Don’t leave mathematics stuck in the past

Mathematicians have always turned to the technologies of their time to push the boundaries of human thinking, fuelled by a mindset of playful rebellion against the rules. So why, then, is the current curriculum setting out to do the opposite of that, asks Junaid Mubeen, who makes the case for a more recreational approach to the subject
20th September 2019, 12:03am
Don't Leave Maths Stuck In The Past


Don’t leave mathematics stuck in the past

School maths is built on a false promise. It goes something like this: learn this huge heap of mathematical procedures so that, one day in the future, they may lead you to gainful employment.

The utilitarian framing is not the problem - as long as maths continues to dominate the education agenda, it had better be tied to enhanced job prospects. The problem is that the particular brand of maths we experience in school, with its heavy emphasis on procedures, is no longer fit for the workplace (or for advanced mathematical study, for that matter).

The most honest answer to a pupil asking “When will I ever need this?” is: “In all likelihood, you won’t. And, what’s more, ploughing through all those practice questions was just a waste of your time and potential.”

In the past, the ability to perform mathematical procedures with speed and unswerving accuracy would have given you a serious edge in the job market. The first “computers” were in fact human, whose flesh-and-blood calculations in the 17th century led to the creation of accounting ledgers and navigational tables. More recently, the human computers of Nasa propelled humankind into space, leaving no margin for error in their handcrafted calculations.

In the half-century since Neil Armstrong’s giant leap, the world has been transformed by technology. The smartphone in your pocket is as potent as the technologies that shot us to the Moon and back. Human computers have become a relic of the past, which means that our education system is preparing students for a workforce that no longer exists.

Mathematicians, for their part, welcome the rise of the machines. They realise that calculation is not a natural human strength and have always sought to relieve its burden by leveraging the tools of their time. Our ancestors marshalled pebble-counting schemes to overcome our difficulty in evaluating large quantities. Many centuries later, the slide rule became a mainstay of laboratories and classrooms, as it eased the burden of complex calculations (by reducing multiplication to addition, for example).

Now, with the advent of digital technologies, mathematicians are liberated from the tired act of wading through complex calculations. It is not that calculation is no longer important to maths - on the contrary, the ability of computers to perform masses of calculations at lightning speed enables mathematicians to peer further into their subject by generating tonnes of data. The issue is around how the cognitive labour is divided: the computers compute, giving human mathematicians space to think more deeply.

Maths has always been inextricably tied to the technologies of its time. In the age of powerful computers, educators should focus on the skills that complement those of highspeed calculation. There are many elements of mathematical thinking that machines cannot yet emulate. These are the ones that will most adequately prepare students for life and work in the digital age. I will focus on three such elements that have been squeezed into the margins of school maths: representations, reasoning and imagination.

The ability to structure bits of information into more meaningful wholes is sometimes referred to as mental representations or schema. As the internet gushes forth exabytes’ worth of data, making sense of all that information is becoming an ever more vital skill. Maths is primed to help because it is replete with patterns and structures that are waiting to be uncovered and connected.

‘Survival skill’

Representations offer a resolution to the enduring hot potato that is times tables. It is not a question of whether automatic recall of such number facts is worthwhile. Of course it is - a lack of basic fluency can overwhelm students’ working memory in the midst of problem solving. It is just as crucial, however, that we do not limit the scope of calculation to blunt acts of memorisation. The human brain seeks to find meaning in bits of information. Thankfully, the number system is a patchwork quilt of patterns and structure. By seeking out the connections that bind numbers together, whole swathes of number facts become so much more digestible.

For instance, the commutative property of multiplication, which says that a x b = b x a, compresses the times-tables grid almost in half. Mathematicians routinely seek ways to compress vast amounts of information into tidier chunks by seeking out big-picture meaning to the objects that they study.

Times tables enjoy prominence in the education discourse because they are seen to represent the basics of maths. Rich representations, however, also guide us to other fundamentals, such as the choice to use a base-10 number system. It took humans thousands of years of grappling with different base systems - the crumbs of which still linger in our everyday lives (bases 60 and 24 govern our timekeeping systems, for example) - before fixing on base 10. Surely we should expend some effort in understanding how such choices came about.

It is no small miracle of the human mind that we devised a system that represents any quantity using just 10 digits. We devote precious little time to understanding the inefficiencies of non-positional systems such as Roman numerals. We show a flagrant disregard for prime numbers, the most fundamental building block of whole numbers. If policymakers see fit to issue a test of multiplication facts, then omitting any mention of primes seems facile.

Reasoning, meanwhile, is the lifeblood of maths: it is the glue that holds together mathematical truths. In a post-truth world mired in fake news, the ability to separate fact from fiction qualifies as a survival skill. Mathematical reasoning conditions the mind to examine and create arguments, or proofs, with unflinching rigour.

The most lamented of all results in school maths is Pythagoras’ theorem, which states that the lengths of the sides of a right-angled triangle are bound by the formula a2 + b2 = c2 (where c is the length of the hypotenuse). The boldness of this claim cannot be overstated: it is an assertion for every conceivable right-angled triangle, small and large. At school we are instructed to verify several instances of the formula, but we shy away from the most profound question of all: why does the formula work every time? Quite remarkably, there are more than 350 proofs of Pythagoras’ theorem - that is, 350 ways of illuminating the same fundamental truth. The essence of maths does not constitute shunting around the symbols of some preordained formula, but probing the arguments that lead to such truths in the first place.

School maths shares a dubious trait with computers: both are confined to fixed rules and regulations. Humans, though, cannot help but tinker with rules. We are daring enough to imagine alternative worlds that snap free of the usual conventions. Just as movies usher us into altogether different realms, maths gives us permission to shake up prescribed rules and to examine the logical consequences of our choices.

For the longest time, the prevailing view in maths was that negative numbers cannot have a square root. That all changed in the 16th century when Rafael Bombelli decided otherwise and willed into existence a new number, i, whose square is -1. At around the same time, mathematicians were indulging the idea of predicting the future. Until then, the future was deemed out of bounds to mathematicians - it was heretical to speculate on events not yet seen. That all changed with the birth of probability, a mathematical toolkit for dealing with uncertainty. As for Pythagoras, did anyone ever consider right-angled triangles on surfaces other than a plane? It turns out we get some rather interesting results on three-dimensional surfaces such as spheres.

Notice how these examples are rooted in a mindset of playfulness: a willingness to rebel against rules, even without a practical goal in mind. Yet all went on to have immense real-world application. The physicist Eugene Wigner wrote of the “unreasonable effectiveness” of maths - concepts that seem inane at first often result in the most practical uses. I believe there is another sense in which maths can be “unreasonably effective”: in training the human mind to think creatively and to solve complex problems.

The usefulness of maths is not felt in specific methods, most of which elude our mental grasp within months of encountering them. Instead, maths exposes us to elegant systems rich in concepts and problems that help to develop the thinking strategies above, which we carry with us to all walks of life.

School maths captures only a sliver of this potential because it is fixated on procedures; it is one particular system of thinking, which has wandered past its expiry date. There will always be a place within any mathematical system for acquiring particular methods, but they should serve the purpose of animating the essential characteristics of that system. Procedures must never be the reason for doing maths, and while some practice and some fluency helps, we must stop being so deferential to them.

For maths to serve its purpose as a system of thinking, we need to reshape topics so that, for example, geometry would be taught primarily as a system for reasoning. We would reduce the bloated diet of procedures and make space for more enriching topics, such as topology and game theory. The whole brand of maths would shift towards a recreational slant. A common theme among maths enthusiasts is that their formative experiences with the subject were situated outside of the classroom, as they plunged headfirst into an array of puzzles. Maths is synonymous with recreational puzzling; the latter must be pulled from the fringes of “popular maths” and permitted to take centre stage in the curriculum.

The most important consequence of any rebrand to maths is in how it orients the learner. As it is currently presented, school maths takes on an authoritarian bent: it is a litany of rules to be passively consumed.

A recreational brand of maths, by contrast, subtly but surely changes students’ attitudes towards learning. Maths is not something that is imposed on them, but something they can assume agency over. It is an act of creation, where ideas are constantly challenged and refined - an art as well as a science.

To understand a mathematical concept is to understand its evolution over time, its origins, its spread across different cultures and civilisations, its controversies, its reasons for prevailing. Students should be encouraged to view themselves as beneficiaries of a rich heritage of ideas and concepts, which they themselves have the freedom to play with and even challenge.

It all adds up

The richest mathematical experiences are also the most social. We come together to decide which problems are worth solving. For the most taxing problems, we pool together our diverse skills and perspectives to achieve the key insight. In a digitally connected world, we can ill afford to conceive of maths as anything other than a shared enterprise.

Lastly, and most importantly, all students, regardless of their perceived abilities, must be granted equitable opportunities to experience the power and beauty of maths. The recreational brand of maths is also the most inclusive because it draws on our most natural human strengths. Never again would we allow any student to abandon their mathematical talents on the grounds that they were unable to emulate archaic skills so readily assumed by computers.

The changes implied here would not come easily to educators. We would have to think much harder, for instance, about how to assess students in meaningful ways. I confess plainly that this brand of maths is also the hardest to pin down in rubrics and assessment criteria. We also know how difficult it is to foster collaborative learning experiences that are truly beneficial to all participants. I see these as challenges to be embraced by teachers, researchers and policymakers, and not as reasons to discard the fundamental tenets of maths.

Our students deserve a brand of maths that plays to their core human strengths, and that prepares them with the most flexible strategies for thinking. At a time when the threat of automation is forcing humans to examine our place in the workplace and in society, recreational maths is the surest and most uplifting educational experience we can offer to students.

Dr Junaid Mubeen is a mathematician and a Countdown champion. He is also director of education at Whizz Education

This article originally appeared in the 20 SEPTEMBER 2019 issue under the headline “It’s a numbers game”

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