 . . . . . . . . . . . . . . . . . . . . . . . . . . . . to teach mathematics to them, generate interest and so on and thirdly to engineering students that is the third difficulty. So, only one saying is there in my Hindi, Karela Meenchara at the time I do not know what to put it, there are three levels of difficulty in this topic but I will try to share with you my views of teaching at IIT Bombay and in general what can be done about these things. Possibly with efforts I think we can do something about it after all it is the responsibility of teachers to somehow make students learn things even if they do not want to learn things right. So, let us so my talk essentially is my views on this that if at any stage you feel you have something to add which can and which my experience of teaching or of some doubts or clarifications or questions some incidents which have come to your knowledge while teaching not only mathematics in general also because relation between student and teacher is almost the same whatever subject it may be right problem many problems or common problems to be resolved. So, let me begin essentially I will talk first about basic teaching methods and what one should do how one should organize and then I will come to specific mathematics a bit later because there are certain things which are common to all. So, basic preparations for teaching a course should be I think there are four in my view one is planning second is teaching methodology that is planning and conducting assessments and the fourth is what we call as tutorials. So, I will take one by one and say something about these things some of these things may not be very directly applicable to your scenario because most of you teach probably in in colleges which are affiliated to a bigger institution called the university probably where everything is a set mode you have very little chance of deviating from the set pattern of teaching that engineering subject or mathematics physics or chemistry or anything there is a set pattern of levels that is set pattern for exams set pattern of valuation you have a very little chance you have to go and fit yourself into that slot of a teacher rest is all fixed by the university. But even then if you get a chance to experiment and probably do something about these things I think that will be nice thing there is a scope for even in a set mode you can add your own touch to teaching I feel. So, little effort is required. So, let me go one by one look at the planning aspect of a course. So, as I said in a university level this is which is not normally done for example, for even if the university is conducting a course or you are affiliated students just come to class you do not know what is going to be taught how long that thing is going to be taught right they just day to day this is kind of thing they come for a lecture you teach them whatever you have prepared on that day right they take down notes or whatever it is and go back and then you say on that day there will be a test or a exam and so on. So, they go through that kind of thing it is like travelling in a train right some people just come and sit in the train and as the stations come they do whatever is required and some people beforehand travelling they get a timetable of the train schedule of the train train is going to pass through which routes which stations it is going to pass through at what station what will be available to eat or not right when that dinner will come when chai will be available such kind of things. So, I think a person who has this knowledge right about timetable and such things is enjoys is only more than a person who just comes and sits in the train. So, that is the analogy I would like to give. So, from my point of view we should try to provide optimal course information which is meaningful with meaningful expectations for students. See what is the course about what are the course contents what you expect the students by the end of the course right in what way they should interact with you or they should gain maximum out of that course if they know beforehand probably some of them will be well prepared to receive your instructions rather than just go through the thing. So, this is more elaboration of this device course outline converting covering objectives schedules assessments texts right your office hours and so on. All this information is a good idea to have it is like the timetable what plan what station how much time the train will stop and so on . See each course it is designed with a certain objective in mind right. For example, the same mathematics course if I am teaching to a size students I would expect them to have more stress on conceptual part and probably less stress on computational part of it. The engineering students probably miss the other way around probably like an engineering course say in micro electronics or something like that you will like to stress more on the diagrams that these are more important than actually writing the things probably. So, whatever it is the course content should be outline detailed more by you keeping what you expect the students to learn after the end of this course. So, the whole thing should go with those objectives in mind. Then the schedules various things like there will be going to be a quiz there is going to be a surprise test sometimes that kind of things. How are you going to evaluate those things assessments right. What are the books which you think they should be reading or what book we will be referring to such kind of things. And then what are called contact hours beyond the lecture hours when all the 50 60 70 whatever number is and roll for that course come together and meet you. Some hour after the beyond these contact hours specified contact hours where if a student is interested to know something he should be able to come and contact you. So, that is called the office hour that means you will be available in your office at that time or whichever place you specify in that particular classroom I will be available if you have doubts questions to ask right about the subject you can come and ask and discuss that is not like personal interaction with the student who is interested in. So, office hour is meant for that and this information not only you should decide you should let the student know this information that is also important as they know they will not come to you if they are interested. So, inform students of the office hours to get help from you. And this is another thing which unfortunately is missing in other university system because students come they write the paper and papers go somewhere somebody is evaluating it and eventually student gets only a mark sheet saying that he has got an out of 180 marks or something like that whereas you are probably expecting 90 right. I know there is a regression system in the university is normally you go and stand in a queue file an application where you want it to be re-evaluated pay some money you will get a photocopy of your answer book and all huge long process is there. But still the student does not know even if he gets his answer book back at what point he lost marks and why he lost marks. Provide clear marking and assessment criteria well in advance to students. Actually beforehand even before the course starts this should be known to them this is the kind of papers which are going to come this is the mode of evaluation there will be objective type or non-objective type papers right quiz will be there quiz will not be there quiz will be of this type this will be the marks marks theme for that and everything should be known to them I feel that is fair enough in fact you will be surprised may be that in IIT for example there in the half year you have Midsum exam is there the solutions complete solutions are given to the students for the Midsum question papers with detailed part marking each one gets a copy in his hand so that he can compare with his answer book at this step I lost this much mark or I gain I have to get a mark so I will come and say that look according to your markings theme I should get you are not given me probably somehow missed it kind of thing so that much transparency is there but probably university system that will be inviting more problems yeah sure bigger I have right all colleges right assessment is being done right nobody bothers to even give the model answers so therefore I get about the model answer reaching the student even the evaluation I will come to that issue also yes that is a very important point yes unfortunately yes I fully agree with you that point see the point is the person who sets the paper if it does not bother to write model answers right there is a possibility some questions could be wrong very high probability because you are not bothered to answering question himself how much time it takes to answer that question moving the subject very well I should sit down myself and try to add answers as I expect the students to write and see how much time it takes I should be able to finish the paper yeah probably half the time specified time then I would think probably it's a right rightly timed question paper right if I myself am taking one hour to answer those questions then I can't expect to answer in one hour right because I have to think try and then answer you already know what is the strategy to answer a question so all these things are part of the paper setting right which as I am saying the bigger the community and the person who is setting the paper one does not know he has not been given the responsibility of writing so next year will not be examined unless you write model answers along with it there should be at least 3 persons who are going to look at your questions and model answers probably they will not be known to you so independent assessors of the question paper and the answers and suitability for that subject should be there otherwise the exam will lose its meaning what you are testing is not going to be effective surely I agree there so provide clear marking assessment criteria well in advance that is what I said to students and tutors and the markers that is what I said so that should be known to all the evaluators who someone is going to evaluate that this is the marking scheme in fact something more should be there that there should be a meeting between the evaluator and and examiner or head evaluator or whatever you want to call it who should explain if a student writes this way then probably you should cut mark if a student doesn't write this way you deduct one mark if he writes probably roughly this way then probably you condone and give marks kind of thing so that makes it more uniform and meaningful evaluation so those kind of things are required and now technology is being advancing I think this is something which we have tried this year at IIT so consider providing a course website for students to access the outlines frequently asked questions maybe you can post some previous years question papers on that bulletin board announcements and so on create a webpage for your course I think every college has a website probably if you request that you want to experiment for your course you want to have a link there where you can make announcements you can ask you can sort of put discussion forum and such kind of things the second part is the assessment part so that also should be planned well in advance so all your assessments should be aligned with that objective of the course right so if you want to test conceptual things then you should be giving test which tests the conceptual part of understanding your course stresses more on the computational part right then accordingly your test should be giving questions which are of computational type so all your assessment should be aligned properly with the objectives that you have set for the course or set by appropriate body so as I said earlier make assessment criteria part marking etc. explains it to students also not only through the evaluators but to students also it should be known beforehand discuss device marking schemes criteria for mark distribution etc. with the evaluators also so that they are very clear what award we are going to give to a student for a particular answer and this is another important thing as far as possible it should be uniform same answer book is being evaluated at one college and another college for the same question you know also mathematical errors plus minus signs in that centre we will ignore that and we will give marks other places we will not grow it we will cut marks for that so same question written differently students are getting different rewards for that so that is what is called uniformity that is possible if it is already discussed beforehand and everybody tries to follow strictly so that is the reason why we feel that they should discuss with the others also and then of course always there is a chance of human error if evaluator is there so there is a possibility of remarking that revaluation chances should be there and actually if possible there should be somebody who should be doing random checking of the answer books random evaluation if they are already evaluated ahead examiner whoever it is you want to call it probably 10% of the answer books you should go through again and see whether that person is doing the job properly or not if not you should go and tell me you are making this kind of mistake in evaluation please correct realign your evaluation process so that is very important after all our education system is exam oriented to whichever career you take it is all exam oriented as students whether he is intelligent or not is measured by how much marks he has gotten in the exam that is the system we have adopted everything is exam oriented teaching there is a coaching everything is exam oriented how to excel in exams how to cross over that hurdle everything is geared towards that whether the subject you are teaching is understood by the person or not exam is there to test how much you are understood exam is the hurdle to be crossed and how well that is whether you understand or not that is not important so for that reason you should be very careful in evaluation also then there is something called tutorials which I do not know how much it is followed in the university system for example in IIT for every 3 hours of lecturing we have 1 hour of tutorials is mandatory it is part of the timetable specified the timetable itself says the course is 3 1 0 8 that means there are 3 contact hours 1 tutorial hour 0 lab if there is a lab then lab hours also will come in that and 8 or 6 means credits for that course so in the timetable the instructor already knows that on Wednesday afternoon there is going to be a tutorial class and what the tutorial class means that is a class in which I will specify more about that so keep separate weekly contact hours as tutorial hours even if it is not specified by the or is not visualized by the course designers if you have 4 contact hours for a particular course per week if not per week keeping 1 hour per week is too much for problem sessions and tutorials because you may not be able to finish the course probably you can keep it once in a fort night probably 1 hour where students can solve problems based on that topic keep some 5-6 problems they can ask doubts and such kind of things so that is the aim of the tutorial hours separate weekly contact hours this is another problem I am suggesting which we follow in IIT but may be difficult to follow in A probably autonomous institute one can do that for the tutorial hours the whole class is divided into smaller batches so that means what if the class is consisting of 100 students and you want to divide them into 3 batches that means you have to have 3 persons who are going to conduct those either you yourself conduct 3 extra hours which will be asking too much from a teacher anyway if you can get help of assistants probably mtext students or if they are in the college or some senior say 4th year students acting as tutors for 1st year or 2nd year kind of things they will also get some experience of they are given a certificate in the end and they have been conducting tutorial hours they will get a certificate probably so a smaller idea smaller batch is the more interaction develops between the person who is conducting and the students students are more free to ask a doubt and get answer from the teacher or the person who is conducting the tutorial class so normally now classes are becoming bigger and bigger the class I am teaching at present has got 300 students so I cannot have interaction in that 1 hour even with the 10% of that probably or if the class is so big I do not know who is going to ask a question from there like 300 students so in a smaller class like you she can ask a question she can ask a question and I can answer probably interact so smaller interaction smaller batch interaction is required and probably this is the way out of that in IIT we do that because we have got a large number of research scholars and mtext students who help in conducting these tutorial batches each batch change assignment so there is a methodology of tutorials problems to be discussed in the tutorial class should be based on the material covered during that week it is like a weekly small bits in a week we add 3 lectures so based on those 3 lectures divide some 3 for problems which should be tried in the problem hour in the tutorial hour which will reinforce the conceptual understanding of the topic so we will revise that also in that hour and try to solve that problem if there is a difficulty the tutor will help them in solving those problems and of course for that the tutor has to be well prepared if the tutor himself doesn't know the answers to the problems that he has specified him to do or solve the new problem so he should be prepared for that so in IIT what we do is we assign a separate teacher see for example for a first year mathematics course 300 students there are 10 batches so around 30 students each and each batch is taken by a separate tutor so there are 10 tutors involved in that so all the 10 tutors are told beforehand that in this coming problem hour or tutorial hour these are the 5 problems to be done in the class done in the class meaning students have to do it where only to help students to solve those problems it's all that they go and solve it on the board students have to try those problems if and when somebody has a difficulty he will ask the tutor how to help and these 5 problems are discussed by that faculty member with those 10 students in one hour so there is a separate class kind of thing for tutors also once in a week where tutors have had chance to understand full solution of those 5 problems understand in and out understand what possible stages are there where a student can get stuck probably what difficulty he may have he can ask the person who is helping the teacher that if the student gets here then how do I help him so that learning process these tutors eventually are going to become teachers also because they are doing intake and research so this is kind of training for them also how to be a good teacher so in that process that is called preparedness for the tutors not only that this process of training should go on over the whole semester or the year whichever duration of the courses then only all these things are meaningful and they will become effective and it has been tested and it works very well in IIT so that's why I am suggesting that if possible if a chance comes you can try to implement this if not at that scale probably once in a month is okay probably we have one tutorial last in a month where a device say 5 or 6 problems on the material covered and how kind of students come and sit and try to do the problems kind of thing so depending on the opportunity one should do that so these are the evolution of how good the tutors are and they also have to be evaluated their performance otherwise they will say nobody bother whether we do well or not in a tutorial class let us just do it that's all so we have what is called a evaluation at the end of a semester for each course in that evaluation form it is anonymous questions are asked from students about the teacher, the course content interaction in the teacher with the students and many other things and about the tutors who was your tutor what do you think how he conducted the classes where he was he helpful or not and kind of things so performance of a teacher tutor is also evaluated well with a teacher at the end of the semester a report comes to the person who is teaching saying that this tutor has performed this well or this well so he gives it to the tutor so look this performance was satisfactory very good or not they also will encourage and that can go into giving a certificate later on that they have helped in this tutorial classes which will help them in getting a job of course in case they go in teaching if not at least it helps them to understand that subject well in any case all this information that I have said about how to plan a course then assessment tutorials these are the things to be done before the course starts all this planning this is all part of planning assessment planning the course contents planning the tutorial hours who are going to be tutors what are we going to do in that tutorial hours we even decide beforehand that in that particular week what are the problems going to be done before the course starts so I will show you the booklet we prepared this semester for the students so this is printed it has a booklet and it is given to the students before the course starts everything all the knowledge all the information so I will show you a sample of that so this is what the booklet looks like the course is MA105 mathematics 1 and it gives the information about the course the course contents followed contact for the instructors and it gives the problem sheets for all the weeks to come in the course and for the tutors we also have solutions written down so those solutions are not given to students of course they are given only to the tutors so all this goes in a booklet and this booklet is prepared beforehand and given to the students on the first day of the instructions so these are given in this booklet that will remain as it is that will remain as it is for one batch then will it be repetition for the coming batches yes after I next year will be next year that depends on the teacher who is going to teach that course he will like to take my set of problems which I have prepared if you don't want to spend too much time on devising new problems because once I take a problem and draw a sheet I will give you a solution also for that for the associates otherwise I am not going to put that problem right so no they may know the answer but they may not know the solutions at least see the idea of the tutorial is answers we are providing anyway right for this problem this is the answer so that students see if we are giving 10 problems in one hour a student may not be able to solve and we would not expect also so we expect that probably 3 and 4 you might have tried beforehand right in his room or hostel or wherever it is and probably one or two has gotten stuck so to check whether he has gotten it right there should be methodology of checking right so answer is given to them that this is the answer if you do it this way this will answer will come out and all that so tutorial is more to clear the doubts rather than actually sort of solving all the 10 problems specific problems based on what is covered in that week right if you are covering say integration of one variable then you give one or two problem which tests whether he knows idea of substitution or not and such kind of things right you do not want to give 10 similar problems like given in books drill problems so we will not do that we will pick up one specimen problem for each main idea covered in that week so that is what is able to realize that idea and reinforce that thing so that is the main idea of tutorials my question is for beachy students okay right beachy students do individual assignments that this tutorial see what you have shown us right that is for the whole group of students may be 300 students this year yes right but sometimes they will work in a team one person will be dominating in a team right he may be getting all the ideas correct good right fine I do not mind that shall I react to that question I do not mind that because learning should not be taken as an individual activity at all there is something called nowadays it is becoming very not a popular it is being recognized as a good idea what is called cooperative learning where actually I am an associate it is not only just calling an associate sometime to faculty also I tell them that this is not a test we are making for you right tutorial class is not a testing ground you can discuss make groups of 2 or 3 each right discuss if you have solved it and if somebody has not solved it you can discuss with him okay in that way you will understand your solution better and you will understand where he is getting stuck so discussion if done properly right is a very useful thing so that idea that I have solved it I should not tell the others because then if a person is bent on not learning then you cannot help it whatever method you device if you want that all should work independently right that is what you are trying to say probably one person will do the problem other will copy right because that is a sort of a easy way out okay this is assignment to be we are not asking him to submit back to us at all solutions see tutorial hour is not we are not testing anything at all in tutorial hour see that is our point see if we are saying that the 5 problems you have to solve and submit back to us then they will try to copy and submit right if I say I am not asking you what you have done in the tutorial class I am saying these are the 5 problems that I will like you to try which will help you to understand the topic whether you want to do it or not it is your choice okay no feedback is asked for whether you have done it actually or not then everybody feels responsible that only I should do 1 or 2 problems if not all 5 that is one way of because there is no pressure of performing now same problem somebody takes 1 hour to solve somebody may take it 5 minutes to solve thinking level of everybody is different right how fast is different so idea is not that you should do all the problems and how fast who does it relax these are the 5 problems all of you try at your own pace if you get stuck somewhere you call me I will come and try to help you yeah teaching methodology as such how to teach kind of thing that is the core of our job planning and everything is okay so first of all become comfortable with the material you are supposed to teach unless you know your subject very well you cannot teach well that is obviously understood sometimes you are given suddenly a new topic to teach right so be sure that you have understood that topic well okay and try to anticipate this is where I think the difference between teachers come that somebody who can anticipate a problem in the class that at this stage student may have the difficulty and may ask a question and what will be my answer to that if you are interested in that then you have to prepare your class very well I would say right so that is one thing how to prepare about so this is another thing like if a class is of 1 hour as towards time span of attention concentration is maximum 10 to 15 minutes right beyond that he will start talking to somebody else or do something and try to disturb the class probably so every 10 to 15 minutes do something which will break the monotonicity of the class that should not become monotone right normally what I do is to stop and say ask a question come on the discussion now suddenly come to a discussion about 2-3 minutes or tell something else so that the mood is broken so you have something different to listen to and then go back to your class right so 10 to 15 minutes if you can show some visuals to them in their class about something do that that depends on the topic but this is a common non fact I think you can see it on TV also say yoga asan program is being shown to 200-300 people you will see that that person does not show the asan continuously but at 10 minutes he will show the asans and how to do it and suddenly he will go into some chat-chit-chat right you know that concentration of the audience you cannot hold it more than 10 minutes 15 minutes on one particular point you have to do a variety right so that kind of thing we should be also doing it in our teaching also to retain their interest if possible in a class of 300 also I try to do it 300 students are there I try to make it interactive I suddenly ask hey red shirt student tell me what this means he does not know the mood I am asking the 300 students who are pointing to but somehow they get the idea and now they start interacting and the activity is one thing because that way the people involved in the lectures if they see most of the time teachers back only that is writing something on the board and students are copying taking notes then the whole idea is gone of lecturing and you should target the average that is what I said you should target the average student in the class but now I will do through a question which will fire a good student not everybody may try it you just say leave an open-ended kind of remark I think I got this question some good students will think and probably come back to after the class this is one thing probably taking your lectures you are following a book probably but what is written in the book word by word if you try to do it then students will the same thing is given in the book why do I come to class you have to add your explanations also to that topic your way of interpreting what is written in the book no touch should be there when only students will feel interested in learning from you otherwise they will go to the book and read it finish carefully prepare your lecture that is what I mean carefully means that in that lecture what is the main idea of that lecture what topic you are going to discuss how many examples you are going to put to illustrate that idea how much you are going to explain all those things that is called preparation and this is what I have not done today for a 60 minute class prepare only for 50 minutes because you think you can do it in 60 minutes but when actually comes you will be able to do only only of 70% or only 50% of what you have planned so probably my thing will go into 3 lectures what I have prepared to show you this is normally because keep 10 minutes for interaction questions and such kind of things and if possible I try to do it quite often when you start a class just 2 minutes so last time in the last lecture we had done the following just revise the continuity kind of thing like in serials now on our TVs they show you a recap before the actual serial starts and in the end they show you what is going to come also 1 minute so that is what I mean so start revising a bit 1 minute to bring in continuity of what we have been doing and by the end of the lecture today we have discussed the following 3 topics just 1 or 2 minutes recapturing what we have done on that day it is not a bad idea so anyway whatever you do the aim of the lecture is to explain the subject in such a way that most of the students are able to understand what you are saying and retain most of it that is how much they understand what you have to retain what is explained so whatever things you think depending on your particular specific topic or subject or the class do that so if that is objective then it will be and final thing is prepare prepare and prepare even if you have prepared the same lecture 10 times there is no harm in preparing for 11th time also I must have got calculus 15 times but whenever I do it again again I prepare for the same class because while doing it topic is same the concept is same but when you are taught it over the years you see that a student got stuck at that problem he asked me that doubt so how is that step so how well can I explain that concept in a different way so I have to think now so it is like elephant being presented that old story so how to let you understand the elephant is not just four billards it is something else or it is not just like a rope it is something else unless a student has asked you question probably you think elephant looks like elephant to everybody so the experience will come that oh he may not look at the legs only I can say it looks like a drum so how to make you realize it is something more than a drum so that kind of thing so there is no end to preparing even if you see the actors who perform on the stage even if you have 100 performances or the same play for 101th also they will go and rehearse even if it is the same play same doll this will rehearse 101th time also if they have to give a show why I think like giving a lecture is also performing on the stage going on the stage performing in front of so many people and the dialogue will be different audience will be different questions will be different so we have more challenging task so we should prepare there is no end to preparing so there is something about max specific I think I have around 10 minutes more okay so first of all it should not be taken further that engineering students understand the need to study mathematics they study mathematics because in the curriculum of engineering there are 5 papers in mathematics which the university has put and they have to clear those 5 papers and hence they have to attend the lectures of those 5 courses that is the attitude taken by students it is not they think that this is required for them to become good engineer mathematics is required physics is required I want to become an electrical engineer why should I study mathematics I am studying electrical engineering I want to be a chemical engineering what mathematics has to do with chemical engineering right they do not know these things at all so somehow before we start I am coming to very specific why to teach mathematics and how to teach mathematics so somehow they should not be made to realize that mathematics is required to become a good engineer right otherwise you will be a mechanic not an engineer see what is the difference between a mechanic who repairs cars on the roadside and an an automobile engineer who works in Maruti Uddhya both are doing similar jobs probably the mechanic will take the car to him so he will say you start and he will see you start and he will go something yes sir your problem is there so he has picked up only techniques of the trade by experience probably he does not know how the carburetor works what is the function of carburetor in the whole region he does not know probably he does not have to know that because his job is only to repair but an engineer has to know to improve the functionality of that kind of a thing so that is the difference between the two kind of scenarios right so an engineer we have to make him realize somehow that mathematics is required and essential for him I think this is my very personal remark do not take it very seriously if you do not like it that many of them come to engineering because they think they would have to study mathematics probably at all or very little of it right if I wanted to study mathematics I would have gone to bsc mathematics rather than coming here so they do not know that they will be studying as much mathematics in engineering as a student of bsc would be studying this is regarding the teaching methods especially subjects like mathematics for the subjects which involved a lot of Brahman solving right in your regular classes what is your general method of education whether you solve the problems or you teach them only the concepts but you ask them to solve the problems and take them to solution right so okay so I think most of it I have discussed I had four parts in my type planning the course the second was assessment how do you assess students then there was tutorials right and fourth I was talking to the fourth one how actually teach mathematics so the planning assessment tutorials all had answers to your question namely how do we assess all methodology I discuss with the people here and secondly problem solving separate tutorial hour for each course mandatory so we had every week for every three hours one hour of problem solving is there that is done how it is done that also I discuss there are tutors smaller batches are there tutors are there and so on so the only thing problem which I have not answered till now is what actually actual actual lectures look like I will show you one I had actually thought of showing you five slides but I cannot show you all five so probably I will show you one of them so we teach a topic called about sequences in mathematics so this is second lecture this is my actual lecture slide I am showing you I am teaching this semester and this is actual lecture slide I am showing you so second lecture starts like this so it will be right so now first I will tell them what I am going to discuss in this lecture that is what they should expect in this lecture to be discussed so this is the kind of thing and then I come to the actual topic right so there is a notion of a sequence in mathematics right what is the sequence whether it is convergent or not and so many things are there so to introduce why the notion of sequences should be at all taught what should be considered so voting at that I will look at this I will tell them that any iterative process experiment that you do which you are repeating again and again give you information every time you get information about that so that is all the collection of observations so that is a sequence of observations that you are getting so a typical example in mathematical example is you want to find the area of a unit circle right now here a bit of interaction also comes in I ask the students if you want to find the area of a unit circle what it will be what comes the answer it is pi so what is there to find the answer is pi then I ask them what is pi this in a 300 class I did that what is pi you cannot determine that many of have heard that it is irrational number then you are not giving me the answer see now you realize what is the area of a unit circle if I say it is pi and what is pi you cannot tell me what is pi so it is a problem somewhere to be understood right that the area of a unit circle is not pi which you can find you can only approximate it and how when you want to approximate that depends on you so you have to iterate your approximations and here is a method of iterating so I show them this clip because for a rectangular figure you can find the area so in the circle you inscribe in the circle rectilinear figures and as you increase the number of sides you see that you are filling more and more of the required concept of area so you are getting better and better approximations so this is a sequence of approximations you are getting first, second, third, fourth and so on and what you are interested in eventually as you increase the number of sides you should fill up the whole circle so you want that these numbers a n, a n is the number of n gone inscribed inside the circle you want to know eventually what happens to that a n physically that eventually it will never take place you can never go to illiterity, physical world you can go only 1,000 approximately 2,000, 3,000, 10 million whatever you like so mathematics helps you to where physically you cannot reach that is the beauty of mathematics so now here I can tell them look there are many physical problems a space shuttle is falling down its temperature is heat it falls on the earth's atmosphere on the surface the temperature starts increasing at some rate it is increasing it becomes probably the thermometers or whatever measuring instruments are there they cannot measure beyond certain range probably what do you want to know what will happen after 1 hour what will the temperature be instrument can measure only upto 200 degrees say whatever something but the temperature will go up much beyond that so how to predict what will be the temperature after some stage after 10 hours of falling after 50 hours of falling you are not going to actually physically go and touch and measure but sitting here you want to predict now so that is beauty of mathematics that is how mathematics come into picture so the simplest example is to define the area of a unit circle which you cannot measure anyway you take a sequence by approximating this and try to understand what is the limit of this sequence as number of observations become large and large I do not spend that much time in my 8th class but this is the actual slide so then you go on doing some examples and so on so I think let me let me not do too much