A Passage to India
Mark Saul
December 2024
India commands the attention of the visitor as no other place I’ve been. Every stroll, every taxi ride, every meal brings new sensations: colors, tastes, sounds, aromas… One friend described her initial visit as ‘sensory overload’.
Dust. Dust on the yellow-and-black motorized rickshaws putting along past the Mercedes in the next lane. Dust on the parkways and the bustling main streets—and the equally bustling side streets. Dust on the broad green leaves of the lush vegetation. Dust on the brightly colored saris billowing from women crowded three to a motorbike. Dust on the sidewalks, the shoes, the kurtas. The dust of winter, to be washed away by the monsoon in later months.
The careful reader may have noticed here a crude imitation of the verb-free opening passage of Dickens’ Bleak House. And in fact a lot of Indian life feels like Victorian England. Labor is cheap. Everyone has servants–even the servants. Enormous wealth abuts extreme poverty. The visitor encounters street urchins, rag pickers, goatherds, letter writers–all professions that have disappeared from the modern world but linger here.
See what I mean by ‘sensory overload’? It’s the fourth paragraph of this essay and I’ve not yet gotten to any discussion of mathematics or education. Or have I? A lot of Indian education, and many educators, are locked into a paradigm left them by the British Raj, and which the British themselves have largely abandoned: private for-profit education, rigid testing, and formal teaching styles. But thoughtful minds in the country seek a way out, and that is why I have been invited here, to give talks and workshops. Reverence for learning and teaching transcends any particular practice that is clung to.
I first visited the country about 20 years ago, to work with several schools, and have returned every two or three years since. My first visit was to a huge private boarding school in Bengaluru (Bangalore), very much in the tradition of English ‘public schools’. The students wear uniforms and pay tuition Their families are well off. Students address teachers as ‘Sir’.
“Sir,” said one twelve year old, in a navy uniform, “Sir, I can square any two-digit number in my head.”
I briefly quizzed him, and found that he was not just boasting. He explained to me his algorithm, and told me, on his level, why it must work. His knowledge was more than procedural.
I gave him the praise due any enthusiastic student who dares approach the teacher. Then I invited him to take the next step: I asked him, “How many perfect squares are there less than 1000?” He thought and thought—and could not answer. He started listing them: 1, 4, 9…. I simply said, “Think about it”.
And he did. The next day he sought me out, gave me the answer, and told me why there are 31 such perfect squares. “Because 322 is too big…” . No one had ever asked him such a question. For him, and probably for his teachers, the endpoint of the learning was the correct execution of the algorithm, and perhaps a correct explanation of why it worked. The next step—turning it backwards, as Piaget would say—was never taken. And he was certainly capable of it.
So what prevents teachers from going further? From asking questions that stretch their students’ minds? In a word, testing. In India, students are tested annually on the national curriculum. The tests are rigorous but predicable. There is no support for asking a ‘stretch’ question, and no incentive for using creatively what the curriculum has delivered. A teacher who wants to spend valuable class time doing this takes a risk. And a student must do it on his own time.
And they do take risks. This year I was the guest of Prodipta Hore, the head of teacher development for the Aditya Birla World Academy in Mumbai. Each year, he organizes an “Infinity Contest,” which draws hundreds of students from across the nation and abroad. The contest problems lie outside the curriculum, but require no more background than the curriculum delivers. Students’ minds are pulled and stretched. And like most mathematical competitions, this effect is not limited just to the winners, or even just the contestants. The team coaches, with whom I worked on mathematics independent of the contest itself, were teachers who could recognize the importance of the subject as more than a catalog of procedures. These teachers would have taken a student who could square two-digit numbers in her head to the next level.
Like water on parched earth, suggestions for expanding or varying the curriculum quickly stimulate pedagogical creativity in such teachers. They respond readily to an invitation to take that next step. And they enjoy it. All they need is permission, or validation. And of course this phenomenon is not peculiar to India: I find it all over the globe, and certainly in the US. An open pedagogy is hard to master. It is much easier to tell students what they ought to know, the test them on what you’ve told them. But teaching is not ‘telling’. It is a much more complicated endeavor.
The effort to open up the classroom to new ideas cannot be limited to the teachers. The administration of a school must likewise be open. The schools I’ve visited in India are typically part of a chain of schools, in different locations, run by the same organization. Each system is huge by US standards, serving tens of thousands of students, analogous to a medium-sized school district in the US. So it is important to influence the administrators of these organizations, and I typically had conversations with groups of principals, some of whom would attend my talk or workshop. These enlightened administrators are preparing the future of their country.
I am not always successful. In one school I happened to see the report of the curriculum leader who observed my workshop: “The session seemed to be rather unplanned and unstructured. It comprised four games/activities based on logical thinking that were fun. That’s about it.” These are the words of someone whose view of pedagogy, and of mathematics, is narrow and rigid. Mathematics, after all, is based on logical thinking. And ‘fun’ is a powerful classroom motivation. (I might add that the teachers and students I worked with in that school had very different reactions to the work.)
Perhaps such views pervade this vast country. I don’t know. I have certainly encountered it in the US. And herein lies some really difficult work. Systemic change may come from below, but must eventually be enacted by the highest levels. How do you change the minds of people who are not in the classroom and unfamiliar with mathematics? An infrequent visitor will not effect such a change.
But an infrequent visitor can plant seeds. I often encountered principals who had met me on previous visits, and who remarked that things they had learned then had borne fruit in their careers. As flattering as these comments were, they brought with them a constraint on my work: I had to be careful not to repeat activities I had used with them before.
Prior knowledge of one’s audience can be very important. I happened to mention to one teacher that I was going to talk about magic squares. “Oh, that’s great!” he said, “Will you show them the ‘knight’s move’ method?” For readers not familiar with the topic, the knight’s move is a procedure for generating magic squares of various types. But there’s not much else it relates to. Students learning about knights’ moves will learn only about how to construct a magic square. Interesting, but not all that useful.
So I simply replied: “We will do some new things with magic squares.” And the activity worked beautifully. The students constructed their own magic squares. They took small slips of paper and wrote the numbers 1 through 9 on them. Then they formed a 3×3 array—at random—and ‘traded’ large numbers for small ones until the rows (just the rows!) had equal sums. They quickly discovered that the sum would be 15, and a few students shared with the others the reason: all the numbers added up to 45, so if we distribute them into rows with equal sums, each sum must be 15. I noted this as “Theorem 1”. Then they got the column sums equal, by trading large numbers for small within each row. Finally—with a little hint—they got the diagonal sums equal, by ‘sorting’ rows and columns. All this is worked out in [1].
But you really had to be there. Discovery was happening all over the room. Students were giggling and smiling, sharing their discoveries. And the teachers were delighted at the involvement of their students in the activity.
And there was more. I called the class together. We proved formally that the number in the center square must be 5, then that the even numbers had to be in the corners. Since the students had generated the squares themselves, they had different ‘versions’ of the same square. So we had a discussion of the symmetries of a square, then (briefly) about composition of symmetries. The ‘playing with numbers’ segued into more serious mathematics—proving theorems. We also lay the groundwork for a deeper investigation: magic squares can be added and scaled, and thus form a vector space. So we can ask about the dimension of the vector space, and construct a basis. But that is for another day.
During my visits to India, mathematicians working there helped me to solve two riddles. The first riddle: In the US, virtually every mathematics faculty includes one or two Indian members. But rarely do they include Pakistani mathematicians. Why not?
A conversation with an American colleague teaching in India (and fluent in Hindi) enlightened me on this point. “The deep mathematical culture we know as Indian,” he told me, “is mostly from South India. There is a cultural divide between North and South in the country. The North was often ruled by foreigners, the South less often. People in the North speak Indo-European languages (Hindi, Bengali, Gujarati), while people in the south speak Dravidian languages, unrelated to Hindi or Sanskrit. And it is in the South of India where the mathematical culture that we know from Ramanujan developed. The most important centers of traditional learning were in South India, and it was out of these that the mathematical culture grew. So it’s not a divide between India and Pakistan, but between the Northern and Southern cultures of the Indian subcontinent. And there are, in fact, numerous mathematicians from North India who have made significant contributions.”
The second riddle concerned the International Mathematical Olympiad. India fields a team of six students every year to this high-level event. The team does moderately well, but not as well as China, the US, or numerous much smaller countries. With its large population, and its tradition of mathematical study, shouldn’t India’s IMO team be among the top?
Alok Kumar, who has had deep experience training students for the IMO, met me in a Delhi suburb. “It’s about college admissions. In the US, success in Olympiad contests is valued by admissions committees. They know the level of the competition, and what the abilities of high-scoring contestants must be. In China, in most years, success in the IMO means that the student is automatically admitted to college: he [usually a boy!] need not take the national college admissions examination that determines the futures of other students.
“But in India, prestigious universities generally don’t value IMO achievement. Even a gold medalist will have to get in line and be evaluated by testing. And the tests are on very standard material, not Olympiad-style questions. So students often must choose between preparing for these tests and preparing for the IMO. Some of our most talented students choose the former, so the IMO team does not include them.”
I have yet to solve one more riddle, which surely has occurred to the reader by now: the issue of caste. I got different responses when asked about it. People in urban areas told me that everyone knows what caste a family belongs to, if only by the family name. But in the modern work place, this detail is not important. An analogy in American life would be nationality. Everyone knows, but it’s just a curiosity, a topic of casual discussion. However, people in smaller cities tell me that there is discrimination, often subtle or unspoken, as racism is often subtle and unspoken in American life. Much has been written, and much legislated. But on a personal level, I would guess that urbanization and professionalization are overtaking even this age-old set of traditions.
India is certainly advancing in many ways. I’ve noticed the changes in the 20 years I’ve been visiting. Indian cities are full of ‘roads to nowhere”: sections of freeways or metros that don’t connect. Piecewise continuous pathways to the future. We can take this as a metaphor for Indian pre-college education as well. The system needs updating. The rich mathematical and scholastic traditions of India have to be integrated into and reconciled with more modern ideas about the classroom. The UN has now informed us that the population of India has surpassed that of China. Almost 1/6 of the world’s people are Indian citizens. Indians have already made significant contributions to the mathematical community. I think their gifts to our community, and to the world, will only get richer.
Works Cited
[1] Peters, Alice and Saul, Mark: A Festival of Mathematics: A Sourcebook. Providence RI: American Mathematical Society and Mathematical Sciences Research Institute, 2022.
The writer wishes to acknowledge Mrs.Neerja Birla, Chairperson, Aditya Birla Education Trust; Mrs.Radhika Sinha, Principal, Aditya Birla World Academy.for making possible my visits to Mumbai. ..
The writer would also like to acknowledge
Ajay Sharma, Birla Academy of Pilani
Satyajit Bag, Pathways School, Noida
Dhirendra Singh, Principal, Birla Academy of Pilani
Geeta Gandhi-Kingdon
Jagdeesh Gandhi (in memoriam)
Bruce Robinson
whose insights into Indian culture and education have been enormously enlightening