Thinking and Problem Solving

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Concepts and Prototypes

One of the many interesting questions that psychologists asks is, "How is thought structured and how do we use thought to solve our everyday problems?" To investigate this let us first begin with how thought is structured. As we grow up we form many concepts in our minds. A concept is a representation of a category of objects, events, or ideas grouped together due to common properties. Anything we experience can be categorized into a concept. I might see an object and categorize it as a pencil. I might see a certain color and categorize it as purple. I might see an event and categorize it as a thunderstorm. I can tell if something is a pencil because they have certain properties that make it a pencil. They are long and thin and of a certain length, have lead running through the core and they are usually sharp on one end. Most pencils have these properties and this makes it easy for me to tell that this particular object is a pencil. As we experience many new things in life, we are constantly forming new concepts. This is a process known as concept formation.

Concepts, however, can sometimes be very fuzzy. If I asked you to imagine a car, you would most likely picture a vehicle with four wheels on the outside and a steering wheel and seats inside. This is because most cars have these properties. They are prototypes of this concept. Prototypes are members of a category that possess all of its characteristic features. They are commonly regarded as the "best examples" of a concept. You may have seen cars with three wheels or six wheels and they are also considered to be cars. But when you explain to a young child what a car is, it would probably be most helpful if you showed him/her a car with four wheels rather than three. Because most cars have four wheels, it makes it easier for the child to understand this concept if you showed him/her a picture of a four wheeled car. This is why a prototype is often considered to be the "best examples" of a concept.

Problem Solving

Algorithms and Heuristics

Using concepts and prototypes we categorize our experiences to understand how the world works and how we should interact with it. When we interact with the world, we are often faced with problems to solve. Having concepts and prototypes not only help us understand problems but also solve them. Let us examine how we solve the many problems we encounter in life. When we face a problem that has a logical solution, there are two ways we can approach the problem. The first is to use an algorithm. An algorithm is a problem solving method that involves rules that guarantee the right solution by using a formula or foolproof method. For example, if my car breaks down, I can check every single part of the car to see if it is working to figure out what is wrong. This would be an algorithm. Sooner or later I will find out what is wrong with the car since I am checking every single possible solution. The problem with algorithms is that in many cases, it takes too much time and effort. Most of us use heuristics, the other problem solving method, in a situation like this. A heuristic is a problem solving method using rules of thumb that we have developed from past experience. They are commonly used because they often allow us to make shortcuts in arriving at the correct solution. In the example with my car breaking down described earlier, I might use my past experience with cars and guess what part may be malfunctioning from the sound it made before it broke down. In this case I am using a heuristic. I am not checking every possible thing in the car but am making an educated guess from my previous experiences. The advantage of heuristics over algorithms is that we may find the solution of a problem faster and with less effort. The disadvantage is that we may not find the solution because we are not checking every possibility. Nevertheless, people commonly use heuristics because they often work and thus save us much time and effort.

Three Common Types of Heuristics

There are many kinds of heuristics. Some of them not only fail to provide the solution sometimes but occasionally lead us to the wrong solution. Let us look at some specific heuristics that help us understand this better. Which is more frequent, words that begin with the letter K or words that have the third letter as K in the English language? Most people say that there are more letters that begin with K because it is easier to recall those words than words that have the third letter as K. In fact, there are three times as many words that have the letter K in the third position than words that start with K (Tversky & Kahneman, 1974). Most of us arrive at the wrong solution with a problem like this because we tend to use a type of heuristic known as availability heuristic. An availability heuristic is used when we judge the likelihood of an event based on how many instances come to mind in our memory. Because it is more difficult to come up with words that have the third letter as K than words that begin with K, we assume it occurs less. This is one example of heuristics leading us to the wrong solution.

Let us look at another example. In a study conducted by Mellers, Hertwig and Kahneman (2001), research participants were given a description of an individual similar to the following sentences: "Linda is 31-years old and she is single. She is considered to be outspoken and very bright. She majored in philosophy in college. As a student, she was very concerned with discrimination and social issues. She participated in several demonstrations." The participants were then asked which statement is more likely to be true: (a) Linda is a bank teller. (b) Linda is a bank teller who is active in the feminist movement. The majority of people who read the above description chose the second answer, although it is statistically more likely that she is just a bank teller. This is because there are more bank tellers in the world than bank tellers who are feminists, no matter what sort of background they may have. This is an example of people using a representativeness heuristic. When we use a representativeness heuristic, we make a judgment on something based on how similar things are to prototypes that we have in our minds. Here again, we find that the use of heuristics can lead to the wrong solution.

We will consider one more example here. Let us say that I asked you the following two questions in the order presented here. (1). Do you think the population of Japan is more or less than 40 million? (2). Estimate the population of Japan. Most people would guess something close to 40 million even though it is actually around 128 million. This happens most likely because most of us use anchoring heuristics. When we use an anchoring heuristic we estimate quantities based on prior information that is perceived to be related to the solution (Tversky & Kahneman, 1974). When people ask us questions like, "Do you think the population of Japan is more or less than 40 million?" we usually assume that they are asking because it must be close to that number. Therefore, when we are asked to estimate the population of Japan right after that, we are most likely going to make an estimate close to 40 million. Even though in many cases, this heuristic is useful, we must be careful because it can also be used to mislead people to estimate numbers that are much higher or lower.

Heuristics can be very useful in solving problems quickly. They work the majority of the time and this is why we use them so often. Sometimes, however, as you can see from the examples above, they can lead to inappropriate solutions. This is one good reason to be very careful when we use heuristics to solve problems.

Fixations

In addition to the problems discussed above, we sometimes encounter other obstacles when trying to solve problems successfully. These obstacles are mostly due to fixations in our thought processes. A fixation can be described as the inability to see a problem from a fresh perspective. For example, consider the following problem. "What occurs once in June, once in July and twice in August?" This is very difficult to solve for most people. The solution is the letter u. The difficulty of this problem is most likely caused by a fixation, the inability to see a problem from a fresh perspective. When we think of what occurs in a month, we tend to think of specific events such as holidays. We rarely think of what "occurs" in the letters of the months when we spell them out. This would be an example of a fixation in problem solving.

There are many specific kinds of fixations. One is called a mental set. A mental set is a fixed mental pattern that make us repeat solutions that have worked in the past. This may be very useful in many cases. Often in life, we encounter similar problems with similar solutions. For example, a mental set may be useful when opening a door at a new place. We encounter many different doors but many of them can be opened in a similar way. If we just do the same thing as we did with similar doors (such as turn the door knob) we might be able to open a door we have never seen before. Mental sets, however, can also be obstacles in problem solving. For example, consider the following riddle. "There are ten eggs in a basket. Ten people take one of the eggs each. How is it that one egg can still be left in the basket?" The difficulty of solving this problem is most probably caused by a mental set. When most of us think of someone taking an egg that is in a basket, we assume they are taking it out of the basket. If this is what we actually imagined, the riddle is impossible to solve. What we wrongly assumed here is that you could not take the basket with the egg in it. In fact, if the tenth person takes the basket with the last egg in it, there is still one egg in the basket. In this example, a mental set may prevent us from solving a problem successfully.

Another specific kind of fixation is called functional fixedness. Functional fixedness is the tendency to perceive the functions of objects as fixed and unchanging. A famous example of this is Maier's (1931) pendulum problem. The problem was to tie together two pieces of string hanging from the ceiling. However, they were too far apart to catch hold of both at once. The participant was told that they could solve the problem only with the use of a pair of pliers. The solution was for the participant to tie the pliers to one string and swing the string like a pendulum, allowing both strings to be reached at the same time. Only 37.7% of the participants were able to solve this problem. This problem was difficult for many participants because we are not used to thinking of using a pair of pliers as a pendulum. When we think of the uses of pliers, we think of gripping things tightly and perhaps turning or pulling things while we are gripping an object. We rarely ever use it as a pendulum even though it could be used for this purpose. This is a classic example of functional fixedness, a kind of fixation that can be an obstacle to successful problem solving.

Other Useful Ideas about Problem Solving

Many of these examples are helpful in reminding us that when we are faced with a problem, it is usually a good idea to engage in two types of thinking. First, we should engage in divergent thinking. Divergent thinking involves widening the range of possibilities and options for a solution. For example, oftentimes when we are faced with a problem we first brainstorm ideas and explore as many options as possible. This encourages us to think "outside the box" to find creative solutions to problems without falling in the trap of fixations. Once we have explored all of the options, we are ready to engage in convergent thinking. Convergent thinking involves narrowing down our options to arrive at a suitable solution. Now that we have considered all of the options, we need to decide on the best one. In many situations in life, we skip the divergent thinking process assuming that we know what our options are and go directly into convergent thinking. This may sometimes save us time and effort but it may also sometimes leave us frustrated in the trap of fixation. Of course in some situations in life, such as in a multiple choice test, all of the possible options are already provided. In this case, we only need to engage in convergent thinking.

From the discussion above, it is clear that even though we all like to think that we are always thinking in logical ways, it is not always so. The discussion on heuristics and fixations tell us that we take mental shortcuts that sometimes fail to help us find the correct solution. Our thinking is sometimes not logical in other ways as well. For one thing, we are often swayed by emotions that lead us away from logic. We see this quite often in our everyday lives. When we go to the supermarket, we often see labels on products saying, "75% lean" in large print. Of course, this can also be expressed as "25% fat" but this is often noted in much smaller print or not at all. This is a common marketing strategy that takes advantage of what is called the framing effect. The framing effect is the emotional effect of the way we present an issue, event or object (Tversky & Kahneman, 1981). It often influences important decisions that we make. Consumers are much more likely to buy a product that says "75% lean" than "25% fat" in large print. Even though both statements have the same logical meaning, depending on which way it is presented, we feel very differently about the product and this influences our decisions about buying it. As you can imagine, people in politics, marketing, advertising as well as lawyers in court are well aware of this effect and often use this to their advantage.

Another situation in which our thinking may not follow logic is when we gain insight. Insight is a sudden understanding with no logical steps to lead to it (Köhler, 1947). You may have experienced times when you were struggling to understand something and suddenly you have a moment of insight. It makes sense all of a sudden. This is often experienced when we are faced with serious challenges in our lives. We spend much time thinking about it both consciously and unconsciously. Even though a solution is not found during this period, it is not a waste of time. Our mind is working on the problem whether we are aware of it or not. This is often referred to as an incubation period. After a while, we may suddenly have an idea that works for us. We find a good solution or something confusing suddenly makes sense. There are no logical steps leading to it but there is a sudden leap in our understanding.

The last topic regarding thinking and problem solving is about a concept that may be useful when we are faced with problems that seem overwhelming to us. It is a concept known as decomposition. Decomposition is a strategy we can use to deal with difficult problems. When we use decomposition, we break down a problem into smaller and more manageable steps. For example, imagine that you have to learn everything in the first three hundred pages in your textbook for a history exam in two weeks. Because this seems overwhelming, many of us are likely to procrastinate and not do anything until the night before the exam and panic or give up at the end. If we use decomposition, however, it may feel more manageable and we are more likely to complete the task. For example, we might decide to read and write notes for fifteen pages in the morning and another fifteen pages in the evening every day. In ten days, you will be finished reading and writing notes for everything you need to know. In the remaining days, you could go over all of your notes once in the morning and once in the evening every day to review and learn the contents. Once we have broken down the task in this way, it feels less overwhelming. Because we are less overwhelmed, we are more likely to start earlier and as we complete one day of work, we feel like we have accomplished something. This makes us feel good about ourselves and motivates us to continue on the next day. This is a classic example of decomposition. It helps us not only manage our time but also our emotions and motivations to complete our tasks. This is why it is considered to be so useful when we feel like our tasks are overwhelming.

References

Köhler, W. (1947). Gestalt Psychology (2nd ed.). New York: Liveright.

Maier, N. R. F. (1931). Reasoning in humans: II. The solution of a problem and its appearance in consciousness. Journal of Comparative Psychology, 12, 181-194.

Mellers, A., Hertwig, R., & Kahneman, D. (2001). Do frequency representations eliminate conjunctioneffects? An exercise in adversarial collaboration. Psychological Science, 12, 269-275.

Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124-1131.

Tversky, A. & Kahneman, D. (1981). The Framing of decisions and the psychology of choice. Science, 211, 453-458.