The best representations of mathematics may not exist yet
Humility. It’s all I can feel after watching 3Blue1Brown’s latest maths video, in which Grant Sanderson and his team provide the most novel of approaches to solving 2D equations, using colour. It is the second time 3Blue1Brown has blown my mind to pieces in just a few weeks, following their earlier video that solved the Basel problem with light.
While I was familiar with the results/methods in both cases, the representations came as a complete surprise. The 3Blue1Brown videos are far from perfect (they proceed at a cadence that sometimes proves frustrating, lack interaction and, in the most recent example, may be of little use to the colour-blind), yet they remind us that our understanding of mathematics will never be complete. However well you think you understand a concept, there will always be new representations to deepen your thinking and force you to confront familiar truths in unfamiliar ways. Mathematics is never a done deal.
Humility is what keeps us open minded to new and novel representations of old concepts. It stands opposed to the arrogance of attaching oneself to singular representations, and to assuming that there is only one way to engage with a given maths concept.
Take bar models: a particular representation that is in vogue here in the UK as the poster child of maths mastery. A key tenet of mastery-based approaches is that mastery is never fully acquired; our understanding of mathematical concepts runs infinitely deep. There are always more problems to solve, more reasoning to be articulated. How sad then, that the implementation of mastery is often restricted to the specific representation of bar models. Don’t get me wrong — I have much love for the bars: Cuisenaire rods and Base 10 blocks are among my most commonly used manipulatives, not least because they can showcase a range of concepts. But bar modelling is often embraced so rigidly that the attachment borders on ideological; I have reluctantly indulged in debates with educators who refuse to entertain any alternatives to the bar model representation.
An exclusive focus on any representation — bar model or otherwise — only defeats the pursuit of mastery because it induces procedural fixedness: students may acquire fluency with a particular representation, only to come unstuck when asked to apply their knowledge to less familiar contexts. Flexible thinking comes from having multiple representations. The more ways we have of seeing a particular concept, the exponentially more likely we are to apply that knowledge in new situations.
The digital medium is poised to serve up a multitude of previously unconceived representations. 3Blue1Brown is an all too rare of EdTech shunning the temptation to imitate static, textbook representations. Instead, Sanderson has illuminated my understanding of mathematical concepts in a manner so dynamic that one can scarcely imagine his content in print form. This is not to discard print representations altogether — Sanderson himself lists a slew of textbooks as his inspiration. But to achieve our deepest understanding of mathematics, we have to embrace digital, print and whatever other media are available to us to build new pathways to mathematical enlightenment.
If the notion of changing up our representations of maths unsettles you, just remember that our mathematical worldview has always been in flux. While mathematical truths are cast in stone, the way in which we interact with those truths depends on the technology available to us. You may, for example, assume that Algebra is necessarily symbolic: it is hard to imagine an algebraic problem that does not involve an x. Yet the original architects of Algebra did not employ such formal expressions. The Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī did not solve quadratic equations in notation that you or I would recognise. He resorted instead to tirelessly writing out his problems of interest (motivated mainly by trade and inheritance laws) in narrative form.
The symbolic form that we so closely associate with Algebra today owes much to the invention of the printing press, which gave mathematicians unprecedented opportunities to share their ideas across cultures and civilisations at rapid speed. The need for a common language, and to avoid ambiguity in translation, became ever more paramount and so entered the x’sand y’s we take for granted today.
Keith Devlin is among those who have argued that the digital technologies of today will occasion a shift towards representations that are both efficient and intuitive, arguing that video games are the natural medium for 21st century mathematics. It’s an idea worth taking seriously given the confusion and anxiety that symbol-rich representations induce.
My greatest hope for EdTech is that it inspires new representations of mathematics. I would wager that for a good chunk of mathematiccal concepts, the most illuminating representations still await our discovery. The ascendance of dynamic modelling apps such as Desmos and GeoGebra, not to mention the breathtaking videos of 3Blue1Brown, are already reshaping the way we see and interact with mathematics.
The mathematics that awaits us may only be limited by our imagination. Thank goodness we have people like Grant Sanderson dreaming up new ways of presenting old ideas.