 Now in this session, we will begin with the first major skill required for research, that is thinking. So we will discuss aspects of thinking such as problem solving, critical thinking and creativity. So in this lecture, you will get answers to questions like, is thinking skill in bond or can I develop it? In how many different ways can I think and how can I develop my thinking and generate ideas? So the last part is the prescription for developing thinking. Let us take the first question, is thinking skill in bond or can I develop it? Answer to this question is crucial because if thinking skill is entirely in bond, then there is not much point in talking about it in detail and teaching because it is not going to serve any purpose. So only if there is some significant aspect of this thinking which can be developed by practice, skill is something that you can develop by practice. So unless there is an element in it which can be developed by practice, it is not worth discussing about it in great detail. So let us see what research has to say on this question. So let us take the highest thinking skill that is creativity. People have done research on this particular topic in last 50 years, in psychology. So these are some of the conclusions from that research. So first one is a statement by Peter Medever. He has not done formal research on thinking process, but this is the conclusion from his own experience. Creativity is beyond analysis. This statement is a romantic illusion. We must now outgrow. That is what he says. The next two statements are conclusions of psychological research. Creativity is a skill which can be developed by practice. Conscious application is needed, not the vagaries of inspiration in order to achieve a creative output. So it is wrong to think that these flashes of brilliance or inspiration, they come without any control to a person. It is true that there is some amount of uncertainty on the timing of these flashes. But apart from that, there is a certain kind of preparation that a mind should have for these flashes and this preparation can be done by practice. Look at the last statement. Creativity is a matter of organizing one's basic skills, not regretting that one was not born with a quick or logical mind. So, since conclusions of research are encouraging, it is worth looking at this particular aspect of thinking in detail. So that we may learn some lessons on how we can develop our own thinking. So what is crucial? It has been recognized for high level of thinking is motivation. This again a conclusion of a psychologist by name Cox, who studied the biographies and personal interactions with 301 geniuses. So he has written the conclusion as follows. Motivation is recognized as a crucial factor in the development of talent and the application of creativity. The importance of intrinsic motivation in driving an individual to practice and work hard to master a specific domain is undisputed. Cox in his study of 301 geniuses wrote, high but not the highest intelligence combined with the greatest degrees of persistence will achieve greater eminence than the highest degree of intelligence with somewhat less persistence. So you look at this conclusion carefully. It is an example of conclusion of a scientific study. Here all the words are read very carefully. He says high but not the highest intelligence combined with the greatest degrees of persistence will achieve greater eminence than highest degree of intelligence with somewhat less persistence. That is a result of a study. So what he is saying is if you study a large number of people who are characterized as genius then it is found that they do not necessarily possess the highest intelligence. They do have a certain level of intelligence but not the highest. But what they have is a very high level of persistence. That is what this statement is saying. So what is the difference between a genius and an average person or a famous scientist and the average scientist? It is not in the level of intelligence. It is not that much in the level of intelligence. The difference lies in the level of persistence, ability to concentrate on one thing for a very long time in spite of any difficulty. So this motivation is crucial for any significant achievement. Let us take up the next question in how many different ways can I think? So we will discuss this with several examples. In this context a quotation of Aurovindo is of great significance. What he has said is education is not about learning diverse subjects. Education is not about learning diverse subjects but about learning diverse ways to the same subject. It is about learning diverse ways to the same subject. It is not about learning diverse subjects. Unfortunately, a lot of our education is about learning diverse subjects. And that is one of the reasons why Aurovindo at least says that people are not creative to the same extent as we would like even after going through an educational process because it focuses on diverse subjects. So in our syllabus we give emphasis to different topics that need to be covered. We don't spend time or discuss at length how the same thing can be done in several different ways. So an example of that would be let us say we are teaching Pythagoras theorem. Now we can teach one or two ways of proving which is what is normally done. How do you prove Pythagoras theorem? And then we move on to the next theorem. Why? Because it is important for us to discuss several theorems in the course. That is what the curriculum says. But if you want to develop creativity or develop higher levels of thinking, it will be more appropriate to discuss ten different ways of proving Pythagoras theorem. Rather than discussing ten different theorems, it will be more useful to discuss ten different ways of proving Pythagoras theorem. That is what is the meaning of this kind of this particular statement. So this point we will develop in further detail. So let us see how many different ways we can do things. So to start with, let us look at the levels of thinking. The lowest level is knowing our memory. I have said this in introduction, I'm repeating it here. Next higher level comes comprehension or understanding. Next higher level is application or problem solving. Then evaluation or critical thinking and finally creativity. To appreciate these various levels of thinking, we will now do an activity. So I will give you a simple looking problem, which you can attempt. And I will give you ten minutes time. It is quite possible that within the ten minutes you may not get a solution. But at the end of ten minutes, what we are going to discuss is, how different people are approaching this problem. So I will call upon a few students to come here and just talk about what is the way they are approaching this problem. And it will be a very interesting study. We will see how different people approach a problem in different ways. The problem is the following. Problem is make five squares of equal size out of a single large square. You are allowed to cut and paste. So imagine that you are given a square piece of paper and you are given a scissor. Nothing else, a square piece of paper and a scissor. And you are asked to divide this square piece of paper into five equal squares. So you are dividing the square piece of paper into five parts. And all the five parts should be equal and each of the parts should be a square. Please understand this very carefully. You have to divide a square piece of paper into five parts. All the five parts should be equal in size and all the five parts should be square. So I am repeating the same question in different ways because people have doubts. They think that the question is divide one paper into five equal parts. That is not the question. The given paper is a square paper. And you have to divide this into five equal squares. Five equal squares, all parts should be square in size. So I will give you ten minutes time. Okay, just try it. Your question was, is it minimum five or you can divide into more parts? Yes, you divide into as many parts as possible but ultimately you must join the parts such that you get five equal parts. So you should get five equal parts but you can divide into as many parts. So for example, if you think you have a strategy where you will divide into 100 some parts and then you will join 20 parts into one square, another 20 parts into another square. And so that ultimately you will get five squares, right? That is what is meaning of the statement, you're allowed to cut and paste. You cut into as many parts as possible and you can paste different sizes of the paper and so on, okay? So let me show this diagrammatically before you start. So this is your square piece of paper. What I want is, ultimately you should have five papers like this. So you can see each of these five papers is a square. And if I add up the areas of these five papers, I should get the area of the bigger square, okay? So that is the question. You can divide this paper into as many parts as possible and if you want then you can go on joining and then make one square. Then make another square like that, you are allowed. I'll just think about it. Here I actually assume square as 5A by 5A. So these are 5U parts. So each part has length A. So its area is 25A square, okay? So here, we want to show that each side is 5A. You can just show on the diagram. 5A, 5A, right. So here we want 5 squares having areas each of 5A square. So finally we want a square having sides root 5A by root 5A. Okay. Now there is a method to get this length root 5. I will just show it. Yeah. If we plot a distance 5 from this point, let's say this is 5U. And if we take, okay, let's say 5A, then if we take a more distance of A from this end point and if we take the center of this length 6A, let's say this A is BC. And if you take a center of AC as O, and if you plot a half circle having O as a center, then if you draw a vertical from B, which will cut the half circle at, let us say D, then BD is length BD will be equal to root 5A because length OD will be 3A, length OB will be 2A because it is center of length AC, which is 6A. So 9A square minus 4A square, which is 5A square. So DB would be root 5A. So in this way, in this figure, I can have a distance of 3A from this point and... The 3A will not be in the middle. Sorry, sorry. Okay. Doesn't matter. Okay. You show it a little bit up. Okay. Yeah. It will be 3A from here. So... You show that 3A. That distance is 3A. Okay. So this is 3A. I can draw a half circle here and where this length would be then root 5A as per BD, length BD. Okay. Okay. Then now I got the length root 5A. So I will cut this square into five parts. Now let us take a part having length 5A by A. Now I know the length root 5A. So I will cut means I will take a length from here, let us say A, A to B. AB is root 5A. Then I know also a length AC which is root 5A where C is a point. Then I will try to fit this, then I will cut this part and I will try to fit it here. Let us see how you do that. It will come like this way. Then I will again cut this part. No, it won't come like that. You see your area, this appears much bigger than this. Actually I have drawn a wrong figure. Okay. So let us draw it to scale because when you draw it to scale you will know where is the difficulty. This is root 5A AB. Then put this part here. So actually incidentally your root 5A, will it be more or less than half of 5A? More than half of 5 is 2.5, so 2.5 square is 6.25. So it will be less than half. So let us show it as less than half. Right. Okay. It will be here. Okay. In fact as we will discuss in problem solving, drawing diagram to scale is an important part of solving problems. We will discuss it later. So length AB is root 5A, then I know A part length C which is equal to AB here. Right. Then I will try to fit the... So that is the now the important question. How will you fit the remaining part? It is true that in terms of area it is alright but how will you get a square? Here I will have to do a lots of cutting. So after n number of cuttings and again... Yes, the question is by doing any number of cuttings will you be able to get? It is quite correct on your part that you need a square of side root 5A. Point number one that is your right. You also shown how you can get root 5 geometrically. That part also is good. It is correct. But now how do you apply this information to the given situation and actually get 5 squares. That is where there seems to be some difficulty. Yeah. Right. At that stage. I will have to have infinite cuttings and again... But moment you say infinite then it is not a practical solution. It doesn't matter. You have come up to some point. Right. Let us see. Let us have one more example. That will tell us how different people are thinking. Then we will discuss the various levels of thinking. Keep it like that. Basically our output should be a square of root 5A each, root 5 each. Right. Okay. This is say 5 and my output will be 5 each. So what I have done is basically, first method is basically I thought of divide into four equal parts. Right. Okay. And this square is actually of 2.5 each. My this thing is actually root 5. So approximately you can... Again if you draw it to scale, you should show less than half of this. Oh. Okay. Then this is fine. Now it is to scale. So basically if I cut this, I get four equal parts of this. So approximately I have... No. Cut which one? No. Basically I need to approximately cut it. Not exactly. Okay. So this length is actually 2.5 minus root 5. Okay. So this is not actually exactly correct. So I have this approximately root 5 and the rest and I have four parts of this kind. Right. So you have... To manage by cutting it and... Yeah. Now that is the question. Now, can you... A remaining part in terms of area it is correct, but can you still get a square by joining the remaining part? Yeah. That again has to be by cutting many parts. Hmm. And one more thing I have done is... Yeah. So one of the things that you gather from here is if you try to cut parallel to the sides in any number of different ways where the cuts are parallel to the sides, there seems to be a difficulty. Okay. You can gather that, you know, you are going up to a point and then getting stuck. Let us see your other approach. Basically in this what it does, I have this is square, I divided into five parts for grid size of 5, 5 cross 5. Okay. So basically I need a size of root 5. Yes. So if I look at it here, this is my root 5. This is your? Root 5. This part is root 5. Can you just explain how it is root 5? Okay. This is one. This part is actually root 5. This is root 5. This is two and this is one. Okay. Good. So this is root 5. Okay. So if I extend it, this is not actually looking like a square, but if you extend it this way, this is actually a square again, this is square, and I got two squares of root 5 each. Okay. Root 5, root 5. Okay. And the remaining part is this. What I do is, I extend again this part, this is, I am going to do this one, okay. This is not exactly coming out, this is my direct spot. So the remaining part also, I actually, I have done it better in my business, but. Fine. You want to take a look at what you have done and just put it back? No, the diagram is not actually proper. Yes. I am not actually getting it to draw, redraw it properly. Okay. So today it was basically, I got squares of size root 5, root 5, root 5 again. Okay. And the remaining is approximately somewhat close to that, but basically I have triangles left out. So you can manage out and put them back in the remaining, I can actually get them somewhere near. Yes. So you have to put the triangles together and then get a square. Yeah. Can you get a square only from triangles? Yeah. So this is a new approach where you are trying to get, now you abandon dividing into smaller squares and then getting squares out of the squares. Instead of that, now you are going to a triangle. It's a change in paradigm, right, change in way of looking at it and you seem to be getting closer and closer. Okay. Basically I have used triangles to construct a square. Okay. I took a square. I joined the midpoints of each of the sides. Okay. Then again I joined the midpoints of the inner square that is formed. Okay. If I cut it out on each triangle, this triangle, this triangle, it will construct a square. One square will be formed by this, other will be by this. The third one will be by this. Fourth one and this one. So this is an interesting approach where you are getting five squares. Now question is are they of equal size? Actually three, three are, right, so that is where there is a problem, okay. So size, that is why I said, you know, each of these conditions should be satisfied. Okay. But you are getting different squares, but not of equal size. Okay. Good. For the square which I need, my target square, the five such squares, so each side will be root five as one of our friends did. So now these are the four squares which I get. Okay. Okay. Of equal size. Now I have these strips remaining. And that corner strip also. And this one also. Okay. So now the dimensions of these are, this root five is 2.23. Whatever, size, okay. And this is about a 0.528. It is five minus two root five. Yes. 0.528 and 0.528. Okay. Now if I take out all this and place it one above the other. This is my area. So one side you will get root five, that is clear. Yeah. But what about the other side? This is root five. And this is four into 0.528, that is 2.111, okay. So now this strip has to be cut into few parts. So as to be added over here. That is the now issue. So how will you do that? So again this is a square so I can cut it. Yes. Any number of parts you cut? No. So if I take this extra dimension as A. Okay. Or whatever A. So A into this length, that is root five. That is 2.236. Okay. So A into 2.236 is equal to root five square. Fine. You can evaluate A from there. So yeah. How will you get that in practice? This is the question from that because ultimately in practice you have to basically cut this square into pieces and then join those pieces so that you get this strip. Yes. Now that is the question. Is it possible? Yeah. In terms of area it looks okay. Yeah. But practically you should show how you will cut. A is 0.125. So how will you get, so it boils down to now your problem boils down to you have a square piece like this and you should get a strip, thin strip, area of this thin strip is same as this. That is fine. Question is how do you cut this in any number of parts and join them so that you get this strip? We can cut with the side of 0.125. Yeah. No. No. But how will you cut this so that you will get one side 0.125 and then all other things you are joining them together so that you get a root five on the other side? Why not? Yeah. Think about it. Is it possible? I think that should be possible. Well, I mean one may say look I will stretch so that I will go on stretching until one side reduces but then you cannot stretch paper, right? In principle one can use such methods but that is not possible here. So here the method is you have to cut and paste that is the only way. So is it possible to cut a square piece and join so that you get a strip like this? This becomes another problem. Like dividing a square piece of paper into five equal parts. So you have to see whether it has a solution at all. This problem I am not saying the original problem. Okay. Yeah. Okay. Think about it. Sir, I will start with rectangles first. Okay. Suppose we have a rectangle like this one would like to see how we can form a square out of this. Okay. How do you form a square from a rectangle? No. A partial square with a hole. This is rectangle with a hole. Okay. Sorry. Square with a hole. Yes. These are filled areas. Okay. And this is the hole. Suppose this is of breadth A and length L. So the total area of the occupied spaces will be four into A L. The shaded area. Oh, sorry. Yes. The shaded area is four into A L. Four. This is the area. And the area which is empty is L minus A square. Okay. So now what we need is can we, we need to construct. We need suppose there are five such parts like this. Okay. So this will be 20 A L plus five into L minus A whole square. Okay. This we want it to be A square of this. This you want to be? A square of certain area, N square. Oh, N square. Okay. N square. So if we can find out the length A L, suppose if we can find it in terms of integer Yeah, but this is nothing on the left hand side. It's nothing but this is A. Yes. And this is L. So your total area is nothing but L plus A square, total area if you take. Now if you add these two, that's what you are doing. And then you are multiplying it by five. Five. So this is nothing but five into L plus A square. Yes. So you are saying five into L plus A square is equal to N square. Yes. So what do you get from here? No, from here we need to find lengths A and L. We need to cut this into strips of length A and L to make this piece. Okay. So how are you getting, so basically you are having five such pieces is it? Yes sir. We want to make five such things. We need to cut. No, you want to make five such things. Yes. Out of this given single square. So which means for constructing each square you will need four such rectangles and a piece inside. Yes. So we need five small squares and then 20 rectangles. Right. So how will you get those? We need solution of this. Yeah, you may get a solution. Solution means practical solution. You should show how it can be cut and see whether it is possible to cut. See that. I don't know how it can be done. You have to try it. Okay. Yeah. We take the midpoints of each of these sites. Okay. And connect the vertices this way. Okay. This is first one. This is another line. Third line by symmetry again. Okay. Fourth line this way. So now this will be one square. Let's say let's call this as A. B, C, D. You can show parts. Now you can number the parts maybe. And then it will be easy for you. Okay. Let's call it as P1, P2, P3, P4 for the inner square. Hmm. Now A, P1, B. A, P1, B. This area. Hmm. Is repeated four times. Correct. So we just need to show that this can be made into a square. Correct. Now since these two lines are parallel. Hmm. And this divided by this, the ratio is the same for, this is one is to two. Hmm. This whole thing, this unit divided by the total length is one is to two. Correct. Therefore this distance will be half of this. Okay. And since this is square, this is 90 degrees. Hmm. This whole thing, this inner, the small triangle can be rotated about this point. Hmm. We get a square like this. Hmm. Similarly it can be done for the other four parts. Okay. So that is. Right. Now you have to show that all these are equal parts. You have got five squares. Okay. At least these, these four outer parts will be equal. Hmm. Four outer parts are equal. That's very clear. Yes. Let's take this triangle A. Let's call this point as let's say X. A, D. There are two lines coming out like this. If you concentrate only on this triangle. Okay. Yes. D, P, C. Right. So let us draw that triangle separately. Yes. So you have a triangle. So this, this will be half of this. Hmm. But this part is here also. Correct. So this whole length can be written as let's call this as some A or something. Okay. So this is two A. Hmm. And now if you concentrate on this triangle. Hmm. It's same as that triangle. Whatever. All these four triangles you have said. Yeah. Are same. Okay. You are, you're almost, you're almost there. What remains is to prove that all parts are now equal. Yeah. Four parts are equal to each other. That is clear. You should show that the fifth part also is equal to the other four. So what remains to be shown is that the square obtained by this. Is the same as. Is the same as the square obtained from any one of the triangles by putting this triangle on this side. Yeah. That is all what remains to be shown. Okay. So. I need to think. Yes. You can think about it. That's fine. You take your time. Right. Because as I said here it is not necessary to get to the solution. But the approach is important. Right. So I mean he has got the solution. But he has to prove it. Yeah. Secondly I said this is my pi cross five square. Correct. This is my pi cross five square. Correct. Okay. So I divided into lines of root five each. Yes. So this is one square I got. Hmm. These are my two squares. Correct. And these are the three parts. Good. This is the third part. Hmm. One, two and three. Hmm. This A, B and C. Okay. Are three which I have got not perfect squares. Fine. So this is root five root five. Okay. Hmm. And this is root five and two and one. So I take this square and adjust it here. Yes. And this one here. Hmm. Now I have to adjust this remaining part. Hmm. In these two squares. Hmm. So one can repeat that. Yes. So I extend this line cut this way. Hmm. I have root five and this can be bought here. Okay. So I can actually bring this area here. Okay. I have this area remaining. Hmm. I can actually adjust it here in this small part. Right. So this area can be put it here. Okay. This is my five squares. One, two, three, four. Good. That's another solution. Where you divide into triangles and squares and then you join the triangles and the squares to get other squares. Okay. So suppose we have this length root five. We got this length root five. Root five, root five. Similarly here root five, root five. Right. So I just draw this length. Now incidentally how did you get, oh I see. Suppose we get this root five. I mean somehow you are getting it. Yeah. Root five by construction. Yeah. But later on you have to explain how will you actually put it on a square piece right. That also you have to explain. Because you have a square and a Caesar. Okay. So how will you measure but let us see now. Supposing you have that. So this root five root five means we have two squares and now this middle area is remaining. Correct. So here also again we measure the length root five, root five, root five and root five. Okay. So we got these four strips and this center square. This area is five minus two root five. Similarly this is five minus two root five. Not area, the side length. Sorry. Length. And these are four strips. Okay. Again this square five minus two root five. Square. Each length is five minus two root five. Right. So we divide into four strips. Okay. It will be five minus two root five divided by four. No, five minus two root five is length. Yeah. So this length, I mean this middle square. Right. You divide. Five minus two root five. So we divide it into four strips. Right. So width of the strip is five minus two root five by four. Five by four. Okay. Now we join this strip to each square. I mean to this each rectangle. How? So each rectangle is five minus two root five. This was root five and this was, this is five minus two root five. Okay. So we, we just joined this strip here. Two one. Then two. If you join like that, then one side will become more than root five. Yeah. Yeah. It will be five plus five minus two root five divided by four. Which will be nothing but five plus two root five divided by four. Now we have four such strips. So multiply by four and their breadth is five minus two root five. So this will be cancelled. This is five by 25 minus two into this is 20. So this is five. Area is five. No. Area is five. But, but how are you getting the square? So, so this strip is attached here. Okay. So to next strip, again this strip is attached here. Third strip, this strip is attached here. Fourth strip, this strip is attached here. So if you do that, now the length of this side is more than root five. And length of this side will obviously be less than root five. No, this will be two, five minus two root five. This side? Yeah. So five minus two root five multiplied by four. That is, so now this side and this side will not be equal. So it won't be a square. I mean you are getting again five parts. But they are not square sides. Yeah. Five minus two root five and five plus five. That's what I'm saying. Five plus two root five. Okay. Now the thing is here also, the point is in the beginning you started with triangles, but then finally again you got into the thing of dividing parallel to the sides. There again you will get a problem. You'll have a problem. Okay. So after this activity, let us look at these levels of thinking. Okay. So definitely as we have seen, if you want to solve this problem, then you have to use Pythagoras theorem because that tells you how geometrically you can realize square root of five in this case. Supposing you teach Pythagoras theorem to people and then you frame a question. You teach Pythagoras theorem meaning you teach what is the statement of the theorem and how it can be proved. And then you set a question in the exam saying state and proof Pythagoras theorem, which is what we encounter in our school. Now what is the question testing? What thinking ability does the question test? It tests the lowest level of thinking. Framing a teaching students the statement and proof of Pythagoras theorem and then asking them a question, state and proof Pythagoras theorem. It only tests the lowest level of thinking that is memory. In fact, an analysis of various questions in the examination shows most of them fall into the category of questions which test only memory. Suppose we modify this question. We say state and proof Pythagoras theorem and explain its significance. Explain the significance of Pythagoras theorem. Now the student has to write a little bit more. For instance, one significance of Pythagoras theorem is that you can get the square root of any integer by a geometrical means. That is one of the significance of Pythagoras theorem. By application you can get. So this question tests slightly higher level of thinking. Now let us look at another question. That is divide a square piece of paper into five equal squares by applying Pythagoras theorem. Supposing you frame the question like this which is different from what I have said. I didn't tell you that you must use Pythagoras theorem. In that case, even in that case you came up to a point where you realized you should use Pythagoras theorem. You did that but I didn't give you the hint in the beginning. Supposing you frame the question like this that applying Pythagoras theorem divide a square piece of paper into five equal squares. Now this is a relatively difficult problem. But still a hint has been given, a starting point has been given and an end point has been told. This particular question tests application or problem solving ability or a higher level of thinking than knowing or comprehension or memory and understanding. Suppose I framed the question differently the way I had done it in the beginning. I do not give you the hint that you must use Pythagoras theorem which is what I have done. So I said that divide a square piece of paper into five equal squares and I didn't give you any starting point. Now it is an even more difficult problem. The level of thinking that this problem tests is evaluation or critical thinking. Now what is critical thinking? Means looking at an issue or a problem from various angles then evaluating each particular angle to see whether it leads due to the solution. And if the evaluation tells you that it will not lead you to the solution then you discard that approach or angle and then take another approach, different angle. So here you are attempting multiple ways to solve the problem. This is the characteristic of critical thinking. Because you do not know, you are not given the starting point. So you are trying to look at things from different angles and try to see whether you know grappling, whether you can get to the solution from this side or from that side. So this kind of thinking where you approach an issue from various angles and evaluate each particular angle or point of view to see whether it leads you to the solution and if it doesn't you discard and choose another approach. This is called critical thinking. And even higher level of thinking is creativity. So in the present context, an example of creativity would be to think of a problem like this and give it to the students and see how far they get to test their various levels of or thinking or ability of thinking. Thinking of a problem like this, right? You ask people to divide a square piece of paper into five equal parts and then see how far do people get. So give them some time. Then after some time give them a hint. You use Pythagoras theorem if you are not getting. Then you see how many more students get to the solution. Then you give further hints like this. So thinking of a problem can be considered an exercise in creativity. It is creativity. So this is how these are the various levels of thinking. In research we have to operate at the three higher levels. That is either application or problem solving or evaluation and critical thinking or creativity. The lowest level knowing our memory and understanding are not sufficient. This point is very important to note because until we have undertaken research our entire examination system and evaluation system is such that it tests mostly memory at most understanding and very rarely application. Very rarely it tests application. I do know that in IIT we do test application. The question papers in IIT do test some level of application. An example of an examination which tests application and to some extent critical thinking and application is the John Tenton's example of IITS. This ability to apply. So unless we are aware of these various levels of thinking we will not be able to know how we should prepare ourselves in our research to do better work. Because we are not geared towards application or critical thinking or creativity. So if you understand this then you will be able to develop these kinds of thinking much better. It does happen that even without knowing these words problem solving, critical thinking, creativity just by going through the process with your guide you may be developing these abilities without your knowledge. But the extent to which this development occurs can be speeded up and the level to which this development occurs can be increased if you are really aware that these are the various levels. So for example supposing I want to evaluate somebody else's research work we have said earlier that what is the achievement of a PhD scholar at the end of going through research. One of the things that a PhD scholar should be able to do is to evaluate others' research. Now how will you be able to evaluate others' research? Basically you have to see in that work how much of problem solving or application is there how much critical thinking is there and how much creativity is there. In a paper if any of you have read some papers can you identify sections in the paper which are related to critical thinking for example where the critical thinking done by the author comes out. Someone can tell me if you have read papers. Results and discussion. Very good. Results and discussion. Supposing you are doing some experimental work you will start with introduction then you will present the apparatus and so on then you will present the methodology used and then you will come to discussion. Methodology used and you will also present the data measurement or whatever and then you will come to results and discussion. This results and discussion section is the one which shows what critical thinking the student has done or the author has done. So by looking at the results and discussion section of the different papers if you are asked to compare you can try to judge the level of research that has been done. So that is one aspect that you can see. There can be creativity for example in design of an apparatus. Someone might have designed an experiment, a new experiment to show something. So an awareness of these levels of thinking also enables you to focus on the strengths of your work when you write a paper. Essentially what you should bring out in your paper or in your technical work or your thesis is application, critical thinking and creativity. So therefore in case you write your thesis and it has a very long review and a very short results and discussion section it will have a negative impact. Even though the thesis may look thick the person who knows how to evaluate research will first try to see the results and discussion section. Let us see what is the person, what is it that the person has done. So awareness of these various levels of thinking is the first important thing in developing our own thinking. Let us look at some of the other words which people have used to describe thinking. So styles of thinking, reproductive thinking means what based on similar problems encountered in the past are taught to solve. Again most of our examination questions and evaluation test reproductive thinking. So something is taught in the class a few examples are given, some tutorials are given and then in the exam you make small modifications of these problems and then see whether the student can solve. So mostly it is reproductive thinking to some extent it may be application level. As against reproductive thinking what is required in research is productive thinking. So productive thinking means generate as many alternative approaches as possible. Person is said to be a productive thinker if he or she can generate as many different approaches as possible. So whenever you look at a problem immediately if in your mind several different approaches come then you can say that you have an ability to think creatively. All these approaches may or may not be correct but the point is at least your mind is able to think of different approaches. Then you may discard one by one if it doesn't work. This ability to generate different approaches is one of the important abilities that a good thinker should have. Then people have always also used words such as fluency. So fluent thinking, what does it mean? Generate large quantity of ideas. A person is said to be a fluent thinker if you can generate large quantity of ideas. Then flexible thinking, what is the meaning of flexible thinking? Flexible thinking is one which goes beyond the ordinary and conventional nature of things. So a good researcher should be a fluent and a flexible thinker. He or she must be a productive thinker. So now let us come to more practical level. Let us see what are the various ways in which problems are solved. We have considered one example of a problem and then we have seen how we approach the solution. Now here we are going to discuss various strategies. These strategies are fairly general in nature. In other words they can be applied to any problem. However it is true that for certain problems certain strategies may give you solutions quickly or they may give you more effective solutions. But first we must be aware of what are the various strategies that one can adopt to solve problems. So here I have listed some of the very commonly used strategies. When I say commonly used, let me maybe correct my statement a little bit. These are strategies which if I utter these words you will say you know what this means. But as I will show you, surprisingly we do not apply these strategies in a given problem because we are not aware that this strategy can be applied and it will lead to a solution. So these strategies are representation. So in a given situation you are given a problem you draw a diagram to represent the information that is available. And from that diagram you try to see how you can approach the solution. By reasoning, another way we solve problems is by reasoning, logical reasoning. Third approach is division into sub-problems. So you have a problem, what you do is you divide into smaller problems and you tackle the smaller problem and then you combine your results that you obtained by tackling each small problem to solve the bigger problem. Stretching to the extreme or limiting approaches, using limiting approaches and then trying to see whether solution exists for them. We will consider examples to illustrate these points. Then verbalization. That is trying to solve a problem by talking about it. It's a very interesting strategy of solving problems. It tells you a psychology of thinking, how we think. If you want to solve a problem, you talk about the problem. After of course you have done some thinking. For example one of our students here came up to give a solution and then as he was explaining suddenly discovered it won't work. The same thing, same discovery didn't happen when the person was thinking for himself. So when you are talking about the problem, then you find a solution. It is a very important aspect of the way our mind works that you should know. That is why research scholars are encouraged to give talks or seminars periodically to their colleagues or to the supervisors. You can solve your problems. You can come up with ideas by talking about them. Provided some amount of thinking has gone into the problem. First a certain amount of thinking should go into the problem. And then you start talking about it. What is interesting about this verbalization strategy of solving problems is the audience may or may not understand what you are talking. The audience, the person sitting in front of you may or may not understand what you are talking. But you will get a solution. You will realize I have got the solution. Something that I didn't have earlier as a result of my talking about the problem. So it's an interesting aspect of thinking that action stimulates thinking. Action, a physical action. That is why you will find sometimes when people are not getting solutions they become restless and start walking. Because you feel that if you walk then your thinking may be speeded up and it may open up and so on. This is another example where you are trying to stimulate thinking by action. So of these various approaches, verbalization is a very powerful stimulant for stimulating your thinking. You can try this approach and see. I have myself got solutions many times while discussing with my student. Or discussing with someone. The student doesn't understand what I am discussing. But I get a solution. And I have seen students getting solutions even when I don't understand what the other person is saying. So I will discuss with some examples the strategies namely representation, logical thinking, division into sub-problems and stretching to the extreme. So verbalization, we have already seen an example in practice.