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Problem Solving

Hello and welcome, we will start talking about problem-solving. Problem-solving is one of the ah very ah complicated it is it is one of the complicated ah you know cognitive ah activities that people do on a daily basis and in some sense lot of times you do not appreciate, people do not really appreciate ah that what are the differentmental processes what are the different cognitive processes that are contributing to somebodyis successfully solving ah particular problem, and those problems can be very simple as tosay for example, choosing between two colors or sometimes very difficult suppose for example,if you have to you know solve amaze with the given particular time or say for example,you have to you know solve the rubiks cube where the time ah boundation.Now at this point in this course and I will ah try and ah link this to the earlier coursethat I have also taken introduction to advance to basic cognitive processes. We have by nowtalked about most of the basic most of the cognitive processes that kind of contributeand kind of you know help in building up ah approaches and help in building up strategiesthat people might use in problem solving. We talked about attention, we have talkedabout perception, we have talked about memory, we have talked about biases, in this courseeven we talked about language, concepts, knowledge, categories. So, all of those things now youwould need to invoke and understand how all of those component cognitive processes arecontributing to people ah finally, achieving you know ah goal states from say for example,initial ah you know state where problem is stated. So, I have already mentioned in the description now, that I have been talking I have already told you what you might expectfrom this week So, we will talk about how people approachah problems, what are the different strategies that people might invoke to solve particularproblems, what are the obstacles in solving problems and what are the factors that mighthelp what are the strategies you know that might help people solve problems. So, thewhole week will be about problem solving, and we we will have ah ah short lecture oncreativity as well. This creativity in in the sense is also a very important aspectof solving problem. People were creative solve problems in ah you know in more efficientin genius ways as compared to ordinary people or as to supposed to people not rated highlycreative, but again both of them achieve the same ah goal state, starting from an initialstate. I have mentioning goal and initial state ahmultiple times. So, I think this is the good time to ah already define what a problem is.So, a problem basically is when there is an obstacle ah that you have to overcome andsurpass, ah in order to reach from initial given state to a goal state. So, the ideais you know somebody or in in some sense. You have given a particular ah statement statementof a problem and the statement of the problem might define an initial state. Suppose forexample, you know somebody could give you an equation that solve for a x, x plus y isequal to ah 5 and x minus y is equal to 2 something like that, and then you basicallyah you know ah have to figure this out. Now again I just arbitrarily picked this up, butthe idea is the statement of equation is the actual state and it is this is what they givenknowledge is when you have to take particular steps to solve for the ah you know to variables.So, the idea is how are you going to do it, how are you basically you know ah going totell us the values of x and y starting from two equations that are given. So, and theobstacles that you have to do some ah calculations you have you know change around ah equationsa little bit, co forms some competition then only then you will reach. ah Simply or ahsay for example, if you take this to a different way, ah somebody could give you that ok thereis a stuffing the cage in you have to go and make ah you know make tea for example, andcome back with the tea. The idea is you go there, you probably notreally ah make tea active ah make tea earlier, but there are these ah you know componentsthere the ingredients are there and then you have the kind of combine them in a particularfashion to come up with tea. So, there are steps that you will have to take, the initialstate is where you have ingredients all separated at final state is that when you have a cupof tea. So, how does one reach from this initial stateto the goal state and what are the obstacles that you really need to surpass or overcomethat is basically what is problem solving involves. Now there could be ah differentkinds of problems that you would come across there would be problems which are very wellstated ah which give you all of the information that is needed which ah indicate in some sensewhat might be the most probable most direct methods that you will take, most directs rulesthat you take to solve these problems, and these problems generally will have a correctanswers. If you spend enough time if you spend ah correct amount of effort you will eventuallysolve these problems if you are applying a particular kind of a method ah there are alsoah these problems wherever are referred to is well defined problems.The other kind of problems sometimes we come across in life or probably a ot of times isthe ill defined problems. Now ill defined problems do not really necessary have a correctanswer you know, what is good, what is bad ah you know is their god in the world thosekind of ah statements are rather ill defined problem. So, if you attempt to answer themwith so much that is unsaid and there is so, much that is assumed or so much that you haveto really you know work way. So, these kind of problems are referred can be referred toa ill defined problem, they do not really have a correct answer for that matter andthe part today a solution ah that to really come up with a response is very difficultits almost unclearance sometimes impossible to really come up with straight forward solutionto such problems. But you will see that people deal both both kinds of problems on a dailybasis, in the kind of you know work around in different ways to ah probable solutionsometimes, correct solutions sometimes incorrect solutions. So, what we will try and do inthis week is to see how people are actually ah achieving these goals.Now, one of the things is ah when you approaching a problem, what is the first thing would needto do? The first thing that you need to do generally is to try and understand the problemfirst. So, the point is that you will not reallybe able to solve any problem, unless you understand what the problem is trying to say and a lotof time we tell ah students that ah even before you start attempting to answer a particularquestion writing your test or exams you should spend a good amount of time reading questionsfirst; unless you understand what the question is actually asking you ah there is no waythat you going to be able to give a correct or a effective answer. So, its kind of youcan take this analogy back to a solving problems in real life if an all those different scenariosis that, we have to understand the problem at first. Now let me elaborate a little bitout understanding. So, understanding basically means, that you have already generated a mentalrepresentation of what the problem is, based on whatever is given in the problem and whateveryour previous experiences have been. Suppose for example, ah somebody is giving you ahagain ah let me go to the ah tea ah things. So, the problem is to make tea, now the ideais you have been given ingredients that is the given information you have some tea andyou have some ah tea leaves, you have some milk you have some water, ah you have ah somesugar and then your previous experience is the that you know that you know these thingshave to be cooked and they have to be mixed in a particular order to finally, make tea.So, you have to first understand what really needs to be done with the problem. And understandbasically means that you have that if created that kind of material representation. Nowproblem solving ah as I was already saying ah in the introduction is ah basically anactivity its a complex mental activity, that builds upon several other cognitive activitysuch things like attention ah perception memory ah reasoning, decision making all of thosethings kind of will contribute to how you are going to approach a particular problem.So, let is let us take a few examples for example, attention is something very important.Attention or ah you kind of in in more ah common parlance way you can call call thisas concentration. Attention of concentration is a is very important ah in ah somebody beingable to understand a particular problem because attention is limited there are limited attentionalresources and you have to allocate them wisely, you have to allocate them in a you know efficientmanner. So, that you kind of are attending the mostrelevant aspects of the problem that is stated and only then you will able to move forwardfrom step zero to step one. Step zero is the initial say for example, first step you taketowards a problem towards solving this particular problem and it has been shown that you knowah say for example, ah if you have ah competing thoughts, if ah you have ah particular kindsof distractions, your attention will be limited and you will not really be able to engagewith the problem correctly. ah One of the studies done by Bransford andStein in 1984 demonstrates this very well, they presented ah high school students ahwith ah story problems ah and they these ah story problems like algebra ah algebra problems,and what happened was what was happening was that the students were asked to recall whateverthey would thinking in their heads ah and they were supposed to write those things downwhile they attempted to solve the problems. So, what happens is, that the students basicallyare having thoughts like you know oh no this is the mathematical problem, do not reallylike Maths, those kind of things they have to you know completely dropdown. Once theystart attending to their thoughts, not really attending to the problem so, much and thesethoughts kept coming ah on and on. These negative ah feelings kept ah comingon and on throughout the ah five minutes that were allocated for solving problem, and theseah things particularly distracted the students attention from the central task. Now whathappens is; obviously, eventually their solving of problems was less efficient, probably theymade more errors, they committed you know they took more time. So, is this is good ahdemonstrational fact that if you distracted if not really concentrating, if your attentionis not there on ah relevant aspects of the particular problem, you will not really beable to achieve the goal that is one. The second is ah again as a subset of attention,ah ah very important ah aspect of ah solving problem is focusing on the appropriate sectionsof the problem. Focusing on ah the problem in such a way that you already get a cluefrom the problem about how to ah you know start approaching it, how do you start solvingit. So, researchers have found that you know effective problem solvers people who are verygood at solving problems, difficult problems, ah they read the description of the problemvery carefully. I was always hinting to how you know studentsshould approach questions papers and stuff, as a research and there is a research Mayerand Hegarty in 1996 have shown that ah effective problem solvers do read the description ofthe problem very carefully. They spend and they allocate a good amount of time in readingwhatever the problem is saying, paying attention to inconsistencies paying attention to patterns,and this attention might contribute in them solving the problem.So, that is that is very very important. Now ah let us move ah further one of the otheraspects that I would like to talk to you about in general ah about problem solving is thatthe way the problem is is stated, we have talked about the framing effect ah in an earlierlecture that how a particular problem worded kind of changes or impacts how people aregoing to react to it. Now ah the way something is ah represented the way the problem is presentedto you ah or the way you represent the problem to yourself, might be reading a ah very difficultequation or you might be having a very difficult passage of a ah text ah in order to understandit, ah the way you can change the way you can represent this ah thing to be yourselfthis problem to yourself determines and has a contributes a lot to your being able tosolve problem. So, there is this importance of choosing appropriateway to represent the problem there is, and that is basically ah effective representationallow you to ah organize the information ah rather efficiently and ah for example, itwill reduce the strain on ah things like working memory. You know ah you will not really needto maintain everything through rehearsal in your working memory. ah If you have organizedsomething very well, suppose if you drawn a picture or something ah all of that informationis out there and you just have to kind a put the pieces together and solve the problem.So, effective representation would increase likelihood of you being able to select a veryuseful and efficiency strategy for encountering the problem. Now moving ahead; the representationof the problem one of the important factors is that, it must ah show it must depict theessential clues essential information that is needed to solve a problem, that is whatis ah an effective representation. Unless it does not ah unless it really ah gives youthat ah relationship, ah it is very difficult for you to ah evolve and start approachingat the problem. ah Let us talk about the few ways in whichpeople represent problem. Suppose for example, ah in high school algebra, you would havebasically ah sometimes be given word problems and word problems something like say for exampleRam is twice as old as Shyam and Rajesh is older than both of them combined by 6 years,and then you have been given Rajesh's age, Rajesh's age is 36 years old.You can just kind of you know spend a minute over it, and come up with the equation whichsays x plus 2, x is plus 6 is equal to 36 and you immediately get to the in a coupleof steps you already get that ok x is equals to 10. So, Ram is basically 10 years old,Ram is only 20 years old and combine both of them adds 6 is the Rajesh is Rajesh's agewhich is 36. Now the idea is in words a lot of times ah people would struggle ah figuringout and I will talk about, ah how people struggle with a statements ah in a bit, but ah thegood way of doing it is just write an equation, you have all three ah ages and then its veryeasy to perform ah mathematical computations and figure out the age.So, this is one example that ah people might use different kinds of ways to represent ahproblems, ah which will help them to solve. Now the major challenge here, a major challengethat ah problem solvers often commit is that the ah commit errors when trying to convertfrom words to abstract symbols you know. ah These kind of algebraic problems the wordproblems you know people solve in standards in third fourth fifth and sixth, the problemin really eventually solving. It is basically there are lot of things peoplemake errors in ah translating the word to the equations. So, Schoenfeld basically showedthis and Schoenfeld showed this at 1982, he was describing how calculus students wereah when they were asked to rephrase simple algebraic ah problem statements, the and theywere basically asked you just make it more understandable. ah It was observed that about10 percent of the students rephrasings how they were actually rephrasing the statement,included information that directly contradicted the input.So, whatever was actually given in the problem statement, when these people rephrase thoseproblems they kind of rephrase is done in such a way it kind of meant opposite to whatthe original problem was. And 20 percent ah of the rephrasings basically ah included ahreally confusing or unintelligible information. So, idea is ah when they are converting theword problem into their own ah representations, ah they are kind of missing out an importantinformation a, and in some sense kind of putting it together in such a way that it is almostimpossible to solve it. Now this is this is something very important the aspect of verbalintelligent the aspect of understanding language also comes in here ah you know and it kindof sequences, in that is why I was saying that you know problem solving process hasa lot of components to it. Now, another common problem just ah somethingthat is ah links to what we are talking about is, in converting sentences to obstacles symbolsis at the sometimes what the problem solver may do is, they might over simplify the sentenceand in that oversimplification, the misrepresenting information. So, Mayer and Hegarty in 1996they did this, and they were asking college students to read a series of algebra wordproblems and then you are asked to recall them later. Now it was seen that a lot oftimes students misrepresented ah things like relational statements, which would have beenvery important to solve those problems. And example is suppose the statement is that youknow the engines rate in still water is 12 miles per hour more than the rate of the currentand ah a lot of students just remembered that you know the engines rate in still water is12 ah miles per hour. So, if you ah kind of miss out on the factthat is 12 hours and 12 miles per hour more than the rate of current, you will you aremissing out of something very important that actually might have been used to solve this.So, that is something you know that is just an example demonstration of how ah peoplemight commit mistakes in ah changing or converting representations. Another way of ah representingah problems might be using matrices. So, one can solve problems effectively by you knowsometimes building just boxes matrices say for example, ah if ah you are reading ah signaldetection, presents detecting, presence and absence of signals. So, there is a high possibilitythat the signal is present there and you will detect it is the high possibility then signalis not present, then you will say no then the signal is not there, but there are alsopossibility to the signal is there and you do not detect it. Also possibility if signalis not there, but you say a false yes. So, in signal detection usually how peopledo it, you can just draw two by two matrix and they said signal present yes no detectedyes no and its very easy if you can just fill in those boxes, you will get what is the percentageof hits, what is the percentage of misses and stuff like that. If there is a big textexplaining that in this is the pattern of formation, just put that in ah the two bytwo matrix, and it will tell you what the pattern of performance of the participantsare. And that is a good way to understand ah a particular problem as well.So, matrices are very useful I mean sciences engineering various cases, matrices are ahvery important to actually look at and they help ah understand and simplify the representationof problems to a great extent. Another way that its a again a very common way of ah peopleah representing the problem is by virtue of diagrams. Now diagrams are very ah usefulwhen you want to put together you 2000 they asked students to construct the origami objectssuppose say for example, billing a paper crow or miniature piano something like that. Andthey were ah supposed to do it using folded papers. People and there were two kinds ofah people who are doing these experiments ah one group basically received only a verbalah description and the other group basically received a verbal description and a step bystep diagram as to how this is done. So, for for example, ah if you ah you knowif you sometimes what ah some electronical appliances, which need ah assembly or somekind of ah we used to have in school ah s u p w kind of things, were you have step bystep demonstration of now fold this then this there then ah now fold this one and then thethis is going to appear. So, students who were actually receiving both the verbal descriptionand a step by step diagram of how to go about it, were actually much more accurate and efficientah than in ah people who just received to the verbal description.Also another very important thing ah interesting thing about ah diagram is, diagrams can beuseful ah you know when somebody wants to represent large amount of information in justone second. Assuming example of matrices say for example, here if borrowed figure fromah matlins book on cognition, ah there you know things this is basically representationof ah the you know different categories of items. For example, you see animals and thebirds and there are two kinds of birds, whatever songbirds and the mammals is ah horses anddogs and cats things like ah that all of those things are organized in a particular way.So, using this diagram we can already understand what are the kind of you know ah hierarchythat is existing in ah these ah set of objects ah.So, diagrams are also very efficient way of representing large ah information that canbe understood and can be used to solve ah particular kinds of problems. Diagrams canall are also very helpful in ah representing ah complicated information in a clear andconcrete form. So, that you have more mental space suppose for example, you are spendinghalf of your effort in just or half of your working you are just maintaining that thing,if you just draw it out sometimes people do it when they are preparing or studying forexams, sometimes would just draw it out ah you know ah in a particular ways. So, thatjust you see the figure when you realize the entire concept.So, these things are very important and they contribute a lot in ah peoples problem solvingefforts. ah Also ah it has been shown is students can really master these ah aids, these differentkinds of representation with relatively little effort. And ah Novick and colleagues theydid this in 1999 and what they did was, they provided students with a brief training sessionon matrices and hierarchical dia[grams] diagrams. And after this training session they actuallytested ah the students on different kinds of problems and they actually found, thatthe students who had attended this training session were much more skilled now in choosingthe most appropriate method for representing a variety of problems. So, ah already sometraining on ah on this ah front and you know for students ah who are watching this canactually practice this little bit and this will help you in understanding complicatedproblems and ah you know; obviously, help you do ah much better in the studies part.So, this is again. So, finally, I will just ah kind of tryingsome of the diagram thing, ah diagrams can also provide additional advantages for example,they would ah very simply just attract peoples eye movements you know. If if there is a lotof things you see presentations if there is a very nice ah presentation with so many diagramsand figures, they would automatically attract our eye movements particularly attract attentionto a relevant areas of the diagram, and in that sense a they are more memorize will andah they are more easy to grasp. You know and in that sense because that iseasy to grasp, ah people will be able to use ah them and solve problems more efficiently.So, there is all about ah these things ah the diagrams, ah finely of the another ahthings that is very important is figures ah visual images and ah it has been shown thatpeople ah have ah been consistently using visual images to solve particular kinds ofproblems. People have found that visual images are veryhelpful in solving problems. There is the demonstration you know from this book, ahis ah the Buddhist Monk problem and lets read this out to you ah little bit. So, the BuddhistMonk problem is is very simplest very ah its rather similar to the you know boat problemthat we have in India, ah ah the boat has to cross the rivers and there is a you knowthere is a boatand the crocodile it is this one is also ah very similar ah.So, the Buddhist Monk problem is that you know exactly at ah sunrise on morning thisMonk is set out to climb a tall mountain, then narrow ah path was not more than a footor too wide it wo ah wound around the mountain to a beautiful glittering temple at the mountainpeak. So, Monk quantity to read the temple and you know it ah early sunrise say the Monkstarts approaching. The mountain ah Monk climbs the path at varying rates of speed he stoppedah at particular time. So, it whatever food is got with himself, and he reaches the templejust before the sunset at the temple he fasted and meditated stayed there for ah severaldays and then he kind of began his journey ah along the same path again starting at sunriseand walking as earlier speeds and then he stops and now what the idea is ah the questionhere is that you have to prove that they must be spot ah along the path coming back andgoing that the Monk will path a pass on both trips at exactly the same time of the day.So, this is this is what the problem is, and ah you know you can kind ah of spend a minuteor do pause this ah think how you will solve the problem ah, but if I have to kind of giveyou the solution or tell you an effective way of solving this is ah something like this.So, ah you know if you kind of assume that ah the start is almost happening at this sametime, and a kind of varying the you know ah the factor in the various speed in sorts etcetera,they must be ah you know at around a half day part that the ah person is going to crossthe same kind of thing, because you are assuming a lot of ah consistencies in in this wholething. So, if you look at the figure, you are easilyable to grasp the solution of the problem and ah this is just one example there areso, many other examples were a use of visual images is encouraged and visual images ahhelp you ah figure out problems much in the way as diagrams do.So, visual image can in assents ah help us escape from the boundaries of traditionalconcrete or verbal representations. And ah people who have good visual imagery skillsalso have an advantage that problem solving requires because if ah certain kinds of ahproblem solving requires to make certain figures if you have a good mental imagery, you canactually use that to immediately draw a figure or immediately think of a ah visual scenario,where you will be able to solve that problem more effetely.So, ah this is about ah the various ways of representing problems and ah you know whatproblems solving is essential, I will very quickly talk to you about a very interestingah approach to a understanding how problems are solved, and this is basically about theimportance of context where are you solving that problem, what is the scenario what isthe setting, what is the place, what is the you know where is ah you know what kind oftask you doing, and those kind of ah you know attention to these kind of details also contributea lot in somebody being able to solve a problem let us talk about a this little bit.So, many psychologies and educators have emphasized ah you know things ah like situated cognitionyou know. Situated cognition is our ability to solve a problem ah is is that is in fact,type through this specific context in which you have learnt to solve their problems. Supposefor a example you have learnt to you know ah do any particular activity in a in a veryspecified set. Suppose for example, you have learnt do a particular task let us say drawah in a classroom or suppose ah a lot of things what happens is the students learn the insciences and engineering students learn, ah how different kinds of engines, where calledhow different kinds of mechanics of ah materials work in their classes you know in their engineeringclassroom. But a lot of kinds you see is they are notable to apply those things outside when they go inside the real world. You have to justtake the concept from the classroom and take it outside and try and apply it to solve theproblem at hand ok and they are very ingenious thing ah suppose for example, ah I am rememberingthere was there was ah movie ah I think ah three idiots, and there in ah the the personwho is ah playing the main lead is using a lot of ah the things that he is learning inclass in real life you know scenarios. So, the idea is you can take whatever youare learning is inside the class outside as well. The fact is that a lot of time peopleare not able to do it, a lot of times our ability to ah perform particular task to ahah achieve particular kinds of goals is tied to the context it is tied to where you havelearnt this. And ah there are there are lot of example so, suppose it if argued that abstractintelligence test, ah often fails to rele ah reveal ah on how competent a person wouldbe in solving the problem in real life setting. So, for example, there is a student who doesvery well in the class, he does very well in all the assignments and tests etcetera,but ah as soon as you take the students out in a real life setting you know of thingslike you know civil engineering or architecture and those kinds of things, students kind ofyou know ah falter a bit and the hesitate and they there are problems of really convertingthis. So, as compared to a particular students notreally done very well in the class and assignments so much, but very quickly he can apply allof that what was learnt in a real life scenario. So, these these kind of examples are there,and there there has been a lot of research on ah situated cognitionand that researchdemo