December 2024<\/p>\n
Mark Saul<\/p>\n
India commands the attention of the visitor as no other place I\u2019ve been. \u00a0Every stroll, every taxi ride, every meal brings new sensations: colors, tastes, sounds, aromas\u2026\u00a0 One friend described her initial visit as \u2018sensory overload\u2019.<\/p>\n
Dust.\u00a0 Dust on the yellow-and-black motorized rickshaws putting along past the Mercedes in the next lane.\u00a0 Dust on the parkways and the bustling main streets\u2014and the equally bustling side streets.\u00a0 Dust on the broad green leaves of the lush vegetation.\u00a0 Dust on the brightly colored saris billowing from women crowded three to a motorbike.\u00a0 Dust on the sidewalks, the shoes, the kurtas.\u00a0 The dust of winter, to be washed away by the monsoon in later months.<\/p>\n
The careful reader may have noticed here a crude imitation of the verb-free opening passage of Dickens\u2019 Bleak House<\/em>.\u00a0 And in fact a lot of Indian life feels like Victorian England.\u00a0 Labor is cheap.\u00a0 Everyone has servants–even the servants.\u00a0 Enormous wealth abuts extreme poverty.\u00a0 The visitor encounters street urchins, rag pickers, goatherds, letter writers–all professions that have disappeared from the modern world but linger here.<\/p>\n See what I mean by \u2018sensory overload\u2019?\u00a0 It\u2019s the fourth paragraph of this essay and I\u2019ve not yet gotten to any discussion of mathematics or education.\u00a0 Or have I?\u00a0 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. \u00a0But thoughtful minds in the country seek a way out, and that is why I have been invited here, to give talks and workshops.\u00a0 Reverence for learning and teaching transcends any particular practice that is clung to.<\/p>\n I first visited the country about 20 years ago, to work with several schools, and have returned every two or three years since.\u00a0 My first visit was to a huge private boarding school in Bengaluru (Bangalore), very much in the tradition of English \u2018public schools\u2019.\u00a0 The students wear uniforms and pay tuition\u00a0 Their families are well off.\u00a0\u00a0 Students address teachers as \u2018Sir\u2019.<\/p>\n \u201cSir,\u201d said one twelve year old, in a navy uniform, \u201cSir, I can square any two-digit number in my head.\u201d<\/p>\n I briefly quizzed him, and found that he was not just boasting.\u00a0 He explained to me his algorithm, and told me, on his level, why it must work.\u00a0 His knowledge was more than procedural.<\/p>\n I gave him the praise due any enthusiastic student who dares approach the teacher.\u00a0 Then I invited him to take the next step:\u00a0 I asked him, \u201cHow many perfect squares are there less than 1000?\u201d\u00a0 He thought and thought\u2014and could not answer.\u00a0 He started listing them:\u00a0 1, 4, 9\u2026.\u00a0 I simply said, \u201cThink about it\u201d.<\/p>\n And he did.\u00a0 The next day he sought me out, gave me the answer, and told me why there are 31 such perfect squares. \u201cBecause 322<\/sup> is too big\u2026\u201d .\u00a0 No one had ever asked him such a question.\u00a0 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.\u00a0 The next step\u2014turning it backwards, as Piaget would say\u2014was never taken.\u00a0 And he was certainly capable of it.<\/p>\n So what prevents teachers from going further?\u00a0 From asking questions that stretch their students\u2019 minds?\u00a0 In a word, testing.\u00a0 In India, students are tested annually on the national curriculum.\u00a0 The tests are rigorous but predicable.\u00a0 There is no support for asking a \u2018stretch\u2019 question, and no incentive for using creatively what the curriculum has delivered.\u00a0 A teacher who wants to spend valuable class time doing this takes a risk.\u00a0 And a student must do it on his own time.<\/p>\n And they do take risks.\u00a0 This year I was the guest of Prodipta Hore, the head of teacher development for the Aditya Birla World Academy in Mumbai.\u00a0 Each year, he organizes an \u201cInfinity Contest,\u201d which draws hundreds of students from across the nation and abroad.\u00a0 The contest problems lie outside the curriculum, but require no more background than the curriculum delivers.\u00a0 Students\u2019 minds are pulled and stretched.\u00a0 And like most mathematical competitions, this effect is not limited just to the winners, or even just the contestants.\u00a0 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.\u00a0 These teachers would have taken a student who could square two-digit numbers in her head to the next level.<\/p>\n Like water on parched earth, suggestions for expanding or varying the curriculum quickly stimulate pedagogical creativity in such teachers.\u00a0 They respond readily to an invitation to take that next step.\u00a0 And they enjoy it.\u00a0 All they need is permission, or validation.\u00a0\u00a0 And of course this phenomenon is not peculiar to India: I find it all over the globe, and certainly in the US.\u00a0 An open pedagogy is hard to master.\u00a0 It is much easier to tell students what they ought to know, the test them on what you\u2019ve told them.\u00a0 But teaching is not \u2018telling\u2019.\u00a0 It is a much more complicated endeavor.<\/p>\n The effort to open up the classroom to new ideas cannot be limited to the teachers.\u00a0 The administration of a school must likewise be open.\u00a0 The schools I\u2019ve visited in India are typically part of a chain of schools, in different locations, run by the same organization.\u00a0 Each system is huge by US standards, serving tens of thousands of students, analogous to a medium-sized school district in the US.\u00a0 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.\u00a0 These enlightened administrators are preparing the future of their country.<\/p>\n I am not always successful.\u00a0 In one school I happened to see the report of the curriculum leader who observed my workshop: \u201cThe session seemed to be rather unplanned and unstructured.\u00a0 It comprised four games\/activities based on logical thinking that were fun.\u00a0 That’s about it.\u201d\u00a0 These are the words of someone whose view of pedagogy, and of mathematics, is narrow and rigid.\u00a0 Mathematics, after all, is based on logical thinking.\u00a0 And \u2018fun\u2019 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.)<\/p>\n Perhaps such views pervade this vast country.\u00a0 I don\u2019t know.\u00a0 I have certainly encountered it in the US.\u00a0 And herein lies some really difficult work.\u00a0 Systemic change may come from below, but must eventually be enacted by the highest levels.\u00a0 How do you change the minds of people who are not in the classroom and unfamiliar with mathematics?\u00a0 An infrequent visitor will not effect such a change.<\/p>\n But an infrequent visitor can plant seeds.\u00a0 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.\u00a0 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.<\/p>\n Prior knowledge of one\u2019s audience can be very important.\u00a0 I happened to mention to one teacher that I was going to talk about magic squares.\u00a0 \u201cOh, that\u2019s great!\u201d he said, \u201cWill you show them the \u2018knight\u2019s move\u2019 method?\u201d\u00a0 For readers not familiar with the topic, the knight\u2019s move is a procedure for generating magic squares of various types.\u00a0 But there\u2019s not much else it relates to.\u00a0 Students learning about knights\u2019 moves will learn only about how to construct a magic square.\u00a0 Interesting, but not all that useful.<\/p>\n So I simply replied: \u201cWe will do some new things with magic squares.\u201d\u00a0 And the activity worked beautifully.\u00a0 The students constructed their own magic squares.\u00a0 They took small slips of paper and wrote the numbers 1 through 9 on them.\u00a0 Then they formed a 3×3 array\u2014at random\u2014and \u2018traded\u2019 large numbers for small ones until the rows (just the rows!) had equal sums.\u00a0 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.\u00a0\u00a0 I noted this as \u201cTheorem 1\u201d.\u00a0\u00a0 Then they got the column sums equal, by trading large numbers for small within each row.\u00a0 Finally\u2014with a little hint\u2014they got the diagonal sums equal, by \u2018sorting\u2019 rows and columns.\u00a0 All this is worked out in [1].<\/p>\n But you really had to be there.\u00a0 Discovery was happening all over the room.\u00a0 Students were giggling and smiling, sharing their discoveries.\u00a0 And the teachers were delighted at the involvement of their students in the activity.<\/p>\n And there was more.\u00a0 I called the class together.\u00a0 We proved formally that the number in the center square must be 5, then that the even numbers had to be in the corners.\u00a0 Since the students had generated the squares themselves, they had different \u2018versions\u2019 of the same square.\u00a0 So we had a discussion of the symmetries of a square, then (briefly) about composition of symmetries.\u00a0 The \u2018playing with numbers\u2019 segued into more serious mathematics\u2014proving theorems.\u00a0 We also lay the groundwork for a deeper investigation: magic squares can be added and scaled, and thus form a vector space.\u00a0 So we can ask about the dimension of the vector space, and construct a basis.\u00a0 But that is for another day.<\/p>\n During my visits to India, mathematicians working there helped me to solve two riddles.\u00a0 The first riddle: In the US, virtually every mathematics faculty includes one or two Indian members.\u00a0 But rarely do they include Pakistani mathematicians.\u00a0 Why not?<\/p>\n A conversation with an American colleague teaching in India (and fluent in Hindi) enlightened me on this point.\u00a0 \u201cThe deep mathematical culture we know as Indian,\u201d he told me, \u201cis mostly from South India.\u00a0 There is a cultural divide between North and South in the country.\u00a0 The North was often ruled by foreigners, the South less often.\u00a0 People in the North speak Indo-European languages (Hindi, Bengali, Gujarati), while people in the south speak Dravidian languages, unrelated to Hindi or Sanskrit.\u00a0 And it is in the South of India where the mathematical culture that we know from Ramanujan developed.\u00a0 The most important centers of traditional learning were in South India, and it was out of these that the mathematical culture grew. So it\u2019s not a divide between India and Pakistan, but between the Northern and Southern cultures of the Indian subcontinent.\u00a0 And there are, in fact, numerous mathematicians from North India who have made significant contributions.\u201d<\/p>\n The second riddle concerned the International Mathematical Olympiad.\u00a0 India fields a team of six students every year to this high-level event.\u00a0 The team does moderately well, but not as well as China, the US, or numerous much smaller countries.\u00a0 With its large population, and its tradition of mathematical study, shouldn\u2019t India\u2019s IMO team be among the top?<\/p>\n Alok Kumar, who has had deep experience training students for the IMO, met me in a Delhi suburb.\u00a0 \u201cIt\u2019s about college admissions.\u00a0 In the US, success in Olympiad contests is valued by admissions committees.\u00a0 They know the level of the competition, and what the abilities of high-scoring contestants must be.\u00a0 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.<\/p>\n \u201cBut in India, prestigious universities generally don\u2019t value IMO achievement.\u00a0 Even a gold medalist will have to get in line and be evaluated by testing.\u00a0 And the tests are on very standard material, not Olympiad-style questions.\u00a0 So students often must choose between preparing for these tests and preparing for the IMO.\u00a0 Some of our most talented students choose the former, so the IMO team does not include them.\u201d<\/p>\n I have yet to solve one more riddle, which surely has occurred to the reader by now: the issue of caste.\u00a0 I got different responses when asked about it.\u00a0 People in urban areas told me that everyone knows what caste a family belongs to, if only by the family name.\u00a0 But in the modern work place, this detail is not important.\u00a0 An analogy in American life would be nationality.\u00a0 Everyone knows, but it\u2019s just a curiosity, a topic of casual discussion.\u00a0 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.\u00a0 Much has been written, and much legislated.\u00a0 But on a personal level, I would guess that urbanization and professionalization are overtaking even this age-old set of traditions.<\/p>\n India is certainly advancing in many ways.\u00a0 I\u2019ve noticed the changes in the 20 years I\u2019ve been visiting.\u00a0 Indian cities are full of \u2018roads to nowhere\u201d: sections of freeways or metros that don\u2019t connect.\u00a0 Piecewise continuous pathways to the future.\u00a0 We can take this as a metaphor for Indian pre-college education as well.\u00a0 The system needs updating.\u00a0 The rich mathematical and scholastic traditions of India have to be integrated into and reconciled with more modern ideas about the classroom.\u00a0 The UN has now informed us that the population of India has surpassed that of China.\u00a0 Almost 1\/6 of the world\u2019s people are Indian citizens.\u00a0 Indians have already made significant contributions to the mathematical community.\u00a0 I think their gifts to our community, and to the world, will only get richer.<\/p>\n Works Cited<\/p>\n [1] Peters, Alice and Saul, Mark: A Festival of Mathematics: A Sourcebook.\u00a0 Providence RI: American Mathematical Society and Mathematical Sciences Research Institute, 2022.<\/p>\n 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.\u00a0 ..<\/p>\n The writer would also like to acknowledge<\/p>\n Ajay Sharma, Birla Academy of Pilani<\/p>\n Satyajit Bag, Pathways School, Noida<\/p>\n Dhirendra Singh, Principal, Birla Academy of Pilani<\/p>\n Geeta Gandhi-Kingdon<\/p>\n Jagdeesh Gandhi (in memoriam)<\/p>\n Bruce Robinson<\/p>\n whose insights into Indian culture and education have been enormously enlightening<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" December 2024 Mark Saul India commands the attention of the visitor as no other place I\u2019ve been. \u00a0Every stroll, every taxi ride, every meal brings new sensations: colors, tastes, sounds, aromas\u2026\u00a0 One friend described her initial visit as \u2018sensory overload\u2019. Dust.\u00a0 Dust on the yellow-and-black motorized rickshaws putting along pastRead More →<\/a><\/span><\/p>\n","protected":false},"author":17,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["entry","author-msaul","post-339","post","type-post","status-publish","format-standard","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts\/339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/comments?post=339"}],"version-history":[{"count":5,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts\/339\/revisions"}],"predecessor-version":[{"id":364,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/posts\/339\/revisions\/364"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/media?parent=339"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/categories?post=339"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/teachingandlearning\/wp-json\/wp\/v2\/tags?post=339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}