How to make every student a maths ‘expert’
Just give them the right problem
Each week in maths club, we recognise a student for their effort. They get to take home a trophy, on the trust that it will be kept safe and returned the following week. As one student hands over the trophy to the next, they are expected to impart advice to their successor. The students’ suggestions typically centre on how to preserve the trophy’s condition. Occasionally, however, the children deliver wisdom beyond their years. One student, all of seven, recently shared her pedagogical outlook by quoting the Hungarian mathematician, Paul Halmos:
“The only way to learn mathematics is to do mathematics.”
Words that soothe a progressive educator’s heart, and that capture the spirit and design of maths club, where our primary objective is for students to experience the joy of mathematics.
Learning by doing seems so obvious and intuitive an approach, yet much of school maths is premised on the opposing model of learning by consuming. Students are presumed novices, incapable of displaying the creative thinking espoused by experts. Problem solving — the most experiential element of mathematics — is relegated to a side-note on the premise that, lacking in prior knowledge, students will be burdened by cognitive overload. It is certainly true that many maths problems, including the vast majority that appear in the curriculum, come loaded with pre-requisite knowledge. But perhaps we are focusing on the wrong problems. Novice and expert are terms that are relative to the underlying activity. With the right problems, all students can become experts.
I am a novice musician, having received no training on how to play an instrument. To date, music is something I have consumed and never thought to create. Enter Yokee’s piano app, which lets you play songs by striking a virtual keyboard according to a sequence of coloured keys. Yokee is strangely uplifting because while I know my practice falls way short of piano playing proper, I cannot help but feel responsible for the melody and rhythm that is achieved from my perfectly times strokes. In this simulated environment, I am no novice; my wife can testify to mistaking my rendition of Dido hits for the real thing.
In providing deliberate scaffolds, Yokee has empowered me to simulate musical expertise. And while I realise that deep expertise remains many thousands of hours of deliberate practice away, the app has gifted me with an unprecedented taste of what it means to play music. To create rather than consume. To be a musician.
Mathematics has always been my music; something to play with. This is not the mathematics of formal curricula, which is more akin to learning music by memorising score sheets. Imagine if we denied piano learners the opportunity to ever put hand to instrument. We would surely call out the approach as cruel and unusual, yet this is the defining pedagogy of much of school maths.
So then: how do we enable students to simulate mathematical expertise, and to experience the joy, emotion and sense of wonder mathematics can evoke?
It is not by limiting maths problems to those that rely on huge swathes of knowledge. Instead we must leave space knowledge-sparse problems that do not come stacked with arduous curriculum pre-requisites.
My own students — including the Halmos supporter — routinely delight in toothpick problems, board games, dotted grid problems, pattern sniffing, the bridge riddle and so many others that require the bare minimum of prior knowledge. Seasoned problem solvers will draw on their repository of mental schemas, plucking ideas from similar problems they have been exposed to in the past. But even the novice has much to go at given the right scope of problem. With experience, they too will nudge along towards expertise as they build their repertorie of problem solving strategies. And they’ll have more fun than they imagined mathematics could be.
I have to yield some faith in trusting my students to engage with each problem. I must believe in their capacity to intuit, invent, predict and hypothesise as they grapple with the unknown. They may even refine and reshape problem if they are in audacious mood. My students are never left unsupported: my role is to judge when they need more guidance, how explicit it should be, and whether the problem is situated within their realm of simulated expertise. My goal is to ensure the challenge of the problem meets their perceived skill, to keep my students in the glorious state of flow.
The nature of knowledge-sparse problems is such that, in moments of struggle, it is not the absence of prior knowledge that holds students back. They can devote their cognitive reserves to the problem itself, triggering all manner of discussion, debate and further exploration.
Most of all, my students get to experience, even if fleetingly and within a limited context, what it means to be a maths expert. They are exposed to an authentic brand of mathematics that is predicated on exploration. They need not wait until some ambiguous delivery date of ‘later on’ to enjoy all that mathematics has to offer. They realise that mathematicians take charge of problems and make them their own.
We can all be maths experts given the right problem.