 Let us look at another method of thinking, so thinking using analogies, so all of you know what analogy means, it is this word analogy means the same as let us say a simile. So you have two situations in which some elements and their relationship to each other is common. So an analogy enables a look at a situation as an interrelated whole, the great strength of thinking using analogies is that it enables to look at a situation as interrelated whole. Supposing you want to contrast analogical approach of solving problems or looking at problems from analytical approach, so analytical approach on the other hand dismembers a whole into parts and may destroy the attributes which may pertain to the phenomenon as a whole. So if in the process of dividing the problem you lose the function of the various parts when they are interconnected and then so this approach will not work, so that is why in many situations analytical approach does not work. Problems are solved and creative works are generated by transfer of existing ideas to new surroundings. So lot of new works or new solutions come from analogical thinking, so transfer of existing ideas to new surrounding, so this happens through analogy. So you have one situation which you are familiar with and you have an idea in this situation and now you are faced with another situation, a problem and you want to get a solution and you find that some elements of this problem are related in the same way as the familiar situation and so use information that you have about the familiar situation in the problem situation. Now let us illustrate this thing with examples. So analogical thinking is in fact the basis of all problems solving that is what some psychologists have said. Some form of analogical thinking is happening at a certain level whenever you solve problems. Now I will give examples of some well known and great discoveries using analogy but it does not mean that the use of analogical thinking is only restricted to great discoveries. Even small innovations and so on at the level of a common researcher can happen through analogical thinking. So two great discoveries namely the discovery of matter waves and the discovery of atomic structure happened through analogical thinking. For example the matter wave was discovered as an analog of the electromagnetic wave and the atomic structure was discovered as an analog of the solar system. Now let me take the simpler of the two first, the solar system and the atomic structure. What is the analogy between them? So the solar system you have the sun as the center and you have planets devolving around the sun. Now the orbits are not necessarily circular, I have drawn them in some fashion, it is only a model, very crude model of the solar system. So people for thousands of years have known about the solar system in this fashion. They have conceived the planets rotating around the sun. So this idea existed earlier. Now they were faced with developing a model for the atom. So when scientists were trying to develop a model based on observations. So they had observed various radiations from the atoms and so on and they wanted to put together a model that will explain all the observations. Now it struck some scientists that hey the big universe is built up like this where there is a central entity like the sun and then you have the planets revolving around the sun. Why cannot a similar scheme exist even at a microscopic level? So at a macroscopic level there is some arrangement of the universe. Why cannot a similar arrangement exist at microscopic level? So let me draw the atomic structure on the side that was conceived as an analogy. So here also you have a central entity in the atom. You know it consists of neutrons and positive charges. So the core of the atom is positively charged and you have the electrons revolving around the nucleus. So starting from a model like this where you have electrons orbiting around a positively charged nucleus people could explain the observations. Now here is an example of a discovery through analogy. Now one may ask but then you know how does one conceive of analogies that itself is an issue. This problem cannot be answered that easily. This question cannot be answered that easily. Well it depends on the intellectual level of our strength of our intellect but we can improve our thinking if we know that analogies can be used to solve problems. Then we will try to see what kind of analogies can be applied for your given situation. So let us look at another important discovery through analogies, electromagnetic wave and matter wave. The fact that light is energy in the form of waves was known much earlier. For instance for light or for electromagnetic energy the wave nature of this energy was known earlier than the particle nature because the wave nature is very evident to our common day to day experience. Through our senses without any adding, without additional equipment we can experience the wave nature of light reflection, reflection, interference and so on. So through our senses without any additional equipment we can experience. Now under special conditions people my scientist showed that the electromagnetic energy which we have experienced commonly as wave also has a particle nature. So the example of that is what is called the photoelectric experiment in which it was shown that light behaves as particles. Now as far as matter is concerned the particle nature of matter is much more evident in day to day world. So two particles colliding with each other and then we can solve this kind of problem by conservation of energy and Newton's laws and so on. So the particle nature of matter was is evident to our day to day experience. So some scientist thought if light can have wave and particle nature surely there could be situations where matter also can behave as a wave. In a common day to day experience we see matter as particles but why not matter also have a wave nature. And then there were some aspects observations related to matter which were not explained and the scientist then suggested that those observations could be explained if you regarded matter as a wave not as a particle. So here is an example of great discovery by analogy. Now according to psychologists who have studied thinking some subtle form of analogies are always at work in any innovation or any problem solving. In fact any higher order thinking seems to happen through analogies. In cases like this that I have discussed in these discoveries the existence of analogy as the basis of the discovery is very very transparent and evident. In some other cases this analogical thinking that we may be using may be much more subtle it may not be that transparent but some form of analogies is used. So all good problem solvers use some form of analogy to solve problems. Now here is an assignment I have not put it on the slide before I go on to the prescriptions for improving our thinking. Assignment is a list out four analogies in your area of study that you might have come across. Ok. So some prescriptions for improving our thinking. In fact you might have got some ideas about how to improve your thinking from our discussions in the morning and also the session in that we had yesterday but let us put them down in some concrete form. Now there is one caution about these prescriptions. Some of these prescriptions which are related to improvement of thinking abilities are not like tablets that can remove your headache. So you take a tablet and within may be matter of hours or even sometimes minutes your headache can be controlled. So these prescriptions if you follow then over a sufficiently long period of time only then they will yield results. So these are more like ayurvedic medicines which work over a longer time scale which take longer time to be effective. So this is a very important point you must note. So whatever prescription is being suggested unless you practice that for may be at least a year or so you will not see a significant change in your thinking. But definitely since thinking is a skill we made this point when we started the discussion on productive thinking that since a significant part of good thinking is a skill that can be developed you can develop your thinking right that is a positive side of it. So it takes time. So I have just repeated some points that we have made in the context of making the prescriptions. So first thing is that creativity is a skill that can be developed by practice and the second creativity is a matter of organizing once basic skills. So two things for developing good thinking is practice and organization. So for instance what do you mean by organization? So we have said for instance if you are representing data you should organize this in the form of table or organize this in the form of graph and so on. So this idea can be further extended. So all your thinking you must try to organize, arrange in some order and another important thing is motivation. So this is also something that we have stressed. So motivation is recognized as a crucial factor in development of creativity unless you have a strong motivation you cannot come up with creative ideas. So just to take an example I mentioned about how a particular person who was a poor farmer and illiterate came up with a very novel scheme of naming his children. So what is important there? If you see the children were named after Shiva who was a god for this particular person. So emotion in this case for example is a strong motivation that was responsible for creativity. So like this there can be different other motivations but the point is some strong motivation should be present passion for intellectual solving difficult intellectual problems. It can be a strong motivation for coming up with creative ideas. So unless there is a strong motivation you cannot come up with new ideas or be creative. Then something more specific, learn different solutions to the same problem. This is a very very important thing I have given an example in an earlier session. So you can read a book like 100 different proofs of Pythagoras theorem how the same theorem can be proved in so many different ways. Now here is one more example that I am giving different solutions to the same problem. I have chosen a particular problem here that namely calculation of pi. Now I have chosen this problem because you can many people can understand this very easily all of us have come across this thing called pi. What is pi? It is a ratio of circumference to diameter of a circle. Now what is so interesting about pi? So you see for thousands of years the human beings have been fascinated by various shapes that we see in nature. So we see triangles, we see polygons, we see rectangles and we see circles. So one thing about circles that fascinated people is that all circles look alike. Why is it that all circles look alike? The small circle and a big circle there is something in the shape that makes you feel that all circles look alike. Now what is the basis of this perception? So that is where people found that this feeling is coming because if you take the aspect ratio of circumference to diameter for all circles this is the same and it is this sameness of the aspect ratio, ratio of circumference to diameter that is responsible for the perception of sameness in the shape of the circle. Now this property is not true for all geometrical figures for instance all rectangles do not look alike, all triangles do not look alike. So therefore then people wanted to calculate what is this ratio of circumference to diameter. That is how lot of people, this problem engaged lot of intelligent minds, calculation of pi. Now it will be interesting to see how so many different methods have been used to calculate this quantity. I want to emphasize here that our goal is not to discuss how pi was calculated but to discuss different ways of doing and solving a problem. So this is only an illustration for this basic principle of learning different ways. So four different ways are illustrated and the results of these methods are illustrated here. Let us take an approximation such as pi equal to 2 root 2, how was this obtained? So let us draw a circle. So I want to find out the ratio of the circumference of this to diameter. Now I can approximate, I do not know the circumference easily. So what I will do is I will approximate the circumference by a different shape namely a square whose perimeter I can use as an approximation for the circumference of the circle. So I say that the perimeter of this square is approximately equal to the circumference. Now it is not a good approximation but you will see how I can improve this approximation. If I and the diameter is this, this is the diameter. So the side of the square is equal to the diameter. So if this is D then the perimeter P is equal to 4 times diameter. So now I am using the approximation circumference C is approximately equal to P. So circumference by diameter is approximately equal to perimeter of the square by diameter that is equal to 4 times D upon D. So that is equal to 4. Now I want to get 2 root 2, I just lost track. Let me do another exercise. Yes, let me put, let me inscribe a square inside the circle. This could be another approximation that I could use. I did not get the 2 root 2 that I wanted but whatever I got will be useful to develop accurate approximation. So let us instead of circumcribing a square, let me inscribe a square and use this approximation. So here if the diameter is D then I know that this side of this square is D upon root 2. So following the same approach the perimeter by diameter is equal to 4 times D upon root 2 upon D which is same as 2 root 2. So if I stick to just this approximation that I have shown where I inscribe a square in a circle I will get pi as approximately equal to P by D and that is approximately equal to 2 root 2. On the other hand if I use an approximation where I circumscribe the square I will get the value of pi as 4 and now I can put these two numbers together because I know that in one case I am overestimating the perimeter when I circumscribe a square I am overestimating the perimeter and when I inscribe the square I am underestimating the perimeter. So I can take the average of the two to cancel out and I will get a even better approximation as pi equal to 4 plus 2 root 2. So I will get pi equal to average of 4 and 2 root 2. So that is equal to 2 plus root 2. Now you know that pi is 3.14159 and so on and 2 plus root 2 is very close to that. So this is how using a geometrical approach I can get an approximation for pi and I can actually go on improving the value that I get by replacing the square by polygon of more number of sides. So instead of a square if I use a pentagon I will get an even better approximation. So what I get is I get a value that is more than pi, accurate value. I get another value which is less than pi whenever I circumscribe the particular polygon around the circle I get a higher value of pi and whenever I inscribe the polygon inside the circle I get a lower value and I did not take the average of these two. I go on increasing the number of sides I get more and more accurate values of pi. So this is a geometrical approach of calculating pi. So that is what is illustrated here in the second line. So pi lies between 2 root 2 and 4 if I use a square for circumscribing and inscribing. If I use a hexagon I get values as 3 and 2 root 3. So this is how my higher and lower values will start moving closer to each other as I increase the number of sides and then the average of these two will become much more precise. So anyway the point is I can use a geometrical approach to calculate pi. Now there is another approach and that is this approach of trigonometric series infinite series. So an Indian mathematician called Madhava came up with a series for tan inverse x. Now this is a little bit more technical so it is possible that some participants may not understand. It does not matter it will take only a minute to explain this if you do not understand just leave it. The next point will be much easily understandable. So you have a series for tan inverse x. So it is x minus and so on that is what is the series. Now in this series you substitute x is equal to 1 and you know that tan inverse 1 is pi by 4. So that is what is your left hand side. Right hand side also in the series x minus x cube by 3 plus x power 5 by 5 you substitute x is equal to 1. So you get a series and then you get more and you take more and more terms and you add them up you will come closer and closer to pi by 4. You multiply the right hand side by 4 you will get the value of pi. So this is an infinite series approach of getting pi trigonometric series approach. Now there is another very interesting method of getting pi the same value. Now there is a little bit of background to this what is called Buffon's needle experiment. So Buffon is the name of the person who came up with this particular scheme of calculating pi. So Buffon was a French person who was intellectually very good and also very wealthy. So it is said that Buffon did not really have to do any work to earn money. So most of the time he spent in leisure. So one such occasion he was sitting on a floor which had tiles of this shape which had tiles you can say parallel lines all equidistant from each other. So he was sitting on a chair and he was smoking a cigar. Accidentally his cigar fell down on this floor so something like this. And then because the person was intelligent a question came up in his mind. He said what is the probability that if a cigar such as his fell on the floor that the cigar will cross a line. And for instance the way I have shown it it crosses this particular horizontal line. Now it is not necessary that it should cross. For instance if it fell like this it would not cross the line. And then he set up out solving this problem. So you have a particular distance between the two parallel lines and all the lines are equidistant. So d is the distance and the length of the cigar is let us say l you know using this information he worked out a solution this is a problem in probability we are not interested in the details of the solution right we are only interested in some features which explain how there can be so many different ways wide variety of ways to solve a problem. So depending on the length and the diameter you get a solution for the particular case when the length and the diameter sorry not the diameter the distance between the parallel lines in the particular case when the length of the cigar is equal to the distance between the lines he obtained the result that is shown on the slide. Now this is more popularly called buffon's needle experiment because instead of the cigar you assume that you have a needle of length l you know smoking is not good for health. So people probably replace the cigar by the needle so what happens if a needle falls on a floor which has parallel lines drawn like this what is the probability that the needle will cross the any of those parallel lines. So here is the answer to the question and the answer is cast in the form of pi equal to the right hand side so actually in this expression here written here the drop indicates the dropping of a needle so you are repeating the experiment because how do you find out the probability in this case you would have to repeat the experiment a number of times. So number of times you throw the needle randomly on the floor and you check whether it is cutting or crossing a horizontal line. So crossing of a horizontal line is called here as a hit so if the needle crossed the line then you will say it is a hit. So find out how many times the hit occurred and how many times did you drop the needle so two times the total number of drops by the number of hits is equal to pi or in other words in fact his formula was number of hits by number of drops is equal to 2 by pi this was his formula that he obtained and then it struck him that actually this formula can be used to find out the value of pi. So if you rewrite this expression as pi in terms of hits and drops then you know you can use it to calculate pi now this is something very interesting because all that you need to do to calculate is just sit and throw the needle randomly and increase the number of times that you do this exercise you will get more and more accurate value of pi so repeating is very simple thing a large number of times. Now since then actually this method of solving mathematical problems or estimating quantities has been generalized and has wide application and today it goes by the name Monte Carlo method of calculation. So you can see how new ideas are generated if you can try to see how a problem can be solved by different methods so in this case we discussed calculation of pi by a geometrical approach or from a trigonometric series or by a statistical approach right where you repeat a very simple thing a number of times and calculate the ratio of two quantities solving like a probability. So here is an example to illustrate learning different ways of solving the same problem. So you should take it as a hint to the kind of things you should be reading or whenever you read your literature kind of things that you must be documenting you must be looking for. Now hopefully after this session whenever you read anything any technical matter you will try to look at the same matter in so many different ways you will try to look for different solutions to the same problem you will try to look for different graphical representations of data and so on okay and make document these ideas now that is what is going to come next the documentation before that this assignment which I gave you earlier learn different ways to the same problem comes is coming here different proofs of Pythagoras theorem okay. Now I will strongly advise you to do these assignments during the course of the next few days because after that the enthusiasm for this sort of things will wane so however if you do these assignments you will be able to sustain your interest in research for a longer time at least your interest in this workshop and you will get more benefit out of it. So continuing with the prescription so solve a problem by different strategies so whenever you are faced with a problem you look at this slide which listed out the various strategies reformulation representation graphical representation tableau representation logical reasoning analysis and so on and try to see which of the strategies will work. So in the beginning you know you will have to do this kind of trials for a number of problems until you develop some sort of intuition to just come up with the right strategy for any given situation so document analogies as and when you come across them this is important described at least two analogies you have come across in your area of interest I think I have already given you assignment four analogies okay so to do that now I come to the very important point of note keeping you want to be creative you must keep notes so what are the advantages of noting so noting ideas as they occur helps you to remember them this is very evident but what is not very evident may not be evident to you speeds of your thinking so noting ideas has an effect on your thinking this is based on research in psychology and this is why people are encouraged to write if they want to think better so in this case we are talking about noting noting is the process of writing right so it speeds of your thinking focuses attention on your subject so your concentration on something that you are thinking about increases okay and very important finally it stimulates cross fertilization of ideas so you keep notes of various ideas and periodically you go over all those ideas and then you can try to see how you can probably combine merge different ideas to solve a problem that you are looking for in your research you whose solution you are looking for in your research if you do not record your ideas you will spend all your mental energy trying to risk resurrect the old ones now this is a common experience of all research scholars who do not develop the habit of note keeping so they will do a lot of literature survey and then first time they read one paper and they get some ideas out of it they feel they will remember then they read another paper and then the third paper and so on but at the end of six months they suddenly realize that there was a particular idea they came across in some paper and now they want to take a relook at this and they spend a lot of time trying to locate that particular reference why because they have not developed the habit of systematically noting all the references that they are reading and all the ideas that they are coming across so this is a very very important habit that people should develop research scholars should develop then we have been saying that we must develop the right attitude for research here is one more attitude for we have talked about motivation now here is another attitude have an open mind this is very very important if you want to get good ideas you want to be creative your mind should be open now what is the meaning of an open mind so here is again a psychologist definition of an open mind so an open mind is one which is receptive to alternate points of view regardless of the present level of commitment to a belief so all of us have our own beliefs our own opinions that is how we are but we must be receptive to alternate points of view so I may have a certain belief and there may be somebody else who may have a completely different or opposite belief now I should be receptive what does receptive mean receptive means I should be willing to listen to the other point of view even if it doesn't match with my own I may not accept right but then I must give a reason why I do not accept now the second definition second aspect of an open mind is also very important it acknowledges areas of common ground with those who hold alternate beliefs and allows dialogue with someone with opposing views without attacking the proponent of those views so normally it is our nature to if someone doesn't agree with us right we don't like him him or her right so we don't like to hear criticism but this is not correct really if you are mind is open you will be open to criticism so openness means open to criticism now there is nothing like openness to praise all of us are always open to praise right what is important is importance is the being important to criticism in fact it is experience of all researchers that the negative comments they get on their papers many times are in when the papers are sent out for review and the reviewer gives a critical comment the critical comments are in many cases responsible for significantly enhancing the quality of work so criticism should always be welcome so some more prescriptions arrange and rearrange what you read or hear from different points of view so we have said that creative thinking is about organizing your skills organizing information and so on so you have to arrange and rearrange from different points of view so information could be arranged in so many different ways you try to arrange in different ways okay to see whether a particular arrangement leads you to some novel pattern then allow opportunities for cross fertilization of ideas so as to generate new problem now what are the means by which cross fertilization can be achieved so one important method is interaction so if a research scholar is one who doesn't like interaction always keeps to herself or himself the chances of getting a new idea okay would be less or in other words if you interact technically then the chances of getting good ideas increases so interact this means discuss answer doubts teach explain and so on therefore it is a very good thing all of us know as teachers that whenever we teach we get new ideas so it is good to be teaching while doing research it is good to be answering doubts so if you are not a teacher at least become a teaching assistant and interact with students and answer their doubts then set aside time to read in other disciplines keeping track of what others are doing that seems original so wide reading even beyond your discipline is encouraged for a good research scholar if possible work in areas outside of areas we are currently learning about okay now with this I would like to have some interaction towards the end of this session and take up the next thing of problem finding in a later session so we have a session divided to how how much and what we should read and then where to publish so I will discuss this thing of problem finding in that session so I want to just remind you that afternoon the first session after lunch will be about communication skills and it is arranged so because there we are going to have an activity where you have participants presenting 10 minute talks now in this activity what we are going to do is we are going to have people present make a presentation and all other participants should carefully pay attention to the presentation and form opinions about strengths and weaknesses where is a scope for improvement what is the strength of the presentation and so on and then we will try to see how the same presentation can be improved for instance we will look at the slides and make suggestion as to how a particular statement that has been made there can be shortened or how it could be rephrased for improvement and things like this and then we are going to list out some common things that should be followed to improve oral or written communication so this activity will be just after lunch we will have a short discussion now with some centers yes Nirmal University Ahmadabad any question or comments it is extrapolating the one of the prescriptions probably the last one like working in area outside you know one's own learning one is on presently one is learning about what I would like to say is that we evolve over a period of time and you know often we end up liking you know identifying what we like to do actually in terms of research and all so would you I take this opportunity for you to probably recommend this to the relevant department to you know evolve a policy to allow people as much as it is possible to do even research apart from this prescription in the areas outside you know their present engagements yes actually whether you can do research areas I mean a PhD in an area outside your research of the same level as you are doing in PhD that is probably a more difficult question but you know at least the IIT's the trend is that we prescribe for our research scholars some subjects some courses which are not directly related to their area and they should undergo these courses okay this is not only for just breadth of knowledge but also exposing to I mean information it is not breadth of information alone because methods used for solving problems and concepts and so on differ in different areas so exposure to these different concepts is important though you may be choose to you know work in a particular area so this is really very important another thing is you know many times the complaint is that you have seminars which are announced and informed widely but not well attended now this is one of the things I have seen that research scholars will waste their time chatting and so on even if there is a seminar going on somewhere technical seminar if they feel it is not in their specific area of interest right so if researchers really know that ideas come from other sources than sources other than in your own field if you know that innovations happen through analogical thinking and so on then you will be much more motivated to attend technical seminars which are not in your area of research with actually a positive intention of getting good ideas right so this is an area where you know this is something that you know people should do if they realize that exposing yourself to areas other than your own actually will improve your chances of doing good research in your own area okay so I think we are getting delayed for lunch we can take questions later also