The Psychology of Problem Solving
Analyzing an individual’s problem solving ability independent from their knowledge of a specific subject matter is a difficult task. Likewise, teaching pure problem solving skills brings challenges of its own. Often the skills are too specific to one field, or too broad to be of any applicable use. Nonetheless, the pursuit of identifying and teaching effective problem solving strategies remains because of its potential benefits. From small businesses to multi-national corporations, all firms’ seek employees more capable of tackling any problem. In a globalized, shape-shifting world, individuals adept at applying their skills to a variety of scenarios are better equipped to succeed.
Despite the many obstacles the problem solving field faces, progress has been made. A better understanding of the general composition of problems has allowed researchers to identify better ways to solve them. At its core all problems have three parts: the start state, the solution path, and the goal state. A big reason people have an easier time solving problems they are familiar with is because they know which process (solution path) will lead to the correct answer (goal state). For example, when faced with the task of multiplying two binomials a math student knows the FOIL process (First Outside Inside Last) will lead to the correct answer. Yet, if the student has never done trinomial multiplication before his chances of correctly multiplying three trinomials is not very high. However, with an understanding of the theorems and reasons behind the FOIL process for binomials, the student could piece together the correct solution path for trinomial multiplication. While this example did require math specific knowledge, it also highlights the difference between using a prior learned process (FOIL), and using prior knowledge to identify the process required for the solution. The student could still get the problem wrong by performing incorrect calculations; however, the scope of this paper is limited to identifying the solution path and not the calculations.
Complex problems inevitably require knowledge of a specific field; however, there are certain factors that can increase the chance of anyone devising, or recognizing the correct solution path. These include focus/attention state, relevancy, direction, strategy preference, and group decision – making.
Problems are divided into categories, with two of the most prominent being analytic and insight. When people are stumped by a problem, and then suddenly have that “aha” moment when they discover the answer, it is called insight problem solving. Following this revelation, the person is usually surprised at how simple the solution really was, and how it took them so long to come to it. One may have had this experience before with brain teasers. Individual insight problems are very often unique, with one solution method unable to be used on multiple problems. This makes them particularly useful to study because one cannot merely know the calculations, or use memorized processes, but must recognize a new solution path to each problem. But, perhaps most importantly, insight problem solving is related to creative thinking (Chu, 2011). In contrast to the large “knowledge leaps” made during insight, analytic problem solving is a more methodical, step-by-step approach.
It may seem obvious that different types of problems require different solution paths. The power rule works great on derivatives, but is of little use when determining the scope of a passage. Yet solution paths are not automatically given, and choosing the wrong one will produce an incorrect answer. Understanding how the brain arrived at which solution path to use is perhaps as important as the solution path itself. Analytic and insight problems have unique characteristics, and likewise the way in which their solution paths are recognized is different. Daniel Aiello et al. (2012) discusses the effects of an individual’s attentional state on problem solving. Unlike analytic problems, a focused, incremental method has little use in a creative problem that requires large leaps in discovery, such as insight problems. When people focus really hard their brain often gets too narrow in its attempt to solve the problem. Unable to see the multitude of options available they will stick to the usual solution paths. To solve a problem when the solution path is abnormal (such as an insight problem) the brain needs to be in a much more flexible state. For example, after consuming an abstemious amount of alcohol the average person has decreased working memory (good for analytic problem solving), but an increased capacity to creative problem solve (Aiello, 2012).
Aiello et al. (2012) hypothesised that a less focused attentional state is more beneficial to solving creative problems than a focused, high attentional state. To test this they used a Remote Associates Task (RAT), an example of a creative problem. If a less focused mindset was beneficial then individuals with this mindset would perform better on the RATs. Two factors were used to influence participants into this less focused mindset: an implicit learning task (in the form of an artificial grammar task) and a “use your gut” instruction (Aiello 118). The experiment was arranged with four conditions: 1) RAT task first, then artificial grammar task 2) Artificial grammar task first without “use your gut” instruction, then RAT task 3) RAT task first with “use your gut” instruction, then artificial grammar task 4) Artificial grammar task first with “use your gut” instruction, then RAT task (Aiello 121). The most significant factor was the “use your gut” instruction. The order the RAT and artificial grammar task were completed did not matter. Participants who received the “use your gut instruction” performed better on the artificial grammar task and the RAT. However, it is believed the “use your gut” instruction caused different reactions in both tasks. In the RAT it shifted participants away from forcing themselves to analyze every detail, to a more passive, opportunistic approach. For the artificial grammar task the direction to “use your gut” seemed to establish a default processing approach that was carried to the next task (Aiello 124).
In a related study Mark Beeman et al. (2011) and his colleagues also examined the relationship between a person’s attentional state and their aptitude for analytic or insight problem solving. This experiment primarily sought to determine whether a person’s focus level could be influenced by visual attention tasks, thereby causing them to solve a problem using insight (creative) or analytic thinking. The test problem was a type of verbal problem known as a Compound Remote Associate (CRA) problem, a closed-ended word problem that can be solved either analytically or with insight (Breeman 98). Participants were either put through a task that broadened their focus (a rapid object identification task), or one that narrowed it (a center-focused task) before performing the CRA problems. It was hypothesized that people who performed the focus narrowing task would solve more problems analytically, and people who performed the focus broadening task would solve more problems creatively (Breeman 101). In corroboration with their prediction, the results showed that participants who completed the focus-enhancing task used analytic problem solving on the second CRA problem set more often than they did in their first. Likewise, participants who completed the rapid object identification task cited using insight problem solving more often on the second set of CRA problems. Participants who completed the focus narrowing task also had an increase in the number of problems they correctly solved in CRA set 2 compared to CRA set 1. However, participants who completed the focus broadening task did not complete more problems in CRA set 2 than in CRA set 1 (Wegbreit 104-108).
Breeman and Aiello showed that a person’s attentional state (wide or narrow focus) affects how they approach solving problems. A highly attentive, narrowly focused state of mind will lead to an analytic approach, whereas a less attentive, broadly focused mindset a creative one. Furthermore, they demonstrated that by previously altering a person’s attentional state they can be influenced into solving a problem either analytically or creatively. Armed with this knowledge a person could manipulate their attentional state depending on what type of problem they were facing. For analytic problems a narrow focus will increase their chance of identifying the correct path to the solution. However, if they knew a creative solution were required they could broaden their focus in their attempt to discover the solution path.
In the start state of a problem there exists a given amount of information. This information must be sifted through and translated into the solution path to reach the goal state. For example, there is a lot of information available to a tourist exiting their hotel in New York City looking for the shortest route to the Empire State Building. The cardinal directions, names of roads, and distances are all important factors to consider. However, what color the car that just passed by was, how many pieces of gum are stuck to the bottom of the bench across the street, or the number of pigeons overhead probably are not going to be of much use. In this scenario determining what information was relevant to finding the fastest route to the Empire State Building was simple, but this is not always the case. For instance, Johan Kwisthout highlights a rather difficult problem many people can unfortunately relate too; when the problem to be solved is “make X love me”, where the current state is (assumed to be) “X doesn’t love me”, it is hard to agree on all the relevant aspects of this problem. What is easier to agree on is what the term relevant means in the context of problem solving. Gorayska and Lindsay produce a fitting definition, where an aspect is relevant towards a goal if there is some action plan from the current state to the goal where this aspect plays an essential role (Kwisthout 2012)
Gathering the relevant information from a problem is crucial to determine the solution path needed to answer the problem correctly. Yet this essential step in the process is often glossed over and taken for granted (Kwisthout 2012). For a given problem there might be several possible ways to solve it. For each solution path there is a minimum amount of information required to solve the problem. Therefore, the known information affects what solution paths are possible. This means the incorrect determination of relevant information can lead to the incorrect solution path. For instance, say an engineer is deciding on what metal to use for the water pipes in the new housing development. He concludes the relevant criteria for choosing a metal is its malleability. Under this criteria the best solution to the problem is lead pipes, but in reality the health hazard of the metal should be a relevant criteria. By taking the wrong information as relevant the engineer unfortunately picked the wrong metal to be used for the water pipes.
Problems become more difficult when much of the information presented appears to be relevant. When the technology or knowledge is not available, accurately narrowing down the amount of relevant information is sometimes not an option. While admitting that this dilemma was outside the scope of his paper, Kwisthout does have useful suggestions. By analyzing the amount of information that appears relevant, the difficulty of determining the solution path of a problem can be known. This means that a problem’s difficultly should not only be based on the complexity of the process or calculations required to get the answer, but also on the difficulty of determining what information is relevant. Kwisthout presents an excellent example by using the nine-dot problem. The problem appears straightforward; nine dots are equally spaced in a 3 x 3 matrix and the goal is to connect all the dots only using three lines. While seemingly irrelevant, the size of the dots is actual the key to solving the problem. Because the dots have area it is possible to connect them without going through their centers. Understanding that the area of the dots was relevant information lead to the formation of the correct solution path.
The total time spent solving a problem can be divided into two components: the time required to determine the solution path and the time required to calculate the answer from that solution path. In the real world problems need to be solved quickly as well as accurately. Shortening the time spent identifying the solution path helps make problem solving more efficient. Yun Chu et. Al. proposes a way to do this is by establishing direction. Direction allows a problem solver to determine the next step without checking alternative ones (Chu 2010). For every problem there exists a problem space: the number of possible ways a problem might be solved. Unfortunately, for complex problems, the amount of short term memory required to search a significant amount of the problem space in a limited amount of time far exceeds that of the average human. A common method employed is to systematically eliminate incorrect solution paths. Chu et. Al suggests utilizing direction to be a superior technique because it does not waste time pondering incorrect solution paths, but instead creates one towards the solution (Chu 2010).
To test their direction concept Chu et. Al created an experiment involving binary and labeled variations of a M + M puzzle.
•Binary: Four X’s and four O’s are displayed in a row:
Start State: XXXXOOOO
Goal State: OXOXOXOX
•Labeled: This version consists of letters with the goal of arranging them in alphabetical order.
Start state: HDJBFGEIAC
Goal state: ABCDEFGHIJ
Participants must rearrange the start state into the goal state by only moving pairs of adjacent X’s and O’s and only moving them to a pair of open spaces that does not affect other tiles.