 Okay. So I think we can start. So, if it is okay, can everyone else use your mics for us? It's muted? No, no. Okay. We are live. Okay. So it's sharp four o'clock and we can begin the 21st VGK Memorial Lecture now. So good afternoon everyone and welcome to the 21st VG Kulkarni Memorial Lecture. She, VG Kulkarni, he's also known as Vigo in HBCSE family. He was the founder, director of Homi Baba Centre for Science Education. And he has played a key role in shaping many initial projects in science education area that were facilitated at HBCSE. And to remember and celebrate VGK's contribution and particularly the mentorship to our centre. Our centre organizes annual VGK Memorial Lecture. And today it is my pleasure to announce that we have gathered for 21st lecture in this series. And it will be delivered by distinguished professor R. Ramanujam. I now request our centre director, Professor Arnab Bhattacharya to welcome our speaker and also the online and in-person audience and share some background information about VGK Memorial Lecture first. So over to you Arnab. Thank you so much Deepa. And good afternoon everyone and a very very warm welcome to each and every one of you whether you're watching it in person, whether you're watching it online or you're watching it on YouTube. This is the 2022 edition of the VGK Memorial Lecture and as Deepa I told you the 21st in this series of HBCSE's flagship annual lecture series dedicated to the memory of our visionary founder director VGK Kulkarni. And over the years we've had a whole list of excellent talks that have dealt with important issues in science, technology, education, society. And I'm sure today's lecture is also going to be equally enthralling. Now last year, if you remember, for those who were there, I had hoped to welcome you all in-person in 2022 for the, you know, this year's VGK Memorial Lecture. And COVID it's on its way out, the restrictions on travel are gone, we can all meet now. So why are we still doing this online? Well, the answer is because the VGK Kulkarni Auditorium in HBCSE where this has traditionally been held is getting a major overhaul. So hence we are online this year but we do hope to welcome you back on campus to a shiny new VGK Memorial Lecture Auditorium in a few months and hopefully we'll do a lecture series there. In any case it's an honor and a privilege to tell you something about our founder director Vinayak Gopal Kulkarni, more popularly known as Vigo Kulkarni if you speak Marathi or just by his initials VGK. He was born in Belgaon in 1932, his early education was in Belgaon then in Ratnagiri, Harawad, eventually coming to Mumbai, joining TIFR after his MSC in 1953 as a scientist in the area of solid state and nuclear physics. And he worked closely with Professor B.M. Udgaonkar, B.B. Tosa, Girish Chandra and others. And in the middle of the science research, VGK Kulkarni along with some of his colleagues realized that there was a need to train science teachers, especially at the school level, and they turned their attention to looking at science education. And in particular, along with Professor B.M. Udgaonkar, they founded and nurtured a center devoted for this purpose, starting in a very small way in a municipal school in Nanachalk in Mumbai in 1974, supported by the Durabji Tata Trust in those days. And the emphasis was on trying to look at improvements in education of science and mathematics, especially to underprivileged students writing many books for this. And over the years, of course, this has grown the center eventually moved to the current location here in Devanar and VGK was the sort of founder director here for two decades, spearheading the activities of the center until he retired in 1994. And, you know, everyone at the center was part of his family, and VGK's family has also been with the center over the past several decades. In fact, this is the first lecture where we will miss the presence of Mrs. Vijaya Kulkarni who passed away earlier this year. Of course, we are very happy that Anita Ji decided to join us in person over here. And Chandu Ji, I'm hoping you're watching this online somewhere with us. Anyway, Professor Vijay Kulkarni was a very cheerful and an approachable individual with a lot of interests in many things, apart from science, be it literature, arts, history, philosophy. And one of the things he really emphasized was the role of language and the importance of being able to convey science to teach science in one's native language and write books in one's native language in a manner which was accurate on the science, but easy to understand. And that has been sort of hallmark of many of the things which we've done here. And, you know, he wrote also extensively in both Hindi, I mean Marathi and English, and, you know, he was a great speaker as well. Of course, received many awards during his career and after that and he was active after for a very long time, writing lots of things in various newspapers, etc. So with this introduction, I know we are all very eagerly listening to, eager to listen to Professor Ramana Jam, Jam as those of us who know him, and let me hand it over back to Deepa, who is going to be the sutradhar for today to moderate the session and introduce the speaker formally. Now, given that this is online, if you would like to ask a question, we will probably take most questions at the end. However, if you have a burning question, please raise your hand. And Deepa will decide to let you can also type in your questions on the chat whether you're watching on YouTube or on the Zoom seminar. So over to you, Deepa. Thank you, it is an honor for me to introduce our speaker, Professor Ramana Jam today. He is a renowned theoretical computer scientist so actually speaking requires no introduction. He's a very well known figure in academic circle for his work particularly in mathematical logic theory of computation and their applications in many other theories like game theories and so on. And Professor Ramana Jam has a long association with Institute of Mathematical Sciences in Chennai, where he served as a faculty member for over 30 years from 1987 to 2021 until his retirement. He is currently a visiting professor at Azim Prenji University in Bangalore. And Professor Ramana Jam has been associated with Tamil Nadu Science Foundation for over 30 years through which he is actively involved in science popularization and science education, which is also the domain of HBCSC at large. He is the editor of a magazine called Thulir since 1992. And it is a monthly magazine for children in Tamil language, which explains science in a very simplistic way. And this magazine has a wide readership amongst children and teachers. And interestingly, like Thulir literally means a tender shoot or two sprout. And this magazine has a section where children can actually contribute in terms of writing. So the vision of this magazine is truly serving to its name. And this initiative is also well recognized in many forums. Professor Ramana Jam has also served in many national committees such as Indian Association for Research and Computing Science, the Association for Logic in India and CS Part Shala initiative that a lot of you have recently heard about. He is also a member of a committee that is set up by Tamil Nadu government very recently to formulate the state education policy where he is actively involved currently. And Professor Amanujan was also a member of steering committee of the National Curriculum Framework NCF-5 in 2005. And he has also chaired the National Focus Group on Teaching of Mathematics of NCRT. And in both of these ventures, a lot of HBCSC members have also participated so that there is a great connection over there. Currently, Professor Amanujan is serving as the President of Mathematics Teacher Association of India. And in 2020, he was awarded Indira Gandhi Award for Science Popularization. He is given by INSA, the Indian National Science Academy. Interestingly, Professor Amanujan also has many ties with HBCSC and PIFR. He has received his doctoral degree in computer science from PIFR. And he was also a team member of ISRC, the Indian scientist response to COVID along with again many HBCSC and PIFR folks. And one of the objective of this group was to act as scientific interpreters for public during COVID pandemic and to explain the COVID related information in a simple yet scientific terms. Professor Amanujan was a member in the discussion group in the initial stages of Rijyan Pratibha program, which is currently ongoing and it is a national level student nurture and teacher capacity building program under the academic leadership of HBCSC. Further, he has edited many books and co-authored book chapters with members of maths education research group particularly. And I would like to also mention Professor K. Subramanyam has collaborated a lot in terms of research with Professor Amanujan. And the book Mathematics Education in India, Status and Outlook is a must to acknowledge here. And I'm sure that we hear a lot about this topic in today's lecture by Professor Amanujan himself. So I think without any further delay, I now request our distinguished speaker Professor Amanujan to deliver the 21st VGK Memorial Lecture. And before we start, I find request as already pointed out by center director. If you have any questions, please do put them in the chat window and I can try to oppose them on your behalf towards the end of this talk. And we'll also try to take some questions from live stream if possible, depending on the time. So I think with that I now hand over the mic to Professor Amanujan and we are eagerly waiting to hear from you. Thank you. Thank you. Thank you very much Deepa for that kind introduction. It's certainly much longer than what I thought you were going to talk about. Thank you. Thank you Arnav. Thank you Savita for this invitation. I consider it an honor to be invited for the VGK Memorial Lecture. Not only because of all the illustrious speakers who have preceded me, but also because of the memory of Professor Kulkarni, who was a source of inspiration to many, including me when I was a graduate student in Tata Institute. I don't know him personally, I didn't know him personally but I heard about his work and as I said there's an inspiring figure. So thank you very much for inviting me here, giving me this opportunity. It's a pleasure to be associated with the Fermi-Verse Hama Center in any way. In fact, much of whatever I have learned from on mathematics education comes from Tamil Nadu Science Forum and work there. And whatever I know of education research is almost entirely coming from Ravi and the students and whatever I have learned from them, from that group. Thank you very much. So let me begin with the anecdote. So the title is very ambitious of course. Mathematics is who knows what is going to happen in the 20th century? Will the planet be around until the end of the 20th century? We don't even know that. And the way we are busily destroying the planet. But okay, there are some, it's a frame on which to pose certain questions, talk about certain things. So some years ago I was giving a math club talk in a school and one girl called Vani, class 9 student came to me afterwards and she was looking a little worried. And now she told me that she found mathematics interesting but difficult. Well, many children say such things. But I was talking to her and she had a very curious question and which made me think a lot about many of the things that you want to think about. She was asking me, will they teach Route 2 plus Route 3 even 50 years from now? So this girl was worried about children 50 years from now and what they should would be going through. So well, I would think that certainly Route 2 plus Route 3 isn't going away. So it doesn't matter whether we are talking about 2754. As I said, if the planet is around, Route 2 plus Route 3 is going to be around. But what she means is basically will the classroom be like what she is going through? And I don't know what agony is that poor child is suffering with Route 2 plus Route 3. So somewhere like that. So there's a good question for all math educators to ponder what it is. And the problem is that Route 2 plus Route 3 is very old. So this is the very old problem. It's a current problem. Will it continue to be a problem? Something for us to think about. Now here is a story that I've told many times. Almost everybody, all my friends in HBCC know the story. Tamil Nadu Science Forum organizes meat to scientist program in villages. And so as a scientist we go and track the children. And I was in this village in Bedinagar district. So we landed in the evening and we were going around. And my guide and chief was one 12 year old girl. And she was showing me around everywhere in the village. Now she knew a lot about trees. She knew names of plants. She was showing me medicinal herbs. She knew the names of birds, insects. And I think I'm a town bread guy. I know very little of all these things. And how many trees I could name in Tamil at the time. In fact, I know more tree names in Tamil than in English anyway. And so very small. So I was going around asking all these things. And I kept saying, look, I don't know. And they were just laughing saying maybe this guy is pretending. I'm the scientist. And she knew about good and making molasses. There is a place in the village where they were doing that. She could identify constellations in the night sky. So next day morning I was leaving and I told Kuroma, oh, you'll make a great scientist someday. And then all the children laughed. And Kuroma says, sir, I never get more than 13 science. Well, everybody around, I'm sure you all had some experience like this. Well, Kuroma can make a good scientist one day. The question is, will she? Now, typically when I give a talk in the lecture fall, I would ask, I would take a vote. How many people think what's the probability? Give me some number and take a poll. And almost everybody agrees that, yeah, well, Chancellor, she will not in the sense of making it to Indian Institute of Science and Tata Institute of Fundamental Research or Math Science or, you know, any of these institutions and ISERS and getting into MPL and NCL and these. Well, question is, what paradigms do we have to address these concerns? Now, I think NCF 2005 made a beginning in this regard. I know HPCSE's own little science books are all addressing this kind of questions. Where are we? How do we see it going into 20 questions in India as well as education goes? Now, of course, the place to start is in the classroom. Now, many speakers speak, I mean, many educators speak of traditional skills. People in the audience here probably know much more about it than me. I was looking at Australia's P-21 framework where they break it up into learning and innovation skills and then information, media and technology skills, life and career skills and then some interdisciplinary skills. Now, there's a lot of buzzwords on this page. Now, I don't want to go into this. I'm not into reading this to point out that they're talking about certain kind of skills. I mean, it's not that I agree with it or not, but I'm just saying this is the sort of thing that people talk about. Now, I've read about some reformulation in the context of math education where, okay, so broadly what people are trying to say is that memory recall of formulas and techniques will not be needed, but finding the right information needed when needed will be a price kill. So it's not about which formula, memorizing the formula, but knowing what to use when, right? That's critical, but that's true today as well. But then there are also these opportunities offered by technology for visualization, data handling, proving. Okay, if you work with systems like Sage and Leel today or Isabelle, there are, you know, these theorem proving systems actually greatly aid the creativity and innovation, but may also hinder critical and independent thought. Okay, so this is a problem to think about. Collaborative work, time management, project planning and completion. Now, where do these things belong in math education, right? They are supposed to become more important than individual theoretical understanding. And the other one that we talk about and educators talk about is coping with uncertainty and setbacks. And this is very important. But where is all this in math education today? How do we see it going ahead? Connections with other disciplines, mathematical approach to other contexts will be important. Now, what does all this mean for classroom? Another very important dimension is resource consciousness and ecological awareness. Now, is it mere fashion to talk about math education in the 21st century? Has anything really changed? Well, we should think about it in terms of changes in children, changes in science and mathematics, changes in ways of learning. And all these, probably there is something to think about. When it comes to children, there is something obvious, right? We are talking about children in the mobile internet generation. They have access to information. There is a very different orientation to knowledge, right? Google has made a big difference. Google Scholar is making a big difference. All this is making a big difference to what we call information and knowledge. Classes are often seen as irrelevant, slow moving and something to be endured. Now, this is a quote. I mean, this is something that I read about in many, many contexts when people talk about 21st century classrooms or 21st century education or the curriculum. They say uncertainty and change are the primary shapers of these young people's values and choices. I mean, this is something from a European Union report that I was reading. Now, there is also change in disciplinary change, right? I mean, the processes of science, you see many changes. I mean, from being a method that yields certainty and exactitude to a process by which complex systems are studied and modeled, knowledge is expressed in terms of increased probability and reduced uncertainty, less in terms of absolutes. The collaborative nature of science and mathematics is more explicit now. I mean, there are projects of international projects of a kind, wiki projects. There are many collaborative research projects that computers have dramatically altered the tools and instruments of science. So, there is something in the discipline itself and the processes of these disciplines are already changing as well. Then, theories of, I mean, changes in theories of learning, right? We talk about working together and communicating ideas as being central to mathematics classrooms as well. Easy access to facts accompanied by difficulty in selection and prioritization of information. And people say that, look, in class, students need to be provided not with new information, but with strategies and skill for selecting, processing, accessing information, making sense of what they already have access to. How is this to be done, right? Of course, the planet matters. And this is a book that Jayashree Subramanian introduced me to, Mathematics, Education, Asset the Planet Matters. Surely the planet matters, does mathematics education matter, right? Now, it's very easy to think of opportunities in content areas in arithmetic, algebra, geometry, data analysis, whatever, to underline the source consciousness and logical awareness. These concerns will become increasingly central, but are we listening? This is the question, right? Now, of course, the role of ICT, you know, you can't talk about all this. NEP 2020 privileges and centralizes the role of information and communication technologies, especially in the context of universalization of calling for education in India. Now, certainly the potential that ICT offers for breaking a lot of entry barriers, right? Linguistic barriers, disciplinary barriers, examination, all immense and this is welcome. In fact, ICT can offer highly flexible modes of navigating educational material. It can also tremendously help in personalizing content, it's all welcome. And I've seen one demonstration, I saw it in a classroom and it was very impressive for me where children had access to the picture of the land and I, you know, this revolving thing. And they were actually taking numbers directly from the application. So these are real numbers that you can use and you can do some calculations and check against the actual data, right? Now, this sort of thing is very hard to do in your classroom, right? I mean, if you didn't have this kind of technology, it's very hard to do. So ICT can provide tools for educational objectives that we actually cannot accomplish without ICT, some of them. For instance, you can ask how would the world look and behave with acceleration on Earth due to gravity, we're just a tiny bit less, right? You can do simulations which, you know, I mean, my favorite question is, can you visualize a cubic polynomial, how a cubic polynomial changes when the coefficient of the square term is double? Now, these are the things that are very hard to do by hand, okay? I mean, GeoGebra is good enough for this, right? Now, and then, yeah, the power of simulation. Now, ICT for teachers, okay? So I'm going to skip through this. There is a lot that ICT can actually provide. It's clear. But the problem is that there are many of the questions here that are relevant here. Now, how does technology help the mathematical purposes that schools need to achieve? What are the areas where it is actually indeed necessary? Does it actually enhance the educational experiences? How can the education system, the math educators, the teachers, contribute to the development of such technology? Without the authorship of math educators, how is that going to serve the mathematical purposes? How do we ensure that the educational purposes are indeed being accomplished? These are, I think, going to become very important questions now, especially with the tremendous push that we see coming for the use of such technology in classrooms. And I think we are pretty much wholly unprepared for the impact of artificial intelligence systems that are shown here. Now, already, if you just look at, you know, systems and their development, right? I'm talking only in the context of math education, already, you know, there is a whole lot of stuff on smart content, you know, this is really technology that takes textbook content, but in a smarter way. There are tools that I'm mentioning. Of course, there are many tools that I don't know about. These references are not to recommend them or anything, but to say that these tools already are doing a lot of these. Like Jill Watson, for instance, is quite a powerful one. Then personalized tutoring, customized learners, interest, attitude and learning style, like Micah, Pearson, there are several ones here. Virtual facilitators and learning environments. You know, there are bots that you can talk to. There is one for calculus that I saw, which is very impressive. I mean, basically, you know, you can talk to that bot, you know, talk about specific problems that you have in calculus and have interesting discussion. Sophia was the one that I was looking at. Now, there are also some tools that combine some of these. Now, there is one experience that I want to single out because, you know, it created an impact on me. This is the squirrel that I saw in Hangzhou and where, when I saw actually how it was being used in, you know, learners that prepare students for the National Gaokao Examination. Now, if you actually go into squirrel and see what it does, it breaks up syllabus units into many small elements and offers intense practice on each and follows a concept graph to navigate the elements. And the concept graph can be humongous. Middle school algebra has, for instance, broken up into something like 10,000 atomic elements. It's a huge web graph and that's what it navigates. Now, as teaching to the test goes, this system is spectacularly successful. But as aims of mathematics education go, it's much less clear what to achieve. Who's going to evaluate these systems that are being developed, especially from educational priorities? Is my voice audible? Is it okay? Jam, sorry to disturb you. Can you hold the mic a little up because we can't hear you so well. Sorry about that. Yeah, yeah, yeah. So tell me if it breaks up. Yeah, it is better, much better. Thanks for that. Sorry, please continue. No problem, yeah. So what about the problems that AI might solve? Like for instance, it can surely help in breaking the tyranny of the textbook, right? Because it can offer multiple ways of navigating textbook content. It can make room for multiple learning styles. And like I said, chatting with the bot, maybe frustrating, but maybe still be less intimidating than a math exam. And that was what the students of calculus in the course in Michigan reported. Now, but I think, for instance, if you want to talk about mathematics with children with disabilities, right? Now, when you talk about visual impairment, tactile techniques currently can be dramatically used to improve using AI. Personalized content, right? Now, for children with multiple disabilities, this can be a tremendous move. There's also bridge content when, you know, when you have a sequential discipline like mathematics and children who are dropping out for a variety of reasons and reentering the system. You know, there are many difficult demands on them and AI can help with some kind of differentiated pace. So there is room for all this. But the point is we need to be very cautious because algorithms can reinforce ways of thinking and can lead you into a well from which it's very hard to climb out. And in the case of children, we should look not only at their interest and style, which is what AI systems typically look at, but also their potential. And algorithms may end up limiting potential by over structuring. And this is a very important danger that algorithms inherently have. And the technology is seductive and addictive. And I would say, I mean, I don't want to say me will work against critical thinking. Much deeper problem, which is already there right now, I would think is the children expect computers to know everything, solve all problems. And that's a grave danger. Part of growing up for children is to, you know, parents transit from people who know everything to people who know nothing. And that's very important part of growing up. Now, okay, let me again skip this. AI can actually help on the other hand for teachers and the students. Because I want to move on to other things. What does all this mean for mathematics curriculum? And what are the implications for classroom processes? The range of approaches being discussed throughout the world today. And I want to describe a couple of areas where I feel that progress in these can make a tremendous difference. One is problem-centered learning. And the other is so-called integrated learning models. Probably, I won't have time for the second one, but I can just talk about one of them. Now, if you want to talk about doing mathematics in classrooms, now this is not about content areas. This is about processes that you want to see. Now, if you're doing mathematics, you know, you talk about representation, coming up with representation, selecting between the representations, right? Deciding which representation is appropriate where? Now, this is very important. A rational number has representation as P by Q, where P and Q are non-negative integers with no common factor. Or it's a point on the real line, right? Now, both are representations and both are important. And there is a decimal representation as well. But deciding when to use what is very important in mathematics. You can think of a linear transformation as a function, you have a definition or you can visualize it as a matrix. And both are very important, right? And a vector is an arrow in space or it's a list. Both are important, right? And this is very important for mathematics, looking for invariances, right? Especially when you can come up with a numerical invariant, you know, like a determinant associated with a linear transformation or chromatic number of a graph, right? Now, these are all very deep notions that, you know, big ideas of mathematics themselves, right? Observing extreme cases and typical ones to come up with conjectures, right? How do you come up with conjectures, right? Now, these are not things that we talk about. Looking actively for counter examples. I mean, mathematicians in general, you don't start looking for a proof, you know, for counter examples. And, you know, proofs come later. Now, this is something unique to mathematics, simplifying or generalizing problems to make them easier to address. When the particular is very difficult, you generalize, find a solution and come back to the particular, you understand it much better. Now, this is something very specific to mathematics. Building on answers to generate new questions for exploration. Now, again, and so on. And these are mostly missing in school. Now, the typical classrooms that we are used to are preset exercises. Like I say, you know, conducted on the blow off the whistle, everyone going through the motions, right? Interactions between students as well as between students and teachers is essential for encouraging the kind of processes that I talk about. Now, curriculum needs, therefore, to be reshaped. So that process such as, you know, and this is all that we've been talking about for quite some time and save 2005 was talking about only these, right? Processes like estimation, approximation, visualization, representation, reasoning, argumentation, making connections between mathematical areas and across the discipline. So, you know, all this requires, you know, reorientations of classrooms, textbooks, systemic expectations. These are all in the background and we are talking about the kind of learning that we want. Now, let me talk about some examples to highlight the sort of things I mean. Now, this is an experience I had, which I liked a lot. So I want to, I'd like to talk about this experience. This is after the floods in December 2015 in Chennai. And we had a project with class 11 children. And it was out of a, you know, the school was actually building, you know, getting material and actually building tents for people. You know, that was a time when something like one and a half million persons were evacuated and housed in temporary shelters. So now, how do you design a tent, an emergency shelter for used by a family? How much height is needed for sleeping or lying flat or on your side? Because you're not going to use the tent very much, you know, during the daytime. Can you actually make use of the floor space? What's a reasonable height at the center? What should be the angle of the walls to achieve the height needed over much of the tent? And do you want a tent to fit the average person? But who's an average person? How do you estimate the dimensions of an average person living in Chennai? What scale do you use for the diagrams and plans? This is not an easy question for children to answer, you know, answer. None of these is actually obvious. And if you start doing it, a whole lot of curricular areas intersect it, right? There is data handling involved because you want to find and analyze body measurement data to plan the dimensions. And you have to think about what it means for sleeping or sitting, geometry and measures, 3D shapes, 2D representations of 3D shapes, nets, construction, angles, scale drawing and measurement, finding lengths and areas to determine the material required. You know, quantities of fabric needed, lengths of poles needed, fair amount of trigonometry, some economics and optimization. So a lot of work is involved in actually designing a tent, right? And this is something that, you know, and as it turned out, dynamic geometry software proved immensely useful for the exploration. Internet was extensively used for information. So now this is an example of what you might call project-based or problem-based learning. In all these, this kind of approach what we are talking about, students are confronted with a problem to solve. There is some looseness to the problem description, necessitating discussion on what needs to be done. And articulation of the problem often involves some discussion and some choices, right? Students get to ask, what do we know? What do we need to know? Access to information comes with the need to select and check for reliability, right? And here is another project that we mentioned where, again, this is in a, you know, Balmela sort of thing, the Children's Science Festival of Thumbnail Science Forum where we actually had students build a structure using bamboo poles. Now they actually built all stacks using blocks and other lightweight objects. They construct catapults. So there were, now the thing is, okay. So this was, you know, my point is here, students get to experiment with notions like stability, right? What kind of structures are stable by actually working with their hands, doing some experience of it, then doing some calculations and then coming up with a formulation, right? Now this catapult slingshot was also a great example of a similar thing. Let me skip back. There is also a project where students were looking at the local building that has stairs but no ramp for wheelchairs, right? So you have, when you start thinking about where to install a ramp, it's not something that is easy to decide. What should be the angle of the ramp? Because, you know, what is convenient for wheelchairs to go over and appropriate slope. And, okay, the switchback is a concept that, in fact, I learned only through that. Because your maximum allowed slope is something that is, it turns out 1 is to 12 and the maximum run allowed is 10 meters. So you have to come up with whatever you need for the particular application. And a detailed understanding of the Pythagorean theorem is essential from this exercise. Here is another one where students explore different ways of tiling a floor using triangular tiles of a single size and shape. Again, you can, you know, cut from paper, experiment with rotations, reflections, translations and compare tiling, you know, different, you know, groups come up with different tiling and you want to say which is the best one. And the, okay, this is something like skip. Okay. There is something called the knackening problem for those who know about it, but let me just mention it. Otherwise, take a look at it. Basically, if you want to show that if a hole of height h is drilled through the center of a sphere, the volume of the portion of the sphere that remains does not depend on the size of the original sphere, but it depends only on h. Now, this is a fairly difficult concept to even get your head around. But you, when you start doing this for a variety of examples, you can start making conjectures for why it should be so, because you see this in examples. And in this particular case, Keith Devlin has a beautiful exposition. So students actually looked at that and responded to it. And they actually explored there are a lot of solutions online. They looked at it and they could actually, the idea was for them to discuss it and arrive at what they considered a good answer. A whole lot of other explorations, let me just mention a few. Here is one to use a computer algebra system to completely factor the polynomial x to the n minus one for positive integer values of n. And very interesting conjectures on the number of factors for each n. Explore, you know, mathematics from the past and columns. This is something that South Indians see on the streets. Now, if you look at, you know, columns can be very complicated mathematical objects to study. How do you define the curvature of a column? That's a beautiful question too. Students create coded messages and decipher each other's messages, break each other's crypto system. You know, statically speaking, right? They understand barcodes and come up with their own codes, right? And here is an exhibition that I want to mention. This is an exhibition that I was part of. This is one called Mathematics You Can Touch, that the Gator Institute and Mathematica in Chennai organized. I think Institute of Mathematical Sciences was also involved in that. Now, these kind of exhibitions are again wonderful things. Here you see this child looking at a soap, you know, film, right? And the kind of mathematics that is involved in this is quite beautiful, visually pleasing, and there is something to discuss and think about. There are these exhibitions like Mathematica in Milano, in Italy. Mathematica in Gazon, that's a group that we were working with. New York Museums Collaborations with Schools. There are like very interesting examples that we can learn from where many of this kind of learning in a collaborative way that is emphasized in a way that you problematize things, you think about it and do. We are not really thinking only in terms of content areas. We are not thinking of, you know, what does it mean to teach a particular problem and, you know, a particular issue in trigonometry? What is it about, you know, a particular issue in algebra? This is really about problem solving of a collaborative nature that I began my talk with. Now, all this, does any of this better Kuruma's prospect of becoming a scientist, right? Now, the point is changing our classrooms into exploratory centers offers far more space for Kuruma and one thing. And whether, yeah, and it is not about some particular curricular issue that you have to master. Of course, there are many important things to master there, but I mean that goes without saying. But classrooms as spaces where you can explore, explore together, looking at knowledge that is available. Textbook is not the sole source of knowledge, right? There is knowledge that you can gain in many ways and knowledge that you gain by problem solving through problem solving. And these become very important. I am not saying that every class can be that, but offering such spaces at all, offering such opportunities at all becomes very important for, you know, many students, especially the ones that I was talking about. The much of math education today has very little organic connection with nature, science and technology in general. And this is a major criticism and if you're going to look at, you know, problems like issues like resource consciousness and ecological awareness, right? How was like I said, mathematics education as if the planet matters as if our social issues matter, right? If that is so, mathematics has to have some organic connection with the, I should have also put the society that you live in, the social problems that you have, citizenship, notions like that. Now, new models in schools that emphasize processes, group learning and open-ended exploration have greater hope of success with children in the century, whatever socio-economic background you are from. Even for inclusion that we talk about, I think, you know, this kind of emphasis has to move from very narrow curricular perspective to processes and processes that emphasize this kind of thing. Exploration, you know, when children are involved in exploration, that already breaks a whole lot of the rigidity in the classroom. It brings in a lot of flexibility. It brings in talk, right? You know, this is another major issue that children talking in the classroom, you know, children rarely talk science, even more rarely do they talk mathematics. And children rarely talk mathematics and they rarely talk mathematics in the classroom. And I think it's very important that you talk mathematics in the classroom. How do you do that? Then language is very important. And, you know, Arnab mentioned Professor Kulkarni's emphasis on language. If language is going to become very important, children have to speak in their language. The mathematics classroom has to offer space for children to speak their own language. Now, where the language of the textbook holds tyranny, right? I mean, this is especially a serious problem in mathematics because the formal textbook language is something that, you know, it's a stylized language. Jargon is very important in mathematics. You have to learn that language. And I think learning that language is very important. Children have to learn the language of science, right? Because it offers the language of quantification. Now, this is not something that comes easily, you know, but to master that language, to talk that language, children need educational opportunities. I think if End of Chapter exercises are the only notion of problem solving, then we are back to, Arnab is worried that about Route 2 plus Route 3, right? It is not about Route 2 plus Route 3. It is about why you should learn about Route 2 plus Route 3, right? And how do I even picture Route 2 plus Route 3, right? Even if I understand Route 2 separately, you know, maybe I can think of this as the diagonal of the unit square, okay? Route 3 is somewhere in the real line between 1 and 2. What is Route 2 plus Route 3, right? But this is not only a matter of answering a question in one particular activity. It is a matter of visualization. It is a matter of engaging with it. You become comfortable with things that you can talk about, right? So, if you are going to give space for children to talk, I think, you know, collaborative learning is very important. Some project-based learning is important. This kind of process is very, very important. This is not easy. How are we to do this, right? Because the social goals of school science and mathematics are rather different from that of the disciplinary goals, right? I mean, disciplinary goals involve understanding trigonometric functions in the secondary school. Now, if you don't know trigonometric functions in the secondary school, you are going to find it very difficult to learn the differential and integral calculus at the higher secondary level or undergraduate level. And calculus is a gatekeeper for a whole lot of science and technology and engineering, right? But that's a disciplinary goal. What are the social goals of teaching trigonometry at the secondary school? Now, school is at the bottom of the science and mathematics pyramid, right? I mean, mathematicians, what do they understand of university education? What does university education understand of high school education? What do high school educators know of primary school, right? Now, if you ask these questions, I mean, clearly, school is not the place that decides these social goals anyway. Schools have to prepare students for a future study and for employment. And therefore, a focus on recall, fluency, accuracy, ways of working are very important and they get emphasized. And this is something that we see in our schools and our mathematics classes. And like I said, in China as well, that whatever I saw, limited time slots, curricular pressures and very rigid assessment regimes, they greatly limit the scope of what can be done by school teachers in class. And the role of educational authority in deciding what they emphasize. And our teachers, like I usually say, are underpaid, underprepared, unsupported teachers. The biggest systemic difficulties, teachers understanding a process in science or mathematics education and lack of systemic means to promote it. In Felix Klein's words, mathematics teachers suffer due to a double discontent. In the sense that you learn certain mathematics in school, you go to university and you realize that that is completely irrelevant. You can completely kick the ladder and you learn a completely new kind of mathematics, right? You learn epsilon delta definition, you learn real analysis, you learn topology, you learn abstract algebra, rings, fields and so on. Nothing to do with school mathematics. And then you become a, you learn some pedagogy perhaps, hopefully, and then you come back to school. It has nothing to do with university teaching, right? University mathematics that you learned. I mean, there is a little bit of overlap with differential calculus, matrices perhaps. Pretty much there isn't much. So this is what Felix Klein calls the double discontent. And many teachers had themselves not negotiated the concepts successfully and lack introspection on these difficulties. Now, so the advocate shift that you are talking about requires knowledgeable teachers, but most teachers do not have personal experience of what it means to do science or do mathematics over time. Not do end of chapter exercises, exploring questions that have intellectual purpose, not pedagogical, right? And all that list that I was talking about what doing mathematics means, teachers haven't had the personal experience of working on such problems, especially problems over some time. And who is to do this? Clearly, work is to be done, but by whom and how? Policymakers who want one quick educational reform and one big sweep. I mean, all over the country over two years, once reform and that's going to solve the problems. Psychologists and educational technicians who want to construct efficient methods of instruction for a particular, you know, one-tenth thousands of a unit and of a syllabus unit. Analysts who convincingly point to impact of sociocultural factors on classrooms and teachers. Specialists with imaginative ways to learn specific items. New age profits would recommend software platforms and global ICT tools for connectivity. They all seem interesting, useful, reasonably important and in the end inadequate. Let me, so I'm not saying this or that, but my point is that we started with people talking about 21st century skills as collaborative learning, using information, etc, etc. That come from documents which say that, you know, this is what 21st century resource consciousness, etc. And then you come back to the classroom. There are problems, you know, mathematics has a very long history, right? Mathematics education has a very long history, a long cultural history, right? People have taught arithmetic for a couple of thousand years in every civilization, right? Now, but individual theoretical learning is at the core of our mathematics law. The text is sacred, right? It might be in the form of slokas or in the form of formulas. It doesn't matter, right? Memorized formulas, you are okay, right? Learn to apply when you look at a problem, ask which formula I should be applying. You are okay. Now, these are messages that you get all through our mathematics classroom, right? Here we are talking about project-based learning. In fact, I didn't even get to integrated models, right? Problem-based learning, project-based learning, collaborative learning, use of information, not memory. How are these things to be brought together? I think classrooms have to become exploratory spaces. Classrooms have to become spaces where children talk, talks, mathematics, do mathematics, do mathematics, not in the sense of saying, you know, what was today's class on? Today's class was on this particular aspect of trigonometry, right? Instead of us talking about it, what is, you know, is there collective problem solving? Are there problems that many can contribute together to find solutions to? Are there problems that admit a multiplicity of solutions? Are there problems that admit a multiplicity of approaches? Again, again, I'm not saying that every class should be of this form. The point is that in one year, we don't even have one class of this form in general, right? Can we ensure that in every batch of students, in every class, every student has some experience of mathematical exploration? Every student is talking mathematics in some context or another, right? It's involved in discovery in some way, where every child is experiencing all this. Are we offering sufficiently many educational opportunities for this kind of educational experiences, right? Now, unless we take these questions seriously, the prospects that are offered by technology will remain ways of using technology to do precisely whatever we have been doing again, right? I think on the other hand, if you're going to do that, the dangers, I think, more than dominate the opportunities that we are talking about, right? So let me end with one of my favorite quotes from Thurston. Writing in 1994, he says, as mathematics teachers, we need to pay much more attention to communicating not just our definitions, theorems and proofs, but also our ways of thinking. The highlighting is mine, not that is. We need to appreciate the value of different ways of thinking about the same mathematical structure. Now, and I mean, he doesn't talk about children talking, students talking, but, you know, that's something implicit in the conversation that he's had. Okay. Now, there are some sources here that I wanted to refer to, that I have referred to in whatever I was saying, some of these policy reports and quotations that I was talking about. I can refer to more. Thank you. Questions, comments, suggestions are welcome. You can also write to me anytime. So let's stop sharing. Yeah. Yeah, I mean, I can. All right. Thank you, Professor Ramanujan. And if there is any question in particular about any particular slide, maybe I'll request you to put up that slide again. Of course. Thank you. So indeed it is. I mean, I was just only in the interest of time we actually have to stop otherwise it was so great to listen and particularly there were several examples. Professor Ramanujan that you brought during the talk and I think all of us. I mean, even educators with where we're really, you know, for a moment, we're tempted to actually start solving those problems. So maybe the triangular, the like the triangular floor slides problem or it could be the emergency problem. So all of these are very realistic problems and I'm sure like the enthusiasm that we had definitely students would have even more and that definitely gives them like a real sense of an exploratory nature of science and mathematics. So, at this point, I request everyone to put your questions in the chat and while we give maybe 10, 20 seconds break to Professor Ramanujan. So please go ahead and put the questions in chat and meanwhile I might just wanted to ask one question to start with Professor Ramanujan. So you talked a lot about nature of science and the language of science and mathematics, particularly like as a teacher or as an educator, we want the students to talk in terms of approximations or errors because probably I also recognizes these words in a much better way. So if you are, if you tend to use AI for maybe a personalized assessment or something, maybe these are the words that it will pick faster. But at the same time, students also have their own vocabulary and they may be doing many of these approximation and error estimation in their own way but probably are not that articulative in terms of the language of science or mathematics. So as a teacher educator, you probably would approach that how I can take the students from what they know or how I can acknowledge what they know already and then take them to something that is maybe well understood or maybe recognized in the community. Yeah. So, yeah, there is also a question about Ayush and Chaap that I want to leave and I think they're related what you just talked about. I can't really hear you much. So is it okay? Yeah, can you just hold it. I forgot for a moment when I was making work here. Yeah. So the, I think what you said, of course, is very important. When I talk about the language of students, right? I'm not talking about language that uses particular words, right? You know, in terms of like for instance, approximation or uncertainty or any of these terms, right? Quantification is very important but students arrive at quantification in their own ways, right? And in some sense, it is only when the language of the student and here I'm not talking about Tamil and Malayalam and Hindi and so on, right? Of course. But what I mean much more is their own articulation, right? Until students own articulation is something that we listen to, something that we allow in the classroom. Now I'm saying that our science classrooms allow very little of it, our mathematics classrooms allow much less of it, right? Listening to that, I'm not sure that, you know, I don't know whether AI can do that or not. Probably AI will be, I mean the kind of strides that I see GPT-3 and you know, the kind of natural language processing systems, probably that's likely to happen. But my point is that it doesn't matter. It is about students talking in the classroom. Now, when do you talk freely? It's only when you feel, and this is tied to what Ayush is talking about, you know, you talk, you raise questions only when you feel free to, right? The sense of freedom is very important. Now the tyranny of the textbook and the tyranny of the formal language, especially when you do not comprehend, when you do comprehend it becomes liberating. When you do not comprehend, it is greatly stifling and oppressive. And in fact, you prefer to be silent, right? The child who's, you know, dropping out of that whole thing mentally is exercising a certain choice, right? In a compulsory education system, I do not have the freedom to walk out, right? I have to suffer through whatever you are telling me, because until the board exam, it's a compulsory thing, right? Now, and I choose to keep my mouth shut, right? Because there is no way you're going to listen to me, what I have to say. That's the point I'm saying, right? So until, and what is it, what kind of educational opportunities can there be where those children feel free to talk, feel free to. Now, there, Ayush's point is to me extremely important. And he asked whether, you know, it's all fine to say that the kind of instructional changes that you are talking about, you know, for Kuruma to learn mathematics, it's fine. But can they substantially change her possibilities of becoming a scientist within course? Yeah, that her access to resources and positions and by her caste, class, gender, sexuality, etc. Now, these are, I mean, of course, you're right. I mean, those are deep systemic issues and these are very important, very deep socio-economic problems, right? My point is that classroom, if you if you now ask classroom as a political space, right? A class, a silent classroom is the death of all critical thinking, right? A silence classroom is the death, you know, silent science classroom is, you know, death in that sense, right? So I'm not saying that this will help, you know, but I'm saying it's a necessary condition. It's not a sufficient condition. A classroom in which Kuruma can talk, can feel that she can engage in. I think has better chance of helping Kuruma see herself as a scientist, right? Now, the other kind of barriers that we are talking about, you know, JEE and the NEET and all that, right? That is something else and also the entire systematic barriers, right? They are of course there, but the point is that negotiating this space, right? Understanding the language of science, mastering the language of mathematics, talking about it, devising it, owning that. I think that requires, you know, change in classroom processes and that's what I was talking about. Thank you, Professor Ramanujam. Sorry, Arush, can you, I think I see your hand raised. So can you also tell your question for our in-person audience real quick and then go ahead with the follow-up question, if you have. Sure, sure. Yeah, my initial question was that these instructional changes that Professor Ramanujam suggested can certainly help Kuruma to learn the mathematics being taught. But I was curious if it was just saying that they can substantially change Kuruma's possibilities of becoming a scientist. That her access to resources, to faculty positions will not be structured and hindered by her class, class, gender, sexuality, etc. And I have a follow-up, which is that I agree with the goal that we want students like Kuruma to have their voice in the classroom, or maybe the classroom can even help them find their voice in a violent society. But if that is the goal, then the pedagogy itself would have to be structured by the class, class, gender, sexuality, ability, divisions on which the society is structured. How can we design a space for Kuruma to be able to speak that doesn't attend to all of these social inequities? And so my broader point is, and I want you to think about this also, is that if the production of mathematics and the access to do mathematics and the teaching of mathematics, all of these are part of a society and our society itself is structured by class, class, gender, sexuality, so on and so forth. Then what does it mean to have a vision of mathematics education that is not attending to those inequities explicitly? Yes. Okay. Now, again, you know, I was talking about in a particular context, let me just mention that. I completely agree with you that the deep inequality, the deep injustice in society is not going to be addressed by any kind of, you know, as you say, instructional changes, right? Of course not. And the kind of access to opportunities, the barriers that exist in society are not going to go away by, no. So let me again make it clear that I was not saying that these things can actually help Kuruma, you know, get into Indian Institute of Science, right? I didn't say that. My point is that it cannot happen until this happens. I meant it not as a sufficient condition, but as a necessary condition. All I was saying was that in any case, our classrooms, silent classrooms, you know, that do not address inequality that does not talk about society. I was talking about relationship with nature, relationship with society, right? As I mentioned, mathematics has very little organic connection with, you know, structure of society and talk about that critiquing that. I agree that these are very much things that I would like to see. But my point was that in any case, classrooms as spaces for talk, as spaces for collaboration are very important. And without that, I do not say. I think, you know, it's particularly one of the things of education as perpetuating inequality, right? Reflecting structures of inequality in the classroom would continue to, now would continue to hold. And if you're looking for ways by which they can be broken, collaborative learning, problem-based learning are become something of importance. That's the context in which I was talking. Thank you, Aayush and thank you, Professor Ramalajam. I know it's a very long question probably and cannot be addressed in such a short time. We can also take it further afterwards, but I would also like to follow up for other questions. In the comment box, I also see Ambat Vijay Kumar, who would like to share some similar experiences. Is it okay? Should I make this room available? Thank you very much. Ambat, go ahead please. Yes, Vijay. Thank you. I've been at Mathematics Education mostly inspired by Jam's work in Chennai. I've been in school math education also in Kerala. And I fully agree with Jam on some of the points and especially he said that math education has little organic connection with nature, science and technology. You don't see mathematics in nature in any of the points of school teachers or school students. They don't see mathematics, they just do mathematics. So that's a very important point and my concern is regarding the role of NCRP to the schools of state board, state CRP connection. I don't know whether you read it. Three days ago there was a press release from NCRP saying that third standard students of Kerala cannot write Malayalam. I read this newspaper item and then immediately I transferred this news to my friends in SCERT. And immediately they replied saying that six months ago NCRP gave a press release saying that the best language students in the entire country is Kerala. So primary one to five. So it's apparent contradiction in their statements. As I understand the SCRT is free to form their own syllabi but what is happening is under the pretext of accreditation both the school teachers and college teachers have absolutely no time to do. They have time to do many things but accept teaching. So there is no pressure of teaching. I was fortunate enough to be in the group of Jam where I enjoyed my teaching in the university. And I prefer to go to schools, several schools still COVID types. Okay so there is something radical change required in the attitude of the management teachers. I will spend one or two minutes more. There's a friend of mine in the university department of education in Central University of Kasarbud who wrote a very critical research article in Madarbhumi weekly about the plight of the school teachers. I have requested him to write the similar article regarding the plight of engineering and arts teachers of the colleges. Okay he said that the teachers they don't have the freedom even to say that they are their own problems and worries. The management people are observing just like the CB officers. They are watching what they are doing during the interval. Now coming to the content level of the students in colleges. You mentioned about Route 2 and you mentioned about Route 3 but Route 2 plus Route 3 is in between. Okay it's absolutely no chance of understanding what is Route 2 plus Route 3. They don't know. Let's agree. More pathetic is that I had been to a college last month and most of the teachers told that if you ask any very good student of BSE third year they cannot draw the graph of Y is equal to X plus 1. They can draw Y is equal to X. They can draw Y is equal to 1 but nothing with X plus 1. This is the third year BSE students situation. So you can imagine what is happening in the colleges. Something radical change has to be done but this problem has been in air and in discussion for several years. But I don't know where these things are going to end. Thank you for that comment. Professor Amaraj, you have been serving on many committees now. Would you like to make any comment about the scenario that Amaraj presented? No, no. I mean I think Vijay knows as much of all this that I know so I couldn't. And also you know committees are just some opportunities to do something. I'm not sure that many of these problems can be solved by you know committees writing reports and as he pointed out contradictory reports as well. So I think yeah. Okay. So I think I also see some comments particularly by Prabha Kulkarni and Shri Rajini. So most of them are about how innovative ways of mathematics teaching is necessary in school level education. Amit Uptay, another Maths teacher educator has also given some meaningful resources that one can try out. So we can try to put some of those and also on the YouTube live streams for our audience. So there is a question by next is by Karen. Karen, would you like to say this question or should I pose it for you? Yeah, I could say it. Can you hear me? Yeah, you are audible Karen. I really appreciate your emphasis on student talk in the classroom. And we've also found that we have a researchers have done observational studies of classrooms they have found that students are just not talking or talking very little compared to how much talk the teacher is doing. And especially students are not asking questions, which I think, you know, those of us who have experienced in mathematics education can also agree with that. You know, we just don't hear students or very few students ever asking a single question the entire year in a classroom. And they're not setting the questions that other students are investigating. So, you know, having discussions with each other rather than just between the teacher and the individual student is important. But as far as solutions go, it seems to me that a lot of things have been tried, but what about just reducing the number of students in each class? Wouldn't that automatically allow more students to talk and and what can we as educators do about it? You know, is there some way we can we can influence this policy? Yeah, I think this is, you know, I couldn't agree more with you, Karan. I think, you know, what you say is really not a question but an assertion. I think I, you know, there is no question here, I think, maybe one, one could argue about, you know, is it one is to 20 or one is to 25 or even one is to 30 can be managed, etc. But certainly one is to 16 makes no sense. Right, if you're talking about, and then we have this move to very large undergraduate classrooms, right? I mean, there are these undergraduate classrooms where you have 150 students and you're teaching calculus. I don't know what even that even means. Even conferences that I give talks and, you know, 150 is a huge number that fine for, you know, let alone an interactive classroom. I think it doesn't make any sense to have large classrooms. I think, you know, people can debate about, as I said, you know, is it 25 or 27 or whatever you can debate. But, you know, one should draw very clear lines and I think the push, what does it mean for policy to go towards that and and this is probably nothing to do with mathematics or science. Any classroom if it has to thrive on discussion and collaboration and working together and I think on open ended questions on discussion. I think, you know, working discussions with each other, like you said, I think classroom size has to be restricted and policy has to work towards that. But that's what I would say. Well, I hope that addresses the question to some extent, though, I mean, it is a long journey and probably we all need to think about whether how this number issue can be dealt with in in a more effective way. So there are a few more questions. Professor, I'm sorry to bother, but can you take the mic a little closer? Oh, yeah, I think at times we lose. Yeah, you have to keep reminding me. No problem. So the next question is from Professor K. Subramaniam. Please go ahead. Let me just read it because I think it's important. It's a long comment and a question. Yes. Would you like to read quickly for our audience as well? Certainly. Yeah. So as he says, clearly curriculum designers of school maths have to take into account broad realities and the major changes both in society and in mathematics. But his question is, in all this, how much guidance comes from the foundations of mathematics, given the spectacular failure of the attempt to build school maths based on said theory, how important are foundations of mathematics and have these foundations substantially change from said theory, something that math math educators need to be aware of. Now, this is a pretty difficult question to answer because the foundations of mathematics continue to rely mainly on said theory, but there are alternative foundations based on arithmetic that exist. But the foundations, basically I'm talking about arithmetical theories and algebraic theories that give foundations on which you can build. And in fact, quite a few of the formal tools that you work with, many of the tools that are being built now are founded on constructive mathematics. What is called the constructable universe? It's based on that. If you look at tools like Lean that I mentioned, that are becoming important today. They're all based on foundations of constructive mathematics. And that is very important because if you take foundations from said theory, if you just take said theoretic foundations, constructions are very hard to find in general. So constructable mathematics is very important and tools are being based on that. This is early days yet to, you know, I would say, I mean, to me it looks like early days to translate them into ways by which educators can make use of immediately. I can see how it can impact on undergraduate mathematics, certainly, certainly at graduate level mathematics and undergraduate mathematics. Clearly, there are ways by which you can use these. In fact, I think the teaching of algebra at undergraduate level can greatly benefit, you know, you use Sage and Lean today, there are ways that you can learn that seem clear. And the direction in which it is going is I think you are going to see more and more of these. Whether, you know, that is rich enough to give the kind of direct experiences that you need for, you know, school mathematics and pedagogy I think is open. I don't see that yet. But the emphasis on constructions and constructability, as opposed to existential arguments, right? I think that I think has a potential for rebuilding the way we think, but I think it has some way to go yet. Thank you very much, Professor Amanachan. In case you had any follow up or you are okay if I move ahead. Is there any follow up question? Oh, thanks, Deepak. I'll follow up. I will probably read and understand, but that's different. I can, yeah, yeah. In fact, I think Lean is a very good place to begin to see what I'm talking about. Coq is also there. Coq is more difficult. Lean is probably easier to get into for what I'm saying and formalization of mathematics and that. Now there is, oh, Kishore, there's a comment from Kishore. Yeah, I think Kishore sees mostly like comment and I'll try to put some of these interesting comments and there are several resources that people have shared in Zoom. So we can try to put some of those also, parallely on the in the YouTube live. So in comment box, you can also take later, take a look at some of these interesting resources. Also, I think Professor Amanujam has brought a wonderful set of resources. For example, during the talks, we'll also try to hyperlink some of those resources which were there in the PDF document. So I guess it is almost the time that we should start wrapping up. So if there are no more questions or a comment, I would like to move to the last segment of this event. And there is a long list of people definitely we would like to thank from HBCSE. So first of all, thanks a lot, Professor Amanujam, for your insightful talk. And it really helped us to understand some very ground level challenges or problem that still very much exists no matter what kind of technology that we are trying to move in mathematics education. And even if we are transitioning to a newer education policy and exploring how this could be implemented in classroom with teacher education and other resources. So it is, I think it is very timely and the talk will inform a lot of educators that are present here and also listening to you in person at HBCSE and also online through the YouTube. So I also take opportunity to thank family members of VGK for your presence. I mean, some of them are present online. Some I also see are there in the auditorium so we can meet them in person later. Unfortunately, we lost Mrs. Kulkarni earlier this year, but I do remember she was taking part even during the online VGK Memorial Lectures every year and she will be truly missed during this year. During this event and a lot of VGK friends and colleagues and other dignitaries are present and I see them in Zoom today. So thanks to all of you for your honorable presence. Thanks to our ex-director Padmashi Professor Arvind Kumar who instituted this VGK Memorial event and all for that matter all the ex-center directors who continued organizing this event with great enthusiasm for past 20 years. And my special thanks also to our ex-center director Professor K. Subramanyam who has a long research association with Professor Ramanujan and which in a way greatly helped us to bring Professor Ramanujan board for this VGK Memorial Lecture. And I thank of course our current center director Professor Arnab Bhattacharya and Dean Professor Savita Ladage for making this event possible in a hybrid mode because we do see some audience in a given and also listening to us on the online platform. Thanks to the entire computer lab team who flawlessly are managing this program on the online platform and we gave you enough challenges today. Thanks to Manoj for developing Manoj Nayar for developing the VGK Memorial poster. And I also would like to thank a few more people particularly Ravindra Savan from Dean's office, Sumana Amin and Gaurav Gidnar and other colleagues from PRO and director's office for email communication. And it is a big event which also initiated a lot of social media publicity. So thanks to all who contributed in this effort and thanks to Pragati Dandekar and all the administrative team for your support. And I hope we hear more hard-working ideas like this which were presented and shared with us in form of a series of wonderful examples that we can very much go ahead and try it out in classrooms by Professor Ramanujan. And we can hear, I hope we can hear such full of excitement talks in the future VGK Memorial events and it really drives the community to understand the real challenges in classroom and connect it better as an education researcher. So on that note I also thank all the online audiences and also the audience listening to us in person for making this event very memorable and very energetic in thought through your comments and questions. So thanks everyone for that. And I think we, with this I definitely would like to thank everyone who contributed in some capacity in this program, but I probably have missed their names. So thanks again, Professor Ramanujan and we hope to see you at HBCSE soon for some more exciting and continued conversation. So Tech Team, you can go ahead and stop the recording now for those who are present in G1. There is a tea arrangement outside G1 in the foyer. So please go ahead and enjoy that. And all those who are listening online, a very good evening to all of you. So thanks for that. I'll have my child later. Okay. Thank you. Thank you all. So we'll keep the chat open for some more time in case if you want to write some comments.