Mathematicians are storytellers. At times it is obvious, as in the case of Charles Dodgson, the 19th century Christ Church maths fellow who is better known for his literary works — you know him by his alias, Lewis Carroll. But storytelling is a natural device for the everyday mathematician too.
My friend Ed once told me that he had spent the afternoon reconstructing every proof from an undergraduate course in Real Analysis, without any aids. Given that he had taken that course seven years prior, and had not been formally practising mathematics for five of those years, I was amazed. Amazed that he had selected this activity among all others to while away his afternoon, and amazed that he had pulled it off.
Mathematical proofs can spill over several pages and often appear as nothing more than a mesh of symbols. I can barely scrape together fragments of those proofs; he had recovered every detail. Ed is not renowned for his memory skills, so I demanded an explanation.
For each proof, Ed had recalled no more than a couple of key ideas, firmly imprinted in his long-term memory from those earlier studies. Taking these ideas as signposts, Ed was then able to fill in the remaining details by leaning on his understanding of how those signposts relate to one another.
To hear Ed tell it, each proof is a story. Think about your favourite novel or film: your recollection depends not on the fine-grained details of every scene or dialogue, but on your high-level understanding of the characters, plots and twists.
My short-term memory skills are superior to Ed’s — I would commit every detail of every proof to memory and reproduce them in exams. But by focusing so much on every detail, I was sometimes oblivious to the signposts within each proof. It is why I am unable to reproduce Ed’s feat: my memory leaned too much towards factual recall, whereas Ed clung to the associations between key ideas.
Ed was able to recover the proofs by combining his recall of the signposts with his ability to link them together. He was effectively problem-solving on the fly, working out the details while I was relying on pure retrieval. Ed’s earlier exposure to the proofs had left a trace of the details, but it was his depth of understanding that allowed him to recapture them in full force several years after the fact.
Stories are a powerful mechanism for organising ideas, so much so that they have earned the label of ‘psychologically privileged.’ Ed Cooke is a Grand Master of Memory (and a different Ed to my friend), who has this to say on how stories aid memory:
Stories make learning connections easier because they make what happens next feel like it’s inevitable. Each item seems to be incomplete without all the others. In this way, stories generate context and momentum, and they bring closure, telling your brain when it’s done…the technique can be applied to more or less any form of information you want to remember. In each case, the best thing to do is weave the items into a compelling story line. The more that this narrative wraps tightly around the available facts and makes each feel like an intuitive part of the whole, the nearer it will come to pure understanding.
Ed Cooke’s advice applies just as well to listing the elements of the periodic table, the wives of Henry VIII and the proofs of Real Analysis. Mathematics may be the most privileged of all domains because it is inherently rich in structure and patterns. Mathematical proofs exist as a collection of joined-up ideas. Weaving them into a storyline is a natural effort because every proof plays out as a logical sequence of events.
The stories may feel abstract, trapped in the particulars of the mathematical universe. That is where good instruction comes in; the educator’s job is to remind the student that ideas do not spring up from nowhere. Each proof should carry a sense of inevitability, which is only felt when the signposts are identified and the relationships between them are emphasised. Just as the best stories delight us with surprising revelations, the most satisfying and memorable proofs are the ones that hook us to their twists and turns.
It requires only imagination to transport abstract mathematical proofs to tangible, real-world stories. This laudable attempt at framing algebraic proofs involving binomial expressions in terms of actual people should give inspiration to those who, like me, tend to get lost in the technical details of a mathematical argument. Stories can help us develop intuition in the midst of symbol-pushing.
But storytelling also has a dark side. It is human nature to wrap ideas into a coherent storyline. The problem is that we pride coherence ahead of accuracy, conveniently slotting in missing details to fit an overarching narrative, without stopping to question their veracity. This is where the discipline of mathematical reasoning, of accounting for every assumption and checking every leap in logic, must shine through. Logic and reasoning can protect against the ‘narrative bias’ that seeps into our daily judgements.
Stories reveal a curious interplay between memory and thinking. ‘Memory is the residue of thought’ according to cognitive psychologist Daniel Willingham. Both Eds featured here are testament to Willingham’s maxim; their ability to recall information is based on their ability to think; to see information as a connected set of ideas. An often overlooked fact in education is that the converse does not hold: thought is not always the residue of memory. Swallowing information as a disjoint collection of facts will not automatically lead to understanding. In fact, an excessive focus on factual memory will only hurt memory in the long run, which is more associative by nature. It’s why Real Analysis proofs always seemed ‘longer’ to me: in the absence of signposts, there are more details to retrieve.
Consider the quadratic formula. Do you remember it? You would be forgiven for not recalling every placement of the symbols. Yet the quadratic formula is nothing more than the generalisation of the method for completing the square. Remembering and understanding this single fact — this signpost — will allow you to derive or ‘rediscover’ the general formula. Understanding the big ideas of mathematics lessens the cognitive burden of memorising every individual detail.
The truths of mathematics may be absolute, but how we arrive at those truths is a matter of pedagogy. A good proof will play out as a compelling narrative, anchored to signposts, full of surprise, and governed by reasoning that integrates the key elements of the story.
Mathematics is an act of storytelling that supports the dual goals of memory and understanding. Students must be supported to become the ultimate storytellers.