Mathematics – possibilities beyond the conventional classroom

Deepa Onkar

On a walk in the park or in the woods, you might notice the symmetries of the flowers, leaves and leaf-clusters, and the bodies of insects, and admire them. You might also stumble upon natural forms with mathematics in them – for example, a fern head, in a spiral, or the florets in a daisy head, in which the Fibonacci sequence – a numerical sequence in which each number is the sum of the previous two numbers (1,1,2,3,5,8,13,21,34,55…) can be traced. The hexagonal grid of a honeycomb divides the surface into regions of equal area with the least total perimeter – which makes it the most efficient way for bees to store honey.

The park or woods might look like an excellent place to discover mathematics, but many mathematics educators will probably view this idea with suspicion – this is not the ‘real business’ of doing mathematics. In schools, mathematics is about learning maths ‘facts’ (such as 8 times 7 equals 56, or 9 times 7 equals 63), which are to be memorized, as are the steps to solve problems. The faster a student recalls the maths facts, or the steps, the more able they are supposed to be. The practice of memorization continues into higher classes. No other subject is thought of as being all about speed and memorization than maths, and this makes it one of the most feared and hated subjects.

In her book “Re-imagining Mathematics: Learning through the Magic of Nature, the Arts and Friendship” Ashna Sen shows that learning mathematics, on the contrary, can be an exciting process. But for that to happen, a shift is needed in educators’ approach to the subject: an understanding that children have an innate numeracy which has to be nourished, rather than overwhelmed by memorization and speed. Sen is a mathematician who has taught undergraduates and children at a Krishnamurti school in England. Citing evidence from schools in forests in Scandinavia and the UK, Sen shows that when children have the leisure to explore the natural world – the sights, sounds, rhythms, visual and sound patterns, symmetries and symmetry within asymmetry, their mathematical intelligence is optimally engaged. The book does not contain methods to learn mathematics from nature, or as the title suggests, from the arts and friendship. Rather, Sen shows that mathematics is inherent to human existence, and can be discovered, played with, and learnt at depth in multiple contexts. Nature is the best place to begin with: she suggests that there are a vast number of natural phenomena that children could become attuned to, whether or not they are near a forest. They could observe the simple rhythms of the natural world – the seasonal flowering of familiar trees and plants, the phases of the moon. They could also learn of more obscure phenomena, such as the appearance of certain cicadas in years that are a prime number away from the previous year in which they appeared. The number pi (the ratio of the circumference and diameter of any circle, 3.14) has a propensity to turn up in the oddest places. The sinuosity of a river is the ratio of the total length of a river to the straight line from source to mouth, and on average, approximates pi.

In practice, not all children are going to be excited enough by mathematics to want to explore it deeply, even if they are attuned to natural phenomena. Sen does mention that students will have to have teachers or mentors, to learn and discuss the formal aspects of mathematics. To Sen, mathematics is “nature’s mother tongue”.

This idea is redolent of the 17^th century mathematician Galileo’s statement: “The book of nature is written in the language of mathematics.” It is worth understanding the historical context of this statement. The 17^th century was a time when philosophers and scientists saw nature, or the universe, as full of magic and mystery. Science consisted of multifarious practices, broadly classifiable into the organic, the magical, and the mechanistic traditions, each of which had their own language or conceptual framework, through which they tried to understand nature. The organic tradition was concerned with the close observation of living things, which developed into what we call biology today, and the magical tradition with a sense of beauty and mystery in nature. The third, mechanistic tradition, dominates science to this day. It has used mathematics as never before – making it the language of that tradition. The mechanistic tradition sees the universe as a machine: stars, planets, the human body, animals, plants, etc.

Sen asks if it is possible to do science and mathematics in an organic way. A concern not just with close natural observation, but also with magic and beauty, runs throughout the book Even that most utilitarian of mathematical tools – the multiplication tables – for example the nine times tables can conjure up magic, as Sen discovered when she was at school: when the multiples of 9 are written in a column, from 9 to 90, the units column reads from 0 to 9, and the tens column from 9 to 0. A ‘table’ with the number 12345679 multiplied by multiples of 9 gives a surprising and magical pattern: 12345679 X 1 X 9 =111,111,111; 12345679 X 2 X 9 = 222,222,222; and so on.

The golden ratio phi (1.618) appears often in nature and the human body – in Yoga-asanas, in the quotients of consecutive Fibonacci numbers, (which are frequently found in nature) and very many more places – evoking a sense of magic and beauty.

The magical tradition’s efforts were directed towards emulating God as the divine artist. The Renaissance artist Leonardo da Vinci’s work seen from today’s perspective blurs the boundaries of science, engineering, mathematics and art. At the time, however, the distinctions were not so rigid. There are other examples of the juxtaposition of art and mathematics in Sen’s book: more than a century after van Gogh painted ‘Starry Night,’ scientists found a correspondence between the emotional turbulence in the painting and fluid turbulence in mathematics. These stories shatter stereotypes of ‘creative’ and ‘mathematical’ students at school and other educational contexts.

The sacred geometries in the ancient world, so entwined with mathematics, are also sites of the meeting of art and mathematics: the trellised jaali patterns in mosques, the self-similar fractal patterns embossed on temple walls in central India, the geometric problems in artistic forms in the Japanese Edo tradition. Stories of civilizations, whether Western, ‘Indic’, or ‘Oriental’, are accounts of the ‘wonders’ that they are; beauty is a value that cuts across all of them. In modern times, beauty is associated with other values such as truth and justice. But when we think of civilization from the point of view of the oppressed, it is far from these values: it is a story of conflict, injustice, and struggle.

The achievements of mechanistic science have made the current civilization the most powerful one in history: one that persists in its domination and control of nature. Even if mathematics and science were to be done in more organic and magical ways, the elites who do science urgently need to introspect about the social hierarchies those disciplines perpetuate. Marginalized indigenous thinkers’ voices urgently need to be heard. The adivasi’s knowledge of nature needs to be conserved.

What role could friendship have in mathematics? It is often thought of as a solitary activity. Sen’s narrative is laced with anecdotes about relationships with other mathematicians in Turkey, India, and England – not just as collaborators but as friends. For Sen, mathematics is an activity to be shared rather than engaged with competitively.

Sen’s book makes an important and necessary point about the possibilities in mathematics education that go beyond the conventional classroom. An historical perspective throwing light on the ways in which mathematics and science came to be the most powerful subjects and the social consequences of that power would have given the book more rigour. From a psychological perspective, the book will be an inspiration for readers to be fearless about mathematics and with luck, to fall in love with it.

The book can be purchased at www.amazon.in.

The writer is a former teacher and journalist. She can be reached at deepa.onkar@gmail.com.