Problem solving

Problem solving in psychology refers to the process of finding solutions to problems encountered in life.[2] Solutions to these problems are usually situation or context-specific. The process starts with problem finding and problem shaping, where the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached and how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis.[3] Mental health professionals study the human problem solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods.[4] Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills.[5]

Problem solving has two major domains: mathematical problem solving and personal problem solving. Both are seen in terms of some difficulty or barrier that is encountered.[6] Empirical research shows many different strategies and factors influence everyday problem solving.[7][8][9] Rehabilitation psychologists studying individuals with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems.[10] Interpersonal everyday problem solving is dependent upon the individual personal motivational and contextual components. One such component is the emotional valence of "real-world" problems and it can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving,[11][12] demonstrating that poor emotional control can disrupt focus on the target task and impede problem resolution and likely lead to negative outcomes such as fatigue, depression, and inertia.[13] In conceptualization, human problem solving consists of two related processes: problem orientation and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. Studies conclude people's strategies cohere with their goals[14] and stem from the natural process of comparing oneself with others.

The early experimental work of the Gestaltists in Germany placed the beginning of problem solving study (e.g., Karl Duncker in 1935 with his book The psychology of productive thinking[15]). Later this experimental work continued through the 1960s and early 1970s with research conducted on relatively simple (but novel for participants) laboratory tasks of problem solving.[16][17] The use of simple, novel tasks was due to the clearly defined optimal solutions and short time for solving, which made it possible for the researchers to trace participants' steps in problem-solving process. Researchers' underlying assumption was that simple tasks such as the Tower of Hanoi correspond to the main properties of "real world" problems and thus the characteristic cognitive processes within participants' attempts to solve simple problems are the same for "real world" problems too; simple problems were used for reasons of convenience and with the expectation that thought generalizations to more complex problems would become possible. Perhaps the best-known and most impressive example of this line of research is the work by Allen Newell and Herbert A. Simon.[18] Other experts have shown that the principle of decomposition improves the ability of the problem solver to make good judgment.[19]

In computer science and in the part of artificial intelligence that deals with algorithms, problem solving includes techniques of algorithms, heuristics and root cause analysis. The amount of resources (e.g. time, memory, energy) required to solve problems is described by computational complexity theory. In more general terms, problem solving is part of a larger process that encompasses problem determination, de-duplication, analysis, diagnosis, repair, and other steps.

Other problem solving tools are linear and nonlinear programming, queuing systems, and simulation.[20]

Much of computer science involves designing completely automatic systems that will later solve some specific problem—systems to accept input data and, in a reasonable amount of time, calculate the correct response or a correct-enough approximation.

In addition, people in computer science spend a surprisingly large amount of human time finding and fixing problems in their programs: Debugging.

Formal logic is concerned with such issues as validity, truth, inference, argumentation and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods, such as the resolution principle developed by John Alan Robinson.

In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. However, already in 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, using a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning.

The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of his approach, emanating from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution,[21] which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving[22] and computational logic to improve human thinking[23]

Problem solving is used when products or processes fail, so corrective action can be taken to prevent further failures. It can also be applied to a product or process prior to an actual failure event—when a potential problem can be predicted and analyzed, and mitigation applied so the problem never occurs. Techniques such as failure mode and effects analysis can be used to proactively reduce the likelihood of problems occurring.

Forensic engineering is an important technique of failure analysis that involves tracing product defects and flaws. Corrective action can then be taken to prevent further failures.

Reverse engineering attempts to discover the original problem-solving logic used in developing a product by taking it apart.[24]

In military science, problem solving is linked to the concept of "end-states", the desired condition or situation that strategists wish to generate.[25]^{:xiii, E-2} The ability to solve problems is important at any military rank, but is highly critical at the command and control level, where it is strictly correlated to the deep understanding of qualitative and quantitative scenarios. Effectiveness of problem solving is used to measure the result of problem solving, tied to accomplishing the goal.[25]^:IV-24 Planning for problem-solving is the process of determining how to achieve the goal[25]^:IV-1

Confirmation bias is an unintentional bias caused by the collection and use of data in a way that favors a preconceived notion. The beliefs affected by confirmation bias do not need to have motivation, the desire to defend or find substantiation for beliefs that are important to that person.[32] Research has found that professionals within scientific fields of study also experience confirmation bias. Andreas Hergovich, Reinhard Schott, and Christoph Burger's experiment conducted online, for instance, suggested that professionals within the field of psychological research are likely to view scientific studies that agree with their preconceived notions more favorably than studies that clash with their established beliefs.[33] According to Raymond Nickerson, one can see the consequences of confirmation bias in real-life situations, which range in severity from inefficient government policies to genocide. Nickerson argued that those who killed people accused of witchcraft demonstrated confirmation bias with motivation. Researcher Michael Allen found evidence for confirmation bias with motivation in school children who worked to manipulate their science experiments in such a way that would produce favorable results.[34] However, confirmation bias does not necessarily require motivation. In 1960, Peter Cathcart Wason conducted an experiment in which participants first viewed three numbers and then created a hypothesis that proposed a rule that could have been used to create that triplet of numbers. When testing their hypotheses, participants tended to only create additional triplets of numbers that would confirm their hypotheses, and tended not to create triplets that would negate or disprove their hypotheses. Thus research also shows that people can and do work to confirm theories or ideas that do not support or engage personally significant beliefs.[35]

Mental set was first articulated by Abraham Luchins in the 1940s and demonstrated in his well-known water jug experiments.[36] In these experiments, participants were asked to fill one jug with a specific amount of water using only other jugs (typically three) with different maximum capacities as tools. After Luchins gave his participants a set of water jug problems that could all be solved by employing a single technique, he would then give them a problem that could either be solved using that same technique or a novel and simpler method. Luchins discovered that his participants tended to use the same technique that they had become accustomed to despite the possibility of using a simpler alternative.[37] Thus mental set describes one's inclination to attempt to solve problems in such a way that has proved successful in previous experiences. However, as Luchins' work revealed, such methods for finding a solution that have worked in the past may not be adequate or optimal for certain new but similar problems. Therefore, it is often necessary for people to move beyond their mental sets in order to find solutions. This was again demonstrated in Norman Maier's 1931 experiment, which challenged participants to solve a problem by using a household object (pliers) in an unconventional manner. Maier observed that participants were often unable to view the object in a way that strayed from its typical use, a phenomenon regarded as a particular form of mental set (more specifically known as functional fixedness, which is the topic of the following section). When people cling rigidly to their mental sets, they are said to be experiencing fixation, a seeming obsession or preoccupation with attempted strategies that are repeatedly unsuccessful.[38] In the late 1990s, researcher Jennifer Wiley worked to reveal that expertise can work to create a mental set in people considered to be experts in their fields, and she gained evidence that the mental set created by expertise could lead to the development of fixation.[38]

Functional fixedness is a specific form of mental set and fixation, which was alluded to earlier in the Maier experiment, and furthermore it is another way in which cognitive bias can be seen throughout daily life. Tim German and Clark Barrett describe this barrier as the fixed design of an object hindering the individual's ability to see it serving other functions. In more technical terms, these researchers explained that "[s]ubjects become "fixed" on the design function of the objects, and problem solving suffers relative to control conditions in which the object's function is not demonstrated."[39] Functional fixedness is defined as only having that primary function of the object itself hinder the ability of it serving another purpose other than its original function. In research that highlighted the primary reasons that young children are immune to functional fixedness, it was stated that "functional fixedness...[is when]subjects are hindered in reaching the solution to a problem by their knowledge of an object's conventional function."[40] Furthermore, it is important to note that functional fixedness can be easily expressed in commonplace situations. For instance, imagine the following situation: a man sees a bug on the floor that he wants to kill, but the only thing in his hand at the moment is a can of air freshener. If the man starts looking around for something in the house to kill the bug with instead of realizing that the can of air freshener could in fact be used not only as having its main function as to freshen the air, he is said to be experiencing functional fixedness. The man's knowledge of the can being served as purely an air freshener hindered his ability to realize that it too could have been used to serve another purpose, which in this instance was as an instrument to kill the bug. Functional fixedness can happen on multiple occasions and can cause us to have certain cognitive biases. If people only see an object as serving one primary focus than they fail to realize that the object can be used in various ways other than its intended purpose. This can in turn cause many issues with regards to problem solving. Common sense seems to be a plausible answer to functional fixedness. One could make this argument because it seems rather simple to consider possible alternative uses for an object. Perhaps using common sense to solve this issue could be the most accurate answer within this context. With the previous stated example, it seems as if it would make perfect sense to use the can of air freshener to kill the bug rather than to search for something else to serve that function but, as research shows, this is often not the case.

Functional fixedness limits the ability for people to solve problems accurately by causing one to have a very narrow way of thinking. Functional fixedness can be seen in other types of learning behaviors as well. For instance, research has discovered the presence of functional fixedness in many educational instances. Researchers Furio, Calatayud, Baracenas, and Padilla stated that "... functional fixedness may be found in learning concepts as well as in solving chemistry problems."[41] There was more emphasis on this function being seen in this type of subject and others.

There are several hypotheses in regards to how functional fixedness relates to problem solving.[42] There are also many ways in which a person can run into problems while thinking of a particular object with having this function. If there is one way in which a person usually thinks of something rather than multiple ways then this can lead to a constraint in how the person thinks of that particular object. This can be seen as narrow minded thinking, which is defined as a way in which one is not able to see or accept certain ideas in a particular context. Functional fixedness is very closely related to this as previously mentioned. This can be done intentionally and or unintentionally, but for the most part it seems as if this process to problem solving is done in an unintentional way.

Functional fixedness can affect problem solvers in at least two particular ways. The first is with regards to time, as functional fixedness causes people to use more time than necessary to solve any given problem. Secondly, functional fixedness often causes solvers to make more attempts to solve a problem than they would have made if they were not experiencing this cognitive barrier. In the worst case, functional fixedness can completely prevent a person from realizing a solution to a problem. Functional fixedness is a commonplace occurrence, which affects the lives of many people.

Unnecessary constraints are another very common barrier that people face while attempting to problem-solve. This particular phenomenon occurs when the subject, trying to solve the problem subconsciously, places boundaries on the task at hand, which in turn forces him or her to strain to be more innovative in their thinking. The solver hits a barrier when they become fixated on only one way to solve their problem, and it becomes increasingly difficult to see anything but the method they have chosen. Typically, the solver experiences this when attempting to use a method they have already experienced success from, and they can not help but try to make it work in the present circumstances as well, even if they see that it is counterproductive.[43]

Groupthink, or taking on the mindset of the rest of the group members, can also act as an unnecessary constraint while trying to solve problems.[44] This is due to the fact that with everybody thinking the same thing, stopping on the same conclusions, and inhibiting themselves to think beyond this. This is very common, but the most well-known example of this barrier making itself present is in the famous example of the dot problem. In this example, there are nine dots lying on a grid three dots across and three dots running up and down. The solver is then asked to draw no more than four lines, without lifting their pen or pencil from the paper. This series of lines should connect all of the dots on the paper. Then, what typically happens is the subject creates an assumption in their mind that they must connect the dots without letting his or her pen or pencil go outside of the square of dots. Standardized procedures like this can often bring mentally invented constraints of this kind,[45] and researchers have found a 0% correct solution rate in the time allotted for the task to be completed.[46] The imposed constraint inhibits the solver to think beyond the bounds of the dots. It is from this phenomenon that the expression "think outside the box" is derived.[47]

This problem can be quickly solved with a dawning of realization, or insight. A few minutes of struggling over a problem can bring these sudden insights, where the solver quickly sees the solution clearly. Problems such as this are most typically solved via insight and can be very difficult for the subject depending on either how they have structured the problem in their minds, how they draw on their past experiences, and how much they juggle this information in their working memories[47] In the case of the nine-dot example, the solver has already been structured incorrectly in their minds because of the constraint that they have placed upon the solution. In addition to this, people experience struggles when they try to compare the problem to their prior knowledge, and they think they must keep their lines within the dots and not go beyond. They do this because trying to envision the dots connected outside of the basic square puts a strain on their working memory.[47]

The solution to the problem becomes obvious as insight occurs following incremental movements made toward the solution. These tiny movements happen without the solver knowing. Then when the insight is realized fully, the "aha" moment happens for the subject.[48] These moments of insight can take a long while to manifest or not so long at other times, but the way that the solution is arrived at after toiling over these barriers stays the same.

Irrelevant information is information presented within a problem that is unrelated or unimportant to the specific problem.[43] Within the specific context of the problem, irrelevant information would serve no purpose in helping solve that particular problem. Often irrelevant information is detrimental to the problem solving process. It is a common barrier that many people have trouble getting through, especially if they are not aware of it. Irrelevant information makes solving otherwise relatively simple problems much harder.[49]

For example: "Fifteen percent of the people in Topeka have unlisted telephone numbers. You select 200 names at random from the Topeka phone book. How many of these people have unlisted phone numbers?"[50]

The people that are not listed in the phone book would not be among the 200 names you selected. The individuals looking at this task would have naturally wanted to use the 15% given to them in the problem. They see that there is information present and they immediately think that it needs to be used. This of course is not true. These kinds of questions are often used to test students taking aptitude tests or cognitive evaluations.[51] They aren't meant to be difficult but they are meant to require thinking that is not necessarily common. Irrelevant Information is commonly represented in math problems, word problems specifically, where numerical information is put for the purpose of challenging the individual.

One reason irrelevant information is so effective at keeping a person off topic and away from the relevant information, is in how it is represented.[51] The way information is represented can make a vast difference in how difficult the problem is to be overcome. Whether a problem is represented visually, verbally, spatially, or mathematically, irrelevant information can have a profound effect on how long a problem takes to be solved; or if it's even possible. The Buddhist monk problem is a classic example of irrelevant information and how it can be represented in different ways:

A Buddhist monk begins at dawn one day walking up a mountain, reaches the top at sunset, meditates at the top for several days until one dawn when he begins to walk back to the foot of the mountain, which he reaches at sunset. Making no assumptions about his starting or stopping or about his pace during the trips, prove that there is a place on the path which he occupies at the same hour of the day on the two separate journeys.

This problem is near impossible to solve because of how the information is represented. Because it is written out in a way that represents the information verbally, it causes us to try and create a mental image of the paragraph. This is often very difficult to do especially with all the irrelevant information involved in the question. This example is made much easier to understand when the paragraph is represented visually. Now if the same problem was asked, but it was also accompanied by a corresponding graph, it would be far easier to answer this question; irrelevant information no longer serves as a road block. By representing the problem visually, there are no difficult words to understand or scenarios to imagine. The visual representation of this problem has removed the difficulty of solving it.

These types of representations are often used to make difficult problems easier.[52] They can be used on tests as a strategy to remove Irrelevant Information, which is one of the most common forms of barriers when discussing the issues of problem solving.[43] Identifying crucial information presented in a problem and then being able to correctly identify its usefulness is essential. Being aware of irrelevant information is the first step in overcoming this common barrier.

Problem solving can also occur without waking consciousness. There are many reports of scientists and engineers who solved problems in their dreams. Elias Howe, inventor of the sewing machine, figured out the structure of the bobbin from a dream.[53]

The chemist August Kekulé was considering how benzene arranged its six carbon and hydrogen atoms. Thinking about the problem, he dozed off, and dreamt of dancing atoms that fell into a snakelike pattern, which led him to discover the benzene ring. As Kekulé wrote in his diary,

One of the snakes seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis.[54]

There also are empirical studies of how people can think consciously about a problem before going to sleep, and then solve the problem with a dream image. Dream researcher William C. Dement told his undergraduate class of 500 students that he wanted them to think about an infinite series, whose first elements were OTTFF, to see if they could deduce the principle behind it and to say what the next elements of the series would be.[55] He asked them to think about this problem every night for 15 minutes before going to sleep and to write down any dreams that they then had. They were instructed to think about the problem again for 15 minutes when they awakened in the morning.

The sequence OTTFF is the first letters of the numbers: one, two, three, four, five. The next five elements of the series are SSENT (six, seven, eight, nine, ten). Some of the students solved the puzzle by reflecting on their dreams. One example was a student who reported the following dream:[55]

I was standing in an art gallery, looking at the paintings on the wall. As I walked down the hall, I began to count the paintings: one, two, three, four, five. As I came to the sixth and seventh, the paintings had been ripped from their frames. I stared at the empty frames with a peculiar feeling that some mystery was about to be solved. Suddenly I realized that the sixth and seventh spaces were the solution to the problem!

With more than 500 undergraduate students, 87 dreams were judged to be related to the problems students were assigned (53 directly related and 34 indirectly related). Yet of the people who had dreams that apparently solved the problem, only seven were actually able to consciously know the solution. The rest (46 out of 53) thought they did not know the solution.

Mark Blechner conducted this experiment and obtained results similar to Dement's.[56] He found that while trying to solve the problem, people had dreams in which the solution appeared to be obvious from the dream, but it was rare for the dreamers to realize how their dreams had solved the puzzle. Coaxing or hints did not get them to realize it, although once they heard the solution, they recognized how their dream had solved it. For example, one person in that OTTFF experiment dreamed:[56]

There is a big clock. You can see the movement. The big hand of the clock was on the number six. You could see it move up, number by number, six, seven, eight, nine, ten, eleven, twelve. The dream focused on the small parts of the machinery. You could see the gears inside.

In the dream, the person counted out the next elements of the series – six, seven, eight, nine, ten, eleven, twelve – yet he did not realize that this was the solution of the problem. His sleeping mindbrain solved the problem, but his waking mindbrain was not aware how.

Albert Einstein believed that much problem solving goes on unconsciously, and the person must then figure out and formulate consciously what the mindbrain has already solved. He believed this was his process in formulating the theory of relativity: "The creator of the problem possesses the solution."[57] Einstein said that he did his problem-solving without words, mostly in images. "The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined."[58]

In Europe, two main approaches have surfaced, one initiated by Donald Broadbent (1977; see Berry & Broadbent, 1995) in the United Kingdom and the other one by Dietrich Dörner (1975, 1985; see Dörner & Wearing, 1995) in Germany. The two approaches share an emphasis on relatively complex, semantically rich, computerized laboratory tasks, constructed to resemble real-life problems. The approaches differ somewhat in their theoretical goals and methodology, however. The tradition initiated by Broadbent emphasizes the distinction between cognitive problem-solving processes that operate under awareness versus outside of awareness, and typically employs mathematically well-defined computerized systems. The tradition initiated by Dörner, on the other hand, has an interest in the interplay of the cognitive, motivational, and social components of problem solving, and utilizes very complex computerized scenarios that contain up to 2,000 highly interconnected variables (e.g., Dörner, Kreuzig, Reither & Stäudel's 1983 LOHHAUSEN project; Ringelband, Misiak & Kluwe, 1990). Buchner (1995) describes the two traditions in detail.

In North America, initiated by the work of Herbert A. Simon on "learning by doing" in semantically rich domains,[59][60] researchers began to investigate problem solving separately in different natural knowledge domains – such as physics, writing, or chess playing – thus relinquishing their attempts to extract a global theory of problem solving (e.g. Sternberg & Frensch, 1991). Instead, these researchers have frequently focused on the development of problem solving within a certain domain, that is on the development of expertise; Chase & Simon, 1973; Chi, Feltovich & Glaser, 1981).[61]

Areas that have attracted rather intensive attention in North America include:

• Reading (Stanovich & Cunningham, 1991)
• Writing (Bryson, Bereiter, Scardamalia & Joram, 1991)
• Calculation (Sokol & McCloskey, 1991)
• Political decision making (Voss, Wolfe, Lawrence & Engle, 1991)
• Managerial problem solving ([62])
• Lawyers' reasoning[63]
• Mechanical problem solving (Hegarty, 1991)
• Problem solving in electronics (Lesgold & Lajoie, 1991)
• Computer skills (Kay, 1991)
• Game playing (Frensch & Sternberg, 1991)
• Personal problem solving (Heppner & Krauskopf, 1987)
• Mathematical problem solving (Pólya, 1945; Schoenfeld, 1985)
• Social problem solving[11][12]
• Problem solving for innovations and inventions: TRIZ[64]