Mental representations in mathematics
What does mathematics look like to you? Do you see a wondrous landscape filled with connected ideas, or a sprawling mess of symbols? The distinction matters a great deal, because your mathematical worldview is inextricably tied to your success in the subject.
We are all familiar with the multiplication grid, a centrepiece of classrooms and home studies the world over:
The standard multiplication grid
You cannot fault this image for accuracy. There is a bluntness to the grid; a seemingly disconnected array of factual truths. They are to be learned, memorised, recalled. To the trained eye, however, patterns and structure abound. Just as Cypher doesn’t even see the code anymore in The Matrix, a mathematician sees much more than 100 solitary numbers.
Perhaps they see this — a scaled drawing of the multiplication grid:
Multiplication grid to scale (source)
With this simple tweak, the grid is beginning to speak to us. It conveys size and proportion rather than just numerical outputs. It connects otherwise disconnected topics, binding together number with geometry; multiplication with area.
With deliberate use of colour, we can extract a different type of structure — here we see the multiplication grid as a nested collection of smaller grids:
Adding some colour to the multiplication grid (source)
The size and shape of each layer reveals new truths. Often, it reveals old truths in new ways — can you see why summing the first n odd integers results in n²?
Having the right mental representations is the key to developing your mathematical potential.
The road to expertise is paved with 10,000 hours of practice, so claims Malcolm Gladwell in Outliers. Gladwell’s claim is a gross simplification. Anders Ericsson, the man behind the original research, has set the record straight in Peak. He duly rebukes Gladwell for imposing an arbitrary threshold of 10,000 hours (it actually varies within and between disciplines). More importantly, Gladwell does not distinguish between different types of practice. Ericsson’s research, and the weight of his book, is premised on the principles of deliberate practice.
The chief virtue of deliberate practice is that it helps us develop rich mental representations: pre-existing patterns of information that sit in our long-term memory.
The stronger and more numerous our representations, the more we can draw on them to connect ideas, develop intuitions and solve problems.
Mental representations anchor us to our worldview of mathematics. The more we have, the better. Sticking with one-word titles, Adam Grant’s Originals lays out the key characteristics of creative geniuses. One such characteristic is that they can form more ideas when approaching a task. In mathematics, it stands to reason: you are more likely to solve a problem if you have different lines of attack. These ideas spring from our mental representations.
But just as not all practice is created equal, some mental representations are stronger than others. This explains why many people leave school fluent in the facts of multiplication but without any real intuition or sense of number. They possess weak representations that privilege recall over understanding. When students are drip-fed multiplication facts as a collection of disconnected truths, it is hardly surprising that they struggle to connect those facts to broader mathematical themes. The best mathematicians understand multiplication as part of a rich tapestry of concepts. For them, fluency and understanding are mutually reinforcing.
Mathematics can seem lifeless when our representations are weak.
Prime numbers do not get the respect they deserve in the curriculum. For many, primes are simply a collection of numbers that happen to have two divisors. Another item on the syllabus, useful for calculating HCFs, LCMs and other arbitrary values that frequent exam papers. To the mathematician, however, primes are the DNA of the subject. This biological metaphor is no accident, nor a triviality.
The Fundamental Theorem of Arithmetic tells us that every whole number is a unique product of primes. The more you dwell on this revelation, the more you will be drawn to the supremacy of primes. How appropriate then, that these objects are shrouded in mystery, giving rise to some of the deepest unresolved problems in mathematics.
The fascination with primes is not the preserve of professional mathematicians. With the right mental representations, they can delight, intrigue and perplex us all at once. So here is the 100-square, as you have never seen it before, courtesy of Daniel Finkel:
Daniel Finkel’s 100-square in primes (source)
As Finkel himself advises, let this grid speak to you. Explore the patterns. Play with its structure. Sink into the depths of multiplication, and discover the starring role primes play in our times tables.
If maths was nothing more than a sprawling mess of symbols for you at school, take solace in the fact that an enthralling, quite separate universe awaits you. Maths is replete with rich and wonderful mental representations — the kinds that foster understanding and forge connections between ideas. This is the maths that mathematicians fall in love with.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
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