 So, let us quickly recapitulate what we did in the previous session, we discussed about various levels of thinking namely memory, application, critical thinking and creativity. Application level thinking is also called problem solving and memory level thinking is also called knowing, sometimes we use the word know, to know is the same as to recall or to use memory. Similarly, critical thinking also is sometimes loosely considered equivalent to evaluation. Then we started off a discussion on, we also give a definition of creativity we said it is an ability to look at the same thing as everyone else and think something different. We gave some examples to illustrate this definition. So, now we are looking at various ways we can think. So, we saw one method of thinking or problem solving in which we change the statement of the problem to simplify or facilitate the solution. Next, we want to discuss graphical and tabular representation. So, in this context we introduce this particular problem where we say that we are measuring the adhesion between the plating and the substrate on which the plating is being applied as a function of different areas and different substrate conditions. From this data we want to derive pattern if any that exists. Does adhesion change in a systematic fashion with area? Does adhesion change in any systematic fashion in terms of the conductivity? Both polarity as well as the magnitude of conductivity. So, now suppose a student has made measurements. So, we say that we said that each measurement is repeated twice. Let us assume in fact in your practical if you are doing a research you may do it repeat the measurement may be four times or even five times. Exactly how many times should you repeat a measurement to reduce the error is something that we are going to discuss later in experimental skills. So, right now we will assume that each measurement is repeated twice. So, now how many data points do we have? So, we have three different areas 0.51 and 2 centimeter square and we have four different substrate conditions. So, four into three twelve and each measurement is repeated twice. So, you have 24 different values and out of this 24 different values you want to derive a pattern. Now, interesting as it may seem I have not found many students using a table to represent the data. So, normally what students would do in this situation is as follows. So, typically a student would come suppose the student is asked to make the measurements and come up with the report the data. So, you would do like this. So, you would write P plus substrate and 0.5 centimeter square area and then you underline it and then you say adhesion equal to so much you know say it is expressed in Newton per meter square. I am getting a little bit technical, but does not matter actually what matters is the value that comes here. You use a particular unit and all values are expressed in the same units all adhesions and then you say P type 0.5 centimeter square and you report it here and so on. So, typically a student I have seen in practice what he does he or she does is come up with 2 or 3 pages of you know write up like this which has all the values and then from here to actually see a pattern is not so easy. So, it does not strike to many students that the first thing that you should do is to report this in a tabular form. So, let us see one form of table could be this. This is a simplest form of table that one can think of it is called list form of table where you list out the various values. So, here you see the first column is the substrate condition called as doping as I have already told you it does not matter whether you understand the meaning of the word doping or not it you can take it as equivalent of substrate condition. So, you have 4 different substrate conditions for each substrate condition you have 3 different areas. So, that is what is listed here then for each substrate condition in area you are repeating the measurement twice. So, you are doing 2 experiments experiment 1 and experiment 2. Now, this is how you have the readings reported in some case let us say you could not obtain a reading for some reason. So, it is reported as dash here. Now, you can look at these numbers and try to ascertain the pattern. For example, if I want to see whether as a function of the substrate condition there is any change in the adhesion systematic change in the adhesion then I must pick the same area for all the substrates because the other variable of the area I should maintain constant. So, what does this mean? This means that I would take this entry p plus substrate 0.5 centimeter square area for example and look at these 2 readings and then I should write actually an average of these 2 something here in the column. Then I should go to the next type of substrate namely p type and I should pick the 0.5 centimeter square area and here I should see the entries take the average of these 2 write them out here. So, I should pick the 0.5 centimeter square values for the 4 different substrates. Now, that alone would not suffice because I should repeat this for 1 centimeter square area and so on. So, you see there are too many numbers here and I should pick out only a few of them out of this. It is not very easy to see the pattern I will have to prepare a separate table in which I just write down these entries next to each other. So, that I can quickly see a pattern, but still putting the values in this form of table is much better than simply reporting the values as I showed you earlier something that you know many students would do. So, you write it out one after the other in this without organizing the data, but there is even better form of representation which is also a table. Now, this is called a matrix form. I could put the data in this form. So, what have I done here? Here I have chosen the variable area and this is written out horizontally the variation in area this is horizontally written out. So, this row indicates the different areas and the column here is the different substrate conditions. Now, each entry in this table has the area above it and the substrate condition towards its left. So, if I take this particular adhesion 10 Newton per meter square actually not 10 Newton it is 10 into 10 power 6 Newton per meter square. So, this adhesion corresponds to an area of 0.5 centimeter square and a substrate condition of p plus. Now, since you are repeating the experiment twice the second value is shown here adjacent to this. Now, you should you will appreciate that this form of table reveals the pattern very easily. For instance, if I want to see as a function of area is there is any pattern I can go to this particular row for p plus substrate I find that as the area of the substrate increases the adhesion falls. So, here you have 10 this is close to around 7 and then here you have about 0.5. So, you see 10 7 5.5 the adhesion is decreasing as the area of the substrate decreases. Now, let me just probably draw a diagram to show what this area means in case some of you do not know what area I am talking about. So, what we are doing is this is your substrate and on this you have a plating. So, let me shade this. So, this is the plating and this is substrate. So, if I draw a cross section here that would be something like this is the substrate and on this you have the plating and the area I am talking about is the area of this substrate in this. So, this area is what I am talking about this is 0.5 centimeter square or 1 centimeter square or 2 centimeter square. And how do I measure the adhesion I basically pull this plating and the substrate. So, I have some mechanism by which I can try to pull these two apart pull this up and pull this down and then try to see when does the plating come off from the substrate. So, in the cross section this is how it looks I am pulling this up pulling this down and trying to see where the breakage happens for what force or tension the breakage happens. So, this is the experiment that is being done here. So, that is the area that we are talking about. So, as I mentioned for this substrate condition clearly as the area increases your adhesion decreases. Then I can see for p type substrate also the same pattern the numbers decrease. So, 8 and 9 average is 8.5 then here it is about 4.5 here it is about 3.05 or something so you see it is decreasing. Now, when you come to the end type you find that actually pattern is not really. So, clear for example, here you get a value of about 4.5 here also even though the area has changed the value has not changed significantly. And in fact here you see further that the value adhesion does not seem to be changing that much. So, this is the pattern that you see for end type substrate. And for n plus you actually see no pattern at all. So, there is a lot of variation in the adhesion measured. So, we can summarize our observation as follows for p plus and p substrates the adhesion decreases as the area increases. For end type does not seem to vary much for n plus there is no pattern. Now, supposing I want to see how the substrate condition is affecting the area. So, I pick let us say 0.5 centimeter square I find that for p plus the adhesion is 10 for p it is 8 for n it is 4.1. So, there seems to be a pattern that as the conductivity changes from p type or positively charged carriers to end type negatively charged carriers the adhesion seems to be decreasing. So, this way if I move vertically I can get one pattern as a function of substrate condition. If I move horizontally I get another pattern that is as a function of area something that was not. So, very evident in the previous form of table that is a list form. Now, there is yet another method of representing this and that is the graphical form. So, the same data has been represented graphically here. So, what is this graph on the vertical axis you show the adhesion in the horizontal axis you have shown the different substrate conditions and for each substrate condition you are making two measurements. So, the two measurements are shown as the end points of this vertical line and these two points are joined by a segment and the length of this segment depicts the variation that is occurring in two different measurements for the same condition. So, you see it gives you a very good feel of the amount of variation in the two measurements for the same condition. So, simply by looking at the length of this vertical segment you can see whether there is a lot of variation or not. This was not very evident in the previous two forms of representation where you have to actually take the two numbers and subtract one from the other to get a feel for what is the difference between the two numbers whereas here the length of the segment this pictorial depiction is much more intuitively satisfying and pick to grasp. So, here when you look at the diagram actually many patterns that we talked about are evident much more quickly for example for p plus substrate you see that the adhesion decreases as the area increases. So, here the three different vertical lines correspond to three different areas in each of these cases similarly for p type adhesion decreases for n type you see adhesion does not seem to be varying much and for n plus you really see no pattern at all there is too much variation between two measurements for the same condition. So, you see graphical form of depiction is always the most powerful this is true in general the important point is to learn what kind of graphs can you draw to represent the data. So, there are many different ways in which graphs can be drawn you can conceive of your own ways of represented data in a graphical form. But please remember that a good researcher whenever he or she collects data particularly large amounts of data they the data should be represented in some table or graphical form preferred graphical form if you cannot think of a graph at least a table this is very very important you must organize the data. So, that the patterns are revealed easily since the graphical form is a very powerful method of representing information let us dwell on this little bit more detail let me take up another problem. So, as far as possible I am choosing problems that can be understood readily by a varied audience and also they are from a variety of different situations. So, that you do not think that creativity or various ways of thinking are restricted to any one particular type of activity like science or engineering or arts because sometimes people feel creativity is only associated with arts it is not with engineering or sciences that is not true creativity is the ability to see the same thing as everyone else and think something different. So, this definition can be applied to any different area all different areas of human endeavour ok. So, let us take this problem how good is a particular classroom teaching learning process. So, let me describe the situation suppose you are asked to look at two classrooms where teaching and learning is happening let us assume that they are side by side and both classrooms have approximately the same number of students and approximately at the same level of intelligence some of these assumptions are important to understand whatever we are talking about and let us say the same topic is being taught in both the classrooms, but the teachers are different ok. Now, you are asked to observe from outside what all is happening in the class and to come up with a conclusion or judgment as to in which classroom the teaching and learning was happening better ok. Now, what would you do? So, typically what you would do is you would observe the activities that are going on and then you will say that if these activities are going on more often in a particular classroom then I would think that you know this classroom the teaching and learning is happening better. On the other hand if you see some other activities happening more often then you would say that that activity is not desirable and therefore if that is happening more in some classroom then that classroom is probably not a good one. Now, what one can do is one can think of representing these activities in a graphical form. Now, let us see what form of graph can be used. So, here is one example I want to emphasize that each of the problems that I discuss could be solved in different ways as far as possible I am trying to hint at the ways that one can think of quickly, but if you spend your own time on this you may come up with the other methods of solving the problems. I have first enumerated the various strategies you can apply the various strategies for each of these problems and see which one works best. So, in this case I want to illustrate the graphical representation. So, what I could do for example is draw a graph like this for each classroom. Now, this graph is drawn as follows all the activities that are possible in a classroom are listed out in this vertical direction and the horizontal axis represents time. So, really this is a graph of the activities in the class as a function of time what is happening sequentially. Now, notice that the activities have been arranged in a careful fashion there is a certain pattern in the arrangement of the activities. For instance activity related to the teacher namely teacher talking and teacher using the chalkboard these activities are listed out very close to the timeline because this will happen often. Now, you cannot have a class where the teacher is not talking like teacher will talk in any class. So, this is something very common teacher using chalkboard this is also quite common. Now, the activities which are rare are placed further and further away from the timeline. Now, in particular activities of students are shown further away from the line as compared to activities of the teacher because in any class teacher is active this activity of the students on the other hand may not be of the same level of the same level. Then further the activities of the students that are rare will be listed further away. Now, student responding to a teacher's question for example is closer to the timeline as compared to student asking a question to the teacher because if the teacher ask the student a question will the student has to give some response whereas student asking a question requires some form of an initiative from the student. So, that is why this is listed out further away from the timeline. Now, similarly if you go to the other direction downward direction then various teaching aids are listed out here and the teaching aid that is used most often is close to the timeline namely the using the chalkboard and use of charts is a little bit further away use of projections is even rare. So, therefore, it is even further away and so on using multimedia that is very rare so that is listed further away. So, you arrange all the activities in this fashion and then you plot a graph. So, the graph would look something as shown in this slide. So, for example, here for some amount of time the teacher talk something then teacher threw a question to the student some student responded and it took some time to respond or other may be not time to respond, but his response was for this much duration and then after this the teacher started using chalkboard to illustrate something some student threw a question suddenly got a doubt. So, threw a question. So, that is why the graph goes up here then teacher responded to the question therefore you have this line and so on. So, teacher use some projections to illustrate some point. So, the graph comes down like this now you can prepare this kind of a graph for the two classrooms you can call them it is something like the electro cardiogram of people. So, something like that and now you place the two graph side by side it would become instantly clear which classroom was better for instance whichever classroom had a higher amplitude for this graph and higher frequency. So, more variations that classroom is better. So, frequency and amplitude of the graph can be compared for the two classrooms. Now, why would higher frequency imply better classroom because higher frequency means students are active and teachers are active changes are happening many activities are taking place. So, it is not monotonous on the other hand higher amplitude means more student activity and further student questioning initiative of the students is higher. So, if the students are getting inspired or stimulated then they would ask questions on their own and so on. So, higher level of student activity in the positive direction actually illustrates or rather implies a much better teaching learning process. So, these are example where we can use graph effectively to convey a point if the same thing you wanted to say in words you know you would say that you know I found that may be lot of students were asking questions in this classroom and the teacher was using aids other than the chalk board and so on. And even if you said in so many words the message would not come out as effectively as when presented in the form of a graph as shown here. So, the point is that it is possible to represent information in graphical form and there are very many different ways of representing information in graphical form different forms of graphs are there. So, it is the ability in a person to choose the appropriate graphical representation for the case at hand and this is one thing that all researchers should spend time on what are the all different possible graphical representations that can be used to represent data. So, that is why this is the assignment that is being given find out different ways of graphical representation of data. So, all these in fact assignments I expect you to use the internet. Another way you could do an assignment like this is we will just go to the library pick a book and then just look at all the graphs from page one to page the last page does not matter what is the area pick a book from any area and then just make a list of the various graphs that you have encountered. Now, many of these things that we are talking about will also help you to develop an appreciation for how whenever you read some material many aspects can be learned. You supposing you go to the library and you come across a new book that is being added to your library. Now, typically our attitude would be that I will see what is the title of the book it is not in my area I do not bother to see it, but no actually you can learn even if the book is not in your area you can read something from the book. For example, this is an here is one example that you can do you can look at all the figures and you can see how the data has been presented in the various figures and from there you could learn something you go to a seminar there are many things you can learn even if the seminar is not on a topic that you are working on. Now, a lot can be learned in this manner and in fact this form of learning is what is expected of a researcher to develop different ways of thinking and to come up with a novel solution to a problem. Let us move on to another thinking strategy namely logical reasoning. So, this is the problem that we will try to solve how many matches should be played on a knockout basis to decide the winner from among 10 teams. So, this is a knockout tournament and you have 10 teams now you have to arrive at a number of matches to be played so that finally a winner will be decided. Now, how could you go about solving the problem? So, one instinctive approach could be graph we just now said that graphical method is powerful you can draw a graph of the information in this situation it would be a pre form of representation for instance. So, what does this representation tell us here the 10 teams are numbered here the numbers here illustrate the 10 teams in this row these are the teams and then suppose team 1 and 2 will play a match then this particular node here the intersection represents a match. So, every node every point of intersection in this tree or in the graph represents a match. So, there is a match being played between teams 1 and 2 there is a graph there is a match being played between teams 3 and 4 and so on. Now, winner of this match and winner of the match between 3 and 4 these 2 winners will then play a match and to decide a winner from among this. So, these are 4 teams for instance 2 matches are played to decide 2 winners and then finally one more match is played to decide 1 winner from among this 4. Now, this is how this graph is constructed. So, you just count the number of nodes in this graph you will get the number of matches to be played. So, in this case I count and I find that there are 9 nodes. Now, you see here for example, that a teams 9 and 10 play a match and a winner is decided particular winner does not play a match actually he gets a buy and you know. So, he plays 1 match less that cannot be avoided you know in knockout tournament some depending on the number of teams here the number of teams is such that some team will get a buy. So, you see that 9 matches need to be played now. So, this is one way of solving the problem you draw a graph of the situation and then from the graph you derive how many matches should be played. Now, what are the disadvantage of this approach? Now, this is something that we need to learn to evaluate a solution what are the advantages and disadvantages. So, any solution you must look for efficiency and effectiveness. Effectiveness means it should give a value that is accurate and efficiency means how much effort do you spend to get the value. So, in this case for example, if I were to solve a problem for 50 teams then I would like my graph would become much more complicated and I will get too many nodes and hopefully I count them correctly to get the number of teams number of matches to be played decide the winner. So, the graph becomes more and more complex as the team number of teams increases. So, that is one of the problems with this solution. What could be another method of solving this problem? So, logical reasoning this is where we want to introduce logical reasoning it is a very powerful method. So, what is logical reasoning? Basically sequence of statements like this shown just below the title illustrate what is logical reasoning. So, there is one premise namely all men are mortal one statement. The next statement is Rama is a man and so there is a conclusion drawn from these two that all men are mortal Rama is a man therefore, Rama is a mortal. So, this chain of statements illustrates what we call as logical reasoning. Now, let us follow the same chain in our problem and see what result we get. So, the first statement is to decide a winner from among ten teams nine matches nine teams had to be eliminated. So, you want one winner. So, you have to eliminate nine teams. Now, every match eliminates one team. So, you need one match to eliminate a team and every match will eliminate a team. Hence logically connecting these two and deriving the conclusion nine matches need to be played for eliminating nine teams. So, you see that by logical reasoning we have come up with the answer that we need to play nine matches. Now, what is the advantage of logical reasoning? Supposing my number of teams were to change I can make the statement more general. So, to decide winner from among n teams n minus one teams were to be eliminated. Every match eliminates one team hence n minus one matches need to be played for eliminating n minus one teams. So, you see the complexity of the solution as a effort required has not changed even though I have changed the number of teams. Now, this is the great advantage of this form of solving problems. So, logical reasoning is extendable whereas, the graphical approach in this case would have resulted in increasing complexity or more and more effort. In fact, rapidly increasing effort as the number of teams increased whereas, the logical reasoning the amount of effort remains the same even though your number of teams is increasing. So, logical reasoning is a very powerful method of solving problems. So, mathematicians use logical reasoning to solve or to provide proofs for theorems. So, you see if you recall you have all of us have in our school days have gone through proofs given by Archimedes and so on for various theorems and you would recall that each proof consisted of just a few statements but all as a arranged in a logical order and just a few statements lead to the proof of the theorem. So, logical reasoning is very very powerful method of solving problems. Now, let us take another problem to illustrate another type of thinking that we use to solve problems. Now, many of us all of us have face this problem sometime or the other we find that our computer is down right it is not working. So, how do you find out what is the reason the computer is down? Now, let me tell you a small personal experience about this there are at least 4 or 5 occasions when I have been called my colleagues by my colleagues in other department. So, I work in the area of electrical engineering. So, for example, from chemistry department and from humanities department and once in fact when I was in IIT Bombay a colleague from the department of atmospheric sciences called me and said you know Sripad my computer is down can you just come and you know just help me to locate the problem. So, you know I asked them you know, but why do you think I should have a solution to this? Now, you are an electrical engineer you know a computer and all these connections and everything it all has to do with electrical engineering. So, you probably will be able to quickly tell me what is the problem meaning that very often we feel that whenever we are faced with a problem we require a specialized knowledge of a particular area to solve the problem. Now, this is a great misconception our discussion in today's this particular session at least should convince you that in fact a large part of the solution for many problems in different areas involves some common thinking strategies this is something very important that is a researcher must understand. So, you should not jump to think that you require specialized knowledge to solve a problem. In fact, you should first attempt to solve a problem by using some of the general strategies. So, for instance in each of these cases when my colleagues called me right what did I do? Now, this is what I did right. So, I first draw a diagram of the situation. So, here is your computer I may draw it on paper at least I have the diagram in my mind right. So, where you have it is not so important, but the fact is that you have to have a diagram of the interconnected various aspects which are there in the situation. So, in this case you have a UPS and then you have a plug point here and you have the wires right. So, this is your picture. Now, what would you do and what did I do? Well, I said the computer is down it could mean maybe the wire here has broken maybe the UPS is not working maybe the wire here between the plug and the UPS something has happened in this or maybe there is no electricity right. So, therefore, you have no computer working. Now, what do you do next? Next you eliminate right. So, you have divided the problem now your problem is the computer is down and then it has you have divided this into maybe smaller problems it could be the each of the wires which could be at fault UPS could be at fault or there will be no electricity and so on. Now, use logical reasoning. So, I find that my tube light is working right and the computer is not working. So, electricity is there because the tube light and the computer are both connected to the same electricity. Now, sometimes it can happen that your tube light is connected to a different phase than your computer in which case one phase may be down right. So, there can be finer details like this. So, now, I find that the electricity is there. So, that is not the reason. Now, is it that the something is wrong with the UPS I need not jump to the conclusion that the computer itself has gone bad right maybe the UPS has gone bad. Now, how do I know whether the UPS has gone bad one method could be if you have another UPS you could put it in this place right another method could be that I remove the computer and put some other appliance here to check whether I am getting the electric output from the UPS. Now, I did this exercise and in fact what I found was that the problem was in the plug. So, because you are removing and putting the plug so many times that is at least that is what these people were doing this colleagues who called me right they were doing this frequently and therefore inside because of the constant movement of the wire the wire had come off from the plug. So, it was not really the problem with the computer, but problem with the wire in the plug. Now, this method of thinking is nothing but what is what goes by the name analysis. So, analysis normally involves using a representation in which all the elements of the problem are put and their interconnected nature is shown then you divide the bigger problem into sub problems into various components then use logical reasoning to solve each small problem and then put the information together. So, more specifically analysis involves separating a problem into parts. So, if I were to represent using a diagram here what analysis means. So, the problem is represented as a whole the bigger problem is separate into parts. So, I have split this problem into smaller parts then you solve each small part. So, understanding the parts and solving them in isolation this is important. So, I separate out for example part 5 is separated out from the other parts and this is solved in isolation and then combining the understanding. So, obtained to understand the whole. So, here in analysis an important issue is how the problem is separated into parts right here lot of thinking good thinking is required because the separation should be such that when I do the separation the parts should not be affected their functioning of the parts when they are joined together and their functioning when they are separated should not be too much different some form of approximation may be involved, but the things should not be too different. So, this is the key point in analysis. So, this approach of solving problems is called analysis separation of the whole into parts understanding or solving the parts in isolation and combining the understanding. So, obtained are the solutions. So, obtained to understand or to solve the whole. So, this is another important form of thinking where actually you are combining various things that we have discussed graphical representation logical reasoning and so on.