An e-mail query in support of a Turkish Mathematics Village more than a year ago lead to a most illuminating, intellectually broadening and exciting experience at the annual Turkish Mathematics Summer School in the idyllic and sleepy but enchanting village of Sirince. A friend had shown me a news story about the world’s first ‘Mathematics village’ on the verge of being ‘shut-down’ because of unnecessary bureaucratic hassles. Left with an un-translated foolscap Turkish news article about what seemed to be an unprecedented experiment—the opening of an Ionian style village, purely for the sake and study of Mathematics—my curiosity was whetted. I wrote a brief note to Ali Nesin, Chairman of the Mathematics Department at Istanbul Bilgi University and the man responsible for this undertaking. Instead of just a ‘thank-you for-your-support’ response I received a long e-mail description of the place and an invitation to be one of the volunteer instructors for an undergraduate level course. Soon after, I received a timetable of classes with my name inscribed for the allotted time. I was scheduled to teach Classical Mechanics while representing Brockwood Park School at Nesin Matematik Koyu.

Thus began my summer rendezvous in the Aegean village of Sirince, a place of ethereal rustic beauty with its stone houses, olive orchards, vineyards and winding dust-roads. As I entered the Mathematics village, a short distance from the main village-centre of Sirince, built on fields belonging to the Nesin Foundation, I was transported to the era of Pythagoras even in the style of the dwellings and classrooms. Incidentally, Pythagoras who hailed from Samos, an island not far from my host village, had many followers devoted solely to the magic of numbers and the study of Mathematics. The Mathematics village was newly built, but in the manner of the ancients, with stone-walled houses, supported by wooden beams and caked with mud. Students in the previous years had helped in its construction and it included accommodation for the 60 odd students who assemble for part of the summer every year. Alongside the living quarters were found two separate hammams with blue domes that provided afternoon respite during the scorching summer months. The motivation for spending time here is not the acquisition of a degree, or any certificate or accolade, nor a diploma or qualification. It is purely for the love of the subject, a motiveless quest for excellence and deep inquiry.

Many of the students lived in dormitory style accommodation while some with a more hardy bent of mind pitched tents in the blistering summer heat. Classes began at 7 7:30 a.m. and carried on until noon, and recommenced at 4 4 p.m. to continue for four additional hours. Not every student attended every class or course offered; they were chosen according to individual levels and areas of interest. Most of the lectures were conducted by practising mathematicians, although some advanced students also delivered lectures or seminars on selected topics. In the evenings the students studied by themselves or in groups, seeking advice on difficult or unsolvable problems, consulting with one another, enjoying each others’ company until the wee hours. I was lucky to join in on the regular evening musical soirees, where the lilting voices of some of my students, singing folk songs on their balamas, a traditional lute-like instrument, filled the air, creating an atmosphere of magical intensity.

Even the classical architecture of the village seemed to inspire mathematical contemplation. Trees grew from the floors of the classrooms, soaring through the thatched roofs. I taught in an Ionian style outdoor theatre with seats of local stone draped with Turkish kilims while cloth pennants fluttered in the breeze overhead. The whole village was laid out in a way that encouraged constant movement; there were no dead-ends. Rather, each area of the village flowed into the next in a most ingenious way, like a scientist’s mind sailing through the sea of mathematical forms and relations. There was no final harbour except truth itself.

Though my husband, who accompanied me on this trip, did not have an official teaching post, he easily weaved his way into the spirit of the village. Impromptu philosophical conversations with several of the professors gradually evolved into more in-depth conversations about the history and development of Mathematics from the philosopher’s point of view. This caused enough excitement in the village and he was invited to teach a course on the Philosophy of Mathematics the following year. He also got on very well with the students, volunteering to teach them English during the mid-day siesta hours and early morning Hatha Yoga on a nearby mountain top at dawn.

This easy spirit of adaptation was one of the main qualities that made the Mathematics village such a wonderful place. The village somehow seemed to bring out the best qualities in everyone: selflessness, care, vigour, a ceaseless spirit of learning and deep curiosity. It made me wonder how all these qualities were achieved without the conscious striving for them that I often encounter in most institutions. The village was a community in the truest sense, a place where all ages were welcome, from families and older relatives to the youngest infant. They added to the tapestry of the place, residing for some weeks or months as accompanying guests. The cook, a cheerful lady (who was reported to be an erstwhile bus driver), charmed us every day with her culinary wizardry. She even started cooking primarily vegetarian dishes as a gesture of hospitality towards the new guests. The students helped in the kitchen and participated in the daily chores, through a rota system that inter alia involved wiping tables, upkeep of the dining facilities, dish washing and peeling vegetables. Much to my surprise, they fully immersed themselves in the general care of the place, including in tending the small vegetable garden with enthusiasm and gusto.

My general teaching philosophy and outlook is that the teacher is also a student. I often learn about effective teaching and learning through my students, who offer me suggestions and tips on how to progress with my lectures (through their questioning and alternate approaches to problems that I myself may have overlooked). My goal as a teacher of Mathematics has always been to express in the simplest terms ideas or concepts that otherwise seem cloaked in difficulty. Thus the art of teaching a notoriously difficult subject like Mathematics, particularly in its more advanced forms, involves a delving together to uncover the symmetries and underlying principles of the subject. Thus, together with the students, we discovered how Newtonian Mathematical Physics, which I introduce at the beginning of my lectures, gives rise to a more powerful approach introduced by Lagrange, Hamilton and Jacobi, mathematicians who paved the way for the beginnings of Quantum Mechanics as well as a more complete understanding of the laws of Physics and the nature of Matter and Energy. The lectures, mainly conducted on a blackboard, using chalk and duster, lasted for about two hours including the question and answer sessions and systematically traversed the journey from Newton’s Laws to the beginnings of Quantum Mechanics. Ultimately, we discovered with the help of some of my colleagues, that ‘symplectic geometry’ creates a bridge between classical Mechanics and Quantum Theory.

Another fortunate encounter during my stay at Nesin Matematik Koyu was with Alexander Borovik, a professor of Mathematics I had first met during my doctoral studies at the University of Manchester. He pointed out that hidden structures and concepts of Elementary Mathematics which frequently remain unnoticed, nevertheless significantly influence students’ perception of Mathematics. As an outcome of our breakfast conversations, he sent me a chapter of his new book, Shadows of the Truth: Metamathematics of Elementary Mathematics, for my feedback and suggestions. While presenting the theme of his book, Professor Borovik extensively analyses real life experiences and episodes relating to the mathematical encounters children have when they are first introduced to the subject. In studying the formation of mathematical concepts in a child, he gains insight into the interplay of mathematical structures within Mathematics itself. He suggests that a philosophically inclined reader would notice a parallel with Plato’s Allegory of the Cave. The children in his book see shadows of the Truth and sometimes find themselves in a psychological trap because their teachers and other adults around them are blinded both to Truth and to the shadows around it. At the start of his book, Borovik relates an illustrative example from his own childhood that pertains to the difficulties encountered in the very elementary structures of the subject that teachers often gloss over, leaving the hapless student frustrated and bewildered. He was instructed by his elementary school teacher that he had to be careful with ‘named’ numbers and not to add apples and people. He remembers asking her: why in that case can we divide apples by people?

10 apples: 5 people = 2 apples.

What is more discomforting, when we distribute 10 apples giving 2 apples to a person, we have

10 apples: 2 apples = 5 people.

“Where do ‘people’ on the right-hand side of the equation come from?

Why do ‘people’ appear and not, say, ‘kids’? There were no ‘people’ on the left-hand side of the operation! How do numbers on the left-hand side know the name of the number on the right-handside?”

In this case the teacher failed to outline the fact that in the first example ‘apples’ divided by ‘people’ should give ‘apples per people’ and in the second case ‘apples’ is divided by ‘apples per people’ to give ‘people’.

I was fascinated to rediscover the hidden mysteries of Elementary Mathematics, realizing that as a teacher one had to possess substantial insights into the structure of the subject even when teaching the basics. I enthusiastically wrote back to the professor offering a few tips on dimensional analysis and its history and a few other suggestions.

As I reflect on a summer well spent, I realize that the Mathematics village is indeed, in the words of Ali Nesin, a ‘beacon of hope’. It not only shines light on a holistic approach to learning Mathematics from the most elementary to the most advanced levels, it also proves that learning and education have no restrictive boundaries. I do hope that a project like the Mathematics village will continue to flourish for years to come, touching the lives of many keen students of the subject and supporters of this enterprise.