild of Chrissy Newell and her daughter Cora. Their t-shirts start conversations about women mathematicians from all parts of the world, including Australia. (More info about the women on the Australian shirt shown below can be found here.)
Another brilliant initiative to change the narrative about who does mathematics is The Mathematicians Project by Annie Perkins, who helpfully includes a large list of not-old-dead-white-dude mathematicians for us to read and tell our students about.
What stories enrich mathematical understanding for you? What stories convey that everyone belongs in mathematics?