knowledge of problem solving and reasoning skills

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7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

• Concepts and inferences (7.1)
• Procedural knowledge (7.1)
• Metacognition (7.1)
• Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
• Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
• Fixation:  functional fixedness, mental set  (7.3)
• Algorithms, heuristics, and the role of confirmation bias (7.3)
• Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

• Identify which type of knowledge a piece of information is (7.1)
• Recognize examples of deductive and inductive reasoning (7.2)
• Recognize judgments that have probably been influenced by the availability heuristic (7.2)
• Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

• Use the principles of critical thinking to evaluate information (7.1)
• Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
• Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

• Take a few minutes to write down everything that you know about dogs.
• Do you believe that:
• Psychic ability exists?
• Hypnosis is an altered state of consciousness?
• Magnet therapy is effective for relieving pain?
• Aerobic exercise is an effective treatment for depression?
• UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

• Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
• Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
• Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

• Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

• What percentage of kidnappings are committed by strangers?
• Which area of the house is riskiest: kitchen, bathroom, or stairs?
• What is the most common cancer in the US?
• What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

• Statement #1: The forecaster on Channel 2 said it is going to rain today.
• Statement #2: The forecaster on Channel 5 said it is going to rain today.
• Statement #3: It is very cloudy and humid.
• Statement #4: You just heard thunder.
• Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

• A particular class will be interesting/useful/difficult
• You will be able to finish writing a paper by next week if you go out tonight
• Your laptop’s battery will last through the next trip to the library
• You will not miss anything important if you skip class tomorrow
• Your instructor will not notice if you skip class tomorrow
• You will be able to find a book that you will need for a paper
• There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

• What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

• Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

• Please take a few minutes to list a number of problems that you are facing right now.
• Now write about a problem that you recently solved.
• What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

• Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
• Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
• Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
• Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
• Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
• Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Article • 8 min read

Critical Thinking

Developing the right mindset and skills.

By the Mind Tools Content Team

We make hundreds of decisions every day and, whether we realize it or not, we're all critical thinkers.

We use critical thinking each time we weigh up our options, prioritize our responsibilities, or think about the likely effects of our actions. It's a crucial skill that helps us to cut out misinformation and make wise decisions. The trouble is, we're not always very good at it!

In this article, we'll explore the key skills that you need to develop your critical thinking skills, and how to adopt a critical thinking mindset, so that you can make well-informed decisions.

What Is Critical Thinking?

Critical thinking is the discipline of rigorously and skillfully using information, experience, observation, and reasoning to guide your decisions, actions, and beliefs. You'll need to actively question every step of your thinking process to do it well.

Collecting, analyzing and evaluating information is an important skill in life, and a highly valued asset in the workplace. People who score highly in critical thinking assessments are also rated by their managers as having good problem-solving skills, creativity, strong decision-making skills, and good overall performance. [1]

Key Critical Thinking Skills

Critical thinkers possess a set of key characteristics which help them to question information and their own thinking. Focus on the following areas to develop your critical thinking skills:

Being willing and able to explore alternative approaches and experimental ideas is crucial. Can you think through "what if" scenarios, create plausible options, and test out your theories? If not, you'll tend to write off ideas and options too soon, so you may miss the best answer to your situation.

To nurture your curiosity, stay up to date with facts and trends. You'll overlook important information if you allow yourself to become "blinkered," so always be open to new information.

But don't stop there! Look for opposing views or evidence to challenge your information, and seek clarification when things are unclear. This will help you to reassess your beliefs and make a well-informed decision later. Read our article, Opening Closed Minds , for more ways to stay receptive.

Logical Thinking

You must be skilled at reasoning and extending logic to come up with plausible options or outcomes.

It's also important to emphasize logic over emotion. Emotion can be motivating but it can also lead you to take hasty and unwise action, so control your emotions and be cautious in your judgments. Know when a conclusion is "fact" and when it is not. "Could-be-true" conclusions are based on assumptions and must be tested further. Read our article, Logical Fallacies , for help with this.

Use creative problem solving to balance cold logic. By thinking outside of the box you can identify new possible outcomes by using pieces of information that you already have.

Self-Awareness

Many of the decisions we make in life are subtly informed by our values and beliefs. These influences are called cognitive biases and it can be difficult to identify them in ourselves because they're often subconscious.

Practicing self-awareness will allow you to reflect on the beliefs you have and the choices you make. You'll then be better equipped to challenge your own thinking and make improved, unbiased decisions.

One particularly useful tool for critical thinking is the Ladder of Inference . It allows you to test and validate your thinking process, rather than jumping to poorly supported conclusions.

Developing a Critical Thinking Mindset

Combine the above skills with the right mindset so that you can make better decisions and adopt more effective courses of action. You can develop your critical thinking mindset by following this process:

Gather Information

First, collect data, opinions and facts on the issue that you need to solve. Draw on what you already know, and turn to new sources of information to help inform your understanding. Consider what gaps there are in your knowledge and seek to fill them. And look for information that challenges your assumptions and beliefs.

Be sure to verify the authority and authenticity of your sources. Not everything you read is true! Use this checklist to ensure that your information is valid:

• Are your information sources trustworthy ? (For example, well-respected authors, trusted colleagues or peers, recognized industry publications, websites, blogs, etc.)
• Is the information you have gathered up to date ?
• Has the information received any direct criticism ?
• Does the information have any errors or inaccuracies ?
• Is there any evidence to support or corroborate the information you have gathered?
• Is the information you have gathered subjective or biased in any way? (For example, is it based on opinion, rather than fact? Is any of the information you have gathered designed to promote a particular service or organization?)

If any information appears to be irrelevant or invalid, don't include it in your decision making. But don't omit information just because you disagree with it, or your final decision will be flawed and bias.

Now observe the information you have gathered, and interpret it. What are the key findings and main takeaways? What does the evidence point to? Start to build one or two possible arguments based on what you have found.

You'll need to look for the details within the mass of information, so use your powers of observation to identify any patterns or similarities. You can then analyze and extend these trends to make sensible predictions about the future.

To help you to sift through the multiple ideas and theories, it can be useful to group and order items according to their characteristics. From here, you can compare and contrast the different items. And once you've determined how similar or different things are from one another, Paired Comparison Analysis can help you to analyze them.

The final step involves challenging the information and rationalizing its arguments.

Apply the laws of reason (induction, deduction, analogy) to judge an argument and determine its merits. To do this, it's essential that you can determine the significance and validity of an argument to put it in the correct perspective. Take a look at our article, Rational Thinking , for more information about how to do this.

Once you have considered all of the arguments and options rationally, you can finally make an informed decision.

Afterward, take time to reflect on what you have learned and what you found challenging. Step back from the detail of your decision or problem, and look at the bigger picture. Record what you've learned from your observations and experience.

Critical thinking involves rigorously and skilfully using information, experience, observation, and reasoning to guide your decisions, actions and beliefs. It's a useful skill in the workplace and in life.

You'll need to be curious and creative to explore alternative possibilities, but rational to apply logic, and self-aware to identify when your beliefs could affect your decisions or actions.

You can demonstrate a high level of critical thinking by validating your information, analyzing its meaning, and finally evaluating the argument.

Critical Thinking Infographic

See Critical Thinking represented in our infographic: An Elementary Guide to Critical Thinking .

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How People Learn II: Learners, Contexts, and Cultures (2018)

Chapter: 5 knowledge and reasoning, 5 knowledge and reasoning.

This chapter examines the development of knowledge as a primary outcome of learning and how learning is affected by accumulating knowledge and expertise. HPL I 1 emphasized these topics as well, but subsequent research has refined and extended understandings in a variety of learning domains. The first section of this chapter describes the problem of knowledge integration from the perspective of learning scientists and illustrates with research findings how people integrate their knowledge at different points in their development and in different learning situations. The second section describes what is known about the effects of accumulated knowledge and expertise on learning. The second half of the chapter discusses strategies for supporting learning. The committee has drawn on both laboratory- and classroom-based research for this chapter.

HPL I noted that the mind works actively to both store and recall information by imposing structure on new perceptions and experiences ( National Research Council, 2000 ). A central focus of HPL I was how experts structure their knowledge of a domain in ways that allow them to readily categorize new information and determine its relevance to what they already know. Because novices lack these frameworks, they have more difficulty assimilating and later recalling new information they encounter. This chapter expands on these themes from HPL I , citing relevant research reported since that study.

___________________

1 As noted in Chapter 1 , this report uses the abbreviation “ HPL I ” for How People Learn: Brain, Mind, Experience, and School: Expanded Edition ( National Research Council, 2000 ).

BUILDING A KNOWLEDGE BASE

Knowledge integration is a process through which learners put together different sorts of information and experiences, identifying and establishing relationships and expanding frameworks for connecting them. Learners must not only accumulate knowledge from individual episodes of experience but also integrate the knowledge they gain across time, location, circumstances, and the various formats in which knowledge appears ( Esposito and Bauer, 2017 ). How knowledge acquired in discrete episodes is integrated has been debated for decades ( Karmiloff-Smith, 1986 , 1990 ; Mandler, 1988 ; Nelson, 1974 ). Some researchers have suggested that infants are born with foundational knowledge that provides the elements necessary for learning and reasoning about their experiences ( Spelke, 2004 ; Spelke and Kinzler, 2007 ) or that infants can build from basic inborn reflexes to actively engage with the world and gradually build skills and knowledge ( Fischer and Bidell, 2006 ). Others have argued that all knowledge is generated through an individual’s direct experience with the world ( Greeno et al., 1996 ; Packer, 1985 ).

More recent work suggests that the integration of knowledge is a natural byproduct of the formation and consolidation of episodic memories ( Bauer, 2009 ; Bauer et al., 2012 ). As described in Chapter 4 , when a memory is consolidated, the learner associates representations of the elements of the experience (e.g., sights, sounds, tactile sensations) and these associations serve to help stabilize that memory. At the same time, these representations may also be linked with older memories from previous experiences that have already been stored in long-term memory ( Zola and Squire, 2000 ). The fact that old and new memory traces can be integrated shows that these traces are not fixed. Instead, elements common to the new and stored memory traces reactivate the old memory and, as the new memory is consolidated, the old memory may be reconstructed and undergo consolidation again ( Nader, 2003 ). When information from either learning episode is later retrieved, elements of both memory traces will be reactivated and will be simultaneously available for reintegration. As memory traces with common elements are simultaneously activated and linked, knowledge is expanded and memories are iteratively reworked. Figure 5-1 illustrates how this happens.

These linked traces may then be integrated with additional new information that comes to the learner later, and another new memory trace undergoes consolidation. Interestingly, it is exactly this process of integration of information from different episodes that may explain why people are sometimes unable to explain when and where they gained particular knowledge. Because the information generated by memory integration was not actually experienced as a single event, the information was not tagged with its origin ( Bauer and Jackson, 2015 ).

The studies of knowledge acquisition in children and college students presented in Box 5-1 illustrate the capacity to integrate unconnected infor-

mation and retain this knowledge starting at a very young age. These studies underscore the active role of the learner; that is, even young children do not simply accrue knowledge from what they have experienced directly but build knowledge from the many things that they have figured out on their own, which, over time, they can do with less repetition and external support.

As discussed in Chapter 2 , adequate sleep is important for integration and learning. The brain continues the work of encoding and consolidation during sleep and facilitates generalizations across learning episodes ( Coutanche et al., 2013 ; Van Kesteren et al., 2010 ). Specifically, activation of the hippocampus (which plays a key role in memory integration) during sleep seems to allow connections between memory traces to be formed across the cortex. This process promotes the integration of new information into existing memory traces, allows for abstraction across episodes ( Lewis and Durant, 2011 ), and leads to the possibility of building novel connections, which may be both creative and insightful or may be bizarre ( Diekelmann and Born, 2010 ).

BOX 5-1 Examples of Developmental Differences in the Process of Knowledge Acquisition

Knowledge and expertise.

When people repeatedly engage with similar situations or topics, they develop mental representations that connect disparate facts and actions into more effective mental structures for acting in the world. For example, when people first move to a new neighborhood, they may learn a set of discrete routes for traveling between pairwise locations, such as from home to school and from home to the grocery store. Over time, people naturally develop a mental representation of spatial relationships, or mental map, that stitches these discrete routes together. Even if they have never traveled between the school and the grocery store, they can figure out the most efficient route by consulting their mental map ( Thorndyke and Hayes-Roth, 1982 ). The observation that experts in a domain have developed frameworks of information and understanding through long experiences in a particular area was a central focus of HPL I . In this section, we briefly describe some of the benefits of expert knowledge (a more detailed discussion of the benefits of expertise appears in HPL I ) and then discuss the knowledge-related biases that may come with expertise.

Benefits of Expertise

One of the most well-documented benefits of the acquisition of knowledge is an increase in the speed and accuracy with which people can complete recurrent tasks: remembering a solution is faster than problem solving. Another benefit is that people who develop expertise can handle increasingly complex problems. One way this occurs is that people master substeps, so that each substep becomes a chunk of knowledge that does not require attention (e.g., Gobet et al., 2001 ). People also learn to handle complexity by developing mental representations that make specific tasks easier to complete. When Hatano and Osawa (1983) studied abacus masters, they found that even without an abacus in front of them, the masters had prodigious memories for numbers and could carry out addition problems with very large numbers because they had developed a mental representation of an abacus, which they manipulated virtually. These abacus masters did not show similarly superior ability to remember or keep track of letters or fruits—tasks that were not aided by manipulating a virtual abacus.

A third benefit is an increase in the ability to extract relevant information from the environment. Experts not only have better-developed knowledge representations than novices have but also can perceive more information that is relevant to those representations. For example, radiologists are able to see telling patterns in an x-ray that appear merely as shadows to a novice ( Myles-Worsley et al., 1988 ). The ability to discern more precise information complements a more-differentiated mental representation of those phenomena.

An implication of this ability is that students need to learn to see the relevant information in the environment to help differentiate concepts, such as the difference between a positive and a negative curvilinear slope ( Kellman et al., 2010 ).

A fourth benefit of acquiring expert knowledge is that it helps people use their environment as a resource. Using what is known as distributed cognition, people can offload some of the cognitive demands of a task onto their environment or other people ( Hollan et al., 2000 ). For instance, a major goal of learning is to develop knowledge of where to look for resources and help, and this is still important in the digital age. Experts typically know which tools are available and who in their network has specialized expertise they can call upon.

Finally, acquiring knowledge helps people gain more knowledge by making it easier to learn new and related information. Although some cognitive abilities related to learning novel information decline, on average, with age, these declines are offset by increases in knowledge accumulated through the life span, which empowers new learning. For example, in a study of young adults and older adults (in their 70s) who listened to a broadcast of a baseball game, the older adults who knew a lot about baseball recalled more of the broadcast than the young adults who knew less about baseball. This occurred despite the fact that the younger adults had superior executive functioning ( Hambrick and Engle, 2002 ).

Bias as a Natural Side Effect of Knowledge

As people’s knowledge develops, their thinking also becomes biased. But the biases may be either useful or detrimental to learning. The word “bias” often has negative connotations, but bias as understood by psychologists is a natural side effect of knowledge acquisition. Learning biases are often implicit and unknown to the individuals who hold them. They appear relatively early in knowledge acquisition, as people begin to form schemas (conceptual frameworks) for how the world operates and their place within it. These schemas help individuals know what to expect and what to attend to in particular situations (e.g., in a doctor’s office versus at a friend’s party) and help them develop a sense of cultural fluency—that is, to know how things work “around here” ( Mourey et al., 2015 ).

Psychologists distinguish two types of bias: one is intrinsic to learning and primarily useful and empowering to the learner; the second occurs when prior experiences or beliefs undermine the acquisition of new knowledge and skills.

An aphorism from the context of medical diagnosis illustrates the two types of bias: “When you hear hoof-beats, think of horses not zebras.” In the United States, horses are much more common than zebras so one is much more likely to encounter the common “horses” than the rare “zebras.” Of course, one should modify assumptions in light of additional evidence: if the

large mammal from which the hoof-beats emanate has black and white stripes, it is much more likely to be a zebra than a horse. Thus, if one sees a striped animal in a zoo but insists that it is a horse and not a zebra, this resistance to new information is a strong form of the limiting effects of bias on learning. A person may fail even to notice the zebra at the zoo because he was so strongly expecting to see a horse instead and was attuned to notice only that kind of animal.

Making matters even more complicated, two people who have different prior levels of expertise, or different beliefs, might legitimately have different interpretations when initially presented with the same information. But if sufficient additional information suggests a particular interpretation, they should converge on an answer, especially if the higher level of expertise is brought to bear.

Beliefs about human-caused global climate change are a good example of the biases that blind individuals to new evidence. Despite nearly universal consensus among climate scientists that global climate change is taking place and that this change is induced by humans’ behavior, a considerable proportion of adults in the United States do not accept these interpretations of the evidence. One might expect that higher levels of science literacy would be associated with greater agreement with the scientific consensus. However, Kahan and colleagues (2012) found that it is among the individuals with the highest levels of science literacy that the most stark polarization is apparent. Those who only seek out and attend to information consistent with their prior beliefs will create an “echo-chamber” that further biases their learning. Often this echo-chamber effect is socially reinforced, as individuals prefer to discuss the topic in question with others whom they know hold beliefs similar to their own.

Stereotypes perpetuate themselves through learned bias, but not all learning biases are considered to have negative consequences. For example, some positive biases promote well-being and mental health ( Taylor and Brown, 1988 ), some may promote accuracy in perceptions of other people ( Funder, 1995 ), and others may be adaptive behaviors—for example, selective attention and action in situations in which errors have a high cost ( Haselton and Buss, 2000 ; Haselton and Funder, 2006 ). Hahn and Harris (2014) have written a useful historical overview of research on bias in human cognition.

Still other biases refine perception and serve to blur distinctions within categories that are not meaningful while highlighting subtle cross-category distinctions that may be important. For example, very young infants respond equally to phonological contrasts that matter in their language (e.g., “r” and “l” if the baby lives in an English-speaking context) and those that do not matter (e.g., “r” and “l” in a Japanese-speaking context). Over time, infants lose this discriminatory capability. This loss is actually a benefit, reflecting the baby’s increasing efficiency in processing his own language context, and is a mark of

learning ( Kuhl et al., 1992 ). In the other direction, dermatologists may learn from experience and formal training to distinguish subtle features of moles and skin growths that signal malignancy, features that to an untrained eye are indistinguishable from those of benign growths.

Biases affect the noncognitive aspects of learning as well. In a variable world, highly stable task environments are not guaranteed and so training to high efficiency may actually create a mindset that makes new learning more difficult, impeding motivation and interest in continuous growth and development. For instance, a person who has learned how to organize her schedule using a specific tool may be reluctant to learn a new tool because of the perception that it will take too much time to learn to use it, even though it may be more efficient in the long run. In this example, it is not that the person is unable to learn the new tool; rather, her beliefs about the amount of effort required affect her motivation and interest in learning. This kind of self-attribution, or prior knowledge of oneself, can have a large influence on how people approach future learning opportunities, which in turn influences what they will learn ( Blackwell et al., 2007 ).

KNOWLEDGE INTEGRATION AND REASONING

We have seen that building a knowledge base requires doing three things: accumulating information (in part by noticing what matters in a situation and is therefore worth attending to); tagging this information as relevant or not; and integrating it across separate episodes. These three activities can happen relatively quickly and automatically, or they can happen slowly through deliberate reflection. However, these processes alone are not sufficient for integrating and extending knowledge. Learners of all ages know many things that were not explicitly taught or directly experienced. They routinely generate their own novel understanding of the information they are accumulating and productively extend their knowledge.

Inferential Reasoning

Inferential reasoning refers to making logical connections between pieces of information in order to organize knowledge for understanding and to drawing conclusions through deductive reasoning, inductive reasoning, and abductive reasoning ( Seel, 2012 ). Inferential thinking is needed for such processes as generalizing, categorizing, and comprehending. The act of reading a text is a good example. To comprehend a text, readers are required to make inferences regarding information that is only implied in the text (see, e.g., Cain and Oakhill, 1999 ; Graesser et al., 1994 ; Paris and Upton, 1976 ). Some types of inferences help readers track the meaning of a text by integrating different information it supplies, for example by recognizing anaphoric

references (words in a text that require the reader to refer back to other ideas in the text for their meaning). Other types of inferences allow a reader to fill in gaps in the text by recruiting information from beyond it (i.e., background knowledge), in order to understand information within the text. Though these types of inferences are essential for understanding, they are thought to survive in working memory only long enough to aid comprehension ( McKoon and Ratcliff, 1992 ).

Other inferences that learners make survive beyond the bounds of working memory and become incorporated into their knowledge base. For example, a person who knows both that liquids expand with heat and that thermometers contain liquid may integrate these two pieces of information and infer that thermometers work because liquid expands as heat increases. In this way, the learner generates understanding through a productive extension of prior learning episodes.

Effective problem solving typically requires retrieved knowledge to be adapted and transformed to fit new situations; therefore, memory retrieval must be coordinated with other cognitive processes. One way to help people realize that something they have learned before is relevant to their current task is to explicitly give them a hint that it is relevant ( Gick and Holyoak, 1980 ). For example, such hints might be embedded in text, provided by a teacher, or incorporated into virtual learning platforms. Another strategy for helping people realize that they already know something useful is to ask people to compare related problems in order to highlight exactly what they have in common, increasing the likelihood that they will recall previously acquired knowledge with similar properties ( Alfieri et al., 2013 ; Gentner et al., 2009 ).

Kolodner et al. (2003) gives the example of an architect trying to build an office building with a naturally lit atrium. She realizes that a familiar library’s design, which includes an exterior wall of glass, could be reused for the office building, but would fit the building’s needs better if translucent glass bricks were used instead of a clear, glass pane. This kind of design-based reasoning is incorporated into problem-based learning ( Hmelo-Silver, 2004 ) activities. Problem-based learning emphasizes that memories are not simply stored to allow future reminiscing, but are formed so that they can be used, reshaped, and flexibly adapted to serve broad reasoning needs. The goal of problem-based learning is to instill in learners flexible knowledge use, effective problem-solving skills, self-directed learning, collaboration, and intrinsic motivation. These goals are in line with several of the goals identified in other contexts as important for success in life and work ( National Research Council, 2012b ).

Age-Related Changes in Knowledge and Reasoning

People’s learning benefits from a steady increase, over many decades, in the accumulation of world knowledge (e.g., Craik and Salthouse, 2008 ;

Hedden and Gabrieli, 2004 ). This accumulation makes it easier for older adults not only to retrieve vocabulary and facts about the world ( Cavanagh and Blanchard-Fields, 2002 ) but also to acquire new information in domains related to their expertise. For example, physicians acquire medical expertise, which enables them to comprehend and remember more information from medical texts than novices can ( Patel et al., 1986 ). It is also thought that older adults can compensate for declines in some abilities by using their extensive world knowledge. For instance, medical experts depend less on working memory because they can draw on their expertise to reconstruct only those facts from long-term memory that are relevant to a current need (e.g., Patel and Groen, 1991 ).

The knowledge learners accumulate throughout the life span is the growing product of the processes of both learning new information from direct experience and generating new information based on reasoning and imagining ( Salthouse, 2010 ). These two cognitive assets together—accumulated knowledge and reasoning ability—are particularly relevant to healthy aging. Reasoning and knowledge abilities tend to be correlated. That is, people who have comparatively higher reasoning capacity are likely to acquire correspondingly more knowledge over the life span than their peers ( Ackerman and Beier, 2006 ; Beier and Ackerman, 2005 ). Reasoning ability is a major determinant of learning throughout life, and it is through reasoning, especially in contexts that allow people to pursue their interests, that people develop knowledge throughout their life span ( Ackerman, 1996 ; Cattell, 1987 ).

On average, however, the trajectories of reasoning and knowledge acquisition are different across the life span. A number of research studies have described the general trajectories of age-related changes in ability, using a variety of measures and research designs (cross-sectional and longitudinal), and have shown a fairly consistent trend in which the development of knowledge remains steady as reasoning capacity (the ability to quickly and accurately manipulate multiple distinct pieces of factual information to make inferences) drops off ( Salthouse, 2010 ). However, there is considerable individual variability in the trajectories, which reflect individual health and other characteristics, as well as educational and experiential opportunities and even social engagement. Yet, even though there is an average decline in inferential reasoning capacity through adulthood, there is not a corresponding decline in the ability to make good decisions—a more colloquial use of the word “reasoning.” In other words, the research does not suggest that the average 14-year-old reasons better about what to do in a complex or emotional real-world situation than would an average 50-year-old. Instead, it describes the 14-year-old’s stronger ability to quickly manipulate multiple distinct pieces of factual information to make logical and combinatorial inferences.

The growth or decline of abilities can be expected to vary not only between individuals but also within the same person over time ( Hertzog et al.,

2008 ). Two 50-year-olds may have extremely different cognitive profiles, such that one may generally have the same ability profile as an average 30-year-old and the other may more closely resemble an average 70-year-old. Within the same person, abilities will decline or grow at varying rates as a function of that individual’s continuing use of some skills and intellectual development in particular domains; losses and declines are associated with disuse of other skills. (Factors that influence cognitive aging are discussed in Chapter 9 .) As mentioned, new learning depends on both reasoning ability and knowledge acquisition ( Ackerman and Beier, 2006 ; Beier and Ackerman, 2005 ). Even though reasoning abilities decline with age, knowledge accumulated throughout the life span facilitates new learning, as long as the information to be learned is aligned with existing domain knowledge. When people select environments for education, work, and hobbies that capitalize on their already-established knowledge and skills as they age, their selectivity allows them to capitalize on their repertoire of knowledge and expertise for learning new information ( Baltes and Baltes, 1990 ).

Cognitive abilities change throughout the life span in a variety of ways that may affect a person’s ability to learn new things (see Hartshorne and Germine, 2015 , for discussion). For instance, as people age, learning may rely more on knowledge and less on reasoning and quick manipulation of factual information. However, examining peoples’ cognitive abilities and learning becomes increasingly complex as people develop past the age of formal education. One reason is that the ways in which people learn become increasingly idiosyncratic outside of a standardized educational curriculum, and understanding this process requires assessing knowledge gained through a wide variety of adult experiences that different individuals amass over a lifetime ( Lubinski, 2000 ). The unique complexities of adult learning and development are discussed in Chapter 8 .

Effects of Culture on Reasoning

As described in Chapter 2 , learning is inherently cultural, given that a person’s experiences in a culture affect biological processes that support learning, perception, and cognition. In the area of reasoning, for example, researchers have explored fundamental differences in peoples’ reasoning about three basic domains of life: physical events (naïve physics), biological events (naïve biology), and social or psychological events (naïve psychology) (see e.g., Carey, 1985 , 2009 ; Goswami, 2002 ; Hirschfeld and Gelman, 1994 ; Spelke and Kinzler, 2007 ; also see Ojalehto and Medin, 2015c , for a review). These distinctions are compelling in the sense that each reflects a set of intuitive principles and inferences. That is, each domain is defined by entities having the same kind of causal properties. These might be marked, for example, by the way they move: physical entities are set into motion by external forces,

while biological entities may propel themselves. These domains are important for understanding cognition because researchers have suggested that whereas the perception of physical causality is universal, causal reasoning in the biological and psychological domains is culturally variable.

Two studies illustrate ways to examine these issues. Morris and Peng (1994) presented two types of animated displays to American and Chinese participants. One set of displays depicted physical interactions (of geometrical shapes), whereas the other set depicted social interactions (among fish). The participants’ answers to questions about what they had seen suggested differences in attention to internal and external causes across the groups, but those differences depended on the domain (social or physical). The authors concluded that attribution of causality in the social domain is susceptible to cultural influences but that causality in the physical domain is not.

Beller and colleagues (2009) asked German, Chinese, and Tongan participants to indicate which entity they regarded as causally most relevant for statements such as “The fact that wood floats on water is basically due to . . . ”. Ratings varied by the cultural background of respondents and also by the phenomena participants were considering. In general, the German and Chinese participants, but not the Tongan participants, considered a carrier’s capability for buoyancy only when the floater was a solid object, such as wood, but not when it was a fluid, such as oil ( Beller et al., 2009 ; see also Bender et al., 2017 ). This is an area of research that has barely been explored, but results to date suggest that the perception of physical causality may in fact not be universal and may be learned in culturally mediated ways.

STRATEGIES TO SUPPORT LEARNING

People are naturally interested in strengthening their ability to acquire and retain knowledge and in ways to improve learning performance. Researchers have explored a variety of strategies to support learning and memory. They have identified several principles for structuring practice and engaging with information to be learned to improve memory, to make sense of new information, and to develop new knowledge.

Several scholars have looked across the research on the effectiveness of specific strategies for supporting learning ( Benassi et al., 2014 ; Dunlosky et al., 2013 ; Pashler et al., 2007 ). The authors of these three studies looked for strategies that (1) have been examined in several studies, using authentic educational materials in classroom settings; (2) show effects that can be generalized across learner characteristics and types of materials; (3) promote learning that is long-lasting; and (4) support comprehension, knowledge application, and problem solving in addition to recall of factual material. These three analyses identified five learning strategies as promising:

• retrieval practice;
• spaced practice;
• interleaved and varied practice;
• summarizing and drawing; and
• explanations: elaborative interrogation, self-explanation, and teaching.

Strategies for Knowledge Retention

The first three strategies are ways of structuring practice that are particularly useful for increasing knowledge retention.

Retrieval Practice

Some evidence shows that the act of retrieval itself enhances learning and that when learners practice retrieval during an initial learning activity, their ability to retrieve and use knowledge again in the future is enhanced ( Karpicke, 2016 ; Roediger and Karpicke, 2006b ). The benefits of retrieval practice in general have been shown to generalize across individual differences in learners, variations in materials, and different assessments of learning. For example, researchers have found effects across learner characteristics in children ( Lipko-Speed et al., 2014 ; Marsh et al., 2012 ). Studies have also suggested that retrieval practice can be a useful memory remediation method among older adults ( Balota et al., 2006 ; Meyer and Logan, 2013 ; also see Dunlosky et al., 2013 , for a review of effective learning techniques). However, most of this research has addressed retrieval of relatively simple information (e.g., vocabulary), rather than deep understanding.

Research has also demonstrated the effects of retrieval practice on recall of texts and other information related to school subjects. For example, Roediger and Karpicke (2006a) had students read brief educational texts and practice recalling them. Students in one condition read the texts four times; students in a second group read three times and recalled the texts once by writing down as much as they could remember; and students in a third group read the material once and then recalled it during three retrieval practice periods. On a final test given 1 week after the initial learning session, students who practiced retrieval one time recalled more of the material than students who only read the texts, and the students who repeatedly retrieved the material performed the best. The results suggest that actively retrieving the material soon after studying it is more productive than spending the same amount of time repeatedly reading.

Attempting retrieval but failing has also been shown to promote learning. Failed retrievals provide feedback signals to learners, signaling that they may not know the information well and should adjust how they encode the material the next time they study it ( Pyc and Rawson, 2010 ). The act of failing to retrieve may thus enhance subsequent encoding ( Kornell, 2014 ).

Such studies suggest that self-testing can be an effective way for students to practice retrieval. However, evidence from surveys of students’ learning strategies and from experiments in which learners are given control over when and how often they can test themselves suggests that students may not test themselves often or effectively enough ( Karpicke et al., 2009 ; Kornell and Son, 2009 ). Many students do not engage in self-testing at all, and when students do test themselves, they often do so as a “knowledge check” to see whether they can or cannot remember what they are learning. While this is an important use of self-testing, few learners self-test because they view the act of retrieval as part of the process of learning. Instead, they are likely to retrieve something once and then, believing they have learned it for the long term, drop the item from further practice.

Spaced Practice

Researchers who have compared spaced and massed practice have shown that the way that learners schedule practice can have an impact on learning ( Carpenter et al., 2012 ; Kang, 2016 ). Massed practice concentrates all of the practice sessions in a short period of time (such as cramming for a test), whereas spaced practice distributes learning events over longer periods of time. Results show greater effects for spacing than for massed practice across learning materials (e.g., vocabulary learning, grammatical rules, history facts, pictures, motor skills) ( Carpenter et al., 2012 ; Dempster, 1996 ), stimulus formats (e.g., audiovisual, text) ( Janiszewski et al., 2003 ), and for both intentional and incidental learning ( Challis, 1993 ; Toppino et al., 2002 ). Studies have shown benefits of spaced practice for learners of ages 4 through 76 ( Balota et al., 1989 ; Rea and Modigliani, 1987 ; Simone et al., 2012 ; Toppino, 1991 ). Cepeda and colleagues (2006) found that spaced practice led to greater recall than massed practice regardless of the size of the lag between practice and recall.

There are many possible reasons why spaced practice might be more effective than massed practice. When an item, concept, or procedure is repeated after a spaced interval, learners have to fully engage in the mental operations they performed the first time because of forgetting that has occurred. But when repetitions are immediate and massed together, learners do not fully engage during repetitions. In the case of reading, one possible reason why massed re-readings do not promote learning is that when people reread immediately, they do not attend to the most informative and meaningful portions of the material during the second reading, as illustrated by Dunlosky and Rawson (2005) in a study of self-paced reading.

A few researchers have attempted to identify the spacing intervals that promote the most memory—a “sweet spot” where spaced practice confers benefits before too much forgetting has occurred ( Cepeda et al., 2008 ; Pavlik and Anderson, 2008 ). For example, a study of vocabulary learning among fifth

graders suggested that a 2-week interval showed the best results ( Sobel et al., 2011 ). Another classroom-based study of spacing effects focused on first-grade children learning to associate letters and sounds during phonics instruction ( Seabrook et al., 2005 ). The children who received spaced practice during the 2-week period significantly outperformed the children who received a single massed practice session each day.

In general, the literature on spaced practice suggests that separating learning episodes by at least 1 day, rather than focusing the learning into a single session, maximizes long-term retention of the material. However, it is important to note that wider spacing is not necessarily always better. The optimal distribution of learning sessions depends at least in part on how long the material needs to be retained in memory (i.e., when the material will be recalled or tested). For example, if the learner will be tested 1 month or more after the last learning session, then the learning should be distributed over weeks or months.

Interleaved and Variable Practice

The way information is presented can significantly affect both what is learned ( Schyns et al., 1998 ) and how well it is learned ( Goldstone, 1996 ). Variable learning generally refers to practicing skills in different ways, while interleaving refers to mixing in different activities. Varying or interleaving different skills, activities, or problems within a learning session—as opposed to focusing on one skill, activity, or problem throughout (called blocked learning)—may better promote learning. Both strategies may also involve spaced practice, and both also present learners with a variety of useful challenges, or “desirable difficulties.” Researchers have identified potential benefits of variable and interleaved practice learning, but they have also found a few benefits for blocked practice.

Several studies have shown benefits for blocking, at least for category learning ( Carpenter and Mueller, 2013 ; Goldstone, 1996 ; Higgins and Ross, 2011 ). Moreover, when given the option, a majority of learners preferred to block their study ( Carvalho et al., 2014 ; Tauber et al., 2013 ). Interleaving can boost learning of the structure of categories; that is, learning that some objects or ideas belong to the same category and others do not ( Birnbaum et al., 2013 ; Carvalho and Goldstone, 2014a , 2014b ; Kornell and Bjork; 2008 ). Other researchers have examined interleaved practice in mathematical problem-solving domains ( Rohrer, 2012 ; Rohrer et al., 2015 ).

Carvalho and Goldstone (2014a) found that the effectiveness of the presentation methods (interleaved or blocked) depended on whether the participant engaged in active or passive study. They also found that interleaving concepts improved students’ capacity to discriminate among different categories, while blocked practice emphasized similarities within each category. These results

suggest that interleaved study improves learning of highly similar categories (by facilitating between-category comparisons), whereas blocked study improves learning of low-similarity categories (by facilitating within-category comparisons).

Interleaved study naturally includes delays between learning blocks and thus easily allows for spaced practice, which has the potential benefits for long-term memory discussed above. However, it may be beneficial because it helps learners to make comparisons among categories, not because it allows time to elapse between learning blocks ( Carvalho and Goldstone, 2014b ). The mechanisms that underlie the benefits of either interleaved or blocked study (e.g, possible effects on attentional processes) are ongoing topics of research. As with other strategies, the optimal way to present material—interleaved or blocked—and the mechanisms most heavily involved will likely depend on the nature of the study task.

Strategies for Understanding and Integration

The other two strategies for which there is strong evidence—summarizing and drawing and developing explanations—draw on inferential processes that research shows to be effective for organizing and integrating information for learning.

Summarizing and Drawing

Summarizing and drawing are two common strategies for elaborating on what has been learned. To summarize is to create a verbal description that distills the most important information from a set of materials. Similarly, when learners create drawings, they use graphic strategies to portray important concepts and relationships. In both activities, learners must take the material they are learning and transform it into a different representation. There are differences between them, but both activities involve identifying important terms and concepts, organizing the information, and using prior knowledge to create verbal or pictorial representations.

Both summarization and drawing have been shown to benefit learning in school-age children ( Gobert and Clement, 1999 ; Van Meter, 2001 ; Van Meter and Garner, 2005 ). Literature reviews by Dunlosky and colleagues (2013) and Fiorella and Mayer (2015a , 2015b ) have identified factors that appear to contribute to the effectiveness of summarization and drawing activities.

A few studies have suggested that the quality of students’ summaries and drawings is directly related to how much they learn from the activities and that learners do these activities more effectively when they are trained and guided ( Bednall and Kehoe, 2011 ; Brown et al., 1983 ; Schmeck et al., 2014 ). For example, the effectiveness of drawing activities is enhanced when learners

compare their drawings to author-generated pictures ( Van Meter et al., 2006 ). Similarly, providing learners with a list of relevant elements to be included in drawings and partial drawings helps learners create more complete drawings and bolsters learning ( Schwamborn et al., 2010 ).

A group of researchers compared summarization and drawing and suggested that their effectiveness depends on the nature of the learning materials. For example, Leopold and Leutner (2012) asked high school students who were studying a science text about water molecules, which contained descriptions of several spatial relations, to either draw diagrams, write a summary of the text, or to re-read the text (the control condition). Those who created drawings performed better on a comprehension test than those who re-read the texts. However, those who created written summaries performed worse than those who re-read. The authors concluded that the drawing was more effective in this case because the learning involved spatial relations.

Note-taking, either writing by hand or typing on a laptop, is a form of summarizing that has also been studied. For example, Mueller and Oppenheimer (2014) found that students who hand-wrote notes learn more than those who typed notes using a laptop computer. The researchers asked students to take notes in these two ways and then tested their recall of factual details, conceptual understanding, and ability to synthesize and generalize the information. They found that students who typed took more voluminous notes than those who wrote by hand, but the hand-writers had a stronger conceptual understanding of the material and were more successful in applying and integrating the material than the typers. The researchers suggested that because writing notes by hand is slower, students doing this cannot take notes verbatim but must listen, digest, and summarize the material, capturing the main points. Students who type notes can do so quickly and without processing the information.

Mueller and Oppenheimer (2014) also examined the contents of notes taken by college students in these two ways across a number of disciplines. They found that the typed notes—which were closer to verbatim transcriptions—were associated with lower retention of the lecture material. Even when study participants using laptops were instructed to think about the information and type the notes in their own words, they were no better at synthesizing material than students who were not given the warning. The authors concluded that typing notes does not promote understanding or application of the information; they suggested that notes in the students’ own words and handwriting may serve as more effective memory prompts by recreating context (e.g., thought processes, conclusions) and content from the original lecture.

Developing Explanations

Encouraging learners to create explanations of what they are learning is a promising method of supporting understanding. Three techniques for doing this have been studied: elaborative interrogation, self-explanation, and teaching.

Elaborative interrogation is a strategy in which learners are asked, or are prompted to ask themselves, questions that invite deep reasoning, such as why, how, what-if, and what-if not (as opposed to shallow questions such as who, what, when, and where) ( Gholson et al., 2009 ). A curious student who applies intelligent elaborative interrogation asks deep-reasoning questions as she strives to comprehend difficult material and solve problems. However, elaborative interrogation does not come naturally to most children and adults; training people to use this skill—and particularly training in asking deep questions—has been shown to have a positive impact on comprehension, learning, and memory ( Gholson et al., 2009 ; Graesser and Lehman, 2012 ; Graesser and Olde, 2003 ; Rosenshine et al., 1996 ). For example, in an early study, people were asked either to provide “why” explanations for several unrelated sentences or to read and study the sentences. Both groups were then tested on their memory of the sentences. Those who asked questions performed better than the group that just studied the sentences ( Pressley et al., 1987 ). Studies with children have also shown benefits of elaborative interrogation ( Woloshyn et al., 1994 ), and the benefits of elaborative interrogation can persist over time (e.g., 1 or 2 weeks after learning), though few studies have examined effects of elaborative interrogation on long-term retention.

Most studies conducted by researchers in experimental psychology have used isolated facts as materials in studying the effects of elaboration and have assessed verbatim retention, but researchers in educational psychology have also looked at more complex text content and assessed inference making ( Dornisch and Sperling, 2006 ; Ozgungor and Guthrie, 2004 ). For example, McDaniel and Donnelly (1996) asked college students to study short descriptions of physics concepts, such as the conservation of angular momentum, and then answer a why question about the concept (e.g., “Why does an object speed up as its radius get smaller, as in conservation of angular momentum?”). A final assessment involved both factual questions and inference questions that tapped into deeper levels of comprehension. The authors found benefits of elaborative interrogation for complex materials and assessments and also found that those who engaged in elaborative interrogation outperformed learners who produced labeled diagrams of the concepts in each brief text.

Self-explanation is a strategy in which learners produce explanations of material or of their thought processes while they are reading, answering questions, or solving problems. In the most general case, learners may simply be asked to explain each step they take as they solve a problem ( Chi et al., 1989b ; McNamara, 2004 ) or explain a text sentence-by-sentence as they read it ( Chi

et al., 1994 ). Self-explanation involves more open-ended prompts than the specific “why” questions used in elaborative interrogation, but both strategies encourage learners to elaborate on the material by generating explanations. Other examples of this work include self-explanations of physics.

An early study of self-explanation was carried out by Chi and colleagues (1994) . Eighth-grade students learned about the circulatory system by reading an expository text. While one group just read the text, a second group of students produced explanations for each sentence in the text. The students who self-explained showed larger gains in comprehension of concepts in the text. A subsequent study showed similar results ( Wylie and Chi, 2014 ). Self-explanation has now been explored in a wide range of contexts, including comprehension of science texts in a classroom setting ( McNamara, 2004 ), learning of chess moves ( de Bruin et al., 2007 ), learning of mathematics concepts ( Rittle-Johnson, 2006 ), and learning from worked examples on problems that require reasoning ( Nokes-Malach et al., 2013 ). Self-explanation prompts have been included in intelligent tutoring systems ( Aleven and Koedinger, 2002 ) and systems with game components ( Jackson and McNamara, 2013 ; Mayer and Johnson, 2010 ). However, relatively few studies have examined the effects of self-explanation on long-term retention or explored the question of how much self-explanation is needed to produce notable results ( Jackson and McNamara, 2013 ).

A few studies have explored the relationship between self-explanation and prior knowledge in learning ( Williams and Lombrozo, 2013 ). For example, Ionas and colleagues (2012) investigated whether self-explanation was beneficial to college students who were asked to do chemistry problems. They found that prior knowledge moderated the effectiveness of self-explanation and that the more prior knowledge of chemistry the students reported having, the more self-explanation appeared to help them learn. Moreover, for students who had just a little prior knowledge, using self-explanation seemed to impede rather than support performance. The researchers suggested that learners search for concepts or processes in their prior knowledge to make sense of new material; when the prior knowledge is weak, the entire process fails. They concluded that educators should thoroughly assess the learners’ prior knowledge and use other cognitive support tools and methods during the early stages of the learning process, as learners strengthen their knowledge base.

Finally, teaching others can be an effective learning experience. When learners prepare to teach they must construct explanations, just as they do in elaborative interrogation and self-explanation activities. However, elaborative interrogation and self-explanation both require that the learner receive fairly specific prompts, whereas the act of preparing to teach can be more open-ended. Teaching others is often an excellent opportunity to hone one’s own knowledge ( Biswas et al., 2005 ; Palincsar and Brown, 1984 ), and learners in this kind of interaction are likely to feel empowered and responsible in a

way that they do not feel when they are the passive recipients of knowledge ( Scardamalia and Bereiter, 1993 ). Peers may be able to express themselves to each other in ways that are particularly relevant, immediate, and informative. Although peer learning and teaching are often quite effective, teachers and instructors typically come closer to injunctive norms and provide better models to observe.

A foundational study of the effects of teaching on learning by Bargh and Schul (1980) has served as a template for subsequent studies. Bargh and Schul asked participants to study a set of materials and either prepare to teach the material to a peer or simply study it for an upcoming test. Both groups were tested on the material without teaching it; only the expectation to teach had been manipulated. Students who prepared to teach others performed better on the assessment than students who simply read and studied the material. Effects of preparing to teach have been replicated in studies since Bargh and Schul’s foundational work (e.g., Fiorella and Mayer, 2014 ).

The benefits of teaching are evident in other contexts. For example, research on tutoring has shown that while students certainly learn by being tutored, the tutors themselves learn from the experience (see Roscoe and Chi, 2007 ). Reciprocal teaching is another strategy, used primarily in improving students’ reading comprehension ( Palincsar, 2013 ; Palincsar and Brown, 1984 ). In reciprocal teaching, students learn by taking turns teaching material to each other. The students are given guidance: training in four strategies to help them recognize and react to signs of comprehension breakdown (questioning, clarifying, summarizing, and predicting) ( Palincsar, 2013 ).

The research suggests several possible reasons why teaching may benefit learners. Preparing to teach requires elaborative processing because learners need to generate, organize, and integrate knowledge. Also, as mentioned, the explanations that people create may promote learning in the same way that elaborative interrogation and self-explanations promote learning. The process of explaining to others is active and generative, and it encourages learners to focus on deeper questions and levels of comprehension. Explaining in a teaching context also involves retrieval practice, as the teacher actively engages in retrieving knowledge in order to explain instructional content and answer questions. Although researchers have documented benefits of explanation, there are cautions to bear in mind. For example, a few researchers in this area have noted that in developing explanations learners may tend to make broad generalizations at the expense of significant specifics ( Lombrozo, 2012 ; Williams and Lombrozo, 2010 ; Williams et al., 2013 ). Children tend to prefer a single explanation for two different phenomena (e.g., a toy that both lights up and spins), even when there are two independent causes ( Bonawitz and Lombrozo, 2012 ). Likewise, when diagnosing diseases based on observable symptoms, adults tend to attribute the two symptoms to a single disease, even when it is more likely that there are two separate diseases ( Lombrozo, 2007 ;

Pacer and Lombrozo, 2017 ). The tendency to prefer simple, broad explanations over more complex ones may affect what people learn and the inferences they draw. For each of the different types of explanation strategies, researchers have noted reasons for educators to plan carefully when and how they can be used most effectively.

CONCLUSIONS

Learners identify and establish relationships among pieces of information and develop increasingly complex structures for using and categorizing what they have learned. Accumulating bodies of knowledge, structuring that knowledge, and developing the capacity to reason about the knowledge one has are key cognitive assets throughout the life span.

Strategies for supporting learning include those that focus on retention and retrieval of knowledge as well as those that support development of deeper and more sophisticated understanding of what is learned. The strategies that have shown promise for promoting learning help learners to develop the mental models they need to retain knowledge so they can use it adaptively and flexibly in making inferences and solving new problems.

CONCLUSION 5-1: Prior knowledge can reduce the attentional demands associated with engaging in well-learned activities, and it can facilitate new learning. However, prior knowledge can also lead to bias by causing people to not attend to new information and to rely on existing schema to solve new problems. These biases can be overcome but only through conscious effort.

CONCLUSION 5-2: Learners routinely generate their own novel understanding of the information they are accumulating and productively extend their knowledge by making logical connections between pieces of information. This capacity to generate novel understanding allows learners to use their knowledge to generalize, categorize, and solve problems.

CONCLUSION 5-3: The learning strategies for which there is evidence of effectiveness include ways to help students retrieve information and encourage them to summarize and explain material they are learning, as well as ways to space and structure the presentation of material. Effective strategies to create organized and distinctive knowledge structures encourage learners to go beyond the explicit material by elaborating

and to enrich their mental representation of information by calling up and applying it in various contexts.

CONCLUSION 5-4: The effectiveness of learning strategies is influenced by such contextual factors as the learner’s existing skills and prior knowledge, the nature of the material, and the goals for learning. Applying these approaches effectively therefore requires careful thought about how their specific mechanisms could be beneficial for particular learners, settings, and learning objectives.

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There are many reasons to be curious about the way people learn, and the past several decades have seen an explosion of research that has important implications for individual learning, schooling, workforce training, and policy.

In 2000, How People Learn: Brain, Mind, Experience, and School: Expanded Edition was published and its influence has been wide and deep. The report summarized insights on the nature of learning in school-aged children; described principles for the design of effective learning environments; and provided examples of how that could be implemented in the classroom.

Since then, researchers have continued to investigate the nature of learning and have generated new findings related to the neurological processes involved in learning, individual and cultural variability related to learning, and educational technologies. In addition to expanding scientific understanding of the mechanisms of learning and how the brain adapts throughout the lifespan, there have been important discoveries about influences on learning, particularly sociocultural factors and the structure of learning environments.

How People Learn II: Learners, Contexts, and Cultures provides a much-needed update incorporating insights gained from this research over the past decade. The book expands on the foundation laid out in the 2000 report and takes an in-depth look at the constellation of influences that affect individual learning. How People Learn II will become an indispensable resource to understand learning throughout the lifespan for educators of students and adults.

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21 Problem Solving

Miriam Bassok, Department of Psychology, University of Washington, Seattle, WA

Laura R. Novick, Department of Psychology and Human Development, Vanderbilt University, Nashville, TN

• Published: 21 November 2012
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This chapter follows the historical development of research on problem solving. It begins with a description of two research traditions that addressed different aspects of the problem-solving process: ( 1 ) research on problem representation (the Gestalt legacy) that examined how people understand the problem at hand, and ( 2 ) research on search in a problem space (the legacy of Newell and Simon) that examined how people generate the problem's solution. It then describes some developments in the field that fueled the integration of these two lines of research: work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. Next, it presents examples of recent work on problem solving in science and mathematics that highlight the impact of visual perception and background knowledge on how people represent problems and search for problem solutions. The final section considers possible directions for future research.

People are confronted with problems on a daily basis, be it trying to extract a broken light bulb from a socket, finding a detour when the regular route is blocked, fixing dinner for unexpected guests, dealing with a medical emergency, or deciding what house to buy. Obviously, the problems people encounter differ in many ways, and their solutions require different types of knowledge and skills. Yet we have a sense that all the situations we classify as problems share a common core. Karl Duncker defined this core as follows: “A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action [i.e., by the performance of obvious operations], then there has to be recourse to thinking” (Duncker, 1945 , p. 1). Consider the broken light bulb. The obvious operation—holding the glass part of the bulb with one's fingers while unscrewing the base from the socket—is prevented by the fact that the glass is broken. Thus, there must be “recourse to thinking” about possible ways to solve the problem. For example, one might try mounting half a potato on the broken bulb (we do not know the source of this creative solution, which is described on many “how to” Web sites).

The above definition and examples make it clear that what constitutes a problem for one person may not be a problem for another person, or for that same person at another point in time. For example, the second time one has to remove a broken light bulb from a socket, the solution likely can be retrieved from memory; there is no problem. Similarly, tying shoes may be considered a problem for 5-year-olds but not for readers of this chapter. And, of course, people may change their goal and either no longer have a problem (e.g., take the guests to a restaurant instead of fixing dinner) or attempt to solve a different problem (e.g., decide what restaurant to go to). Given the highly subjective nature of what constitutes a problem, researchers who study problem solving have often presented people with novel problems that they should be capable of solving and attempted to find regularities in the resulting problem-solving behavior. Despite the variety of possible problem situations, researchers have identified important regularities in the thinking processes by which people (a) represent , or understand, problem situations and (b) search for possible ways to get to their goal.

A problem representation is a model constructed by the solver that summarizes his or her understanding of the problem components—the initial state (e.g., a broken light bulb in a socket), the goal state (the light bulb extracted), and the set of possible operators one may apply to get from the initial state to the goal state (e.g., use pliers). According to Reitman ( 1965 ), problem components differ in the extent to which they are well defined . Some components leave little room for interpretation (e.g., the initial state in the broken light bulb example is relatively well defined), whereas other components may be ill defined and have to be defined by the solver (e.g., the possible actions one may take to extract the broken bulb). The solver's representation of the problem guides the search for a possible solution (e.g., possible attempts at extracting the light bulb). This search may, in turn, change the representation of the problem (e.g., finding that the goal cannot be achieved using pliers) and lead to a new search. Such a recursive process of representation and search continues until the problem is solved or until the solver decides to abort the goal.

Duncker ( 1945 , pp. 28–37) documented the interplay between representation and search based on his careful analysis of one person's solution to the “Radiation Problem” (later to be used extensively in research analogy, see Holyoak, Chapter 13 ). This problem requires using some rays to destroy a patient's stomach tumor without harming the patient. At sufficiently high intensity, the rays will destroy the tumor. However, at that intensity, they will also destroy the healthy tissue surrounding the tumor. At lower intensity, the rays will not harm the healthy tissue, but they also will not destroy the tumor. Duncker's analysis revealed that the solver's solution attempts were guided by three distinct problem representations. He depicted these solution attempts as an inverted search tree in which the three main branches correspond to the three general problem representations (Duncker, 1945 , p. 32). We reproduce this diagram in Figure 21.1 . The desired solution appears on the rightmost branch of the tree, within the general problem representation in which the solver aims to “lower the intensity of the rays on their way through healthy tissue.” The actual solution is to project multiple low-intensity rays at the tumor from several points around the patient “by use of lens.” The low-intensity rays will converge on the tumor, where their individual intensities will sum to a level sufficient to destroy the tumor.

A search-tree representation of one subject's solution to the radiation problem, reproduced from Duncker ( 1945 , p. 32).

Although there are inherent interactions between representation and search, some researchers focus their efforts on understanding the factors that affect how solvers represent problems, whereas others look for regularities in how they search for a solution within a particular representation. Based on their main focus of interest, researchers devise or select problems with solutions that mainly require either constructing a particular representation or finding the appropriate sequence of steps leading from the initial state to the goal state. In most cases, researchers who are interested in problem representation select problems in which one or more of the components are ill defined, whereas those who are interested in search select problems in which the components are well defined. The following examples illustrate, respectively, these two problem types.

The Bird-and-Trains problem (Posner, 1973 , pp. 150–151) is a mathematical word problem that tends to elicit two distinct problem representations (see Fig. 21.2a and b ):

Two train stations are 50 miles apart. At 2 p.m. one Saturday afternoon two trains start toward each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead to the front of the second train. When the bird reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet? Fig. 21.2 Open in new tab Download slide Alternative representations of Posner's ( 1973 ) trains-and-bird problem. Adapted from Novick and Hmelo ( 1994 ).

Some solvers focus on the back-and-forth path of the bird (Fig. 21.2a ). This representation yields a problem that would be difficult for most people to solve (e.g., a series of differential equations). Other solvers focus on the paths of the trains (Fig. 21.2b ), a representation that yields a relatively easy distance-rate-time problem.

The Tower of Hanoi problem falls on the other end of the representation-search continuum. It leaves little room for differences in problem representations, and the primary work is to discover a solution path (or the best solution path) from the initial state to the goal state .

There are three pegs mounted on a base. On the leftmost peg, there are three disks of differing sizes. The disks are arranged in order of size with the largest disk on the bottom and the smallest disk on the top. The disks may be moved one at a time, but only the top disk on a peg may be moved, and at no time may a larger disk be placed on a smaller disk. The goal is to move the three-disk tower from the leftmost peg to the rightmost peg.

Figure 21.3 shows all the possible legal arrangements of disks on pegs. The arrows indicate transitions between states that result from moving a single disk, with the thicker gray arrows indicating the shortest path that connects the initial state to the goal state.

The division of labor between research on representation versus search has distinct historical antecedents and research traditions. In the next two sections, we review the main findings from these two historical traditions. Then, we describe some developments in the field that fueled the integration of these lines of research—work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. In the fifth section, we present some examples of recent work on problem solving in science and mathematics. This work highlights the role of visual perception and background knowledge in the way people represent problems and search for problem solutions. In the final section, we consider possible directions for future research.

Our review is by no means an exhaustive one. It follows the historical development of the field and highlights findings that pertain to a wide variety of problems. Research pertaining to specific types of problems (e.g., medical problems), specific processes that are involved in problem solving (e.g., analogical inferences), and developmental changes in problem solving due to learning and maturation may be found elsewhere in this volume (e.g., Holyoak, Chapter 13 ; Smith & Ward, Chapter 23 ; van Steenburgh et al., Chapter 24 ; Simonton, Chapter 25 ; Opfer & Siegler, Chapter 30 ; Hegarty & Stull, Chapter 31 ; Dunbar & Klahr, Chapter 35 ; Patel et al., Chapter 37 ; Lowenstein, Chapter 38 ; Koedinger & Roll, Chapter 40 ).

All possible problem states for the three-disk Tower of Hanoi problem. The thicker gray arrows show the optimum solution path connecting the initial state (State #1) to the goal state (State #27).

Problem Representation: The Gestalt Legacy

Research on problem representation has its origins in Gestalt psychology, an influential approach in European psychology during the first half of the 20th century. (Behaviorism was the dominant perspective in American psychology at this time.) Karl Duncker published a book on the topic in his native German in 1935, which was translated into English and published 10 years later as the monograph On Problem-Solving (Duncker, 1945 ). Max Wertheimer also published a book on the topic in 1945, titled Productive Thinking . An enlarged edition published posthumously includes previously unpublished material (Wertheimer, 1959 ). Interestingly, 1945 seems to have been a watershed year for problem solving, as mathematician George Polya's book, How to Solve It , also appeared then (a second edition was published 12 years later; Polya, 1957 ).

The Gestalt psychologists extended the organizational principles of visual perception to the domain of problem solving. They showed that various visual aspects of the problem, as well the solver's prior knowledge, affect how people understand problems and, therefore, generate problem solutions. The principles of visual perception (e.g., proximity, closure, grouping, good continuation) are directly relevant to problem solving when the physical layout of the problem, or a diagram that accompanies the problem description, elicits inferences that solvers include in their problem representations. Such effects are nicely illustrated by Maier's ( 1930 ) nine-dot problem: Nine dots are arrayed in a 3x3 grid, and the task is to connect all the dots by drawing four straight lines without lifting one's pencil from the paper. People have difficulty solving this problem because their initial representations generally include a constraint, inferred from the configuration of the dots, that the lines should not go outside the boundary of the imaginary square formed by the outer dots. With this constraint, the problem cannot be solved (but see Adams, 1979 ). Without this constraint, the problem may be solved as shown in Figure 21.4 (though the problem is still difficult for many people; see Weisberg & Alba, 1981 ).

The nine-dot problem is a classic insight problem (see van Steenburgh et al., Chapter 24 ). According to the Gestalt view (e.g., Duncker, 1945 ; Kohler, 1925 ; Maier, 1931 ; see Ohlsson, 1984 , for a review), the solution to an insight problem appears suddenly, accompanied by an “aha!” sensation, immediately following the sudden “restructuring” of one's understanding of the problem (i.e., a change in the problem representation): “The decisive points in thought-processes, the moments of sudden comprehension, of the ‘Aha!,’ of the new, are always at the same time moments in which such a sudden restructuring of the thought-material takes place” (Duncker, 1945 , p. 29). For the nine-dot problem, one view of the required restructuring is that the solver relaxes the constraint implied by the perceptual form of the problem and realizes that the lines may, in fact, extend past the boundary of the imaginary square. Later in the chapter, we present more recent accounts of insight.

The entities that appear in a problem also tend to evoke various inferences that people incorporate into their problem representations. A classic demonstration of this is the phenomenon of functional fixedness , introduced by Duncker ( 1945 ): If an object is habitually used for a certain purpose (e.g., a box serves as a container), it is difficult to see

A solution to the nine-dot problem.

that object as having properties that would enable it to be used for a dissimilar purpose. Duncker's basic experimental paradigm involved two conditions that varied in terms of whether the object that was crucial for solution was initially used for a function other than that required for solution.

Consider the candles problem—the best known of the five “practical problems” Duncker ( 1945 ) investigated. Three candles are to be mounted at eye height on a door. On the table, for use in completing this task, are some tacks and three boxes. The solution is to tack the three boxes to the door to serve as platforms for the candles. In the control condition, the three boxes were presented to subjects empty. In the functional-fixedness condition, they were filled with candles, tacks, and matches. Thus, in the latter condition, the boxes initially served the function of container, whereas the solution requires that they serve the function of platform. The results showed that 100% of the subjects who received empty boxes solved the candles problem, compared with only 43% of subjects who received filled boxes. Every one of the five problems in this study showed a difference favoring the control condition over the functional-fixedness condition, with average solution rates across the five problems of 97% and 58%, respectively.

The function of the objects in a problem can be also “fixed” by their most recent use. For example, Birch and Rabinowitz ( 1951 ) had subjects perform two consecutive tasks. In the first task, people had to use either a switch or a relay to form an electric circuit. After completing this task, both groups of subjects were asked to solve Maier's ( 1931 ) two-ropes problem. The solution to this problem requires tying an object to one of the ropes and making the rope swing as a pendulum. Subjects could create the pendulum using either the object from the electric-circuit task or the other object. Birch and Rabinowitz found that subjects avoided using the same object for two unrelated functions. That is, those who used the switch in the first task made the pendulum using the relay, and vice versa. The explanations subjects subsequently gave for their object choices revealed that they were unaware of the functional-fixedness constraint they imposed on themselves.

In addition to investigating people's solutions to such practical problems as irradiating a tumor, mounting candles on the wall, or tying ropes, the Gestalt psychologists examined how people understand and solve mathematical problems that require domain-specific knowledge. For example, Wertheimer ( 1959 ) observed individual differences in students' learning and subsequent application of the formula for finding the area of a parallelogram (see Fig. 21.5a ). Some students understood the logic underlying the learned formula (i.e., the fact that a parallelogram can be transformed into a rectangle by cutting off a triangle from one side and pasting it onto the other side) and exhibited “productive thinking”—using the same logic to find the area of the quadrilateral in Figure 21.5b and the irregularly shaped geometric figure in Figure 21.5c . Other students memorized the formula and exhibited “reproductive thinking”—reproducing the learned solution only to novel parallelograms that were highly similar to the original one.

The psychological study of human problem solving faded into the background after the demise of the Gestalt tradition (during World War II), and problem solving was investigated only sporadically until Allen Newell and Herbert Simon's ( 1972 ) landmark book Human Problem Solving sparked a flurry of research on this topic. Newell and Simon adopted and refined Duncker's ( 1945 ) methodology of collecting and analyzing the think-aloud protocols that accompany problem solutions and extended Duncker's conceptualization of a problem solution as a search tree. However, their initial work did not aim to extend the Gestalt findings

Finding the area of ( a ) a parallelogram, ( b ) a quadrilateral, and ( c ) an irregularly shaped geometric figure. The solid lines indicate the geometric figures whose areas are desired. The dashed lines show how to convert the given figures into rectangles (i.e., they show solutions with understanding).

pertaining to problem representation. Instead, as we explain in the next section, their objective was to identify the general-purpose strategies people use in searching for a problem solution.

Search in a Problem Space: The Legacy of Newell and Simon

Newell and Simon ( 1972 ) wrote a magnum opus detailing their theory of problem solving and the supporting research they conducted with various collaborators. This theory was grounded in the information-processing approach to cognitive psychology and guided by an analogy between human and artificial intelligence (i.e., both people and computers being “Physical Symbol Systems,” Newell & Simon, 1976 ; see Doumas & Hummel, Chapter 5 ). They conceptualized problem solving as a process of search through a problem space for a path that connects the initial state to the goal state—a metaphor that alludes to the visual or spatial nature of problem solving (Simon, 1990 ). The term problem space refers to the solver's representation of the task as presented (Simon, 1978 ). It consists of ( 1 ) a set of knowledge states (the initial state, the goal state, and all possible intermediate states), ( 2 ) a set of operators that allow movement from one knowledge state to another, ( 3 ) a set of constraints, and ( 4 ) local information about the path one is taking through the space (e.g., the current knowledge state and how one got there).

We illustrate the components of a problem space for the three-disk Tower of Hanoi problem, as depicted in Figure 21.3 . The initial state appears at the top (State #1) and the goal state at the bottom right (State #27). The remaining knowledge states in the figure are possible intermediate states. The current knowledge state is the one at which the solver is located at any given point in the solution process. For example, the current state for a solver who has made three moves along the optimum solution path would be State #9. The solver presumably would know that he or she arrived at this state from State #5. This knowledge allows the solver to recognize a move that involves backtracking. The three operators in this problem are moving each of the three disks from one peg to another. These operators are subject to the constraint that a larger disk may not be placed on a smaller disk.

Newell and Simon ( 1972 ), as well as other contemporaneous researchers (e.g., Atwood & Polson, 1976 ; Greeno, 1974 ; Thomas, 1974 ), examined how people traverse the spaces of various well-defined problems (e.g., the Tower of Hanoi, Hobbits and Orcs). They discovered that solvers' search is guided by a number of shortcut strategies, or heuristics , which are likely to get the solver to the goal state without an extensive amount of search. Heuristics are often contrasted with algorithms —methods that are guaranteed to yield the correct solution. For example, one could try every possible move in the three-disk Tower of Hanoi problem and, eventually, find the correct solution. Although such an exhaustive search is a valid algorithm for this problem, for many problems its application is very time consuming and impractical (e.g., consider the game of chess).

In their attempts to identify people's search heuristics, Newell and Simon ( 1972 ) relied on two primary methodologies: think-aloud protocols and computer simulations. Their use of think-aloud protocols brought a high degree of scientific rigor to the methodology used by Duncker ( 1945 ; see Ericsson & Simon, 1980 ). Solvers were required to say out loud everything they were thinking as they solved the problem, that is, everything that went through their verbal working memory. Subjects' verbalizations—their think-aloud protocols—were tape-recorded and then transcribed verbatim for analysis. This method is extremely time consuming (e.g., a transcript of one person's solution to the cryptarithmetic problem DONALD + GERALD = ROBERT, with D = 5, generated a 17-page transcript), but it provides a detailed record of the solver's ongoing solution process.

An important caveat to keep in mind while interpreting a subject's verbalizations is that “a protocol is relatively reliable only for what it positively contains, but not for that which it omits” (Duncker, 1945 , p. 11). Ericsson and Simon ( 1980 ) provided an in-depth discussion of the conditions under which this method is valid (but see Russo, Johnson, & Stephens, 1989 , for an alternative perspective). To test their interpretation of a subject's problem solution, inferred from the subject's verbal protocol, Newell and Simon ( 1972 ) created a computer simulation program and examined whether it solved the problem the same way the subject did. To the extent that the computer simulation provided a close approximation of the solver's step-by-step solution process, it lent credence to the researcher's interpretation of the verbal protocol.

Newell and Simon's ( 1972 ) most famous simulation was the General Problem Solver or GPS (Ernst & Newell, 1969 ). GPS successfully modeled human solutions to problems as different as the Tower of Hanoi and the construction of logic proofs using a single general-purpose heuristic: means-ends analysis . This heuristic captures people's tendency to devise a solution plan by setting subgoals that could help them achieve their final goal. It consists of the following steps: ( 1 ) Identify a difference between the current state and the goal (or subgoal ) state; ( 2 ) Find an operator that will remove (or reduce) the difference; (3a) If the operator can be directly applied, do so, or (3b) If the operator cannot be directly applied, set a subgoal to remove the obstacle that is preventing execution of the desired operator; ( 4 ) Repeat steps 1–3 until the problem is solved. Next, we illustrate the implementation of this heuristic for the Tower of Hanoi problem, using the problem space in Figure 21.3 .

As can be seen in Figure 21.3 , a key difference between the initial state and the goal state is that the large disk is on the wrong peg (step 1). To remove this difference (step 2), one needs to apply the operator “move-large-disk.” However, this operator cannot be applied because of the presence of the medium and small disks on top of the large disk. Therefore, the solver may set a subgoal to move that two-disk tower to the middle peg (step 3b), leaving the rightmost peg free for the large disk. A key difference between the initial state and this new subgoal state is that the medium disk is on the wrong peg. Because application of the move-medium-disk operator is blocked, the solver sets another subgoal to move the small disk to the right peg. This subgoal can be satisfied immediately by applying the move-small-disk operator (step 3a), generating State #3. The solver then returns to the previous subgoal—moving the tower consisting of the small and medium disks to the middle peg. The differences between the current state (#3) and the subgoal state (#9) can be removed by first applying the move-medium-disk operator (yielding State #5) and then the move-small-disk operator (yielding State #9). Finally, the move-large-disk operator is no longer blocked. Hence, the solver moves the large disk to the right peg, yielding State #11.

Notice that the subgoals are stacked up in the order in which they are generated, so that they pop up in the order of last in first out. Given the first subgoal in our example, repeated application of the means-ends analysis heuristic will yield the shortest-path solution, indicated by the large gray arrows. In general, subgoals provide direction to the search and allow solvers to plan several moves ahead. By assessing progress toward a required subgoal rather than the final goal, solvers may be able to make moves that otherwise seem unwise. To take a concrete example, consider the transition from State #1 to State #3 in Figure 21.3 . Comparing the initial state to the goal state, this move seems unwise because it places the small disk on the bottom of the right peg, whereas it ultimately needs to be at the top of the tower on that peg. But comparing the initial state to the solver-generated subgoal state of having the medium disk on the middle peg, this is exactly where the small disk needs to go.

Means-ends analysis and various other heuristics (e.g., the hill-climbing heuristic that exploits the similarity, or distance, between the state generated by the next operator and the goal state; working backward from the goal state to the initial state) are flexible strategies that people often use to successfully solve a large variety of problems. However, the generality of these heuristics comes at a cost: They are relatively weak and fallible (e.g., in the means-ends solution to the problem of fixing a hole in a bucket, “Dear Liza” leads “Dear Henry” in a loop that ends back at the initial state; the lyrics of this famous song can be readily found on the Web). Hence, although people use general-purpose heuristics when they encounter novel problems, they replace them as soon as they acquire experience with and sufficient knowledge about the particular problem space (e.g., Anzai & Simon, 1979 ).

Despite the fruitfulness of this research agenda, it soon became evident that a fundamental weakness was that it minimized the importance of people's background knowledge. Of course, Newell and Simon ( 1972 ) were aware that problem solutions require relevant knowledge (e.g., the rules of logical proofs, or rules for stacking disks). Hence, in programming GPS, they supplemented every problem they modeled with the necessary background knowledge. This practice highlighted the generality and flexibility of means-ends analysis but failed to capture how people's background knowledge affects their solutions. As we discussed in the previous section, domain knowledge is likely to affect how people represent problems and, therefore, how they generate problem solutions. Moreover, as people gain experience solving problems in a particular knowledge domain (e.g., math, physics), they change their representations of these problems (e.g., Chi, Feltovich, & Glaser, 1981 ; Haverty, Koedinger, Klahr, & Alibali, 2000 ; Schoenfeld & Herrmann, 1982 ) and learn domain-specific heuristics (e.g., Polya, 1957 ; Schoenfeld, 1979 ) that trump the general-purpose strategies.

It is perhaps inevitable that the two traditions in problem-solving research—one emphasizing representation and the other emphasizing search strategies—would eventually come together. In the next section we review developments that led to this integration.

The Two Legacies Converge

Because Newell and Simon ( 1972 ) aimed to discover the strategies people use in searching for a solution, they investigated problems that minimized the impact of factors that tend to evoke differences in problem representations, of the sort documented by the Gestalt psychologists. In subsequent work, however, Simon and his collaborators showed that such factors are highly relevant to people's solutions of well-defined problems, and Simon ( 1986 ) incorporated these findings into the theoretical framework that views problem solving as search in a problem space.

In this section, we first describe illustrative examples of this work. We then describe research on insight solutions that incorporates ideas from the two legacies described in the previous sections.

Relevance of the Gestalt Ideas to the Solution of Search Problems

In this subsection we describe two lines of research by Simon and his colleagues, and by other researchers, that document the importance of perception and of background knowledge to the way people search for a problem solution. The first line of research used variants of relatively well-defined riddle problems that had the same structure (i.e., “problem isomorphs”) and, therefore, supposedly the same problem space. It documented that people's search depended on various perceptual and conceptual inferences they tended to draw from a specific instantiation of the problem's structure. The second line of research documented that people's search strategies crucially depend on their domain knowledge and on their prior experience with related problems.

Problem Isomorphs

Hayes and Simon ( 1977 ) used two variants of the Tower of Hanoi problem that, instead of disks and pegs, involved monsters and globes that differed in size (small, medium, and large). In both variants, the initial state had the small monster holding the large globe, the medium-sized monster holding the small globe, and the large monster holding the medium-sized globe. Moreover, in both variants the goal was for each monster to hold a globe proportionate to its own size. The only difference between the problems concerned the description of the operators. In one variant (“transfer”), subjects were told that the monsters could transfer the globes from one to another as long as they followed a set of rules, adapted from the rules in the original Tower of Hanoi problem (e.g., only one globe may be transferred at a time). In the other variant (“change”), subjects were told that the monsters could shrink and expand themselves according to a set of rules, which corresponded to the rules in the transfer version of the problem (e.g., only one monster may change its size at a time). Despite the isomorphism of the two variants, subjects conducted their search in two qualitatively different problem spaces, which led to solution times for the change variant being almost twice as long as those for the transfer variant. This difference arose because subjects could more readily envision and track an object that was changing its location with every move than one that was changing its size.

Recent work by Patsenko and Altmann ( 2010 ) found that, even in the standard Tower of Hanoi problem, people's solutions involve object-bound routines that depend on perception and selective attention. The subjects in their study solved various Tower of Hanoi problems on a computer. During the solution of a particular “critical” problem, the computer screen changed at various points without subjects' awareness (e.g., a disk was added, such that a subject who started with a five-disc tower ended with a six-disc tower). Patsenko and Altmann found that subjects' moves were guided by the configurations of the objects on the screen rather than by solution plans they had stored in memory (e.g., the next subgoal).

The Gestalt psychologists highlighted the role of perceptual factors in the formation of problem representations (e.g., Maier's, 1930 , nine-dot problem) but were generally silent about the corresponding implications for how the problem was solved (although they did note effects on solution accuracy). An important contribution of the work on people's solutions of the Tower of Hanoi problem and its variants was to show the relevance of perceptual factors to the application of various operators during search for a problem solution—that is, to the how of problem solving. In the next section, we describe recent work that documents the involvement of perceptual factors in how people understand and use equations and diagrams in the context of solving math and science problems.

Kotovsky, Hayes, and Simon ( 1985 ) further investigated factors that affect people's representation and search in isomorphs of the Tower of Hanoi problem. In one of their isomorphs, three disks were stacked on top of each other to form an inverted pyramid, with the smallest disc on the bottom and the largest on top. Subjects' solutions of the inverted pyramid version were similar to their solutions of the standard version that has the largest disc on the bottom and the smallest on top. However, the two versions were solved very differently when subjects were told that the discs represent acrobats. Subjects readily solved the version in which they had to place a small acrobat on the shoulders of a large one, but they refrained from letting a large acrobat stand on the shoulders of a small one. In other words, object-based inferences that draw on people's semantic knowledge affected the solution of search problems, much as they affect the solution of the ill-defined problems investigated by the Gestalt psychologists (e.g., Duncker's, 1945 , candles problem). In the next section, we describe more recent work that shows similar effects in people's solutions to mathematical word problems.

The work on differences in the representation and solution of problem isomorphs is highly relevant to research on analogical problem solving (or analogical transfer), which examines when and how people realize that two problems that differ in their cover stories have a similar structure (or a similar problem space) and, therefore, can be solved in a similar way. This research shows that minor differences between example problems, such as the use of X-rays versus ultrasound waves to fuse a broken filament of a light bulb, can elicit different problem representations that significantly affect the likelihood of subsequent transfer to novel problem analogs (Holyoak & Koh, 1987 ). Analogical transfer has played a central role in research on human problem solving, in part because it can shed light on people's understanding of a given problem and its solution and in part because it is believed to provide a window onto understanding and investigating creativity (see Smith & Ward, Chapter 23 ). We briefly mention some findings from the analogy literature in the next subsection on expertise, but we do not discuss analogical transfer in detail because this topic is covered elsewhere in this volume (Holyoak, Chapter 13 ).

Expertise and Its Development

In another line of research, Simon and his colleagues examined how people solve ecologically valid problems from various rule-governed and knowledge-rich domains. They found that people's level of expertise in such domains, be it in chess (Chase & Simon, 1973 ; Gobet & Simon, 1996 ), mathematics (Hinsley, Hayes, & Simon, 1977 ; Paige & Simon, 1966 ), or physics (Larkin, McDermott, Simon, & Simon, 1980 ; Simon & Simon, 1978 ), plays a crucial role in how they represent problems and search for solutions. This work, and the work of numerous other researchers, led to the discovery (and rediscovery, see Duncker, 1945 ) of important differences between experts and novices, and between “good” and “poor” students.

One difference between experts and novices pertains to pattern recognition. Experts' attention is quickly captured by familiar configurations within a problem situation (e.g., a familiar configuration of pieces in a chess game). In contrast, novices' attention is focused on isolated components of the problem (e.g., individual chess pieces). This difference, which has been found in numerous domains, indicates that experts have stored in memory many meaningful groups (chunks) of information: for example, chess (Chase & Simon, 1973 ), circuit diagrams (Egan & Schwartz, 1979 ), computer programs (McKeithen, Reitman, Rueter, & Hirtle, 1981 ), medicine (Coughlin & Patel, 1987 ; Myles-Worsley, Johnston, & Simons, 1988 ), basketball and field hockey (Allard & Starkes, 1991 ), and figure skating (Deakin & Allard, 1991 ).

The perceptual configurations that domain experts readily recognize are associated with stored solution plans and/or compiled procedures (Anderson, 1982 ). As a result, experts' solutions are much faster than, and often qualitatively different from, the piecemeal solutions that novice solvers tend to construct (e.g., Larkin et al., 1980 ). In effect, experts often see the solutions that novices have yet to compute (e.g., Chase & Simon, 1973 ; Novick & Sherman, 2003 , 2008 ). These findings have led to the design of various successful instructional interventions (e.g., Catrambone, 1998 ; Kellman et al., 2008 ). For example, Catrambone ( 1998 ) perceptually isolated the subgoals of a statistics problem. This perceptual chunking of meaningful components of the problem prompted novice students to self-explain the meaning of the chunks, leading to a conceptual understanding of the learned solution. In the next section, we describe some recent work that shows the beneficial effects of perceptual pattern recognition on the solution of familiar mathematics problems, as well as the potentially detrimental effects of familiar perceptual chunks to understanding and reasoning with diagrams depicting evolutionary relationships among taxa.

Another difference between experts and novices pertains to their understanding of the solution-relevant problem structure. Experts' knowledge is highly organized around domain principles, and their problem representations tend to reflect this principled understanding. In particular, they can extract the solution-relevant structure of the problems they encounter (e.g., meaningful causal relations among the objects in the problem; see Cheng & Buehner, Chapter 12 ). In contrast, novices' representations tend to be bound to surface features of the problems that may be irrelevant to solution (e.g., the particular objects in a problem). For example, Chi, Feltovich, and Glaser ( 1981 ) examined how students with different levels of physics expertise group mechanics word problems. They found that advanced graduate students grouped the problems based on the physics principles relevant to the problems' solutions (e.g., conservation of energy, Newton's second law). In contrast, undergraduates who had successfully completed an introductory course in mechanics grouped the problems based on the specific objects involved (e.g., pulley problems, inclined plane problems). Other researchers have found similar results in the domains of biology, chemistry, computer programming, and math (Adelson, 1981 ; Kindfield, 1993 / 1994 ; Kozma & Russell, 1997 ; McKeithen et al., 1981 ; Silver, 1979 , 1981 ; Weiser & Shertz, 1983 ).

The level of domain expertise and the corresponding representational differences are, of course, a matter of degree. With increasing expertise, there is a gradual change in people's focus of attention from aspects that are not relevant to solution to those that are (e.g., Deakin & Allard, 1991 ; Hardiman, Dufresne, & Mestre, 1989 ; McKeithen et al., 1981 ; Myles-Worsley et al., 1988 ; Schoenfeld & Herrmann, 1982 ; Silver, 1981 ). Interestingly, Chi, Bassok, Lewis, Reimann, and Glaser ( 1989 ) found similar differences in focus on structural versus surface features among a group of novices who studied worked-out examples of mechanics problems. These differences, which echo Wertheimer's ( 1959 ) observations of individual differences in students' learning about the area of parallelograms, suggest that individual differences in people's interests and natural abilities may affect whether, or how quickly, they acquire domain expertise.

An important benefit of experts' ability to focus their attention on solution-relevant aspects of problems is that they are more likely than novices to recognize analogous problems that involve different objects and cover stories (e.g., Chi et al., 1989 ; Novick, 1988 ; Novick & Holyoak, 1991 ; Wertheimer, 1959 ) or that come from other knowledge domains (e.g., Bassok & Holyoak, 1989 ; Dunbar, 2001 ; Goldstone & Sakamoto, 2003 ). For example, Bassok and Holyoak ( 1989 ) found that, after learning to solve arithmetic-progression problems in algebra, subjects spontaneously applied these algebraic solutions to analogous physics problems that dealt with constantly accelerated motion. Note, however, that experts and good students do not simply ignore the surface features of problems. Rather, as was the case in the problem isomorphs we described earlier (Kotovsky et al., 1985 ), they tend to use such features to infer what the problem's structure could be (e.g., Alibali, Bassok, Solomon, Syc, & Goldin-Meadow, 1999 ; Blessing & Ross, 1996 ). For example, Hinsley et al. ( 1977 ) found that, after reading no more than the first few words of an algebra word problem, expert solvers classified the problem into a likely problem category (e.g., a work problem, a distance problem) and could predict what questions they might be asked and the equations they likely would need to use.

Surface-based problem categorization has a heuristic value (Medin & Ross, 1989 ): It does not ensure a correct categorization (Blessing & Ross, 1996 ), but it does allow solvers to retrieve potentially appropriate solutions from memory and to use them, possibly with some adaptation, to solve a variety of novel problems. Indeed, although experts exploit surface-structure correlations to save cognitive effort, they have the capability to realize that a particular surface cue is misleading (Hegarty, Mayer, & Green, 1992 ; Lewis & Mayer, 1987 ; Martin & Bassok, 2005 ; Novick 1988 , 1995 ; Novick & Holyoak, 1991 ). It is not surprising, therefore, that experts may revert to novice-like heuristic methods when solving problems under pressure (e.g., Beilock, 2008 ) or in subdomains in which they have general but not specific expertise (e.g., Patel, Groen, & Arocha, 1990 ).

Relevance of Search to Insight Solutions

We introduced the notion of insight in our discussion of the nine-dot problem in the section on the Gestalt tradition. The Gestalt view (e.g., Duncker, 1945 ; Maier, 1931 ; see Ohlsson, 1984 , for a review) was that insight problem solving is characterized by an initial work period during which no progress toward solution is made (i.e., an impasse), a sudden restructuring of one's problem representation to a more suitable form, followed immediately by the sudden appearance of the solution. Thus, solving problems by insight was believed to be all about representation, with essentially no role for a step-by-step solution process (i.e., search). Subsequent and contemporary researchers have generally concurred with the Gestalt view that getting the right representation is crucial. However, research has shown that insight solutions do not necessarily arise suddenly or full blown after restructuring (e.g., Weisberg & Alba, 1981 ); and even when they do, the underlying solution process (in this case outside of awareness) may reflect incremental progress toward the goal (Bowden & Jung-Beeman, 2003 ; Durso, Rea, & Dayton, 1994 ; Novick & Sherman, 2003 ).

“Demystifying insight,” to borrow a phrase from Bowden, Jung-Beeman, Fleck, and Kounios ( 2005 ), requires explaining ( 1 ) why solvers initially reach an impasse in solving a problem for which they have the necessary knowledge to generate the solution, ( 2 ) how the restructuring occurred, and ( 3 ) how it led to the solution. A detailed discussion of these topics appears elsewhere in this volume (van Steenburgh et al., Chapter 24 ). Here, we describe briefly three recent theories that have attempted to account for various aspects of these phenomena: Knoblich, Ohlsson, Haider, and Rhenius's ( 1999 ) representational change theory, MacGregor, Ormerod, and Chronicle's ( 2001 ) progress monitoring theory, and Bowden et al.'s ( 2005 ) neurological model. We then propose the need for an integrated approach to demystifying insight that considers both representation and search.

According to Knoblich et al.'s ( 1999 ) representational change theory, problems that are solved with insight are highly likely to evoke initial representations in which solvers place inappropriate constraints on their solution attempts, leading to an impasse. An impasse can be resolved by revising one's representation of the problem. Knoblich and his colleagues tested this theory using Roman numeral matchstick arithmetic problems in which solvers must move one stick to a new location to change a false numerical statement (e.g., I = II + II ) into a statement that is true. According to representational change theory, re-representation may occur through either constraint relaxation or chunk decomposition. (The solution to the example problem is to change II + to III – , which requires both methods of re-representation, yielding I = III – II ). Good support for this theory has been found based on measures of solution rate, solution time, and eye fixation (Knoblich et al., 1999 ; Knoblich, Ohlsson, & Raney, 2001 ; Öllinger, Jones, & Knoblich, 2008 ).

Progress monitoring theory (MacGregor et al., 2001 ) was proposed to account for subjects' difficulty in solving the nine-dot problem, which has traditionally been classified as an insight problem. According to this theory, solvers use the hill-climbing search heuristic to solve this problem, just as they do for traditional search problems (e.g., Hobbits and Orcs). In particular, solvers are hypothesized to monitor their progress toward solution using a criterion generated from the problem's current state. If solvers reach criterion failure, they seek alternative solutions by trying to relax one or more problem constraints. MacGregor et al. found support for this theory using several variants of the nine-dot problem (also see Ormerod, MacGregor, & Chronicle, 2002 ). Jones ( 2003 ) suggested that progress monitoring theory provides an account of the solution process up to the point an impasse is reached and representational change is sought, at which point representational change theory picks up and explains how insight may be achieved. Hence, it appears that a complete account of insight may require an integration of concepts from the Gestalt (representation) and Newell and Simon's (search) legacies.

Bowden et al.'s ( 2005 ) neurological model emphasizes the overlap between problem solving and language comprehension, and it hinges on differential processing in the right and left hemispheres. They proposed that an impasse is reached because initial processing of the problem produces strong activation of information irrelevant to solution in the left hemisphere. At the same time, weak semantic activation of alternative semantic interpretations, critical for solution, occurs in the right hemisphere. Insight arises when the weakly activated concepts reinforce each other, eventually rising above the threshold required for conscious awareness. Several studies of problem solving using compound remote associates problems, involving both behavioral and neuroimaging data, have found support for this model (Bowden & Jung-Beeman, 1998 , 2003 ; Jung-Beeman & Bowden, 2000 ; Jung-Beeman et al., 2004 ; also see Moss, Kotovsky, & Cagan, 2011 ).

Note that these three views of insight have received support using three quite distinct types of problems (Roman numeral matchstick arithmetic problems, the nine-dot problem, and compound remote associates problems, respectively). It remains to be established, therefore, whether these accounts can be generalized across problems. Kershaw and Ohlsson ( 2004 ) argued that insight problems are difficult because the key behavior required for solution may be hindered by perceptual factors (the Gestalt view), background knowledge (so expertise may be important; e.g., see Novick & Sherman, 2003 , 2008 ), and/or process factors (e.g., those affecting search). From this perspective, solving visual problems (e.g., the nine-dot problem) with insight may call upon more general visual processes, whereas solving verbal problems (e.g., anagrams, compound remote associates) with insight may call upon general verbal/semantic processes.

The work we reviewed in this section shows the relevance of problem representation (the Gestalt legacy) to the way people search the problem space (the legacy of Newell and Simon), and the relevance of search to the solution of insight problems that require a representational change. In addition to this inevitable integration of the two legacies, the work we described here underscores the fact that problem solving crucially depends on perceptual factors and on the solvers' background knowledge. In the next section, we describe some recent work that shows the involvement of these factors in the solution of problems in math and science.

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

Although the use of puzzle problems continues in research on problem solving, especially in investigations of insight, many contemporary researchers tackle problem solving in knowledge-rich domains, often in academic disciplines (e.g., mathematics, biology, physics, chemistry, meteorology). In this section, we provide a sampling of this research that highlights the importance of visual perception and background knowledge for successful problem solving.

The Role of Visual Perception

We stated at the outset that a problem representation (e.g., the problem space) is a model of the problem constructed by solvers to summarize their understanding of the problem's essential nature. This informal definition refers to the internal representations people construct and hold in working memory. Of course, people may also construct various external representations (Markman, 1999 ) and even manipulate those representations to aid in solution (see Hegarty & Stull, Chapter 31 ). For example, solvers often use paper and pencil to write notes or draw diagrams, especially when solving problems from formal domains (e.g., Cox, 1999 ; Kindfield, 1993 / 1994 ; S. Schwartz, 1971 ). In problems that provide solvers with external representation, such as the Tower of Hanoi problem, people's planning and memory of the current state is guided by the actual configurations of disks on pegs (Garber & Goldin-Meadow, 2002 ) or by the displays they see on a computer screen (Chen & Holyoak, 2010 ; Patsenko & Altmann, 2010 ).

In STEM (science, technology, engineering, and mathematics) disciplines, it is common for problems to be accompanied by diagrams or other external representations (e.g., equations) to be used in determining the solution. Larkin and Simon ( 1987 ) examined whether isomorphic sentential and diagrammatic representations are interchangeable in terms of facilitating solution. They argued that although the two formats may be equivalent in the sense that all of the information in each format can be inferred from the other format (informational equivalence), the ease or speed of making inferences from the two formats might differ (lack of computational equivalence). Based on their analysis of several problems in physics and math, Larkin and Simon further argued for the general superiority of diagrammatic representations (but see Mayer & Gallini, 1990 , for constraints on this general conclusion).

Novick and Hurley ( 2001 , p. 221) succinctly summarized the reasons for the general superiority of diagrams (especially abstract or schematic diagrams) over verbal representations: They “(a) simplify complex situations by discarding unnecessary details (e.g., Lynch, 1990 ; Winn, 1989 ), (b) make abstract concepts more concrete by mapping them onto spatial layouts with familiar interpretational conventions (e.g., Winn, 1989 ), and (c) substitute easier perceptual inferences for more computationally intensive search processes and sentential deductive inferences (Barwise & Etchemendy, 1991 ; Larkin & Simon, 1987 ).” Despite these benefits of diagrammatic representations, there is an important caveat, noted by Larkin and Simon ( 1987 , p. 99) at the very end of their paper: “Although every diagram supports some easy perceptual inferences, nothing ensures that these inferences must be useful in the problem-solving process.” We will see evidence of this in several of the studies reviewed in this section.

Next we describe recent work on perceptual factors that are involved in people's use of two types of external representations that are provided as part of the problem in two STEM disciplines: equations in algebra and diagrams in evolutionary biology. Although we focus here on effects of perceptual factors per se, it is important to note that such factors only influence performance when subjects have background knowledge that supports differential interpretation of the alternative diagrammatic depictions presented (Hegarty, Canham, & Fabricant, 2010 ).

In the previous section, we described the work of Patsenko and Altmann ( 2010 ) that shows direct involvement of visual attention and perception in the sequential application of move operators during the solution of the Tower of Hanoi problem. A related body of work documents similar effects in tasks that require the interpretation and use of mathematical equations (Goldstone, Landy, & Son, 2010 ; Landy & Goldstone, 2007a , b). For example, Landy and Goldstone ( 2007b ) varied the spatial proximity of arguments to the addition (+) and multiplication (*) operators in algebraic equations, such that the spatial layout of the equation was either consistent or inconsistent with the order-of-operations rule that multiplication precedes addition. In consistent equations , the space was narrower around multiplication than around addition (e.g., g*m + r*w = m*g + w*r ), whereas in inconsistent equations this relative spacing was reversed (e.g., s * n+e * c = n * s+c * e ). Subjects' judgments of the validity of such equations (i.e., whether the expressions on the two sides of the equal sign are equivalent) were significantly faster and more accurate for consistent than inconsistent equations.

In discussing these findings and related work with other external representations, Goldstone et al. ( 2010 ) proposed that experience with solving domain-specific problems leads people to “rig up” their perceptual system such that it allows them to look at the problem in a way that is consistent with the correct rules. Similar logic guides the Perceptual Learning Modules developed by Kellman and his collaborators to help students interpret and use algebraic equations and graphs (Kellman et al., 2008 ; Kellman, Massey, & Son, 2009 ). These authors argued and showed that, consistent with the previously reviewed work on expertise, perceptual training with particular external representations supports the development of perceptual fluency. This fluency, in turn, supports students' subsequent use of these external representations for problem solving.

This research suggests that extensive experience with particular equations or graphs may lead to perceptual fluency that could replace the more mindful application of domain-specific rules. Fisher, Borchert, and Bassok ( 2011 ) reported results from algebraic-modeling tasks that are consistent with this hypothesis. For example, college students were asked to represent verbal statements with algebraic equations, a task that typically elicits systematic errors (e.g., Clement, Lochhead, & Monk, 1981 ). Fisher et al. found that such errors were very common when subjects were asked to construct “standard form” equations ( y = ax ), which support fluent left-to-right translation of words to equations, but were relatively rare when subjects were asked to construct nonstandard division-format equations (x = y/a) that do not afford such translation fluency.

In part because of the left-to-right order in which people process equations, which mirrors the linear order in which they process text, equations have traditionally been viewed as sentential representations. However, Landy and Goldstone ( 2007a ) have proposed that equations also share some properties with diagrammatic displays and that, in fact, in some ways they are processed like diagrams. That is, spatial information is used to represent and to support inferences about syntactic structure. This hypothesis received support from Landy and Goldstone's ( 2007b ) results, described earlier, in which subjects' judgments of the validity of equations were affected by the Gestalt principle of grouping: Subjects did better when the grouping was consistent rather than inconsistent with the underlying structure of the problem (order of operations). Moreover, Landy and Goldstone ( 2007a ) found that when subjects wrote their own equations they grouped numbers and operators (+, *, =) in a way that reflected the hierarchical structure imposed by the order-of-operations rule.

In a recent line of research, Novick and Catley ( 2007 ; Novick, Catley, & Funk, 2010 ; Novick, Shade, & Catley, 2011 ) have examined effects of the spatial layout of diagrams depicting the evolutionary history of a set of taxa on people's ability to reason about patterns of relationship among those taxa. We consider here their work that investigates the role of another Gestalt perceptual principle—good continuation—in guiding students' reasoning. According to this principle, a continuous line is perceived as a single entity (Kellman, 2000 ). Consider the diagrams shown in Figure 21.6 . Each is a cladogram, a diagram that depicts nested sets of taxa that are related in terms of levels of most recent common ancestry. For example, chimpanzees and starfish are more closely related to each other than either is to spiders. The supporting evidence for their close relationship is their most recent common ancestor, which evolved the novel character of having radial cleavage. Spiders do not share this ancestor and thus do not have this character.

Cladograms are typically drawn in two isomorphic formats, which Novick and Catley ( 2007 ) referred to as trees and ladders. Although these formats are informationally equivalent (Larkin & Simon, 1987 ), Novick and Catley's ( 2007 ) research shows that they are not computationally equivalent (Larkin & Simon, 1987 ). Imagine that you are given evolutionary relationships in the ladder format, such as in Figure 21.6a (but without the four characters—hydrostatic skeleton, bilateral symmetry, radial cleavage, and trocophore larvae—and associated short lines indicating their locations on the cladogram), and your task is to translate that diagram to the tree format. A correct translation is shown in Figure 21.6b . Novick and Catley ( 2007 ) found that college students were much more likely to get such problems correct when the presented cladogram was in the nested circles (e.g., Figure 21.6d ) rather than the ladder format. Because the Gestalt principle of good continuation makes the long slanted line at the base of the ladder appear to represent a single hierarchical level, a common translation error for the ladder to tree problems was to draw a diagram such as that shown in Figure 21.6c .

The difficulty that good continuation presents for interpreting relationships depicted in the ladder format extends to answering reasoning questions as well. Novick and Catley (unpublished data) asked comparable questions about relationships depicted in the ladder and tree formats. For example, using the cladograms depicted in Figures 21.6a and 21.6b , consider the following questions: (a) Which taxon—jellyfish or earthworm—is the closest evolutionary relation to starfish, and what evidence supports your answer? (b) Do the bracketed taxa comprise a clade (a set of taxa consisting of the most recent common ancestor and all of its descendants), and what evidence supports your answer? For both such questions, students had higher accuracy and evidence quality composite scores when the relationships were depicted in the tree than the ladder format.

Four cladograms depicting evolutionary relationships among six animal taxa. Cladogram ( a ) is in the ladder format, cladograms ( b ) and ( c ) are in the tree format, and cladogram ( d ) is in the nested circles format. Cladograms ( a ), ( b ), and ( d ) are isomorphic.

If the difficulty in extracting the hierarchical structure of the ladder format is due to good continuation (which leads problem solvers to interpret continuous lines that depict multiple hierarchical levels as depicting only a single level), then a manipulation that breaks good continuation at the points where a new hierarchical level occurs should improve understanding. Novick et al. ( 2010 ) tested this hypothesis using a translation task by manipulating whether characters that are the markers for the most recent common ancestor of each nested set of taxa were included on the ladders. Figure 21.6a shows a ladder with such characters. As predicted, translation accuracy increased dramatically simply by adding these characters to the ladders, despite the additional information subjects had to account for in their translations.

The Role of Background Knowledge

As we mentioned earlier, the specific entities in the problems people encounter evoke inferences that affect how people represent these problems (e.g., the candle problem; Duncker, 1945 ) and how they apply the operators in searching for the solution (e.g., the disks vs. acrobats versions of the Tower of Hanoi problem; Kotovsky et al., 1985 ). Such object-based inferences draw on people's knowledge about the properties of the objects (e.g., a box is a container, an acrobat is a person who can be hurt). Here, we describe the work of Bassok and her colleagues, who found that similar inferences affect how people select mathematical procedures to solve problems in various formal domains. This work shows that the objects in the texts of mathematical word problems affect how people represent the problem situation (i.e., the situation model they construct; Kintsch & Greeno, 1985 ) and, in turn, lead them to select mathematical models that have a corresponding structure. To illustrate, a word problem that describes constant change in the rate at which ice is melting off a glacier evokes a model of continuous change, whereas a word problem that describes constant change in the rate at which ice is delivered to a restaurant evokes a model of discrete change. These distinct situation models lead subjects to select corresponding visual representations (e.g., Bassok & Olseth, 1995 ) and solutions methods, such as calculating the average change over time versus adding the consecutive changes (e.g., Alibali et al., 1999 ).

In a similar manner, people draw on their general knowledge to infer how the objects in a given problem are related to each other and construct mathematical solutions that correspond to these inferred object relations. For example, a word problem that involves doctors from two hospitals elicits a situation model in which the two sets of doctors play symmetric roles (e.g., work with each other), whereas a mathematically isomorphic problem that involves mechanics and cars elicits a situation model in which the sets play asymmetric roles (e.g., mechanics fix cars). The mathematical solutions people construct to such problems reflect this difference in symmetry (Bassok, Wu, & Olseth, 1995 ). In general, people tend to add objects that belong to the same taxonomic category (e.g., doctors + doctors) but divide functionally related objects (e.g., cars ÷ mechanics). People establish this correspondence by a process of analogical alignment between semantic and arithmetic relations, which Bassok and her colleagues refer to as “semantic alignment” (Bassok, Chase, & Martin, 1998 ; Doumas, Bassok, Guthormsen, & Hummel, 2006 ; Fisher, Bassok, & Osterhout, 2010 ).

Semantic alignment occurs very early in the solution process and can prime arithmetic facts that are potentially relevant to the problem solution (Bassok, Pedigo, & Oskarsson, 2008 ). Although such alignments can lead to erroneous solutions, they have a high heuristic value because, in most textbook problems, object relations indeed correspond to analogous mathematical relations (Bassok et al., 1998 ). Interestingly, unlike in the case of reliance on specific surface-structure correlations (e.g., the keyword “more” typically appears in word problems that require addition; Lewis & Mayer, 1987 ), people are more likely to exploit semantic alignment when they have more, rather than less modeling experience. For example, Martin and Bassok ( 2005 ) found very strong semantic-alignment effects when subjects solved simple division word problems, but not when they constructed algebraic equations to represent the relational statements that appeared in the problems. Of course, these subjects had significantly more experience with solving numerical word problems than with constructing algebraic models of relational statements. In a subsequent study, Fisher and Bassok ( 2009 ) found semantic-alignment effects for subjects who constructed correct algebraic models, but not for those who committed modeling errors.

Conclusions and Future Directions

In this chapter, we examined two broad components of the problem-solving process: representation (the Gestalt legacy) and search (the legacy of Newell and Simon). Although many researchers choose to focus their investigation on one or the other of these components, both Duncker ( 1945 ) and Simon ( 1986 ) underscored the necessity to investigate their interaction, as the representation one constructs for a problem determines (or at least constrains) how one goes about trying to generate a solution, and searching the problem space may lead to a change in problem representation. Indeed, Duncker's ( 1945 ) initial account of one subject's solution to the radiation problem was followed up by extensive and experimentally sophisticated work by Simon and his colleagues and by other researchers, documenting the involvement of visual perception and background knowledge in how people represent problems and search for problem solutions.

The relevance of perception and background knowledge to problem solving illustrates the fact that, when people attempt to find or devise ways to reach their goals, they draw on a variety of cognitive resources and engage in a host of cognitive activities. According to Duncker ( 1945 ), such goal-directed activities may include (a) placing objects into categories and making inferences based on category membership, (b) making inductive inferences from multiple instances, (c) reasoning by analogy, (d) identifying the causes of events, (e) deducing logical implications of given information, (f) making legal judgments, and (g) diagnosing medical conditions from historical and laboratory data. As this list suggests, many of the chapters in the present volume describe research that is highly relevant to the understanding of problem-solving behavior. We believe that important advancements in problem-solving research would emerge by integrating it with research in other areas of thinking and reasoning, and that research in these other areas could be similarly advanced by incorporating the insights gained from research on what has more traditionally been identified as problem solving.

As we have described in this chapter, many of the important findings in the field have been established by a careful investigation of various riddle problems. Although there are good methodological reasons for using such problems, many researchers choose to investigate problem solving using ecologically valid educational materials. This choice, which is increasingly common in contemporary research, provides researchers with the opportunity to apply their basic understanding of problem solving to benefit the design of instruction and, at the same time, allows them to gain a better understanding of the processes by which domain knowledge and educational conventions affect the solution process. We believe that the trend of conducting educationally relevant research is likely to continue, and we expect a significant expansion of research on people's understanding and use of dynamic and technologically rich external representations (e.g., Kellman et al., 2008 ; Mayer, Griffith, Jurkowitz, & Rothman, 2008 ; Richland & McDonough, 2010 ; Son & Goldstone, 2009 ). Such investigations are likely to yield both practical and theoretical payoffs.

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Problem Solving, Critical Thinking, and Analytical Reasoning Skills Sought by Employers

In this section:

Problem Solving

• Critical Thinking

Analytical Reasoning

View the content on this page in a Word document.

Critical thinking, analytical reasoning, and problem-solving skills are required to perform well on tasks expected by employers. 1 Having good problem-solving and critical thinking skills can make a major difference in a person’s career. 2

Every day, from an entry-level employee to the Chairman of the Board, problems need to be resolved. Whether solving a problem for a client (internal or external), supporting those who are solving problems, or discovering new problems to solve, the challenges faced may be simple/complex or easy/difficult.

A fundamental component of every manager's role is solving problems. So, helping students become a confident problem solver is critical to their success; and confidence comes from possessing an efficient and practiced problem-solving process.

Employers want employees with well-founded skills in these areas, so they ask four questions when assessing a job candidate 3 :

• Evaluation of information: How well does the applicant assess the quality and relevance of information?
• Analysis and Synthesis of information: How well does the applicant analyze and synthesize data and information?
• Drawing conclusions: How well does the applicant form a conclusion from their analysis?
• Acknowledging alternative explanations/viewpoints: How well does the applicant consider other options and acknowledge that their answer is not the only perspective?

When an employer says they want employees who are good at solving complex problems, they are saying they want employees possessing the following skills:

• Analytical Thinking — A person who can use logic and critical thinking to analyze a situation.
• Critical Thinking – A person who makes reasoned judgments that are logical and well thought out.
• Initiative — A person who will step up and take action without being asked. A person who looks for opportunities to make a difference.
• Creativity — A person who is an original thinker and have the ability to go beyond traditional approaches.
• Resourcefulness — A person who will adapt to new/difficult situations and devise ways to overcome obstacles.
• Determination — A person who is persistent and does not give up easily.
• Results-Oriented — A person whose focus is on getting the problem solved.

Two of the major components of problem-solving skills are critical thinking and analytical reasoning.  These two skills are at the top of skills required of applicants by employers.

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Critical Thinking 4

“Mentions of critical thinking in job postings have doubled since 2009, according to an analysis by career-search site Indeed.com.” 5 Making logical and reasoned judgments that are well thought out is at the core of critical thinking. Using critical thinking an individual will not automatically accept information or conclusions drawn from to be factual, valid, true, applicable or correct. “When students are taught how to use critical thinking to tap into their creativity to solve problems, they are more successful than other students when they enter management-training programs in large corporations.” 6

A strong applicant should question and want to make evidence-based decisions. Employers want employees who say things such as: “Is that a fact or just an opinion? Is this conclusion based on data or gut feel?” and “If you had additional data could there be alternative possibilities?” Employers seek employees who possess the skills and abilities to conceptualize, apply, analyze, synthesize, and evaluate information to reach an answer or conclusion.

Employers require critical thinking in employees because it increases the probability of a positive business outcome. Employers want employees whose thinking is intentional, purposeful, reasoned, and goal directed.

Recruiters say they want applicants with problem-solving and critical thinking skills. They “encourage applicants to prepare stories to illustrate their critical-thinking prowess, detailing, for example, the steps a club president took to improve attendance at weekly meetings.” 7

Employers want students to possess analytical reasoning/thinking skills — meaning they want to hire someone who is good at breaking down problems into smaller parts to find solutions. “The adjective, analytical, and the related verb analyze can both be traced back to the Greek verb, analyein — ‘to break up, to loosen.’ If a student is analytical, you are good at taking a problem or task and breaking it down into smaller elements in order to solve the problem or complete the task.” 9

Analytical reasoning connotes a person's general aptitude to arrive at a logical conclusion or solution to given problems. Just as with critical thinking, analytical thinking critically examines the different parts or details of something to fully understand or explain it. Analytical thinking often requires the person to use “cause and effect, similarities and differences, trends, associations between things, inter-relationships between the parts, the sequence of events, ways to solve complex problems, steps within a process, diagraming what is happening.” 10

Analytical reasoning is the ability to look at information and discern patterns within it. “The pattern could be the structure the author of the information uses to structure an argument, or trends in a large data set. By learning methods of recognizing these patterns, individuals can pull more information out of a text or data set than someone who is not using analytical reasoning to identify deeper patterns.” 11

Employers want employees to have the aptitude to apply analytical reasoning to problems faced by the business. For instance, “a quantitative analyst can break down data into patterns to discern information, such as if a decrease in sales is part of a seasonal pattern of ups and downs or part of a greater downward trend that a business should be worried about. By learning to recognize these patterns in both numbers and written arguments, an individual gains insights into the information that someone who simply takes the information at face value will miss.” 12

Managers with excellent analytical reasoning abilities are considered good at, “evaluating problems, analyzing them from more than one angle and finding a solution that works best in the given circumstances”. 13 Businesses want managers who can apply analytical reasoning skills to meet challenges and keep a business functioning smoothly

A person with good analytical reasoning and pattern recognition skills can see trends in a problem much easier than anyone else.

How to Train Your Problem-Solving Skills

From the hiccups that disrupt your morning routines to the hurdles that define your professional paths, there is always a problem to be solved. 

The good news is that every obstacle is an opportunity to develop problem-solving skills and become the best version of yourself. That’s right: It turns out you can get better at problem-solving, which will help you increase success in daily life and long-term goals.  

Read on to learn how to improve your problem-solving abilities through scientific research and practical strategies.

Understanding Problem-Solving Skills

You may be surprised to learn that your problem-solving skills go beyond just trying to find a solution. Problem-solving skills involve cognitive abilities such as analytical thinking, creativity, decision-making, logical reasoning, and memory. 

Strong problem-solving skills boost critical thinking, spark creativity, and hone decision-making abilities. For you or anyone looking to improve their mental fitness , these skills are necessary for career advancement, personal growth, and positive interpersonal relationships. 

Core Components of Problem-Solving Skills Training

To effectively train your problem-solving skills, it’s important to practice all of the steps required to solve the problem. Think of it this way: Before attempting to solve a problem, your brain has already been hard at work evaluating the situation and picking the best action plan. After you’ve worked hard preparing, you’ll need to implement your plan and assess the outcome by following these steps:  

• Identify and define problems: Recognizing and clearly articulating issues is the foundational step in solving them.
• Generate solutions: Employing brainstorming techniques helps you develop multiple potential solutions.
• Evaluate and select solutions: Using specific criteria to assess solutions helps you choose the most effective one.
• Implement solutions: Developing and executing action plans, including preparing for potential obstacles, guides you to positive outcomes.
• Review and learn from outcomes: Assessing the success of solutions and learning from the results for future improvement facilitates future success. 

Strategies for Developing Problem-Solving Skills

There are many practical exercises and activities that can improve problem-solving abilities.

Cultivate a Problem-Solving Mindset

• Adopt a growth mindset: A growth mindset involves transforming phrases like “I can’t” into “I can’t yet.” Believing in the capacity to improve your skills through effort and perseverance can lead to greater success in problem-solving.
• Practice mindfulness: Mindfulness can enhance cognitive flexibility , allowing you to view problems from multiple perspectives and find creative solutions.

Enhance Core Cognitive Skills 

• Strengthen your memory: Engage in activities that challenge your memory since accurately recalling information is crucial in problem-solving. Techniques such as mnemonic devices or memory palaces can be particularly effective.
• Build your critical thinking: Regularly question assumptions, evaluate arguments, and engage in activities that require reasoning, such as strategy games or debates.

Apply Structured Problem-Solving Techniques

• Use the STOP method: This stands for Stop , Think , Observe , and Plan . It's a simple yet effective way to approach any problem methodically, ensuring you consider all aspects before taking action.
• Try reverse engineering: Start with the desired outcome and work backward to understand the steps needed to achieve that result. This approach can be particularly useful for complex problems with unclear starting points.

Incorporate Technology into Your Training

• Engage with online courses and workshops: Many platforms offer courses specifically designed to enhance problem-solving skills, ranging from critical thinking to creative problem-solving techniques.
• Use cognitive training apps: Apps like Elevate provide targeted, research-backed games and workouts to improve cognitive skills including attention, processing speed, and more. 

Practice with Real-World Applications and Learn from Experience

• Tackle daily challenges: Use everyday issues as opportunities to practice problem-solving. Whether figuring out a new recipe or managing a tight budget, applying your skills in real-world situations can reinforce learning.
• Keep a problem-solving journal: Record the challenges you face, the strategies you employ, and the outcomes you achieve. Reflecting on your problem-solving process over time can provide insights into your strengths and areas for improvement.

Embracing Problem-Solving as a Lifelong Journey

Since problems arise daily, it’s important to feel confident in solving them. 

And you can do just that by downloading the Elevate brain training app. Elevate offers 40+ games and activities designed to improve problem-solving, communication, and other cognitive skills in a personalized way that’s backed by science. Pretty cool, right? 

Consider downloading the Elevate app on Android or iOS now—it’ll be the easiest problem you solve all day. 

Related Articles

How Problem-Solving Games Can Boost Your Brain

• Discover why problem-solving games are fun and effective ways to train your brain. 

Improving Your Problem-Solving Skills

• Discover how to improve your problem-solving skills and make logical, informed decisions.  

Best Ways to Boost Your Mental Fitness

• Mental fitness refers to your ability to sustain your overall well-being. Learn tips to improve yours.  

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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Maths Lesson

Neil Almond

Fluency reasoning and problem solving have been central to the new maths national curriculum for primary schools introduced in 2014. Here we look at how these three approaches or elements of maths can be interwoven in a child’s maths education through KS1 and KS2. We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help primary school teachers think carefully about their practice and the pedagogical choices they make around the teaching of reasoning and problem solving in particular.

Before we can think about what this would look like in practice however, we need to understand the background tothese terms.

What is fluency in maths?

Fluency in maths is a fairly broad concept. The basics of mathematical fluency – as defined by the KS1 / KS2 National Curriculum for maths – involve knowing key mathematical facts and being able to recall them quickly and accurately.

But true fluency in maths (at least up to Key Stage 2) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in maths lessons means we teach the content using a range of representations, to ensure that all pupils understand and have sufficient time to practise what is taught.

Read more: How the best schools develop maths fluency at KS2 .

What is reasoning in maths?

Reasoning in maths is the process of applying logical thinking to a situation to derive the correct problem solving strategy for a given question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows pupils to use the former to accurately carry out the latter.

Read more: Developing maths reasoning at KS2: the mathematical skills required and how to teach them .

What is problem solving in maths?

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in maths. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in maths is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Maths problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some pupils can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognising faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

Mathematical problem solving is a  learned skill

As you might have guessed, the domain of mathematics is far from innate. Maths doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of maths) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like maths) can only be improved by practising elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that pupils can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018)So what is the best method of teaching problem solving to primary maths pupils?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case maths. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Centre at Loughborough University, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support pupils in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

Performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving

I mean to make no sweeping generalisation here; this was my experience both at university when training and from working in schools.

At some point schools become obsessed with the ridiculous notion of ‘accelerated progress’. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘ accelerated progress in maths ’ in this lesson,’ ‘Ofsted will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get pupils onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of pupils and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the pupils to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; set the pupils some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

What IS performance vs learning’?

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those pupils were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same maths content as the fluency exercises, making it more likely that pupils would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers – not exactly high level problem solving skills.

Teaching to “cover the curriculum” hinders development of strong problem solving skills.

This is one of my worries with ‘maths mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content just not happening in the classroom.

Pupils are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

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Fluency and reasoning – Best practice in a lesson, a unit, and a term

By now I hope you have realised that when it comes to problem solving, fluency is king. As such we should look to mastery maths based teaching to ensure that the fluency that pupils need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the year group you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental maths strategies pupils should learn in each year group is a good place to start when thinking about the core aspects of fluency that pupils should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get pupils practicing.

They could chant multiplications when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz pupils on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach times tables KS1 and KS2 for total recall .

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

LY = Last Year

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all maths lessons.

When we think about a term, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that term.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

Best practice for problem solving in a lesson, a unit, and a term 

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practise before we can make connections, reason and problem solve with it.

The same is true for pupils. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practise with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

Now given that much of the content of the KS2 SATs will come from years 5 and 6 it can be hard to stick to this two-year idea as pupils will need to solve problems with content that can be only weeks old to them.

But certainly in other year groups, the argument could be made that content should come from previous years.

You could get pupils in Year 4 to solve complicated place value problems with the numbers they should know from Year 2 or 3. This would lessen the cognitive load, freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Read more: Cognitive load theory in the classroom

Increase complexity gradually.

Once they practise solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move pupils from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

Fluency, Reasoning and Problem Solving should NOT be taught by rote 

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your pupils (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering pupils on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Maths Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Year 6 Maths Reasoning Questions and Answers
• Get to Grips with Maths Problem Solving KS2
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Maths Challenges To Really Stretch Your More Able Pupils
• Maths Reasoning and Problem Solving CPD Powerpoint
• Why You Should Be Incorporating Stem Sentences Into Your Primary Maths Teaching

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How We Use Abstract Thinking

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

MoMo Productions / Getty Images

• How It Develops

Abstract thinking, also known as abstract reasoning, involves the ability to understand and think about complex concepts that, while real, are not tied to concrete experiences, objects, people, or situations.

Abstract thinking is considered a type of higher-order thinking, usually about ideas and principles that are often symbolic or hypothetical. This type of thinking is more complex than the type of thinking that is centered on memorizing and recalling information and facts.

Examples of Abstract Thinking

Examples of abstract concepts include ideas such as:

• Imagination

While these things are real, they aren't concrete, physical things that people can experience directly via their traditional senses.

You likely encounter examples of abstract thinking every day. Stand-up comedians use abstract thinking when they observe absurd or illogical behavior in our world and come up with theories as to why people act the way they do.

You use abstract thinking when you're in a philosophy class or when you're contemplating what would be the most ethical way to conduct your business. If you write a poem or an essay, you're also using abstract thinking.

With all of these examples, concepts that are theoretical and intangible are being translated into a joke, a decision, or a piece of art. (You'll notice that creativity and abstract thinking go hand in hand.)

Abstract Thinking vs. Concrete Thinking

One way of understanding abstract thinking is to compare it with concrete thinking. Concrete thinking, also called concrete reasoning, is tied to specific experiences or objects that can be observed directly.

Research suggests that concrete thinkers tend to focus more on the procedures involved in how a task should be performed, while abstract thinkers are more focused on the reasons why a task should be performed.

It is important to remember that you need both concrete and abstract thinking skills to solve problems in day-to-day life. In many cases, you utilize aspects of both types of thinking to come up with solutions.

Other Types of Thinking

Depending on the type of problem we face, we draw from a number of different styles of thinking, such as:

• Creative thinking : This involves coming up with new ideas, or using existing ideas or objects to come up with a solution or create something new.
• Convergent thinking : Often called linear thinking, this is when a person follows a logical set of steps to select the best solution from already-formulated ideas.
• Critical thinking : This is a type of thinking in which a person tests solutions and analyzes any potential drawbacks.
• Divergent thinking : Often called lateral thinking, this style involves using new thoughts or ideas that are outside of the norm in order to solve problems.

How Abstract Thinking Develops

While abstract thinking is an essential skill, it isn’t something that people are born with. Instead, this cognitive ability develops throughout the course of childhood as children gain new abilities, knowledge, and experiences.

The psychologist Jean Piaget described a theory of cognitive development that outlined this process from birth through adolescence and early adulthood. According to his theory, children go through four distinct stages of intellectual development:

• Sensorimotor stage : During this early period, children's knowledge is derived primarily from their senses.
• Preoperational stage : At this point, children develop the ability to think symbolically.
• Concrete operational stage : At this stage, kids become more logical but their understanding of the world tends to be very concrete.
• Formal operational stage : The ability to reason about concrete information continues to grow during this period, but abstract thinking skills also emerge.

This period of cognitive development when abstract thinking becomes more apparent typically begins around age 12. It is at this age that children become more skilled at thinking about things from the perspective of another person. They are also better able to mentally manipulate abstract ideas as well as notice patterns and relationships between these concepts.

Uses of Abstract Thinking

Abstract thinking is a skill that is essential for the ability to think critically and solve problems. This type of thinking is also related to what is known as fluid intelligence , or the ability to reason and solve problems in unique ways.

Fluid intelligence involves thinking abstractly about problems without relying solely on existing knowledge.

Abstract thinking is used in a number of ways in different aspects of your daily life. Some examples of times you might use this type of thinking:

• When you describe something with a metaphor
• When you talk about something figuratively
• When you come up with creative solutions to a problem
• When you analyze a situation
• When you notice relationships or patterns
• When you form a theory about why something happens
• When you think about a problem from another point of view

Research also suggests that abstract thinking plays a role in the actions people take. Abstract thinkers have been found to be more likely to engage in risky behaviors, where concrete thinkers are more likely to avoid risks.

Impact of Abstract Thinking

People who have strong abstract thinking skills tend to score well on intelligence tests. Because this type of thinking is associated with creativity, abstract thinkers also tend to excel in areas that require creativity such as art, writing, and other areas that benefit from divergent thinking abilities.

Abstract thinking can have both positive and negative effects. It can be used as a tool to promote innovative problem-solving, but it can also lead to problems in some cases:

• Bias : Research also suggests that it can sometimes promote different types of bias . As people seek to understand events, abstract thinking can sometimes cause people to seek out patterns, themes, and relationships that may not exist.
• Catastrophic thinking : Sometimes these inferences, imagined scenarios, and predictions about the future can lead to feelings of fear and anxiety. Instead of making realistic predictions, people may catastrophize and imagine the worst possible potential outcomes.
• Anxiety and depression : Research has also found that abstract thinking styles are sometimes associated with worry and rumination . This thinking style is also associated with a range of conditions including depression , anxiety, and post-traumatic stress disorder (PTSD) .

Conditions That Impact Abstract Thinking

The presence of learning disabilities and mental health conditions can affect abstract thinking abilities. Conditions that are linked to impaired abstract thinking skills include:

• Learning disabilities
• Schizophrenia
• Traumatic brain injury (TBI)

The natural aging process can also have an impact on abstract thinking skills. Research suggests that the thinking skills associated with fluid intelligence peak around the ages of 30 or 40 and begin to decline with age.

Tips for Reasoning Abstractly

While some psychologists believe that abstract thinking skills are a natural product of normal development, others suggest that these abilities are influenced by genetics, culture, and experiences. Some people may come by these skills naturally, but you can also strengthen these abilities with practice.

Some strategies that you might use to help improve your abstract thinking skills:

• Think about why and not just how : Abstract thinkers tend to focus on the meaning of events or on hypothetical outcomes. Instead of concentrating only on the steps needed to achieve a goal, consider some of the reasons why that goal might be valuable or what might happen if you reach that goal.
• Reframe your thinking : When you are approaching a problem, it can be helpful to purposefully try to think about the problem in a different way. How might someone else approach it? Is there an easier way to accomplish the same thing? Are there any elements you haven't considered?
• Consider the big picture : Rather than focusing on the specifics of a situation, try taking a step back in order to view the big picture. Where concrete thinkers are more likely to concentrate on the details, abstract thinkers focus on how something relates to other things or how it fits into the grand scheme of things.

Abstract thinking allows people to think about complex relationships, recognize patterns, solve problems, and utilize creativity. While some people tend to be naturally better at this type of reasoning, it is a skill that you can learn to utilize and strengthen with practice. 

It is important to remember that both concrete and abstract thinking are skills that you need to solve problems and function successfully. 

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American Psychological Association. Creative thinking .

American Psychological Association. Convergent thinking .

American Psychological Association. Critical thinking .

American Psychological Association. Divergent thinking .

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By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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TIMSS 2019 Mathematics Framework

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Mathematics Cognitive Domains–Fourth and Eighth Grades

In order to respond correctly to TIMSS test items, students need to be familiar with the mathematics content being assessed, but they also need to draw on a range of cognitive skills. Describing these skills plays a crucial role in the development of an assessment like TIMSS 2019, because they are vital in ensuring that the survey covers the appropriate range of cognitive skills across the content domains already outlined.

The first domain, knowing , covers the facts, concepts, and procedures students need to know, while the second, applying , focuses on the ability of students to apply knowledge and conceptual understanding to solve problems or answer questions. The third domain, reasoning , goes beyond the solution of routine problems to encompass unfamiliar situations, complex contexts, and multistep problems.

Knowing, applying, and reasoning are exercised in varying degrees when students display their mathematical competency, which goes beyond content knowledge. These TIMSS cognitive domains encompass the competencies of problem solving, providing a mathematical argument to support a strategy or solution, representing a situation mathematically (e.g., using symbols and graphs), creating mathematical models of a problem situation, and using tools such as a ruler or a calculator to help solve problems.

The three cognitive domains are used for both grades, but the balance of testing time differs, reflecting the difference in age and experience of students in the two grades. For the fourth and eighth grades, each content domain will include items developed to address each of the three cognitive domains. For example, the number domain will include knowing, applying, and reasoning items as will the other content domains.

Exhibit 1.4 shows the target percentages of testing time devoted to each cognitive domain for the fourth and eighth grade assessments.

Exhibit 1.4: Target Percentages of the TIMSS 2019 Mathematics Assessment Devoted to Cognitive Domains at the Fourth and Eighth Grades

Facility in applying mathematics, or reasoning about mathematical situations, depends on familiarity with mathematical concepts and fluency in mathematical skills. The more relevant knowledge a student is able to recall and the wider the range of concepts he or she understands, the greater the potential for engaging in a wide range of problem solving situations.

Without access to a knowledge base that enables easy recall of the language and basic facts and conventions of number, symbolic representation, and spatial relations, students would find purposeful mathematical thinking impossible. Facts encompass the knowledge that provides the basic language of mathematics, as well as the essential mathematical concepts and properties that form the foundation for mathematical thought.

Procedures form a bridge between more basic knowledge and the use of mathematics for solving problems, especially those encountered by many people in their daily lives. In essence, a fluent use of procedures entails recall of sets of actions and how to carry them out. Students need to be efficient and accurate in using a variety of computational procedures and tools. They need to see that particular procedures can be used to solve entire classes of problems, not just individual problems.

The applying domain involves the application of mathematics in a range of contexts. In this domain, the facts, concepts, and procedures as well as the problems should be familiar to the student. In some items aligned with this domain, students need to apply mathematical knowledge of facts, skills, and procedures or understanding of mathematical concepts to create representations. Representation of ideas forms the core of mathematical thinking and communication, and the ability to create equivalent representations is fundamental to success in the subject.

Problem solving is central to the applying domain, with an emphasis on more familiar and routine tasks. Problems may be set in real life situations, or may be concerned with purely mathematical questions involving, for example, numeric or algebraic expressions, functions, equations, geometric figures, or statistical data sets.

Reasoning mathematically involves logical, systematic thinking. It includes intuitive and inductive reasoning based on patterns and regularities that can be used to arrive at solutions to problems set in novel or unfamiliar situations. Such problems may be purely mathematical or may have real life settings. Both types of items involve transferring knowledge and skills to new situations; and interactions among reasoning skills usually are a feature of such items.

Even though many of the cognitive skills listed in the reasoning domain may be drawn on when thinking about and solving novel or complex problems, each by itself represents a valuable outcome of mathematics education, with the potential to influence learners’ thinking more generally. For example, reasoning involves the ability to observe and make conjectures. It also involves making logical deductions based on specific assumptions and rules, and justifying results.

Does Problem Solving = Prior Knowledge + Reasoning Skills in Earth Science? An Exploratory Study

• Published: 04 October 2008
• Volume 40 , pages 103–116, ( 2010 )

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• Chun-Yen Chang 1  

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This study examined the interrelationship between tenth-grade students’ problem solving ability (PSA) and their domain-specific knowledge (DSK) as well as reasoning skills (RS) in a secondary school of Taiwan. The PSA test was designed to emphasize students’ divergent-thinking ability (DTA) and convergent-thinking ability (CTA) subscales in the area of Earth science. Two hundred and sixty tenth graders who were enrolled in six Earth science classes at a public senior high school located in the eastern region of Taiwan were participants. Major findings are as follows: (a) A significantly positive correlation existed between students’ PSA and their DSK and RS, approaching large effect sizes; (b) Both students’ DSK and RS significantly explained the variance of their PSA with large effect sizes; (c) Students’ RS could more significantly explain the variance of their DTA subscale with medium effect size while DSK might more significantly explain the variance of their CTA, approaching large effect size. The research suggests that more emphasis should be placed on the reasoning skills when developing students’ divergent-thinking abilities, while stressing more domain-specific knowledge when students’ convergent-thinking ability is considered.

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Chang, CY. Does Problem Solving = Prior Knowledge + Reasoning Skills in Earth Science? An Exploratory Study. Res Sci Educ 40 , 103–116 (2010). https://doi.org/10.1007/s11165-008-9102-0

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DOI : https://doi.org/10.1007/s11165-008-9102-0

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