what problem solving cognitive level entails in relation to the skills to be demonstrated

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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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• Published: 11 January 2023

The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature

• Enwei Xu   ORCID: orcid.org/0000-0001-6424-8169 1 ,
• Wei Wang 1 &
• Qingxia Wang 1  

Humanities and Social Sciences Communications volume  10 , Article number:  16 ( 2023 ) Cite this article

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Collaborative problem-solving has been widely embraced in the classroom instruction of critical thinking, which is regarded as the core of curriculum reform based on key competencies in the field of education as well as a key competence for learners in the 21st century. However, the effectiveness of collaborative problem-solving in promoting students’ critical thinking remains uncertain. This current research presents the major findings of a meta-analysis of 36 pieces of the literature revealed in worldwide educational periodicals during the 21st century to identify the effectiveness of collaborative problem-solving in promoting students’ critical thinking and to determine, based on evidence, whether and to what extent collaborative problem solving can result in a rise or decrease in critical thinking. The findings show that (1) collaborative problem solving is an effective teaching approach to foster students’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]); (2) in respect to the dimensions of critical thinking, collaborative problem solving can significantly and successfully enhance students’ attitudinal tendencies (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI[0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI[0.58, 0.82]); and (3) the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have an impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. On the basis of these results, recommendations are made for further study and instruction to better support students’ critical thinking in the context of collaborative problem-solving.

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Introduction

Although critical thinking has a long history in research, the concept of critical thinking, which is regarded as an essential competence for learners in the 21st century, has recently attracted more attention from researchers and teaching practitioners (National Research Council, 2012 ). Critical thinking should be the core of curriculum reform based on key competencies in the field of education (Peng and Deng, 2017 ) because students with critical thinking can not only understand the meaning of knowledge but also effectively solve practical problems in real life even after knowledge is forgotten (Kek and Huijser, 2011 ). The definition of critical thinking is not universal (Ennis, 1989 ; Castle, 2009 ; Niu et al., 2013 ). In general, the definition of critical thinking is a self-aware and self-regulated thought process (Facione, 1990 ; Niu et al., 2013 ). It refers to the cognitive skills needed to interpret, analyze, synthesize, reason, and evaluate information as well as the attitudinal tendency to apply these abilities (Halpern, 2001 ). The view that critical thinking can be taught and learned through curriculum teaching has been widely supported by many researchers (e.g., Kuncel, 2011 ; Leng and Lu, 2020 ), leading to educators’ efforts to foster it among students. In the field of teaching practice, there are three types of courses for teaching critical thinking (Ennis, 1989 ). The first is an independent curriculum in which critical thinking is taught and cultivated without involving the knowledge of specific disciplines; the second is an integrated curriculum in which critical thinking is integrated into the teaching of other disciplines as a clear teaching goal; and the third is a mixed curriculum in which critical thinking is taught in parallel to the teaching of other disciplines for mixed teaching training. Furthermore, numerous measuring tools have been developed by researchers and educators to measure critical thinking in the context of teaching practice. These include standardized measurement tools, such as WGCTA, CCTST, CCTT, and CCTDI, which have been verified by repeated experiments and are considered effective and reliable by international scholars (Facione and Facione, 1992 ). In short, descriptions of critical thinking, including its two dimensions of attitudinal tendency and cognitive skills, different types of teaching courses, and standardized measurement tools provide a complex normative framework for understanding, teaching, and evaluating critical thinking.

Cultivating critical thinking in curriculum teaching can start with a problem, and one of the most popular critical thinking instructional approaches is problem-based learning (Liu et al., 2020 ). Duch et al. ( 2001 ) noted that problem-based learning in group collaboration is progressive active learning, which can improve students’ critical thinking and problem-solving skills. Collaborative problem-solving is the organic integration of collaborative learning and problem-based learning, which takes learners as the center of the learning process and uses problems with poor structure in real-world situations as the starting point for the learning process (Liang et al., 2017 ). Students learn the knowledge needed to solve problems in a collaborative group, reach a consensus on problems in the field, and form solutions through social cooperation methods, such as dialogue, interpretation, questioning, debate, negotiation, and reflection, thus promoting the development of learners’ domain knowledge and critical thinking (Cindy, 2004 ; Liang et al., 2017 ).

Collaborative problem-solving has been widely used in the teaching practice of critical thinking, and several studies have attempted to conduct a systematic review and meta-analysis of the empirical literature on critical thinking from various perspectives. However, little attention has been paid to the impact of collaborative problem-solving on critical thinking. Therefore, the best approach for developing and enhancing critical thinking throughout collaborative problem-solving is to examine how to implement critical thinking instruction; however, this issue is still unexplored, which means that many teachers are incapable of better instructing critical thinking (Leng and Lu, 2020 ; Niu et al., 2013 ). For example, Huber ( 2016 ) provided the meta-analysis findings of 71 publications on gaining critical thinking over various time frames in college with the aim of determining whether critical thinking was truly teachable. These authors found that learners significantly improve their critical thinking while in college and that critical thinking differs with factors such as teaching strategies, intervention duration, subject area, and teaching type. The usefulness of collaborative problem-solving in fostering students’ critical thinking, however, was not determined by this study, nor did it reveal whether there existed significant variations among the different elements. A meta-analysis of 31 pieces of educational literature was conducted by Liu et al. ( 2020 ) to assess the impact of problem-solving on college students’ critical thinking. These authors found that problem-solving could promote the development of critical thinking among college students and proposed establishing a reasonable group structure for problem-solving in a follow-up study to improve students’ critical thinking. Additionally, previous empirical studies have reached inconclusive and even contradictory conclusions about whether and to what extent collaborative problem-solving increases or decreases critical thinking levels. As an illustration, Yang et al. ( 2008 ) carried out an experiment on the integrated curriculum teaching of college students based on a web bulletin board with the goal of fostering participants’ critical thinking in the context of collaborative problem-solving. These authors’ research revealed that through sharing, debating, examining, and reflecting on various experiences and ideas, collaborative problem-solving can considerably enhance students’ critical thinking in real-life problem situations. In contrast, collaborative problem-solving had a positive impact on learners’ interaction and could improve learning interest and motivation but could not significantly improve students’ critical thinking when compared to traditional classroom teaching, according to research by Naber and Wyatt ( 2014 ) and Sendag and Odabasi ( 2009 ) on undergraduate and high school students, respectively.

The above studies show that there is inconsistency regarding the effectiveness of collaborative problem-solving in promoting students’ critical thinking. Therefore, it is essential to conduct a thorough and trustworthy review to detect and decide whether and to what degree collaborative problem-solving can result in a rise or decrease in critical thinking. Meta-analysis is a quantitative analysis approach that is utilized to examine quantitative data from various separate studies that are all focused on the same research topic. This approach characterizes the effectiveness of its impact by averaging the effect sizes of numerous qualitative studies in an effort to reduce the uncertainty brought on by independent research and produce more conclusive findings (Lipsey and Wilson, 2001 ).

This paper used a meta-analytic approach and carried out a meta-analysis to examine the effectiveness of collaborative problem-solving in promoting students’ critical thinking in order to make a contribution to both research and practice. The following research questions were addressed by this meta-analysis:

What is the overall effect size of collaborative problem-solving in promoting students’ critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills)?

How are the disparities between the study conclusions impacted by various moderating variables if the impacts of various experimental designs in the included studies are heterogeneous?

This research followed the strict procedures (e.g., database searching, identification, screening, eligibility, merging, duplicate removal, and analysis of included studies) of Cooper’s ( 2010 ) proposed meta-analysis approach for examining quantitative data from various separate studies that are all focused on the same research topic. The relevant empirical research that appeared in worldwide educational periodicals within the 21st century was subjected to this meta-analysis using Rev-Man 5.4. The consistency of the data extracted separately by two researchers was tested using Cohen’s kappa coefficient, and a publication bias test and a heterogeneity test were run on the sample data to ascertain the quality of this meta-analysis.

Data sources and search strategies

There were three stages to the data collection process for this meta-analysis, as shown in Fig. 1 , which shows the number of articles included and eliminated during the selection process based on the statement and study eligibility criteria.

This flowchart shows the number of records identified, included and excluded in the article.

First, the databases used to systematically search for relevant articles were the journal papers of the Web of Science Core Collection and the Chinese Core source journal, as well as the Chinese Social Science Citation Index (CSSCI) source journal papers included in CNKI. These databases were selected because they are credible platforms that are sources of scholarly and peer-reviewed information with advanced search tools and contain literature relevant to the subject of our topic from reliable researchers and experts. The search string with the Boolean operator used in the Web of Science was “TS = (((“critical thinking” or “ct” and “pretest” or “posttest”) or (“critical thinking” or “ct” and “control group” or “quasi experiment” or “experiment”)) and (“collaboration” or “collaborative learning” or “CSCL”) and (“problem solving” or “problem-based learning” or “PBL”))”. The research area was “Education Educational Research”, and the search period was “January 1, 2000, to December 30, 2021”. A total of 412 papers were obtained. The search string with the Boolean operator used in the CNKI was “SU = (‘critical thinking’*‘collaboration’ + ‘critical thinking’*‘collaborative learning’ + ‘critical thinking’*‘CSCL’ + ‘critical thinking’*‘problem solving’ + ‘critical thinking’*‘problem-based learning’ + ‘critical thinking’*‘PBL’ + ‘critical thinking’*‘problem oriented’) AND FT = (‘experiment’ + ‘quasi experiment’ + ‘pretest’ + ‘posttest’ + ‘empirical study’)” (translated into Chinese when searching). A total of 56 studies were found throughout the search period of “January 2000 to December 2021”. From the databases, all duplicates and retractions were eliminated before exporting the references into Endnote, a program for managing bibliographic references. In all, 466 studies were found.

Second, the studies that matched the inclusion and exclusion criteria for the meta-analysis were chosen by two researchers after they had reviewed the abstracts and titles of the gathered articles, yielding a total of 126 studies.

Third, two researchers thoroughly reviewed each included article’s whole text in accordance with the inclusion and exclusion criteria. Meanwhile, a snowball search was performed using the references and citations of the included articles to ensure complete coverage of the articles. Ultimately, 36 articles were kept.

Two researchers worked together to carry out this entire process, and a consensus rate of almost 94.7% was reached after discussion and negotiation to clarify any emerging differences.

Eligibility criteria

Since not all the retrieved studies matched the criteria for this meta-analysis, eligibility criteria for both inclusion and exclusion were developed as follows:

The publication language of the included studies was limited to English and Chinese, and the full text could be obtained. Articles that did not meet the publication language and articles not published between 2000 and 2021 were excluded.

The research design of the included studies must be empirical and quantitative studies that can assess the effect of collaborative problem-solving on the development of critical thinking. Articles that could not identify the causal mechanisms by which collaborative problem-solving affects critical thinking, such as review articles and theoretical articles, were excluded.

The research method of the included studies must feature a randomized control experiment or a quasi-experiment, or a natural experiment, which have a higher degree of internal validity with strong experimental designs and can all plausibly provide evidence that critical thinking and collaborative problem-solving are causally related. Articles with non-experimental research methods, such as purely correlational or observational studies, were excluded.

The participants of the included studies were only students in school, including K-12 students and college students. Articles in which the participants were non-school students, such as social workers or adult learners, were excluded.

The research results of the included studies must mention definite signs that may be utilized to gauge critical thinking’s impact (e.g., sample size, mean value, or standard deviation). Articles that lacked specific measurement indicators for critical thinking and could not calculate the effect size were excluded.

Data coding design

In order to perform a meta-analysis, it is necessary to collect the most important information from the articles, codify that information’s properties, and convert descriptive data into quantitative data. Therefore, this study designed a data coding template (see Table 1 ). Ultimately, 16 coding fields were retained.

The designed data-coding template consisted of three pieces of information. Basic information about the papers was included in the descriptive information: the publishing year, author, serial number, and title of the paper.

The variable information for the experimental design had three variables: the independent variable (instruction method), the dependent variable (critical thinking), and the moderating variable (learning stage, teaching type, intervention duration, learning scaffold, group size, measuring tool, and subject area). Depending on the topic of this study, the intervention strategy, as the independent variable, was coded into collaborative and non-collaborative problem-solving. The dependent variable, critical thinking, was coded as a cognitive skill and an attitudinal tendency. And seven moderating variables were created by grouping and combining the experimental design variables discovered within the 36 studies (see Table 1 ), where learning stages were encoded as higher education, high school, middle school, and primary school or lower; teaching types were encoded as mixed courses, integrated courses, and independent courses; intervention durations were encoded as 0–1 weeks, 1–4 weeks, 4–12 weeks, and more than 12 weeks; group sizes were encoded as 2–3 persons, 4–6 persons, 7–10 persons, and more than 10 persons; learning scaffolds were encoded as teacher-supported learning scaffold, technique-supported learning scaffold, and resource-supported learning scaffold; measuring tools were encoded as standardized measurement tools (e.g., WGCTA, CCTT, CCTST, and CCTDI) and self-adapting measurement tools (e.g., modified or made by researchers); and subject areas were encoded according to the specific subjects used in the 36 included studies.

The data information contained three metrics for measuring critical thinking: sample size, average value, and standard deviation. It is vital to remember that studies with various experimental designs frequently adopt various formulas to determine the effect size. And this paper used Morris’ proposed standardized mean difference (SMD) calculation formula ( 2008 , p. 369; see Supplementary Table S3 ).

Procedure for extracting and coding data

According to the data coding template (see Table 1 ), the 36 papers’ information was retrieved by two researchers, who then entered them into Excel (see Supplementary Table S1 ). The results of each study were extracted separately in the data extraction procedure if an article contained numerous studies on critical thinking, or if a study assessed different critical thinking dimensions. For instance, Tiwari et al. ( 2010 ) used four time points, which were viewed as numerous different studies, to examine the outcomes of critical thinking, and Chen ( 2013 ) included the two outcome variables of attitudinal tendency and cognitive skills, which were regarded as two studies. After discussion and negotiation during data extraction, the two researchers’ consistency test coefficients were roughly 93.27%. Supplementary Table S2 details the key characteristics of the 36 included articles with 79 effect quantities, including descriptive information (e.g., the publishing year, author, serial number, and title of the paper), variable information (e.g., independent variables, dependent variables, and moderating variables), and data information (e.g., mean values, standard deviations, and sample size). Following that, testing for publication bias and heterogeneity was done on the sample data using the Rev-Man 5.4 software, and then the test results were used to conduct a meta-analysis.

Publication bias test

When the sample of studies included in a meta-analysis does not accurately reflect the general status of research on the relevant subject, publication bias is said to be exhibited in this research. The reliability and accuracy of the meta-analysis may be impacted by publication bias. Due to this, the meta-analysis needs to check the sample data for publication bias (Stewart et al., 2006 ). A popular method to check for publication bias is the funnel plot; and it is unlikely that there will be publishing bias when the data are equally dispersed on either side of the average effect size and targeted within the higher region. The data are equally dispersed within the higher portion of the efficient zone, consistent with the funnel plot connected with this analysis (see Fig. 2 ), indicating that publication bias is unlikely in this situation.

This funnel plot shows the result of publication bias of 79 effect quantities across 36 studies.

Heterogeneity test

To select the appropriate effect models for the meta-analysis, one might use the results of a heterogeneity test on the data effect sizes. In a meta-analysis, it is common practice to gauge the degree of data heterogeneity using the I 2 value, and I 2  ≥ 50% is typically understood to denote medium-high heterogeneity, which calls for the adoption of a random effect model; if not, a fixed effect model ought to be applied (Lipsey and Wilson, 2001 ). The findings of the heterogeneity test in this paper (see Table 2 ) revealed that I 2 was 86% and displayed significant heterogeneity ( P  < 0.01). To ensure accuracy and reliability, the overall effect size ought to be calculated utilizing the random effect model.

The analysis of the overall effect size

This meta-analysis utilized a random effect model to examine 79 effect quantities from 36 studies after eliminating heterogeneity. In accordance with Cohen’s criterion (Cohen, 1992 ), it is abundantly clear from the analysis results, which are shown in the forest plot of the overall effect (see Fig. 3 ), that the cumulative impact size of cooperative problem-solving is 0.82, which is statistically significant ( z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]), and can encourage learners to practice critical thinking.

This forest plot shows the analysis result of the overall effect size across 36 studies.

In addition, this study examined two distinct dimensions of critical thinking to better understand the precise contributions that collaborative problem-solving makes to the growth of critical thinking. The findings (see Table 3 ) indicate that collaborative problem-solving improves cognitive skills (ES = 0.70) and attitudinal tendency (ES = 1.17), with significant intergroup differences (chi 2  = 7.95, P  < 0.01). Although collaborative problem-solving improves both dimensions of critical thinking, it is essential to point out that the improvements in students’ attitudinal tendency are much more pronounced and have a significant comprehensive effect (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]), whereas gains in learners’ cognitive skill are slightly improved and are just above average. (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

The analysis of moderator effect size

The whole forest plot’s 79 effect quantities underwent a two-tailed test, which revealed significant heterogeneity ( I 2  = 86%, z  = 12.78, P  < 0.01), indicating differences between various effect sizes that may have been influenced by moderating factors other than sampling error. Therefore, exploring possible moderating factors that might produce considerable heterogeneity was done using subgroup analysis, such as the learning stage, learning scaffold, teaching type, group size, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, in order to further explore the key factors that influence critical thinking. The findings (see Table 4 ) indicate that various moderating factors have advantageous effects on critical thinking. In this situation, the subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), learning scaffold (chi 2  = 9.03, P  < 0.01), and teaching type (chi 2  = 7.20, P  < 0.05) are all significant moderators that can be applied to support the cultivation of critical thinking. However, since the learning stage and the measuring tools did not significantly differ among intergroup (chi 2  = 3.15, P  = 0.21 > 0.05, and chi 2  = 0.08, P  = 0.78 > 0.05), we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving. These are the precise outcomes, as follows:

Various learning stages influenced critical thinking positively, without significant intergroup differences (chi 2  = 3.15, P  = 0.21 > 0.05). High school was first on the list of effect sizes (ES = 1.36, P  < 0.01), then higher education (ES = 0.78, P  < 0.01), and middle school (ES = 0.73, P  < 0.01). These results show that, despite the learning stage’s beneficial influence on cultivating learners’ critical thinking, we are unable to explain why it is essential for cultivating critical thinking in the context of collaborative problem-solving.

Different teaching types had varying degrees of positive impact on critical thinking, with significant intergroup differences (chi 2  = 7.20, P  < 0.05). The effect size was ranked as follows: mixed courses (ES = 1.34, P  < 0.01), integrated courses (ES = 0.81, P  < 0.01), and independent courses (ES = 0.27, P  < 0.01). These results indicate that the most effective approach to cultivate critical thinking utilizing collaborative problem solving is through the teaching type of mixed courses.

Various intervention durations significantly improved critical thinking, and there were significant intergroup differences (chi 2  = 12.18, P  < 0.01). The effect sizes related to this variable showed a tendency to increase with longer intervention durations. The improvement in critical thinking reached a significant level (ES = 0.85, P  < 0.01) after more than 12 weeks of training. These findings indicate that the intervention duration and critical thinking’s impact are positively correlated, with a longer intervention duration having a greater effect.

Different learning scaffolds influenced critical thinking positively, with significant intergroup differences (chi 2  = 9.03, P  < 0.01). The resource-supported learning scaffold (ES = 0.69, P  < 0.01) acquired a medium-to-higher level of impact, the technique-supported learning scaffold (ES = 0.63, P  < 0.01) also attained a medium-to-higher level of impact, and the teacher-supported learning scaffold (ES = 0.92, P  < 0.01) displayed a high level of significant impact. These results show that the learning scaffold with teacher support has the greatest impact on cultivating critical thinking.

Various group sizes influenced critical thinking positively, and the intergroup differences were statistically significant (chi 2  = 8.77, P  < 0.05). Critical thinking showed a general declining trend with increasing group size. The overall effect size of 2–3 people in this situation was the biggest (ES = 0.99, P  < 0.01), and when the group size was greater than 7 people, the improvement in critical thinking was at the lower-middle level (ES < 0.5, P  < 0.01). These results show that the impact on critical thinking is positively connected with group size, and as group size grows, so does the overall impact.

Various measuring tools influenced critical thinking positively, with significant intergroup differences (chi 2  = 0.08, P  = 0.78 > 0.05). In this situation, the self-adapting measurement tools obtained an upper-medium level of effect (ES = 0.78), whereas the complete effect size of the standardized measurement tools was the largest, achieving a significant level of effect (ES = 0.84, P  < 0.01). These results show that, despite the beneficial influence of the measuring tool on cultivating critical thinking, we are unable to explain why it is crucial in fostering the growth of critical thinking by utilizing the approach of collaborative problem-solving.

Different subject areas had a greater impact on critical thinking, and the intergroup differences were statistically significant (chi 2  = 13.36, P  < 0.05). Mathematics had the greatest overall impact, achieving a significant level of effect (ES = 1.68, P  < 0.01), followed by science (ES = 1.25, P  < 0.01) and medical science (ES = 0.87, P  < 0.01), both of which also achieved a significant level of effect. Programming technology was the least effective (ES = 0.39, P  < 0.01), only having a medium-low degree of effect compared to education (ES = 0.72, P  < 0.01) and other fields (such as language, art, and social sciences) (ES = 0.58, P  < 0.01). These results suggest that scientific fields (e.g., mathematics, science) may be the most effective subject areas for cultivating critical thinking utilizing the approach of collaborative problem-solving.

The effectiveness of collaborative problem solving with regard to teaching critical thinking

According to this meta-analysis, using collaborative problem-solving as an intervention strategy in critical thinking teaching has a considerable amount of impact on cultivating learners’ critical thinking as a whole and has a favorable promotional effect on the two dimensions of critical thinking. According to certain studies, collaborative problem solving, the most frequently used critical thinking teaching strategy in curriculum instruction can considerably enhance students’ critical thinking (e.g., Liang et al., 2017 ; Liu et al., 2020 ; Cindy, 2004 ). This meta-analysis provides convergent data support for the above research views. Thus, the findings of this meta-analysis not only effectively address the first research query regarding the overall effect of cultivating critical thinking and its impact on the two dimensions of critical thinking (i.e., attitudinal tendency and cognitive skills) utilizing the approach of collaborative problem-solving, but also enhance our confidence in cultivating critical thinking by using collaborative problem-solving intervention approach in the context of classroom teaching.

Furthermore, the associated improvements in attitudinal tendency are much stronger, but the corresponding improvements in cognitive skill are only marginally better. According to certain studies, cognitive skill differs from the attitudinal tendency in classroom instruction; the cultivation and development of the former as a key ability is a process of gradual accumulation, while the latter as an attitude is affected by the context of the teaching situation (e.g., a novel and exciting teaching approach, challenging and rewarding tasks) (Halpern, 2001 ; Wei and Hong, 2022 ). Collaborative problem-solving as a teaching approach is exciting and interesting, as well as rewarding and challenging; because it takes the learners as the focus and examines problems with poor structure in real situations, and it can inspire students to fully realize their potential for problem-solving, which will significantly improve their attitudinal tendency toward solving problems (Liu et al., 2020 ). Similar to how collaborative problem-solving influences attitudinal tendency, attitudinal tendency impacts cognitive skill when attempting to solve a problem (Liu et al., 2020 ; Zhang et al., 2022 ), and stronger attitudinal tendencies are associated with improved learning achievement and cognitive ability in students (Sison, 2008 ; Zhang et al., 2022 ). It can be seen that the two specific dimensions of critical thinking as well as critical thinking as a whole are affected by collaborative problem-solving, and this study illuminates the nuanced links between cognitive skills and attitudinal tendencies with regard to these two dimensions of critical thinking. To fully develop students’ capacity for critical thinking, future empirical research should pay closer attention to cognitive skills.

The moderating effects of collaborative problem solving with regard to teaching critical thinking

In order to further explore the key factors that influence critical thinking, exploring possible moderating effects that might produce considerable heterogeneity was done using subgroup analysis. The findings show that the moderating factors, such as the teaching type, learning stage, group size, learning scaffold, duration of the intervention, measuring tool, and the subject area included in the 36 experimental designs, could all support the cultivation of collaborative problem-solving in critical thinking. Among them, the effect size differences between the learning stage and measuring tool are not significant, which does not explain why these two factors are crucial in supporting the cultivation of critical thinking utilizing the approach of collaborative problem-solving.

In terms of the learning stage, various learning stages influenced critical thinking positively without significant intergroup differences, indicating that we are unable to explain why it is crucial in fostering the growth of critical thinking.

Although high education accounts for 70.89% of all empirical studies performed by researchers, high school may be the appropriate learning stage to foster students’ critical thinking by utilizing the approach of collaborative problem-solving since it has the largest overall effect size. This phenomenon may be related to student’s cognitive development, which needs to be further studied in follow-up research.

With regard to teaching type, mixed course teaching may be the best teaching method to cultivate students’ critical thinking. Relevant studies have shown that in the actual teaching process if students are trained in thinking methods alone, the methods they learn are isolated and divorced from subject knowledge, which is not conducive to their transfer of thinking methods; therefore, if students’ thinking is trained only in subject teaching without systematic method training, it is challenging to apply to real-world circumstances (Ruggiero, 2012 ; Hu and Liu, 2015 ). Teaching critical thinking as mixed course teaching in parallel to other subject teachings can achieve the best effect on learners’ critical thinking, and explicit critical thinking instruction is more effective than less explicit critical thinking instruction (Bensley and Spero, 2014 ).

In terms of the intervention duration, with longer intervention times, the overall effect size shows an upward tendency. Thus, the intervention duration and critical thinking’s impact are positively correlated. Critical thinking, as a key competency for students in the 21st century, is difficult to get a meaningful improvement in a brief intervention duration. Instead, it could be developed over a lengthy period of time through consistent teaching and the progressive accumulation of knowledge (Halpern, 2001 ; Hu and Liu, 2015 ). Therefore, future empirical studies ought to take these restrictions into account throughout a longer period of critical thinking instruction.

With regard to group size, a group size of 2–3 persons has the highest effect size, and the comprehensive effect size decreases with increasing group size in general. This outcome is in line with some research findings; as an example, a group composed of two to four members is most appropriate for collaborative learning (Schellens and Valcke, 2006 ). However, the meta-analysis results also indicate that once the group size exceeds 7 people, small groups cannot produce better interaction and performance than large groups. This may be because the learning scaffolds of technique support, resource support, and teacher support improve the frequency and effectiveness of interaction among group members, and a collaborative group with more members may increase the diversity of views, which is helpful to cultivate critical thinking utilizing the approach of collaborative problem-solving.

With regard to the learning scaffold, the three different kinds of learning scaffolds can all enhance critical thinking. Among them, the teacher-supported learning scaffold has the largest overall effect size, demonstrating the interdependence of effective learning scaffolds and collaborative problem-solving. This outcome is in line with some research findings; as an example, a successful strategy is to encourage learners to collaborate, come up with solutions, and develop critical thinking skills by using learning scaffolds (Reiser, 2004 ; Xu et al., 2022 ); learning scaffolds can lower task complexity and unpleasant feelings while also enticing students to engage in learning activities (Wood et al., 2006 ); learning scaffolds are designed to assist students in using learning approaches more successfully to adapt the collaborative problem-solving process, and the teacher-supported learning scaffolds have the greatest influence on critical thinking in this process because they are more targeted, informative, and timely (Xu et al., 2022 ).

With respect to the measuring tool, despite the fact that standardized measurement tools (such as the WGCTA, CCTT, and CCTST) have been acknowledged as trustworthy and effective by worldwide experts, only 54.43% of the research included in this meta-analysis adopted them for assessment, and the results indicated no intergroup differences. These results suggest that not all teaching circumstances are appropriate for measuring critical thinking using standardized measurement tools. “The measuring tools for measuring thinking ability have limits in assessing learners in educational situations and should be adapted appropriately to accurately assess the changes in learners’ critical thinking.”, according to Simpson and Courtney ( 2002 , p. 91). As a result, in order to more fully and precisely gauge how learners’ critical thinking has evolved, we must properly modify standardized measuring tools based on collaborative problem-solving learning contexts.

With regard to the subject area, the comprehensive effect size of science departments (e.g., mathematics, science, medical science) is larger than that of language arts and social sciences. Some recent international education reforms have noted that critical thinking is a basic part of scientific literacy. Students with scientific literacy can prove the rationality of their judgment according to accurate evidence and reasonable standards when they face challenges or poorly structured problems (Kyndt et al., 2013 ), which makes critical thinking crucial for developing scientific understanding and applying this understanding to practical problem solving for problems related to science, technology, and society (Yore et al., 2007 ).

Suggestions for critical thinking teaching

Other than those stated in the discussion above, the following suggestions are offered for critical thinking instruction utilizing the approach of collaborative problem-solving.

First, teachers should put a special emphasis on the two core elements, which are collaboration and problem-solving, to design real problems based on collaborative situations. This meta-analysis provides evidence to support the view that collaborative problem-solving has a strong synergistic effect on promoting students’ critical thinking. Asking questions about real situations and allowing learners to take part in critical discussions on real problems during class instruction are key ways to teach critical thinking rather than simply reading speculative articles without practice (Mulnix, 2012 ). Furthermore, the improvement of students’ critical thinking is realized through cognitive conflict with other learners in the problem situation (Yang et al., 2008 ). Consequently, it is essential for teachers to put a special emphasis on the two core elements, which are collaboration and problem-solving, and design real problems and encourage students to discuss, negotiate, and argue based on collaborative problem-solving situations.

Second, teachers should design and implement mixed courses to cultivate learners’ critical thinking, utilizing the approach of collaborative problem-solving. Critical thinking can be taught through curriculum instruction (Kuncel, 2011 ; Leng and Lu, 2020 ), with the goal of cultivating learners’ critical thinking for flexible transfer and application in real problem-solving situations. This meta-analysis shows that mixed course teaching has a highly substantial impact on the cultivation and promotion of learners’ critical thinking. Therefore, teachers should design and implement mixed course teaching with real collaborative problem-solving situations in combination with the knowledge content of specific disciplines in conventional teaching, teach methods and strategies of critical thinking based on poorly structured problems to help students master critical thinking, and provide practical activities in which students can interact with each other to develop knowledge construction and critical thinking utilizing the approach of collaborative problem-solving.

Third, teachers should be more trained in critical thinking, particularly preservice teachers, and they also should be conscious of the ways in which teachers’ support for learning scaffolds can promote critical thinking. The learning scaffold supported by teachers had the greatest impact on learners’ critical thinking, in addition to being more directive, targeted, and timely (Wood et al., 2006 ). Critical thinking can only be effectively taught when teachers recognize the significance of critical thinking for students’ growth and use the proper approaches while designing instructional activities (Forawi, 2016 ). Therefore, with the intention of enabling teachers to create learning scaffolds to cultivate learners’ critical thinking utilizing the approach of collaborative problem solving, it is essential to concentrate on the teacher-supported learning scaffolds and enhance the instruction for teaching critical thinking to teachers, especially preservice teachers.

Implications and limitations

There are certain limitations in this meta-analysis, but future research can correct them. First, the search languages were restricted to English and Chinese, so it is possible that pertinent studies that were written in other languages were overlooked, resulting in an inadequate number of articles for review. Second, these data provided by the included studies are partially missing, such as whether teachers were trained in the theory and practice of critical thinking, the average age and gender of learners, and the differences in critical thinking among learners of various ages and genders. Third, as is typical for review articles, more studies were released while this meta-analysis was being done; therefore, it had a time limit. With the development of relevant research, future studies focusing on these issues are highly relevant and needed.

Conclusions

The subject of the magnitude of collaborative problem-solving’s impact on fostering students’ critical thinking, which received scant attention from other studies, was successfully addressed by this study. The question of the effectiveness of collaborative problem-solving in promoting students’ critical thinking was addressed in this study, which addressed a topic that had gotten little attention in earlier research. The following conclusions can be made:

Regarding the results obtained, collaborative problem solving is an effective teaching approach to foster learners’ critical thinking, with a significant overall effect size (ES = 0.82, z  = 12.78, P  < 0.01, 95% CI [0.69, 0.95]). With respect to the dimensions of critical thinking, collaborative problem-solving can significantly and effectively improve students’ attitudinal tendency, and the comprehensive effect is significant (ES = 1.17, z  = 7.62, P  < 0.01, 95% CI [0.87, 1.47]); nevertheless, it falls short in terms of improving students’ cognitive skills, having only an upper-middle impact (ES = 0.70, z  = 11.55, P  < 0.01, 95% CI [0.58, 0.82]).

As demonstrated by both the results and the discussion, there are varying degrees of beneficial effects on students’ critical thinking from all seven moderating factors, which were found across 36 studies. In this context, the teaching type (chi 2  = 7.20, P  < 0.05), intervention duration (chi 2  = 12.18, P  < 0.01), subject area (chi 2  = 13.36, P  < 0.05), group size (chi 2  = 8.77, P  < 0.05), and learning scaffold (chi 2  = 9.03, P  < 0.01) all have a positive impact on critical thinking, and they can be viewed as important moderating factors that affect how critical thinking develops. Since the learning stage (chi 2  = 3.15, P  = 0.21 > 0.05) and measuring tools (chi 2  = 0.08, P  = 0.78 > 0.05) did not demonstrate any significant intergroup differences, we are unable to explain why these two factors are crucial in supporting the cultivation of critical thinking in the context of collaborative problem-solving.

Data availability

All data generated or analyzed during this study are included within the article and its supplementary information files, and the supplementary information files are available in the Dataverse repository: https://doi.org/10.7910/DVN/IPFJO6 .

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Acknowledgements

This research was supported by the graduate scientific research and innovation project of Xinjiang Uygur Autonomous Region named “Research on in-depth learning of high school information technology courses for the cultivation of computing thinking” (No. XJ2022G190) and the independent innovation fund project for doctoral students of the College of Educational Science of Xinjiang Normal University named “Research on project-based teaching of high school information technology courses from the perspective of discipline core literacy” (No. XJNUJKYA2003).

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Xu, E., Wang, W. & Wang, Q. The effectiveness of collaborative problem solving in promoting students’ critical thinking: A meta-analysis based on empirical literature. Humanit Soc Sci Commun 10 , 16 (2023). https://doi.org/10.1057/s41599-023-01508-1

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Cognitive Development Theory: What Are the Stages?

Sensorimotor stage, preoperational stage, concrete operational stage, formal operational stage.

Cognitive development is the process by which we come to acquire, understand, organize, and learn to use information in various ways. Cognitive development helps a child obtain the skills needed to live a productive life and function as an independent adult.

The late Swiss psychologist Jean Piaget was a major figure in the study of cognitive development theory in children. He believed that it occurs in four stages—sensorimotor, preoperational, concrete operational, and formal operational.

This article discusses Piaget’s stages of cognitive development, including important concepts and principles.

FatCamera / Getty Images

History of Cognitive Development

During the 1920s, the psychologist Jean Piaget was given the task of translating English intelligence tests into French. During this process, he observed that children think differently than adults do and have a different view of the world. He began to study children from birth through the teenage years—observing children who were too young to talk, and interviewing older children while he also observed their development.

Piaget published his theory of cognitive development in 1936. This theory is based on the idea that a child’s intelligence changes throughout childhood and cognitive skills—including memory, attention, thinking, problem-solving, logical reasoning, reading, listening, and more—are learned as a child grows and interacts with their environment.

Stages of Cognitive Development

Piaget’s theory suggests that cognitive development occurs in four stages as a child ages. These stages are always completed in order, but last longer for some children than others. Each stage builds on the skills learned in the previous stage.

The four stages of cognitive development include:

• Sensorimotor
• Preoperational
• Concrete operational
• Formal operational

The sensorimotor stage begins at birth and lasts until 18 to 24 months of age. During the sensorimotor stage, children are physically exploring their environment and absorbing information through their senses of smell, sight, touch, taste, and sound.

The most important skill gained in the sensorimotor stage is object permanence, which means that the child knows that an object still exists even when they can't see it anymore. For example, if a toy is covered up by a blanket, the child will know the toy is still there and will look for it. Without this skill, the child thinks that the toy has simply disappeared.

Language skills also begin to develop during the sensorimotor stage.

Activities to Try During the Sensorimotor Stage

Appropriate activities to do during the sensorimotor stage include:

• Playing peek-a-boo
• Reading books
• Providing toys with a variety of textures
• Singing songs
• Playing with musical instruments
• Rolling a ball back and forth

The preoperational stage of Piaget's theory of cognitive development occurs between ages 2 and 7 years. Early on in this stage, children learn the skill of symbolic representation. This means that an object or word can stand for something else. For example, a child might play "house" with a cardboard box.

At this stage, children assume that other people see the world and experience emotions the same way they do, and their main focus is on themselves. This is called egocentrism .

Centrism is another characteristic of the preoperational stage. This means that a child is only able to focus on one aspect of a problem or situation. For example, a child might become upset that a friend has more pieces of candy than they do, even if their pieces are bigger.

During this stage, children will often play next to each other—called parallel play—but not with each other. They also believe that inanimate objects, such as toys, have human lives and feelings.

Activities to Try During the Preoperational Stage

Appropriate activities to do during the preoperational stage include:

• Playing "house" or "school"
• Building a fort
• Playing with Play-Doh
• Building with blocks
• Playing charades

The concrete operational stage occurs between the ages of 7 and 11 years. During this stage, a child develops the ability to think logically and problem-solve but can only apply these skills to objects they can physically see—things that are "concrete."

Six main concrete operations develop in this stage. These include:

• Conservation : This skill means that a child understands that the amount of something or the number of a particular object stays the same, even when it looks different. For example, a cup of milk in a tall glass looks different than the same amount of milk in a short glass—but the amount did not change.
• Classification : This skill is the ability to sort items by specific classes, such as color, shape, or size.
• Seriation : This skill involves arranging objects in a series, or a logical order. For example, the child could arrange blocks in order from smallest to largest.
• Reversibility : This skill is the understanding that a process can be reversed. For example, a balloon can be blown up with air and then deflated back to the way it started.
• Decentering : This skill allows a child to focus on more than one aspect of a problem or situation at the same time. For example, two candy bars might look the same on the outside, but the child knows that they have different flavors on the inside.
• Transitivity : This skill provides an understanding of how things relate to each other. For example, if John is older than Susan, and Susan is older than Joey, then John is older than Joey.

Activities to Try During the Concrete Operational Stage

Appropriate activities to do during the concrete operational stage include:

• Using measuring cups (for example, demonstrate how one cup of water fills two half-cups)
• Solving simple logic problems
• Practicing basic math
• Doing crossword puzzles
• Playing board games

The last stage in Piaget's theory of cognitive development occurs during the teenage years into adulthood. During this stage, a person learns abstract thinking and hypothetical problem-solving skills.

Deductive reasoning—or the ability to make a conclusion based on information gained from a person's environment—is also learned in this stage. This means, for example, that a person can identify the differences between dogs of various breeds, instead of putting them all in a general category of "dogs."

Activities to Try During the Formal Operational Stage

Appropriate activities to do during the formal operational stage include:

• Learning to cook
• Solving crossword and logic puzzles
• Exploring hobbies
• Playing a musical instrument

Piaget's theory of cognitive development is based on the belief that a child gains thinking skills in four stages: sensorimotor, preoperational, concrete operational, and formal operational. These stages roughly correspond to specific ages, from birth to adulthood. Children progress through these stages at different paces, but according to Piaget, they are always completed in order.

National Library of Medicine. Cognitive testing . MedlinePlus.

Oklahoma State University. Cognitive development: The theory of Jean Piaget .

SUNY Cortland. Sensorimotor stage .

Marwaha S, Goswami M, Vashist B. Prevalence of principles of Piaget’s theory among 4-7-year-old children and their correlation with IQ . J Clin Diagn Res. 2017;11(8):ZC111-ZC115. doi:10.7860%2FJCDR%2F2017%2F28435.10513

Börnert-Ringleb M, Wilbert J. The association of strategy use and concrete-operational thinking in primary school . Front Educ. 2018;0. doi:10.3389/feduc.2018.00038

By Aubrey Bailey, PT, DPT, CHT Dr, Bailey is a Virginia-based physical therapist and professor of anatomy and physiology with over a decade of experience.

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12 Problem Solving

Stephen K. Reed, Department of Psychology, San Diego State University

• Published: 05 December 2014
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Solving a problem results in obtaining a desired goal through the use of higher mental functions, including reasoning and planning. Problems—such as those requiring arrangement, transformation, and inducing structure—can be classified based on the cognitive skills that are required to solve them. Although general heuristics are sufficient for solving knowledge-lean problems, organized knowledge structures (schemas) are needed to solve knowledge-rich problems. Using analogous solutions is often helpful for both types of problems. Mappings across concepts, problem states, and operations relate the structure of analogous problems and of different solutions to the same problem. EUREKA, CLARION, and ACT are examples of cognitive architectures that apply to problem solving. Underinvestigated topics include problems with insufficient information, estimated answers, complex problem solving, and collaborative problem solving.

The APA Dictionary of Psychology ( VandenBoss, 2006 ) defines problem solving as:

The process by which individuals attempt to overcome difficulties, achieve plans that move them from a starting situation to a desired goal, or reach conclusions through the use of higher mental functions such as reasoning and creative thinking.

Reviewing problem-solving research and theories is a challenge because this definition is so inclusive. Our task is made easier, however, because of previous reviews. In particular, I have built on problem-solving chapters by Bassok and Novick (2012) and by VanLehn (1989) . The Bassok and Novick chapter appears in the Oxford Handbook of Thinking and Reasoning and emphasizes research by cognitive psychologists. The VanLehn chapter appears in Foundations of Cognitive Science and includes a computational approach to problem solving. My objective here is to present both research findings and computational models while extending the contributions made in these previous chapters.

There are many different kinds of problems that fit the definition of problem solving in the first paragraph. The first section of this chapter therefore includes a taxonomy that partitions problems into categories based on the skills required to solve them. This section also describes major historical approaches. The second section discusses the role of organized knowledge structures, labeled schemas , in supporting the development of expertise. The third section explores relations between different problems and between different solutions to the same problem. The fourth section illustrates how cognitive architectures have enhanced our understanding through embedding problem solving within broad theoretical frameworks. The final section proposes future directions by identifying underresearched and developing topics.

Kinds of Problems

To make a review of problem solving more manageable, Greeno (1978) divided problems into three categories based on the cognitive skills required to solve them. He labeled the categories arrangement problems, transformation problems, and inducing structure problems. Arrangement problems require rearranging parts to satisfy some criterion, such as creating a word from the letters ARAGMAN. Transformation problems require transforming an initial state into a goal state, such as moving four rings from peg A to peg C under the constraint that a larger ring can not be placed on a smaller ring. The goal state is known in transformation problems: the stack of rings on peg C varies from the largest on the bottom to the smallest on the top. Inducing structure problems require identifying relations among the parts of a problem and then using that structure to produce the solution. Examples include series completion problems such as producing the next four letters in the series r s c d s t d e t u e f . Discovering the relations among parts is crucial for solving both arrangement and inducing structure problems, but parts cannot be rearranged in inducing structure problems.

I begin by discussing problems that fall into each of these three categories because it provides an opportunity to write about two important movements during the 20th-century work on problem solving. Gestalt psychologists focused on arrangement problems during the earlier half of the century because these problems fit into their theoretical framework that a pattern is more than the sum of its parts. Stimulated by applications of computer science to human problem solving ( Newell, Shaw, & Simon, 1958 ), transformation problems began to play an important role in the second half of the century. Inducing structure has a more eclectic history, with psychometric, artificial intelligence, and information processing approaches all making significant contributions.

The classification of problems into one of the three categories should not imply that all problems fit into a single category. Greeno (1978) admitted that complex problems require multiple skills. For instance, playing chess requires arranging chess pieces to accomplish the goal of winning the game, moving (transforming) pieces toward particular arrangements for executing a plan, and inducing structure to analyze the opponent’s plan.

Arrangement Problems and Gestalt Psychology

Gestalt psychologists were primarily interested in problems that required arranging the parts to find new relations that achieved a goal. Kohler (1925) described an early example in his book The Mentality of Apes . A cage of a chimpanzee contained fruit hanging from the top and sticks and crates on the floor. The chimp could obtain the fruit by using a stick after stacking and climbing on the crates. Solving the problem, according to the Gestalt analysis, depended on reorganizing the objects into a new structure.

Another famous example is Duncker’s (1945) radiation problem. A medical procedure required using radiation to destroy a tumor without destroying the healthy tissue that surrounds it. A solution is to divide the radiation into multiple rays that converge on the location of the tumor. Intense radiation occurs only at the point of convergence so does not harm healthy tissue.

Gestalt psychologists used the term insight to describe the sudden discovery of a correct arrangement of parts following a succession of incorrect arrangements ( Kohler, 1947 ). Metcalfe and Wiebe (1987) empirically evaluated this concept by giving students nonroutine problems such as planting four trees exactly the same distance from the others. Every 15 seconds, the participants had to indicate on a 7-point scale how close they believed they were to solving the problem. Although the highest rating was the most frequent rating at the solution, the lowest rating was the most frequent rating 15 seconds before the solution. The findings support the construct of insight in which solutions occur very suddenly following a perceived lack of progress.

One interpretation of such findings is that insight occurs when solvers remove self-imposed constraints ( Knoblich, Ohlsson, Haider, & Rhenius, 1999 ). For example, people typically attempt to solve the four-trees problem in two dimensions although this constraint is not mentioned in the problem. The solution requires a three-dimensional arrangement.

Knoblich and his coauthors evaluated their theory of constraint relaxation by asking participants to rearrange matchsticks, including the ones shown in Figure 12.1 . The objective is to move a single stick to turn an incorrect arithmetic statement into a correct one. The stick cannot be discarded but must occupy a new position in the equation. The findings confirmed the predictions based on constraint relaxation. Type a problems are solved by modifying the numerals (changing IV to VI) and were the easiest. Type b problems are solved by modifying arithmetic operations (moving a match stick from the equals sign to the minus sign). Type c problems are solved by creating more than one equals sign and were the most difficult. The three types of problems became equally easy after participants realized that they could modify and create equal signs.

Matchstick problems.

A question regarding the restructuring that leads to insight is whether restructuring involves controlled search processes or whether it involves an automatic redistribution of activation in long-term memory. Ash and Wiley (2006) investigated this question by determining whether individual differences in working memory span predicted performance on the initial search and restructuring phases. Working memory span did predict success on problems that required both initial search and restructuring but did not predict success on problems that isolated the restructuring phase. The findings are consistent with the interpretation that restructuring involves an automatic redistribution of activation.

Although the Gestalt approach emphasized problem representations, and Newell and Simon (1972) emphasized searching for a solution, both search and representation are important in solving problems ( Bassok & Novick, 2012 ). The next section focuses on the search process.

Transformation Problems and Search

The study of transformation problems—the transformation of an initial problem state into a goal state—became an important area of research as computers became a source of symbol manipulation. Newell, Shaw, and Simon (1958) made the connection between computers and human problem solving in their Psychological Review article “Elements of a Theory of Human Problem Solving.” Their theory proposed (1) a control system consisting of a number of memories containing symbols interconnected by various relations, (2) primitive information processes that operate on information in the memories, and (3) a set of rules for combining these processes into whole programs.

A program constitutes a theory that can make precise predictions. As stated by Newell, Shaw, and Simon:

The ability to specify programs precisely, and to infer accurately the behavior they produce, drives from the use of high-speed digital computers. Each specific theory—each program of information processes that purports to describe some human behavior—is coded for a computer. That is, each primitive information process is coded to be a separate computer routine, and a “master” routine is written that allows these primitive processes to be assembled into any system we wish to specify. Once this has been done, we can find out exactly what behavior the purported theory predicts by having the computer “simulate” the system. (pp. 152–153)

The authors designed the programs to simulate human problem solving by comparing the behavior predicted by the program with actual behavior observed in experimental settings. The promise of the digital computer was that it provided a device for determining what behavior is implied by a program and for subsequently modifying the program if the predictions failed. Programming required a detailed specification of the operations, which enabled theorists to evaluate whether the operations were sufficient to produce the behavior. It thereby avoided the vagueness that limited other theories of higher mental processes ( Newell et al., 1958 ).

The interest in writing simulation programs was accompanied by an interest in writing artificial intelligence (AI) programs to enable computers to produce, rather than simulate, intelligent behavior. Early examples of these activities are provided in the book GPS: A Case Study in Generality and Problem Solving ( Ernst & Newell, 1969 ). The General Problem Solver (GPS) had the objective of using generic principles to solve a variety of problems including Tower of Hanoi, Missionaries and Cannibals, integration, and logical proofs. The GPS was not completely general because it solved only transformation problems by using the means-end analysis heuristic. Means-end analysis attempts to successively eliminate differences between the initial state and the goal state until the program arrives at the goal state. This heuristic is often successful on transformation problems because these problems have a well-defined goal state: all rings are moved from peg A to peg C or all missionaries and cannibals are moved across the river. In contrast, producing the goal state is typically required in arrangement problems such as solving an anagram or Dunker’s radiation problem.

Solution of a logic problem using means-end analysis.

These initial AI programs provided possible theories for how people solve problems. One example is the construction of logical proofs, a task that Newell and Simon (1972) extensively analyzed in their classic book Human Problem Solving . The problem solver was given 12 rules for manipulating letters connected by dots •, wedges v, horseshoes ⊃, and tildes ~. These connectives are used to represent and, or, implies , and no t in logic but were not interpreted for the participants. The 12 rules enable problem solvers to modify logical expressions until they have constructed a proof by transforming the initial state into the goal state. For instance, the initial state could be A ⊃ B and the goal state could be ~ B ⊃ ~ A .

Means-end analysis was implemented in the GPS for logic problems by including a table of connections in the program that showed which of six differences could be modified by each of the 12 rules. Figure 12.2 shows a simplified table of connections consisting of three rules and three differences for solving the A ⊃ B problem.

Transforming the initial state A ⊃ B into the goal state ~ B ⊃ ~ A requires eliminating differences in both the sign and position of the letters. The table of connections reveals that both Rules 2 and 3 can change a sign, but Rule 2 cannot be applied because it has different connectives than the initial state. Application of Rule 3 to the initial state changes the sign of A to negation. However, the resulting problem state, ~ A v B , now differs from the goal state in the sign of B , position of the letters, and connective. The application of Rule 1 to Line 2 produces an expression that can be changed to the goal state through the reapplication of Rule 3. Newell and Simon (1972) asked their participants to verbalize their thoughts as they worked on the problems. Many aspects of their thinking corresponded to means-end analysis used in the GPS.

A difference between arrangement and transformation problems is that solvers of transformation problems should realize that they are making progress as they gradually reduce differences between the current problem state and the goal state. Metcalfe and Wiebe (1987) confirmed this difference by finding higher ratings of approaching the solution as their participants continued to work on the transformation problems. Both arrangement and transformation problems received a high rating at the solution, but only transformation problems received a high rating 15 seconds before the solution.

An important theoretical component of Newell and Simon’s (1972) theory of problem solving is the problem space . The search space specifies the permissible actions (legal moves) at each problem state. Figure 12.3 shows the problem space for the five missionaries-cannibals problem ( Simon & Reed, 1976 ):

Five missionaries and five cannibals who have to cross a river find a boat, but the boat is so small that it can hold no more than three persons. If the cannibals outnumber the missionaries on either bank of the river or in the boat at any time, the missionaries will be eaten. Find the simplest schedule of crossings that will allow everyone to cross safely. At least one person must be in the boat at each crossing.

Problem space for the five Missionaries and Cannibals problems.

Each oval in Figure 12.3 is a problem state of the form MC/MC* in which the first MC is the number of missionaries (M) and cannibals (C) on the initial bank, and the second MC is the number of missionaries and cannibals across the river. The asterisk shows the location of the boat, and the links show the number of missionaries and cannibals in the boat. Solving the problem requires transforming the initial state A into the goal state Z . The problem space reveals a number of important characteristics of the problem, such as there are four legal moves at the initial state, state J is the end of a blind alley that requires reversing the two previous moves, and the minimal solution requires 11 moves.

The search space differs from the problem space because it reveals which moves are considered by the problem solver ( Newell & Simon, 1972 ). For instance, undergraduates required an average of 30 moves to solve the problem without a hint and 20 moves to solve the problem when given a subgoal that at some point there will be 3 cannibals and 0 missionaries across the river without the boat (state L ). Simon and Reed (1976) proposed a strategy-shift model to predict the average number of times students in each group would visit each of the problem states in Figure 12.3 . The model assumes that students begin with a balance strategy in which they attempt to equalize the number of missionaries and cannibals across the river, as in state D . They then switch to a means-end strategy in which they attempt to take as many people across the river as possible (3) and bring back as few as possible (1). The probability of switching strategies is higher for the subgoal group, which helps them avoid the blind alley ending in state J . The strategy-shift model is consistent with the “unbalanced” subgoal—3 cannibals and 0 missionaries across the river.

Inducing Structure and Reasoning

The sections on arrangement and transformation tasks contained research that is typically included in problem-solving chapters. In contrast, tasks that require inducing structure might appear in reasoning chapters. An inclusive definition of problem solving, such as the one at the beginning of this chapter, includes reasoning, but there is a distinction between reasoning and problem solving. Holyoak and Morrison (2012) state that reasoning places an emphasis on drawing inferences (conclusions) from some initial information (premises) and has a foundation in logic. Problem solving involves a course of action to achieve a goal.

I focus on a particular reasoning task (the four-card selection problem) in this section for three reasons. First, the task has been one of the most widely studied tasks in the reasoning literature. Second, it illustrates how inducing structure differs from arranging and transforming components. Inducing structure is similar to arrangement problems because it is necessary to discover the relations among the components of the problem ( Greeno, 1978 ). However, unlike arrangement problems, these components are static and cannot be rearranged. Third, research on this task illustrates the challenge of identifying the extent to which reasoning depends on general knowledge. The arrangement and transformation problems in the previous two sections consisted primarily of puzzles that did require extensive knowledge about a particular domain. The four-card selection problem illustrates how our familiarity with the content of information in rules influences our ability to evaluate those rules.

The four-card selection problem ( Wason & Johnson-Laird, 1972 ) requires deciding which one of four cards needs to be turned over to evaluate a conditional rule; for example, if there is a D on one side of the card, then there is a 3 on the other side. The four cards in this example either display the letter D , the letter K , the number 3 , or the number 7 . The experimenter informs participants that each of the cards contains a letter on one side and a number on the other side. The answer is that it is necessary to turn over the D card and the 7 card but only 5 of 128 participants turned over only the two correct cards ( Wason & Shapiro, 1971 ).

Wason and Shapiro (1971) hypothesized that performance would dramatically improve if the conditional rules had realistic rather than abstract content, a prediction that was confirmed in a letter-sorting task ( Johnson-Laird, Legrenzi, & Legrenzi, 1972 ). The task consisted of four envelopes. Two were face up, revealing either a 50 lira or a 40 lira stamp. Two were face down, revealing either a sealed or an unsealed envelope. Participants were told to imagine that they worked in a post office and had to enforce the rule “If a letter is sealed then it has a 50-lira stamp on it.” Most participants (17 of 24) accurately selected the two envelopes required to enforce the rule.

Although Wason and Shapiro (1971) argued that conditional reasoning is vastly improved with realistic content, Griggs and Cox (1982) questioned whether the letter task required conditional reasoning. Their memory-retrieval explanation proposed that the British participants did the task by recalling their experience in placing more postage on sealed envelopes. Griggs and Cox therefore predicted that their American students, who lacked such experience, would do poorly on the task. As predicted, American students did poorly on the unfamiliar letter task but excelled in evaluating a familiar drinking-age rule “If a person is drinking beer then the person must be over 19 years of age.”

Griggs and Cox’s findings are discouraging because they support the conclusion that people are very limited in evaluating conditional rules unless the rules contain familiar content, in which case reasoning is not required. A more optimistic view of reasoning is that people do well at conditional reasoning if the content is familiar at a general, schematic level. For instance, pragmatic reasoning schemata are organized knowledge structures that enable us to evaluate practical situations such as seeking permission or fulfilling an obligation ( Cheng, Holyoak, Nisbett, & Oliver, 1986 ).

Imagine that you are hired to enforce the rule “If a passenger wishes to enter the country, then he or she must have an inoculation against cholera.” Four cards identify a passenger who wishes to enter, a passenger who does not wish to enter, a passenger who has been inoculated, and a passenger who has not been inoculated. The pragmatic reasoning hypothesis predicts that you can use your schematic knowledge about seeking permission to evaluate this rule even if you have no experience with this particular task. More information is required for the passenger who wishes to enter and for the passenger who has not been inoculated. Research supports the hypothesis that people do much better in evaluating conditional statements involving permission or obligation than in evaluating conditional statements involving arbitrary relations ( Cheng et al., 1986 ).

In summary, the evolution of research on the four-card selection problem reveals the relative influence of concrete and familiar experiences on reasoning. People did very poorly in evaluating the implications of conditional rules involving arbitrary relations between letters and numbers. Performance dramatically improved on concrete versions of the rules but raised the question of whether retrieving experiences from memory removed the need to reason. An intermediate level of abstractness is provided by schemas that generalize the commonality among individual experiences, such as seeking permission or fulfilling an obligation. People can effectively reason about unfamiliar experiences if those experiences can be linked to a familiar schema. Schemas also play an important role in problem solving, as discussed in the next section.

Much of the research on problem solving during the 1970s was influenced by Newell and Simon’s (1972) book in which general strategies (heuristics) such as using means-end analysis or forming subgoals guided the search process. VanLehn (1989) refers to problems such as Missionaries and Cannibals or the Tower of Hanoi as knowledge-lean tasks because they can be solved without prior experience. In contrast, research in the 1980s began to focus on problems from algebra, physics, geometry, and computer programming. These are knowledge-rich tasks that require many hours of instruction ( VanLehn, 1989 ).

Schemas as a Theoretical Construct

Organized knowledge structures called schemas are an effective method for organizing this knowledge. Brewer and Nakamura (1984) described the characteristics of schemas by contrasting them with learning based on stimulus-response (S-R) associations.

S-R learning is based on small units of knowledge. A schema is a larger unit in which knowledge is combined into clusters.

S-R learning requires learning an association between a stimulus and a response. A schema provides a knowledge structure for interpreting and encoding aspects of particular experiences.

S-R learning involves a particular stimulus and response. A schema is more general and represents a variety of experiences.

The association between a stimulus and a response can be learned in a passive manner. Invoking a schema is a more active process in which a particular experience is matched to the schema that best fits the experience.

In her book, Marshall (1995) began by reviewing the historic development of schemas as a theoretical construct by tracing the ideas of Plato, Aristotle, Kant, Bartlett, and Piaget. In her working definition, a schema is a memory organization that can (1) recognize similar experiences; (2) access a general framework that contains essential elements of those experiences; (3) use the framework to draw inferences, create goals, and develop plans; and (4) provide skills and procedures for solving problems in which the framework is relevant.

Marshall then described her research that built on the analysis of addition and subtraction problems. Riley, Greeno, and Heller (1983) had analyzed elementary word problems into change, combine, and compare problems. Kintsch and Greeno (1985) further developed these distinctions as a set schema in which the slots consisted of objects <noun>; quantity <number>; specification <owner>, <location>, <time>; and role <start, transfer, result, superset, subset, largeset, smallset, difference>. Marshall added two additional schematic situations (labeled restate and vary ) and constructed a computer tutor to help students learn to solve multistep arithmetic word problems.

Learning these schematic components is important because they form the building blocks of more complex problems, as in physics ( Sherin, 2001 ) and algebra word problems ( Reed et al., 2012 ). Research shows that algebra word problems are difficult for university students, not only because of algebra, but because students have not adequately learned the change, combine, and compare schema that are the components of both arithmetic and algebra word problems ( Reed et al., 2012 ). Learning these elementary and more advanced schemas supports the development of expertise.

Schemas in Experts

The transition from the study of domain-lean problems in the 1970s to domain-rich problems in the 1980s resulted in investigations of how domain knowledge influenced problem solving. Silver (1981) asked good, average, and poor problem solvers to sort arithmetic word problems into groups based on common solution procedures. The better problem solvers excelled at this assignment, but the weaker problem solvers sorted by story content. For example, they placed problems about hens and rabbits into the same category although the problems required different solutions.

Silver’s finding has been confirmed for many domains and for many levels of expertise. Chi, Glaser, and Reese (1982) asked eight undergraduates and eight advanced physics doctoral students to sort 24 physics problems into categories based on similar solutions. Novices tended to classify problems on the basis of common objects such as inclined planes and springs. Experts tended to classify problems based on physics principles such as the conservation of energy or Newton’s second law (F = MA).

Although such expert-defined schemas are usually very helpful, they can occasionally constrain innovative solutions. Dane (2010) defines cognitive entrenchment as a high level of stability in domain schemas that can cause experts to be inflexible in their thinking. Cognitive entrenchment increases the likelihood of problem-solving fixation and blocks the generation of novel ideas. However, Dane proposes two factors that can reduce cognitive entrenchment. The first is working in a dynamic environment in which one must remain open to a wide range of possibilities and options. The second is focusing attention on outside-domain tasks in which counterexamples and exceptions can increase the flexibility of one’s beliefs.

Schema Abstraction

The ability to see structural commonalities in situations that appear quite different can be very helpful, as illustrated by the use of pragmatic reasoning schema to reason about conditional rules ( Cheng et al., 1986 ); the use of change, combine, and compare schema to classify arithmetic word problems ( Silver, 1981 ); and the use of physics principles to classify physics problems ( Chi et al., 1982 ). All of these situations can be aided by schema abstraction , in which problem solvers focus on the structural relations among the objects (inoculation, cholera, hens, rabbits, springs, inclined planes) rather than on the objects.

A challenge is to encourage noticing these structural relations through schema abstraction—a challenge that was met in a classic study by Gick and Holyoak (1983) . Three years earlier, they published research that demonstrated the difficulty of spontaneously noticing analogous solutions ( Gick & Holyoak, 1980 ). Their goal in this earlier research was to increase the number of convergence solutions to Duncker’s (1945) radiation problem. Participants read an analogous problem in which a general wanted to capture a fortress but could not attack along one road because it was mined. The general therefore divided his army into small groups that simultaneously converged on the fortress from different roads. Very few participants, however, used the analogy unless they were given a hint that the military problem would help them solve the radiation problem.

To spontaneously notice an analogy, people need to think about analogous solutions at a more abstract level so that differences in the objects, such as a fortress and a tumor, would not be a hindrance. Gick and Holyoak (1983) therefore asked participants to compare the similarities between two stories, the military problem and a story about Red Adair whose crew put out fires in oil derricks by using multiple hoses that converged on the site of the fire. Comparing two stories helped participants spontaneously notice the analogy to the radiation problem by creating the more abstract convergence schema shown in Table 12.1 . Simply reading the two stories was insufficient; abstraction depended on the comparison ( Catrambone & Holyoak, 1989 ).

A productive application of this finding occurred in a negotiation training program for management consultants who had approximately 15 years of work experience ( Gentner, Lowenstein, Thompson, & Forbus, 2009 ). The consultants studied two cases of a contingent contract that depended on the outcome of some future event. One group studied the two cases separately, and another group compared the similarities of the two cases. As found in laboratory studies ( Catrambone & Holyoak, 1989 ), the comparison aided schema abstraction. The comparison group was more successful in describing the principles of a contingent contract and in recalling examples of contingent contracts from their own experiences.

Mapping Across Problems and Solutions

Using the solution of the military problem to find a solution to the radiation problem requires finding corresponding objects and relations in the two solutions. As shown in Table 12.1 , the fortress in the military problem corresponds to the tumor in the radiation problem, the large army corresponds to powerful rays, the inability to use a single road corresponds to the inability to use a single pathway, and dividing the army corresponds to dividing the radiation. Establishing these correspondences requires mapping the objects and relations in the military problem to objects and relations in the radiation problem.

Illustration of one-to-one, one-to-many, and partial mappings across knowledge states.

There have been a number of detailed computational models of analogical mappings, including one by Hummel and Holyoak (1997) . Mappings in their model are guided by three constraints:

Structural consistency implies a one-to-one mapping between an element in the source and an element in the target.

Semantic similarity implies that elements with prior semantic similarity (such as joint membership in a taxonomic category) should tend to map to each other.

Pragmatic centrality implies that mappings should give preference to elements that are important for goal attainment.

Structural consistency in the fortress-tumor analogy is illustrated by the one-to-one mapping between objects in the two problems, semantic similarity is illustrated by similar actions (dividing the army and the tumor), and pragmatic centrality is illustrated by the principle of converging forces in both solutions.

Reed (2012) has extended this one-to-one mapping across problems to construct a taxonomy consisting of three types of mappings (one-to-one, partial, and one-to-many, as illustrated in Figure 12.4 ) and four types of situations (problems, solutions, representations, and sociocultural contexts). Mappings across problems and mappings across solutions—different solutions to the same problem—illustrate parts of the taxonomy.

Mapping Across Problems

Most computational models of transfer, including the one proposed by Hummel and Holyoak (1997) , have emphasized one-to-one mappings across isomorphic problems. In contrast, Reed, Ernst, and Banerji (1974) investigated transfer between two problems in which the problem states and moves in one problem had a one-to-many mapping to the problem states and moves in the other problem. One of the problems was the Missionaries and Cannibals (MC) problem, in which three missionaries and three cannibals cross a river using a boat that can hold two people under the constraint that cannibals can never outnumber missionaries. The other problem was the Jealous Husbands (JH) problem:

Three jealous husbands, and their wives, having to cross a river, find a boat. However, the boat is so small that it can hold no more than two persons. Find the simplest schedule of crossings that will permit all six persons to cross the river so that no woman is left in the company of any other woman’s husband unless her own husband is present.

We anticipated, based on our perceived similarity of the two problems, that there would be substantial transfer from one problem to the other. Our first experiment found no transfer, but our second experiment found some transfer when students were informed about the mapping between the two problems: husbands correspond to missionaries and wives correspond to cannibals. However, even with this hint, there was evidence of transfer only from the JH problem to the MC problem. The asymmetrical transfer is consistent with a one-to-many mapping from the MC to the JH problem because moving a missionary does not specify which husband to move and moving a cannibal does not specify which wife to move. Although all missionaries and cannibals are equivalent, all husbands and wives are not because they are paired with each other. For example, the three circles in Figure 12.2 representing one-to-many mappings might represent moving husbands A and B, B and C , and A and C . Each of these moves maps onto moving two missionaries, but moving two missionaries does not specify which two husbands to move. It should therefore be more difficult to map moves from the MC problem to the JH problem because this mapping does not specify a unique move.

Partial mappings are similar to isomorphic mappings because both specify one-to-one mappings between the source and the target. The difference is that isomorphic mappings are sufficient for solving the target problem, whereas partial mappings are not. It may therefore be helpful to use more than one analogy ( Gentner & Gentner, 1983 ).

The Garden Border Problem. From Greeno & van de Sande (2007) .

The Gentners identified two analogies (flowing waters and teeming crowds) for helping students understand electric circuits. They predicted that students who used the flowing waters analogy (pressure of water, flow in a pipe) should do well on questions about voltage and current because serial and parallel reservoirs combine in the same manner as serial and parallel batteries. In contrast, students with the moving crowd model should do better on resistors because of the analogy to gates. The results supported their predictions in the first experiment. In the second experiment, the analogy to flowing waters was not as helpful as expected because students lacked knowledge in this area.

Spiro, Feltovich, Coulson, and Anderson (1989) discuss practical implications of partial mappings. They propose that simple analogies help beginners gain a preliminary understanding of complex concepts but can later block fuller understanding if learners never progress beyond the simple analogy. One consequence is that instructors need to pay closer attention to how analogies can fail. The authors discuss eight possible failures of simple analogies including misleading properties, missing properties, a focus on surfaces descriptions, and wrong grain size. Their remedy is to use multiple analogies to convey the complexity of difficult ideas.

Mapping Across Solutions

Most teachers and researchers are delighted if problem solvers find one solution to a problem. However, Alan Schoenfeld is more demanding. After students in his math classes at Berkeley solve the problem, he asks them to find another solution. Then a third. The reason is that any one of these solutions might prove helpful in solving future problems ( Schoenfeld, 1985 ).

Studying mapping across solutions attempts to establish how one solution to a problem is related to an alternative solution ( Reed, 2012 ). Rittle-Johnson, Star, and Durkin (2009) discovered that asking seventh- and eighth-grade students to compare two solutions for solving the same problem was helpful when they had the appropriate prior knowledge. One of the solutions showed a short-cut method. In the example given here, the first method requires multiplication, subtraction, and division. The second method requires only division and subtraction:

Students who were familiar with one of the two methods typically noticed that one method required fewer steps or was more efficient than the other. Comparing solutions for these students produced flexible knowledge of procedures. In contrast, students who were not familiar with either method benefited more from the sequential presentation of the solutions.

Comparing alternative solutions can be particularly rewarding when the solutions are generated by different people. Greeno and van de Sande’s (2007) analysis of the Garden Border problem in Figure 12.5 illustrates how a shift in a teacher’s perspective helped her understand that a student’s different approach to the problem could provide an alternative solution. The key difference between the two solutions was how the teacher and student used the phrase “an even border of flowers.” The teacher represented the width of this border by the unknown variable (such as w ) and constructed an equation to represent the area of the inner rectangle by multiplying the length of this rectangle by its width:

This equation followed a previous calculation that the area of the inner rectangle is 1,680 square feet. Because the border is even, 2 w can be subtracted from both the length and width of the outer rectangle.

However, this one-to-one mapping from the text to a variable did not occur to a student who represented the border’s width by two variables: w 1 = ( 72 − y ) / 2 with respect to the length of the outer rectangle and w 2 = ( 40 − x ) / 2 with respect to the width of the outer rectangle. Another student understood how this representation could work by using two equations. The teacher then encouraged the class to figure out the values of x and y by generating the two equations. This second solution was not as efficient because it requires two equations to solve for the two unknown variables. However, the teacher not only encouraged the students in their attempts to generate an alternative solution but recognized that the alternative solution provided a learning opportunity for the class to practice solving for two unknown values.

In summary, one-to-one, one-to-many, and partial mappings across knowledge states provides a basis for analyzing both analogical transfer across problems and the relations between different solutions to the same problem.

Cognitive Architectures

Computational models have contributed to our theoretical understanding of topics such as exploring a problem space ( Simon & Reed, 1976 ) and using analogous solutions ( Hummel & Holyoak, 1997 ). Embedding computational models within cognitive architectures increases their generality by modeling a greater range of activities. EUREKA ( Jones & Langley, 2005 ), CLARION ( Heile & Son, 2010 ; Sun & Zhang, 2006 ), and ACT ( Anderson, Byrne, Douglass, Lebeire, & Qin, 2004 ) illustrate how cognitive architectures have been used to model problem solving.

VanLehn (1989) identified 10 robust findings in his problem-solving chapter that provided a test bed for the design and evaluation of a problem-solving architecture called EUREKA ( Jones & Langley, 2005 ). EUREKA attempts to solve all problems by using analogical reasoning. By incorporating human memory constraints, EUREKA strives to qualitatively replicate VanLehn’s (1989) reported findings. To find solutions to problems, analogies are created to map the operators used in a previous problem to the new problem. The degree of mapping can differ, depending on the degree of the match of the two situations and the level of activation of relevant retrieval patterns.

EUREKA uses means-ends analysis ( Newell & Simon, 1972 ) to divide problem solution into two tasks. The transform task transforms the current state into a desired state. The apply task satisfies preconditions of operators. If the current state satisfies the operators’ preconditions, then the operators are applied to generate a desired state. Otherwise, another transform task is necessary to change the current state into a new state that satisfies the preconditions.

The transformations and applications are stored in EUREKA’s long-term memory in the form of a semantic network of concepts and relations. To make helpful retrievals, Jones and Langley (2005) use a spreading activation framework similar to Anderson’s (1983) early ACT models. EUREKA activates links of concepts in proportions to the trace strengths attached to the links. By increasing or decreasing the trace strengths, the retrieval patterns are strengthened or weakened, respectively.

Table 12.2 lists the 10 psychological findings identified in VanLehn’s (1989) literature review. The first three describe practice effects. Item 1 refers to how people automate the problem-solving process with practice. The rate of learning is fastest at the beginning but slows with more practice, as described in items 2–3. To evaluate EUREKA’s practice effects, Jones and Langley (2005) presented the system with Towers of Hanoi and Blocks World problems. Similar to humans, graphs of EUREKA’s performance showed a rapid decrease on several measures (number of attempts, total search effort, and productive search effort) after the first and second trials, and it remained fairly constant for the remaining trials. The data indicate that EUREKA had difficulty solving the problems on the first trial but quickly improved after the first trial. Once productive trace links are strengthened, the problem-solving process becomes more automatic.

Item 4 describes differences in improvement across intradomain problems that vary in complexity. Assuming that difficult problems are a composite of simple problems, transfer can occur from simple problems to difficult problems, but not vice versa. To test this prediction, EUREKA again was given Towers of Hanoi and Blocks World problems that became increasingly difficult to solve. In the control condition, each trial was run separately. In the test condition, trials were run continuously, allowing EUREKA to store information from previous trials. EUREKA struggled to solve the problems as they became more difficult in the control condition. However, for the test condition, the system was able to solve even the most difficult problems by using analogy to previous solutions.

Based on VanLehn (1989) .

Negative transfer, in which previous learning makes new learning more difficult, rarely occurs, as stated in item 5 of Table 12.2 . An exception, however, is the set effect or Einstellung (item 6) demonstrated in Luchins’s (1942) water jug task that requires obtaining a specified amount of water by filling and emptying jugs of varying sizes. Luchins found that when people solved practice problems with complex solutions, they failed to discover simpler solutions for the test problems. Similar to the human subjects, EUREKA failed to find simpler solutions after the system solved more complex problems. The results can be explained by the fact that EUREKA continues to use operators that have been successful.

Analogy can also be used to solve isomorphic problems from different domains. To evaluate how EUREKA represents interdomain transfer, Jones and Langley (2005) gave it Holyoak and Koh’s (1987) radiation and broken-light problems. The test group was presented with Duncker’s (1945) radiation problem in which a patient with a tumor must be saved by using X-rays. The transfer problem consisted of a broken light bulb that had to be repaired by using laser beams. The test group was able to solve the transfer problem more successfully than a control group that did not receive the light bulb problem.

In EUREKA’s simulation, the light bulb problem was given first, and the radiation problem was used as the analogous problem. The results showed that EUREKA successfully solved the radiation problem 50% of the time in the control condition and 80% of the time in the test condition. This improvement suggests the use of analogical reasoning to transfer solutions, similar to intradomain transfer.

Intradomain and interdomain transfer differ in that concepts and relations in the current problem and the analogous problem are semantically further apart for interdomain transfer. Therefore, for interdomain transfer, retrieval may be more difficult because activation of abstract nodes may be necessary to find solutions. This assumption explains why semantically similar isomorphic problems, such as the tumor and light bulb problems, are easier to solve than semantically dissimilar isomorphic problems (items 7 and 8). The more semantically similar the problems are, the more likely the “correct” activation will occur.

Items 9 and 10 in Table 12.2 state that spontaneous retrieval of solutions is rare and usually only occurs when people use analogies based on surface similarities. Although spontaneous noticing of analogies is uncommon, it can occur with hints. To simulate the effect of hints, the semantic network nodes describing the broken bulb were activated before the system attempted to solve the radiation problem. Compared to the previous trials in which the hint was not given, the activation of relevant nodes greatly reduced the number of attempts and search efforts. The strengthened activation of relevant nodes helped improve the system’s problem-solving performance, as stated in items 9 and 10.

CLARION is an integrative cognitive architecture that consists of a top-level explicit representation and a bottom-level implicit representation ( Sun & Zhang, 2006 ). Explicit knowledge is represented by easily interpretable symbols that have clear conceptual meaning. Implicit knowledge is represented by a subsymbolic distributed representation within a back-propagation network. In contrast to an explicit memory that encodes rules as all or none, implicit memory supports a more gradual accumulation of knowledge

Heile and Sun (2010) subsequently developed the explicit-implicit interaction (EII) theory based on CLARION to analyze the four stages of problem solving proposed in Wallas’s (1926) influential book The Art of Thought . Preparation is the initial search for a solution, incubation is a period of inactivity following an impasse, illumination (or insight) is a sudden discovery of a possible solution, and verification is a determination of whether the discovered solution is valid.

The EII theory distinguishes between explicit processing based on well-defined rules and implicit processing based on associations. Most problems elicit both implicit and explicit processing. The integration of conclusions from both types of processing influences an internal confidence level that measures the probability of finding the solution.

The theory postulates that the initial preparation phase is predominately rule-based processing as people respond to verbal instructions, form representation of the problem, and establish goals. In contrast, the second incubation state is predominately implicit processing in which people may not consciously think about the problem. The third stage, insight, occurs when the internal confidence level crosses a threshold that makes the output available for verbal report. The final verification stage, like the initial stage, requires primarily explicit processing to evaluate the potential of the discovered solution.

The importance of implicit processes in solving insight problems is illustrated by the success of solving the following problem from Schooler, Ohlsson, and Brooks (1993) :

A dealer in antique coins got an offer to buy a beautiful bronze coin. The coin had an emperor’s head on one side and the date 544 B.C. stamped on the other. The dealer examined the coin, but instead of buying it, he called the police. Why?

After working on the problem for 2 minutes in Schooler’s experiment, half of the participants verbalized their strategies while the remainder worked on an unrelated task. After returning to the problem, 36% of the former group and 46% of the latter group solved the problem. CLARION simulates these findings by assuming that the explicit process of verbalizing strategies disrupts the implicit process that can result in insight.

The goal of CLARION and EUREKA is to propose computational models that can provide theoretical explanations of research on human problem solving. In contrast, an early objective in the evaluation of ACT was to evaluate its theoretical assumptions by designing cognitive tutors to improve instruction ( Anderson, Boyle, & Reiser, 1985 ). An extensive ongoing project at Carnegie Mellon University has continued to design intelligent tutoring systems for teaching topics such as algebra, high school geometry, genetics, and computer programming ( Koedinger & Corbett, 2006 ).

ACT consists of a set of assumptions about both declarative and procedural knowledge. The assumptions about declarative knowledge emphasize the representation and organization of factual information. The assumptions about procedural knowledge emphasize how people use this knowledge to carry out various tasks. This part of the theory consists of production rules that specify which action should be performed under a particular set of conditions and have the form IF <condition> THEN <action>. The condition typically states a goal, and the action specifies a potential way to achieve the goal. Production rules were formulated by Newell and used in his cognitive architecture SOAR ( Laird, Newell, & Rosenbloom, 1987 ). The goal of the production rules in ACT, however, is to model human cognition.

One of the initial cognitive tutors helped students learn the programming language LISP. The major theoretical assumptions underlying the construction of the LISP tutor include the following ( Anderson, 1990 ):

Production rules . A skill such as programming can be decomposed into a set of production rules.

Skill complexity . Hundreds of production rules are required to learn a complex skill. This assumption is consistent with the domain-specific view of knowledge.

Hierarchical goal organization . All productions are organized by a hierarchical goal structure in which subgoals are helpful in accomplishing goals.

Declarative origins of knowledge . All knowledge begins in some declarative representation, typically acquired from instruction or example. Before people practice solving problems, they are instructed in how to solve problems.

Compilation of procedural knowledge . Solving problems requires more than being told about how to solve problems. Problem solvers have to convert this declarative knowledge into efficient procedures for solving specific problems.

The LISP tutor consisted of 1,200 production rules that model student performances on programming problems. It covered all the basic concepts of LISP during a full-semester, self-paced course at Carnegie Mellon University. Students who worked on problems with the LISP tutor generally received one letter grade higher on exams than did students who had not worked with the tutor.

Both ACT theory ( Anderson, 2007 ) and cognitive tutors have continued to evolve. The most extensive application of the cognitive tutors has been to mathematics classes, and, by 2007, data had been collected from more than 7,000 students in pre-algebra classes ( Ritter, Anderson, Koedinger, & Corbett, 2007 ). The curriculum includes both a textbook and software so students can divide their time between the classroom (typically 3 days a week) and a computer lab (typically 2 days a week).

The primary source of declarative knowledge is worked examples that show problem solutions ( Anderson & Fincham, 1994 ). Although the presentation of worked examples has typically occurred in the classroom rather than in the computer lab, interweaving worked examples with practice problems has been particularly effective ( Pashler et al., 2007 ). This can be achieved by adding worked examples to the cognitive tutor and requiring that students solve a practice problem on the cognitive tutor after studying each worked example ( Reed, Corbett, Hoffman, Wagner, & MacLaren, 2013 ).

Other recent work to improve the cognitive tutor provides support for seeking help. Students can request hints but occasionally either do not take advantage of this feature or exploit it by requesting so many hints that the tutor does most of the problem solving. Ideally, learners should develop strong metacognitive skills in which they become proficient at requesting the appropriate amount of help. Such training is provided by the help tutor , which has been integrated into the geometry cognitive tutor ( Roll, Aleven, McLaren, & Koedinger, 2011 ). Results showed that this additional assistance not only improved help-seeking skills for solving geometry problems but transferred to a different topic a month later ( Roll et al., 2011 ).

Future Directions

The typical research paradigm for studying problem solving requires individuals to find a single solution. We still have much to learn by using this paradigm, but our knowledge of problem solving would be broadened by investigating a greater variety of topics such as the value of multiple solutions to a problem ( Reed, 2012 ; Rittle-Johnson et al., 2009 ; Schoenfeld, 1985 ), including understanding an alternative solution ( Greeno & van de Sande, 2007 ). Alternative solutions to a problem reveal the problem space of possible solutions. Other underinvestigated topics include (1) problems with insufficient information, (2) estimated answers, (3) complex problem solving, and (4) collaborative problem solving.

A useful skill outside the classroom is the ability to identify problems that have missing information required for a solution. Perhaps because students do not expect to be assigned such problems, they require a hint to identify them. The hint helped high math ability students discover the missing information, but students with moderate ability required familiar cover stories ( Rehder, 1999 ). Other problems provide only enough information to constrain correct answers:

Morita has five friends and Takeda has seven friends. They decide to throw a party together and invite all their friends. All friends are present. How many friends are there at the party?

An instructional example was moderately helpful in reducing single answers and increasing the number of two answers or, more appropriately, a range of possible answers ( Kinda, 2012 ).

Some problems provide enough information for an estimate:

An athlete’s best time to run a mile is 4 minutes and 7 seconds. About how long would it take him to run 3 miles? ( Greer, 1993 )

Research on these kinds of problems has focused on the incorrect application of proportional reasoning ( Verschaffel, Greer, & de Corte, 2000 ), but we need more information on how people use proportional reasoning as a first step toward making reasonable estimates. Estimated answers are important because people often base their decisions on estimates rather than on precise calculations. Estimates are also helpful in evaluating whether a calculated answer is correct. Checking a calculation is recommended when the answer appears unreasonable.

We also need more information on how to help people improve their estimates. The animation tutor provides simulations of people’s estimates so they can improve their estimates of the time to fill a tank, paint a fence, or complete a round trip ( Reed, 2005 ). Although both the American Association for the Advancement of Science ( AAAS, 1993 ) and the National Council of Teachers of Mathematics ( NCTM, 2000 ) have stressed the importance of estimated answers, their recommendations have had a limited impact on instruction.

The topic of complex problem solving is slowly becoming integrated with the more mainstream research and theory discussed in this chapter. Complex problem solving emerged approximately 30 years ago in Europe as a new topic of investigation ( Funke, 2010 ). The problems are formulated in computer-simulated microworlds (MicroDYN) that require discovering causal relations between input and output variables. In one application labeled Handball Training, the input variables were three different training procedures, and the output variables were motivation, power of the throw, and exhaustion ( Wustenberg, Greiff, & Funke, 2012 ). Participants attempted to reach specified target goals in the output variables by adjusting the values of the three training procedures. Performance on this task explained variance in grade point average beyond reasoning ability as measured by scores on Raven’s Advanced Progressive Matrices ( Wustenberg et al., 2012 ).

Another topic that is receiving increased attention is collaborative problem solving, in part assisted by the 2006 launch of the International Journal of Computer-supported Collaborative Learning ( Stahl & Hesse, 2006 ). An example article from this journal is the Engelmann and Hesse (2010) study in which three group members, working at separate computers, had to determine which pesticide and fertilizer to use to rescue a spruce forest. Each member of the group was given both relevant and irrelevant information to construct a concept map of shared information. Groups who initially had access to the knowledge of other group members started significantly earlier in discussing the problems and solved the fertilizer problem significantly sooner.

In conclusion, although research on the individual solutions of individual problem solvers will continue to be a major focus, research on multiple solutions, problems with insufficient information, estimated answers, complex problem solving, and collaborative problem solving will expand and enrich our knowledge.

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Piaget’s Theory and Stages of Cognitive Development

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

Key Takeaways

• Jean Piaget is famous for his theories regarding changes in cognitive development that occur as we move from infancy to adulthood.
• Cognitive development results from the interplay between innate capabilities (nature) and environmental influences (nurture).
• Children progress through four distinct stages , each representing varying cognitive abilities and world comprehension: the sensorimotor stage (birth to 2 years), the preoperational stage (2 to 7 years), the concrete operational stage (7 to 11 years), and the formal operational stage (11 years and beyond).
• A child’s cognitive development is not just about acquiring knowledge, the child has to develop or construct a mental model of the world, which is referred to as a schema .
• Piaget emphasized the role of active exploration and interaction with the environment in shaping cognitive development, highlighting the importance of assimilation and accommodation in constructing mental schemas.

Stages of Development

Jean Piaget’s theory of cognitive development suggests that children move through four different stages of intellectual development which reflect the increasing sophistication of children’s thought

Each child goes through the stages in the same order (but not all at the same rate), and child development is determined by biological maturation and interaction with the environment.

At each stage of development, the child’s thinking is qualitatively different from the other stages, that is, each stage involves a different type of intelligence.

Although no stage can be missed out, there are individual differences in the rate at which children progress through stages, and some individuals may never attain the later stages.

Piaget did not claim that a particular stage was reached at a certain age – although descriptions of the stages often include an indication of the age at which the average child would reach each stage.

The Sensorimotor Stage

Ages: Birth to 2 Years

The first stage is the sensorimotor stage , during which the infant focuses on physical sensations and learning to coordinate its body.

Major Characteristics and Developmental Changes:

• The infant learns about the world through their senses and through their actions (moving around and exploring their environment).
• During the sensorimotor stage, a range of cognitive abilities develop. These include: object permanence; self-recognition (the child realizes that other people are separate from them); deferred imitation; and representational play.
• They relate to the emergence of the general symbolic function, which is the capacity to represent the world mentally
• At about 8 months, the infant will understand the permanence of objects and that they will still exist even if they can’t see them and the infant will search for them when they disappear.

During the beginning of this stage, the infant lives in the present. It does not yet have a mental picture of the world stored in its memory therefore it does not have a sense of object permanence.

If it cannot see something, then it does not exist. This is why you can hide a toy from an infant, while it watches, but it will not search for the object once it has gone out of sight.

The main achievement during this stage is object permanence – knowing that an object still exists, even if it is hidden. It requires the ability to form a mental representation (i.e., a schema) of the object.

Towards the end of this stage the general symbolic function begins to appear where children show in their play that they can use one object to stand for another. Language starts to appear because they realise that words can be used to represent objects and feelings.

The child begins to be able to store information that it knows about the world, recall it, and label it.

Individual Differences

• Cultural Practices : In some cultures, babies are carried on their mothers’ backs throughout the day. This constant physical contact and varied stimuli can influence how a child perceives their environment and their sense of object permanence.
• Gender Norms : Toys assigned to babies can differ based on gender expectations. A boy might be given more cars or action figures, while a girl might receive dolls or kitchen sets. This can influence early interactions and sensory explorations.

Learn More: The Sensorimotor Stage of Cognitive Development

The Preoperational Stage

Ages: 2 – 7 Years

Piaget’s second stage of intellectual development is the preoperational stage . It takes place between 2 and 7 years. At the beginning of this stage, the child does not use operations, so the thinking is influenced by the way things appear rather than logical reasoning.

A child cannot conserve which means that the child does not understand that quantity remains the same even if the appearance changes.

Furthermore, the child is egocentric; he assumes that other people see the world as he does. This has been shown in the three mountains study.

As the preoperational stage develops, egocentrism declines, and children begin to enjoy the participation of another child in their games, and let’s pretend play becomes more important.

Toddlers often pretend to be people they are not (e.g. superheroes, policemen), and may play these roles with props that symbolize real-life objects. Children may also invent an imaginary playmate.

• Toddlers and young children acquire the ability to internally represent the world through language and mental imagery.
• During this stage, young children can think about things symbolically. This is the ability to make one thing, such as a word or an object, stand for something other than itself.
• A child’s thinking is dominated by how the world looks, not how the world is. It is not yet capable of logical (problem-solving) type of thought.
• Moreover, the child has difficulties with class inclusion; he can classify objects but cannot include objects in sub-sets, which involves classifying objects as belonging to two or more categories simultaneously.
• Infants at this stage also demonstrate animism. This is the tendency for the child to think that non-living objects (such as toys) have life and feelings like a person’s.

By 2 years, children have made some progress toward detaching their thoughts from the physical world. However, have not yet developed logical (or “operational”) thought characteristics of later stages.

Thinking is still intuitive (based on subjective judgments about situations) and egocentric (centered on the child’s own view of the world).

• Cultural Storytelling : Different cultures have unique stories, myths, and folklore. Children from diverse backgrounds might understand and interpret symbolic elements differently based on their cultural narratives.
• Race & Representation : A child’s racial identity can influence how they engage in pretend play. For instance, a lack of diverse representation in media and toys might lead children of color to recreate scenarios that don’t reflect their experiences or background.

Learn More: The Preoperational Stage of Cognitive Development

The Concrete Operational Stage

Ages: 7 – 11 Years

By the beginning of the concrete operational stage , the child can use operations (a set of logical rules) so they can conserve quantities, realize that people see the world in a different way (decentring), and demonstrate improvement in inclusion tasks. Children still have difficulties with abstract thinking.

• During this stage, children begin to think logically about concrete events.
• Children begin to understand the concept of conservation; understanding that, although things may change in appearance, certain properties remain the same.
• During this stage, children can mentally reverse things (e.g., picture a ball of plasticine returning to its original shape).
• During this stage, children also become less egocentric and begin to think about how other people might think and feel.

The stage is called concrete because children can think logically much more successfully if they can manipulate real (concrete) materials or pictures of them.

Piaget considered the concrete stage a major turning point in the child’s cognitive development because it marks the beginning of logical or operational thought. This means the child can work things out internally in their head (rather than physically try things out in the real world).

Children can conserve number (age 6), mass (age 7), and weight (age 9). Conservation is the understanding that something stays the same in quantity even though its appearance changes.

But operational thought is only effective here if the child is asked to reason about materials that are physically present. Children at this stage will tend to make mistakes or be overwhelmed when asked to reason about abstract or hypothetical problems.

• Cultural Context in Conservation Tasks : In a society where resources are scarce, children might demonstrate conservation skills earlier due to the cultural emphasis on preserving and reusing materials.
• Gender & Learning : Stereotypes about gender abilities, like “boys are better at math,” can influence how children approach logical problems or classify objects based on perceived gender norms.

Learn More: The Concrete Operational Stage of Development

The Formal Operational Stage

Ages: 12 and Over

The formal operational period begins at about age 11. As adolescents enter this stage, they gain the ability to think in an abstract manner, the ability to combine and classify items in a more sophisticated way, and the capacity for higher-order reasoning.

Adolescents can think systematically and reason about what might be as well as what is (not everyone achieves this stage). This allows them to understand politics, ethics, and science fiction, as well as to engage in scientific reasoning.

Adolescents can deal with abstract ideas: e.g. they can understand division and fractions without having to actually divide things up, and solve hypothetical (imaginary) problems.

• Concrete operations are carried out on things whereas formal operations are carried out on ideas. Formal operational thought is entirely freed from physical and perceptual constraints.
• During this stage, adolescents can deal with abstract ideas (e.g. no longer needing to think about slicing up cakes or sharing sweets to understand division and fractions).
• They can follow the form of an argument without having to think in terms of specific examples.
• Adolescents can deal with hypothetical problems with many possible solutions. E.g. if asked ‘What would happen if money were abolished in one hour’s time? they could speculate about many possible consequences.

From about 12 years children can follow the form of a logical argument without reference to its content. During this time, people develop the ability to think about abstract concepts, and logically test hypotheses.

This stage sees the emergence of scientific thinking, formulating abstract theories and hypotheses when faced with a problem.

• Culture & Abstract Thinking : Cultures emphasize different kinds of logical or abstract thinking. For example, in societies with a strong oral tradition, the ability to hold complex narratives might develop prominently.
• Gender & Ethics : Discussions about morality and ethics can be influenced by gender norms. For instance, in some cultures, girls might be encouraged to prioritize community harmony, while boys might be encouraged to prioritize individual rights.

Learn More: The Formal Operational Stage of Development

Piaget’s Theory

• Piaget’s theory places a strong emphasis on the active role that children play in their own cognitive development.
• According to Piaget, children are not passive recipients of information; instead, they actively explore and interact with their surroundings.
• This active engagement with the environment is crucial because it allows them to gradually build their understanding of the world.

1. How Piaget Developed the Theory

Piaget was employed at the Binet Institute in the 1920s, where his job was to develop French versions of questions on English intelligence tests. He became intrigued with the reasons children gave for their wrong answers to the questions that required logical thinking.

He believed that these incorrect answers revealed important differences between the thinking of adults and children.

Piaget branched out on his own with a new set of assumptions about children’s intelligence:

• Children’s intelligence differs from an adult’s in quality rather than in quantity. This means that children reason (think) differently from adults and see the world in different ways.
• Children actively build up their knowledge about the world . They are not passive creatures waiting for someone to fill their heads with knowledge.
• The best way to understand children’s reasoning is to see things from their point of view.

Piaget did not want to measure how well children could count, spell or solve problems as a way of grading their I.Q. What he was more interested in was the way in which fundamental concepts like the very idea of number , time, quantity, causality , justice , and so on emerged.

Piaget studied children from infancy to adolescence using naturalistic observation of his own three babies and sometimes controlled observation too. From these, he wrote diary descriptions charting their development.

He also used clinical interviews and observations of older children who were able to understand questions and hold conversations.

2. Piaget’s Theory Differs From Others In Several Ways:

Piaget’s (1936, 1950) theory of cognitive development explains how a child constructs a mental model of the world. He disagreed with the idea that intelligence was a fixed trait, and regarded cognitive development as a process that occurs due to biological maturation and interaction with the environment.

Children’s ability to understand, think about, and solve problems in the world develops in a stop-start, discontinuous manner (rather than gradual changes over time).

• It is concerned with children, rather than all learners.
• It focuses on development, rather than learning per se, so it does not address learning of information or specific behaviors.
• It proposes discrete stages of development, marked by qualitative differences, rather than a gradual increase in number and complexity of behaviors, concepts, ideas, etc.

The goal of the theory is to explain the mechanisms and processes by which the infant, and then the child, develops into an individual who can reason and think using hypotheses.

To Piaget, cognitive development was a progressive reorganization of mental processes as a result of biological maturation and environmental experience.

Children construct an understanding of the world around them, then experience discrepancies between what they already know and what they discover in their environment.

Piaget claimed that knowledge cannot simply emerge from sensory experience; some initial structure is necessary to make sense of the world.

According to Piaget, children are born with a very basic mental structure (genetically inherited and evolved) on which all subsequent learning and knowledge are based.

Schemas are the basic building blocks of such cognitive models, and enable us to form a mental representation of the world.

Piaget (1952, p. 7) defined a schema as: “a cohesive, repeatable action sequence possessing component actions that are tightly interconnected and governed by a core meaning.”

In more simple terms, Piaget called the schema the basic building block of intelligent behavior – a way of organizing knowledge. Indeed, it is useful to think of schemas as “units” of knowledge, each relating to one aspect of the world, including objects, actions, and abstract (i.e., theoretical) concepts.

Wadsworth (2004) suggests that schemata (the plural of schema) be thought of as “index cards” filed in the brain, each one telling an individual how to react to incoming stimuli or information.

When Piaget talked about the development of a person’s mental processes, he was referring to increases in the number and complexity of the schemata that a person had learned.

When a child’s existing schemas are capable of explaining what it can perceive around it, it is said to be in a state of equilibrium, i.e., a state of cognitive (i.e., mental) balance.

Operations are more sophisticated mental structures which allow us to combine schemas in a logical (reasonable) way.

As children grow they can carry out more complex operations and begin to imagine hypothetical (imaginary) situations.

Apart from the schemas we are born with schemas and operations are learned through interaction with other people and the environment.

Piaget emphasized the importance of schemas in cognitive development and described how they were developed or acquired.

A schema can be defined as a set of linked mental representations of the world, which we use both to understand and to respond to situations. The assumption is that we store these mental representations and apply them when needed.

Examples of Schemas

A person might have a schema about buying a meal in a restaurant. The schema is a stored form of the pattern of behavior which includes looking at a menu, ordering food, eating it and paying the bill.

This is an example of a schema called a “script.” Whenever they are in a restaurant, they retrieve this schema from memory and apply it to the situation.

The schemas Piaget described tend to be simpler than this – especially those used by infants. He described how – as a child gets older – his or her schemas become more numerous and elaborate.

Piaget believed that newborn babies have a small number of innate schemas – even before they have had many opportunities to experience the world. These neonatal schemas are the cognitive structures underlying innate reflexes. These reflexes are genetically programmed into us.

For example, babies have a sucking reflex, which is triggered by something touching the baby’s lips. A baby will suck a nipple, a comforter (dummy), or a person’s finger. Piaget, therefore, assumed that the baby has a “sucking schema.”

Similarly, the grasping reflex which is elicited when something touches the palm of a baby’s hand, or the rooting reflex, in which a baby will turn its head towards something which touches its cheek, are innate schemas. Shaking a rattle would be the combination of two schemas, grasping and shaking.

4. The Process of Adaptation

Piaget also believed that a child developed as a result of two different influences: maturation, and interaction with the environment. The child develops mental structures (schemata) which enables him to solve problems in the environment.

Adaptation is the process by which the child changes its mental models of the world to match more closely how the world actually is.

Adaptation is brought about by the processes of assimilation (solving new experiences using existing schemata) and accommodation (changing existing schemata in order to solve new experiences).

The importance of this viewpoint is that the child is seen as an active participant in its own development rather than a passive recipient of either biological influences (maturation) or environmental stimulation.

When our existing schemas can explain what we perceive around us, we are in a state of equilibration . However, when we meet a new situation that we cannot explain it creates disequilibrium, this is an unpleasant sensation which we try to escape, and this gives us the motivation to learn.

According to Piaget, reorganization to higher levels of thinking is not accomplished easily. The child must “rethink” his or her view of the world. An important step in the process is the experience of cognitive conflict.

In other words, the child becomes aware that he or she holds two contradictory views about a situation and they both cannot be true. This step is referred to as disequilibrium .

Jean Piaget (1952; see also Wadsworth, 2004) viewed intellectual growth as a process of adaptation (adjustment) to the world. This happens through assimilation, accommodation, and equilibration.

To get back to a state of equilibration, we need to modify our existing schemas to learn and adapt to the new situation.

This is done through the processes of accommodation and assimilation . This is how our schemas evolve and become more sophisticated. The processes of assimilation and accommodation are continuous and interactive.

5. Assimilation

Piaget defined assimilation as the cognitive process of fitting new information into existing cognitive schemas, perceptions, and understanding. Overall beliefs and understanding of the world do not change as a result of the new information.

Assimilation occurs when the new experience is not very different from previous experiences of a particular object or situation we assimilate the new situation by adding information to a previous schema.

This means that when you are faced with new information, you make sense of this information by referring to information you already have (information processed and learned previously) and trying to fit the new information into the information you already have.

• Imagine a young child who has only ever seen small, domesticated dogs. When the child sees a cat for the first time, they might refer to it as a “dog” because it has four legs, fur, and a tail – features that fit their existing schema of a dog.
• A person who has always believed that all birds can fly might label penguins as birds that can fly. This is because their existing schema or understanding of birds includes the ability to fly.
• A 2-year-old child sees a man who is bald on top of his head and has long frizzy hair on the sides. To his father’s horror, the toddler shouts “Clown, clown” (Siegler et al., 2003).
• If a baby learns to pick up a rattle he or she will then use the same schema (grasping) to pick up other objects.

6. Accommodation

Accommodation: when the new experience is very different from what we have encountered before we need to change our schemas in a very radical way or create a whole new schema.

Psychologist Jean Piaget defined accommodation as the cognitive process of revising existing cognitive schemas, perceptions, and understanding so that new information can be incorporated.

This happens when the existing schema (knowledge) does not work, and needs to be changed to deal with a new object or situation.

In order to make sense of some new information, you actually adjust information you already have (schemas you already have, etc.) to make room for this new information.

• A baby tries to use the same schema for grasping to pick up a very small object. It doesn’t work. The baby then changes the schema by now using the forefinger and thumb to pick up the object.
• A child may have a schema for birds (feathers, flying, etc.) and then they see a plane, which also flies, but would not fit into their bird schema.
• In the “clown” incident, the boy’s father explained to his son that the man was not a clown and that even though his hair was like a clown’s, he wasn’t wearing a funny costume and wasn’t doing silly things to make people laugh. With this new knowledge, the boy was able to change his schema of “clown” and make this idea fit better to a standard concept of “clown”.
• A person who grew up thinking all snakes are dangerous might move to an area where garden snakes are common and harmless. Over time, after observing and learning, they might accommodate their previous belief to understand that not all snakes are harmful.

7. Equilibration

Piaget believed that all human thought seeks order and is uncomfortable with contradictions and inconsistencies in knowledge structures. In other words, we seek “equilibrium” in our cognitive structures.

Equilibrium occurs when a child’s schemas can deal with most new information through assimilation. However, an unpleasant state of disequilibrium occurs when new information cannot be fitted into existing schemas (assimilation).

Piaget believed that cognitive development did not progress at a steady rate, but rather in leaps and bounds. Equilibration is the force which drives the learning process as we do not like to be frustrated and will seek to restore balance by mastering the new challenge (accommodation).

Once the new information is acquired the process of assimilation with the new schema will continue until the next time we need to make an adjustment to it.

Equilibration is a regulatory process that maintains a balance between assimilation and accommodation to facilitate cognitive growth. Think of it this way: We can’t merely assimilate all the time; if we did, we would never learn any new concepts or principles.

Everything new we encountered would just get put in the same few “slots” we already had. Neither can we accommodate all the time; if we did, everything we encountered would seem new; there would be no recurring regularities in our world. We’d be exhausted by the mental effort!

Applications to Education

Think of old black and white films that you’ve seen in which children sat in rows at desks, with ink wells, would learn by rote, all chanting in unison in response to questions set by an authoritarian old biddy like Matilda!

Children who were unable to keep up were seen as slacking and would be punished by variations on the theme of corporal punishment. Yes, it really did happen and in some parts of the world still does today.

Piaget is partly responsible for the change that occurred in the 1960s and for your relatively pleasurable and pain-free school days!

“Children should be able to do their own experimenting and their own research. Teachers, of course, can guide them by providing appropriate materials, but the essential thing is that in order for a child to understand something, he must construct it himself, he must re-invent it. Every time we teach a child something, we keep him from inventing it himself. On the other hand that which we allow him to discover by himself will remain with him visibly”. Piaget (1972, p. 27)

Plowden Report

Piaget (1952) did not explicitly relate his theory to education, although later researchers have explained how features of Piaget’s theory can be applied to teaching and learning.

Piaget has been extremely influential in developing educational policy and teaching practice. For example, a review of primary education by the UK government in 1966 was based strongly on Piaget’s theory. The result of this review led to the publication of the Plowden Report (1967).

In the 1960s the Plowden Committee investigated the deficiencies in education and decided to incorporate many of Piaget’s ideas into its final report published in 1967, even though Piaget’s work was not really designed for education.

The report makes three Piaget-associated recommendations:

• Children should be given individual attention and it should be realized that they need to be treated differently.
• Children should only be taught things that they are capable of learning
• Children mature at different rates and the teacher needs to be aware of the stage of development of each child so teaching can be tailored to their individual needs.

“The report’s recurring themes are individual learning, flexibility in the curriculum, the centrality of play in children’s learning, the use of the environment, learning by discovery and the importance of the evaluation of children’s progress – teachers should “not assume that only what is measurable is valuable.”

Discovery learning – the idea that children learn best through doing and actively exploring – was seen as central to the transformation of the primary school curriculum.

How to teach

Within the classroom learning should be student-centered and accomplished through active discovery learning. The role of the teacher is to facilitate learning, rather than direct tuition.

Because Piaget’s theory is based upon biological maturation and stages, the notion of “readiness” is important. Readiness concerns when certain information or concepts should be taught.

According to Piaget’s theory, children should not be taught certain concepts until they have reached the appropriate stage of cognitive development.

According to Piaget (1958), assimilation and accommodation require an active learner, not a passive one, because problem-solving skills cannot be taught, they must be discovered.

Therefore, teachers should encourage the following within the classroom:

• Educational programs should be designed to correspond to Piaget’s stages of development. Children in the concrete operational stage should be given concrete means to learn new concepts e.g. tokens for counting.
• Devising situations that present useful problems, and create disequilibrium in the child.
• Focus on the process of learning, rather than the end product of it. Instead of checking if children have the right answer, the teacher should focus on the student’s understanding and the processes they used to get to the answer.
• Child-centered approach. Learning must be active (discovery learning). Children should be encouraged to discover for themselves and to interact with the material instead of being given ready-made knowledge.
• Accepting that children develop at different rates so arrange activities for individual children or small groups rather than assume that all the children can cope with a particular activity.
• Using active methods that require rediscovering or reconstructing “truths.”
• Using collaborative, as well as individual activities (so children can learn from each other).
• Evaluate the level of the child’s development so suitable tasks can be set.
• Adapt lessons to suit the needs of the individual child (i.e. differentiated teaching).
• Be aware of the child’s stage of development (testing).
• Teach only when the child is ready. i.e. has the child reached the appropriate stage.
• Providing support for the “spontaneous research” of the child.
• Using collaborative, as well as individual activities.
• Educators may use Piaget’s stages to design age-appropriate assessment tools and strategies.

Classroom Activities

Sensorimotor stage (0-2 years):.

Although most kids in this age range are not in a traditional classroom setting, they can still benefit from games that stimulate their senses and motor skills.

• Object Permanence Games : Play peek-a-boo or hide toys under a blanket to help babies understand that objects still exist even when they can’t see them.
• Sensory Play : Activities like water play, sand play, or playdough encourage exploration through touch.
• Imitation : Children at this age love to imitate adults. Use imitation as a way to teach new skills.

Preoperational Stage (2-7 years):

• Role Playing : Set up pretend play areas where children can act out different scenarios, such as a kitchen, hospital, or market.
• Use of Symbols : Encourage drawing, building, and using props to represent other things.
• Hands-on Activities : Children should interact physically with their environment, so provide plenty of opportunities for hands-on learning.
• Egocentrism Activities : Use exercises that highlight different perspectives. For instance, having two children sit across from each other with an object in between and asking them what the other sees.

Concrete Operational Stage (7-11 years):

• Classification Tasks : Provide objects or pictures to group, based on various characteristics.
• Hands-on Experiments : Introduce basic science experiments where they can observe cause and effect, like a simple volcano with baking soda and vinegar.
• Logical Games : Board games, puzzles, and logic problems help develop their thinking skills.
• Conservation Tasks : Use experiments to showcase that quantity doesn’t change with alterations in shape, such as the classic liquid conservation task using different shaped glasses.

Formal Operational Stage (11 years and older):

• Hypothesis Testing : Encourage students to make predictions and test them out.
• Abstract Thinking : Introduce topics that require abstract reasoning, such as algebra or ethical dilemmas.
• Problem Solving : Provide complex problems and have students work on solutions, integrating various subjects and concepts.
• Debate and Discussion : Encourage group discussions and debates on abstract topics, highlighting the importance of logic and evidence.
• Feedback and Questioning : Use open-ended questions to challenge students and promote higher-order thinking. For instance, rather than asking, “Is this the right answer?”, ask, “How did you arrive at this conclusion?”

While Piaget’s stages offer a foundational framework, they are not universally experienced in the same way by all children.

Social identities play a critical role in shaping cognitive development, necessitating a more nuanced and culturally responsive approach to understanding child development.

Piaget’s stages may manifest differently based on social identities like race, gender, and culture:

• Race & Teacher Interactions : A child’s race can influence teacher expectations and interactions. For example, racial biases can lead to children of color being perceived as less capable or more disruptive, influencing their cognitive challenges and supports.
• Racial and Cultural Stereotypes : These can affect a child’s self-perception and self-efficacy . For instance, stereotypes about which racial or cultural groups are “better” at certain subjects can influence a child’s self-confidence and, subsequently, their engagement in that subject.
• Gender & Peer Interactions : Children learn gender roles from their peers. Boys might be mocked for playing “girl games,” and girls might be excluded from certain activities, influencing their cognitive engagements.
• Language : Multilingual children might navigate the stages differently, especially if their home language differs from their school language. The way concepts are framed in different languages can influence cognitive processing. Cultural idioms and metaphors can shape a child’s understanding of concepts and their ability to use symbolic representation, especially in the pre-operational stage.

Curriculum Development

According to Piaget, children’s cognitive development is determined by a process of maturation which cannot be altered by tuition so education should be stage-specific.

For example, a child in the concrete operational stage should not be taught abstract concepts and should be given concrete aid such as tokens to count with.

According to Piaget children learn through the process of accommodation and assimilation so the role of the teacher should be to provide opportunities for these processes to occur such as new material and experiences that challenge the children’s existing schemas.

Furthermore, according to this theory, children should be encouraged to discover for themselves and to interact with the material instead of being given ready-made knowledge.

Curricula need to be developed that take into account the age and stage of thinking of the child. For example there is no point in teaching abstract concepts such as algebra or atomic structure to children in primary school.

Curricula also need to be sufficiently flexible to allow for variations in the ability of different students of the same age. In Britain, the National Curriculum and Key Stages broadly reflect the stages that Piaget laid down.

For example, egocentrism dominates a child’s thinking in the sensorimotor and preoperational stages. Piaget would therefore predict that using group activities would not be appropriate since children are not capable of understanding the views of others.

However, Smith et al. (1998), point out that some children develop earlier than Piaget predicted and that by using group work children can learn to appreciate the views of others in preparation for the concrete operational stage.

The national curriculum emphasizes the need to use concrete examples in the primary classroom.

Shayer (1997), reported that abstract thought was necessary for success in secondary school (and co-developed the CASE system of teaching science). Recently the National curriculum has been updated to encourage the teaching of some abstract concepts towards the end of primary education, in preparation for secondary courses. (DfEE, 1999).

Child-centered teaching is regarded by some as a child of the ‘liberal sixties.’ In the 1980s the Thatcher government introduced the National Curriculum in an attempt to move away from this and bring more central government control into the teaching of children.

So, although the British National Curriculum in some ways supports the work of Piaget, (in that it dictates the order of teaching), it can also be seen as prescriptive to the point where it counters Piaget’s child-oriented approach.

However, it does still allow for flexibility in teaching methods, allowing teachers to tailor lessons to the needs of their students.

Social Media (Digital Learning)

Jean Piaget could not have anticipated the expansive digital age we now live in.

Today, knowledge dissemination and creation are democratized by the Internet, with platforms like blogs, wikis, and social media allowing for vast collaboration and shared knowledge. This development has prompted a reimagining of the future of education.

Classrooms, traditionally seen as primary sites of learning, are being overshadowed by the rise of mobile technologies and platforms like MOOCs (Passey, 2013).

The millennial generation, defined as the first to grow up with cable TV, the internet, and cell phones, relies heavily on technology.

They view it as an integral part of their identity, with most using it extensively in their daily lives, from keeping in touch with loved ones to consuming news and entertainment (Nielsen, 2014).

Social media platforms offer a dynamic environment conducive to Piaget’s principles. These platforms allow for interactions that nurture knowledge evolution through cognitive processes like assimilation and accommodation.

They emphasize communal interaction and shared activity, fostering both cognitive and socio-cultural constructivism. This shared activity promotes understanding and exploration beyond individual perspectives, enhancing social-emotional learning (Gehlbach, 2010).

A standout advantage of social media in an educational context is its capacity to extend beyond traditional classroom confines. As the material indicates, these platforms can foster more inclusive learning, bridging diverse learner groups.

This inclusivity can equalize learning opportunities, potentially diminishing biases based on factors like race or socio-economic status, resonating with Kegan’s (1982) concept of “recruitability.”

However, there are challenges. While the potential of social media in learning is vast, its practical application necessitates intention and guidance. Cuban, Kirkpatrick, and Peck (2001) note that certain educators and students are hesitant about integrating social media into educational contexts.

This hesitancy can stem from technological complexities or potential distractions. Yet, when harnessed effectively, social media can provide a rich environment for collaborative learning and interpersonal development, fostering a deeper understanding of content.

In essence, the rise of social media aligns seamlessly with constructivist philosophies. Social media platforms act as tools for everyday cognition, merging daily social interactions with the academic world, and providing avenues for diverse, interactive, and engaging learning experiences.

Applications to Parenting

Parents can use Piaget’s stages to have realistic developmental expectations of their children’s behavior and cognitive capabilities.

For instance, understanding that a toddler is in the pre-operational stage can help parents be patient when the child is egocentric.

Play Activities

Recognizing the importance of play in cognitive development, many parents provide toys and games suited for their child’s developmental stage.

Parents can offer activities that are slightly beyond their child’s current abilities, leveraging Vygotsky’s concept of the “Zone of Proximal Development,” which complements Piaget’s ideas.

• Peek-a-boo : Helps with object permanence.
• Texture Touch : Provide different textured materials (soft, rough, bumpy, smooth) for babies to touch and feel.
• Sound Bottles : Fill small bottles with different items like rice, beans, bells, and have children shake and listen to the different sounds.
• Memory Games : Using cards with pictures, place them face down, and ask students to find matching pairs.
• Role Playing and Pretend Play : Let children act out roles or stories that enhance symbolic thinking. Encourage symbolic play with dress-up clothes, playsets, or toy cash registers. Provide prompts or scenarios to extend their imagination.
• Story Sequencing : Give children cards with parts of a story and have them arranged in the correct order.
• Number Line Jumps : Create a number line on the floor with tape. Ask students to jump to the correct answer for math problems.
• Classification Games : Provide a mix of objects and ask students to classify them based on different criteria (e.g., color, size, shape).
• Logical Puzzle Games : Games that involve problem-solving using logic, such as simple Sudoku puzzles or logic grid puzzles.
• Debate and Discussion : Provide a topic and let students debate on pros and cons. This promotes abstract thinking and logical reasoning.
• Hypothesis Testing Games : Present a scenario and have students come up with hypotheses and ways to test them.
• Strategy Board Games : Games like chess, checkers, or Settlers of Catan can help in developing strategic and forward-thinking skills.

Critical Evaluation

• The influence of Piaget’s ideas on developmental psychology has been enormous. He changed how people viewed the child’s world and their methods of studying children.

He was an inspiration to many who came after and took up his ideas. Piaget’s ideas have generated a huge amount of research which has increased our understanding of cognitive development.

• Piaget (1936) was one of the first psychologists to make a systematic study of cognitive development. His contributions include a stage theory of child cognitive development, detailed observational studies of cognition in children, and a series of simple but ingenious tests to reveal different cognitive abilities.
• His ideas have been of practical use in understanding and communicating with children, particularly in the field of education (re: Discovery Learning). Piaget’s theory has been applied across education.
• According to Piaget’s theory, educational programs should be designed to correspond to the stages of development.
• Are the stages real? Vygotsky and Bruner would rather not talk about stages at all, preferring to see development as a continuous process. Others have queried the age ranges of the stages. Some studies have shown that progress to the formal operational stage is not guaranteed.

For example, Keating (1979) reported that 40-60% of college students fail at formal operation tasks, and Dasen (1994) states that only one-third of adults ever reach the formal operational stage.

The fact that the formal operational stage is not reached in all cultures and not all individuals within cultures suggests that it might not be biologically based.

• According to Piaget, the rate of cognitive development cannot be accelerated as it is based on biological processes however, direct tuition can speed up the development which suggests that it is not entirely based on biological factors.
• Because Piaget concentrated on the universal stages of cognitive development and biological maturation, he failed to consider the effect that the social setting and culture may have on cognitive development.

Cross-cultural studies show that the stages of development (except the formal operational stage) occur in the same order in all cultures suggesting that cognitive development is a product of a biological process of maturation.

However, the age at which the stages are reached varies between cultures and individuals which suggests that social and cultural factors and individual differences influence cognitive development.

Dasen (1994) cites studies he conducted in remote parts of the central Australian desert with 8-14-year-old Indigenous Australians. He gave them conservation of liquid tasks and spatial awareness tasks. He found that the ability to conserve came later in the Aboriginal children, between ages of 10 and 13 (as opposed to between 5 and 7, with Piaget’s Swiss sample).

However, he found that spatial awareness abilities developed earlier amongst the Aboriginal children than the Swiss children. Such a study demonstrates cognitive development is not purely dependent on maturation but on cultural factors too – spatial awareness is crucial for nomadic groups of people.

Vygotsky , a contemporary of Piaget, argued that social interaction is crucial for cognitive development. According to Vygotsky the child’s learning always occurs in a social context in cooperation with someone more skillful (MKO). This social interaction provides language opportunities and Vygotsky considered language the foundation of thought.

• Piaget’s methods (observation and clinical interviews) are more open to biased interpretation than other methods. Piaget made careful, detailed naturalistic observations of children, and from these, he wrote diary descriptions charting their development. He also used clinical interviews and observations of older children who were able to understand questions and hold conversations.

Because Piaget conducted the observations alone the data collected are based on his own subjective interpretation of events. It would have been more reliable if Piaget conducted the observations with another researcher and compared the results afterward to check if they are similar (i.e., have inter-rater reliability).

Although clinical interviews allow the researcher to explore data in more depth, the interpretation of the interviewer may be biased.

For example, children may not understand the question/s, they have short attention spans, they cannot express themselves very well, and may be trying to please the experimenter. Such methods meant that Piaget may have formed inaccurate conclusions.

• As several studies have shown Piaget underestimated the abilities of children because his tests were sometimes confusing or difficult to understand (e.g., Hughes , 1975).

Piaget failed to distinguish between competence (what a child is capable of doing) and performance (what a child can show when given a particular task). When tasks were altered, performance (and therefore competence) was affected. Therefore, Piaget might have underestimated children’s cognitive abilities.

For example, a child might have object permanence (competence) but still not be able to search for objects (performance). When Piaget hid objects from babies he found that it wasn’t till after nine months that they looked for it.

However, Piaget relied on manual search methods – whether the child was looking for the object or not.

Later, researchers such as Baillargeon and Devos (1991) reported that infants as young as four months looked longer at a moving carrot that didn’t do what it expected, suggesting they had some sense of permanence, otherwise they wouldn’t have had any expectation of what it should or shouldn’t do.

• The concept of schema is incompatible with the theories of Bruner (1966) and Vygotsky (1978). Behaviorism would also refute Piaget’s schema theory because is cannot be directly observed as it is an internal process. Therefore, they would claim it cannot be objectively measured.
• Piaget studied his own children and the children of his colleagues in Geneva to deduce general principles about the intellectual development of all children. His sample was very small and composed solely of European children from families of high socio-economic status. Researchers have, therefore, questioned the generalisability of his data.
• For Piaget, language is considered secondary to action, i.e., thought precedes language. The Russian psychologist Lev Vygotsky (1978) argues that the development of language and thought go together and that the origin of reasoning has more to do with our ability to communicate with others than with our interaction with the material world.

Piaget’s Theory vs Vygotsky

Piaget maintains that cognitive development stems largely from independent explorations in which children construct knowledge of their own.

Whereas Vygotsky argues that children learn through social interactions, building knowledge by learning from more knowledgeable others such as peers and adults. In other words, Vygotsky believed that culture affects cognitive development.

These factors lead to differences in the education style they recommend: Piaget would argue for the teacher to provide opportunities that challenge the children’s existing schemas and for children to be encouraged to discover for themselves.

Alternatively, Vygotsky would recommend that teachers assist the child to progress through the zone of proximal development by using scaffolding.

However, both theories view children as actively constructing their own knowledge of the world; they are not seen as just passively absorbing knowledge.

They also agree that cognitive development involves qualitative changes in thinking, not only a matter of learning more things.

What is cognitive development?

Cognitive development is how a person’s ability to think, learn, remember, problem-solve, and make decisions changes over time.

This includes the growth and maturation of the brain, as well as the acquisition and refinement of various mental skills and abilities.

Cognitive development is a major aspect of human development, and both genetic and environmental factors heavily influence it. Key domains of cognitive development include attention, memory, language skills, logical reasoning, and problem-solving.

Various theories, such as those proposed by Jean Piaget and Lev Vygotsky, provide different perspectives on how this complex process unfolds from infancy through adulthood.

What are the 4 stages of Piaget’s theory?

Piaget divided children’s cognitive development into four stages; each of the stages represents a new way of thinking and understanding the world.

He called them (1) sensorimotor intelligence , (2) preoperational thinking , (3) concrete operational thinking , and (4) formal operational thinking . Each stage is correlated with an age period of childhood, but only approximately.

According to Piaget, intellectual development takes place through stages that occur in a fixed order and which are universal (all children pass through these stages regardless of social or cultural background).

Development can only occur when the brain has matured to a point of “readiness”.

What are some of the weaknesses of Piaget’s theory?

Cross-cultural studies show that the stages of development (except the formal operational stage) occur in the same order in all cultures suggesting that cognitive development is a product of a biological maturation process.

However, the age at which the stages are reached varies between cultures and individuals, suggesting that social and cultural factors and individual differences influence cognitive development.

What are Piaget’s concepts of schemas?

Schemas are mental structures that contain all of the information relating to one aspect of the world around us.

According to Piaget, we are born with a few primitive schemas, such as sucking, which give us the means to interact with the world.

These are physical, but as the child develops, they become mental schemas. These schemas become more complex with experience.

Baillargeon, R., & DeVos, J. (1991). Object permanence in young infants: Further evidence . Child development , 1227-1246.

Bruner, J. S. (1966). Toward a theory of instruction. Cambridge, Mass.: Belkapp Press.

Cuban, L., Kirkpatrick, H., & Peck, C. (2001). High access and low use of technologies in high school classrooms: Explaining an apparent paradox.  American Educational Research Journal ,  38 (4), 813-834.

Dasen, P. (1994). Culture and cognitive development from a Piagetian perspective. In W .J. Lonner & R.S. Malpass (Eds.), Psychology and culture (pp. 145–149). Boston, MA: Allyn and Bacon.

Gehlbach, H. (2010). The social side of school: Why teachers need social psychology.  Educational Psychology Review ,  22 , 349-362.

Hughes, M. (1975). Egocentrism in preschool children . Unpublished doctoral dissertation. Edinburgh University.

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence . New York: Basic Books.

Keating, D. (1979). Adolescent thinking. In J. Adelson (Ed.), Handbook of adolescent psychology (pp. 211-246). New York: Wiley.

Kegan, R. (1982).  The evolving self: Problem and process in human development . Harvard University Press.

Nielsen. 2014. “Millennials: Technology = Social Connection.” http://www.nielsen.com/content/corporate/us/en/insights/news/2014/millennials-technology-social-connecti on.html.

Passey, D. (2013).  Inclusive technology enhanced learning: Overcoming cognitive, physical, emotional, and geographic challenges . Routledge.

Piaget, J. (1932). The moral judgment of the child . London: Routledge & Kegan Paul.

Piaget, J. (1936). Origins of intelligence in the child. London: Routledge & Kegan Paul.

Piaget, J. (1945). Play, dreams and imitation in childhood . London: Heinemann.

Piaget, J. (1957). Construction of reality in the child. London: Routledge & Kegan Paul.

Piaget, J., & Cook, M. T. (1952). The origins of intelligence in children . New York, NY: International University Press.

Piaget, J. (1981).  Intelligence and affectivity: Their relationship during child development.(Trans & Ed TA Brown & CE Kaegi) . Annual Reviews.

Plowden, B. H. P. (1967). Children and their primary schools: A report (Research and Surveys). London, England: HM Stationery Office.

Siegler, R. S., DeLoache, J. S., & Eisenberg, N. (2003). How children develop . New York: Worth.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes . Cambridge, MA: Harvard University Press.

Wadsworth, B. J. (2004). Piaget’s theory of cognitive and affective development: Foundations of constructivism . New York: Longman.

Further Reading

• BBC Radio Broadcast about the Three Mountains Study
• Piagetian stages: A critical review
• Bronfenbrenner’s Ecological Systems Theory

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Child cognitive development is a fascinating and complex process that entails the growth of a child’s mental abilities, including their ability to think, learn, and solve problems. This development occurs through a series of stages that can vary among individuals. As children progress through these stages, their cognitive abilities and skills are continuously shaped by a myriad of factors such as genetics, environment, and experiences. Understanding the nuances of child cognitive development is essential for parents, educators, and professionals alike, as it provides valuable insight into supporting the growth of the child’s intellect and overall well-being.

Throughout the developmental process, language and communication play a vital role in fostering a child’s cognitive abilities . As children acquire language skills, they also develop their capacity for abstract thought, reasoning, and problem-solving. It is crucial for parents and caregivers to be mindful of potential developmental delays, as early intervention can greatly benefit the child’s cognitive development. By providing stimulating environments, nurturing relationships, and embracing diverse learning opportunities, adults can actively foster healthy cognitive development in children.

Key Takeaways

• Child cognitive development involves the growth of mental abilities and occurs through various stages.
• Language and communication are significant factors in cognitive development , shaping a child’s ability for abstract thought and problem-solving.
• Early intervention and supportive environments can play a crucial role in fostering healthy cognitive development in children.

Child Cognitive Development Stages

Child cognitive development is a crucial aspect of a child’s growth and involves the progression of their thinking, learning, and problem-solving abilities. Swiss psychologist Jean Piaget developed a widely recognized theory that identifies four major stages of cognitive development in children.

Sensorimotor Stage

The Sensorimotor Stage occurs from birth to about 2 years old. During this stage, infants and newborns learn to coordinate their senses (sight, sound, touch, etc.) with their motor abilities. Their understanding of the world begins to develop through their physical interactions and experiences. Some key milestones in this stage include object permanence, which is the understanding that an object still exists even when it’s not visible, and the development of intentional actions.

Preoperational Stage

The Preoperational Stage takes place between the ages of 2 and 7 years old. In this stage, children start to think symbolically, and their language capabilities rapidly expand. They also develop the ability to use mental images, words, and gestures to represent the world around them. However, their thinking is largely egocentric, which means they struggle to see things from other people’s perspectives. During this stage, children start to engage in pretend play and begin to grasp the concept of conservation, recognizing that certain properties of objects (such as quantity or volume) remain the same even if their appearance changes.

Concrete Operational Stage

The Concrete Operational Stage occurs between the ages of 7 and 12 years old. At this stage, children’s cognitive development progresses to more logical and organized ways of thinking. They can now consider multiple aspects of a problem and better understand the relationship between cause and effect . Furthermore, children become more adept at understanding other people’s viewpoints, and they can perform basic mathematical operations and understand the principles of classification and seriation.

Formal Operational Stage

Lastly, the Formal Operational Stage typically begins around 12 years old and extends into adulthood. In this stage, children develop the capacity for abstract thinking and can consider hypothetical situations and complex reasoning. They can also perform advanced problem-solving and engage in systematic scientific inquiry. This stage allows individuals to think about abstract concepts, their own thought processes, and understand the world in deeper, more nuanced ways.

By understanding these stages of cognitive development, you can better appreciate the complex growth process that children undergo as their cognitive abilities transform and expand throughout their childhood.

Key Factors in Cognitive Development

Genetics and brain development.

Genetics play a crucial role in determining a child’s cognitive development. A child’s brain development is heavily influenced by genetic factors, which also determine their cognitive potential , abilities, and skills. It is important to understand that a child’s genes do not solely dictate their cognitive development – various environmental and experiential factors contribute to shaping their cognitive abilities as they grow and learn.

Environmental Influences

The environment in which a child grows up has a significant impact on their cognitive development. Exposure to various experiences is essential for a child to develop essential cognitive skills such as problem-solving, communication, and critical thinking. Factors that can have a negative impact on cognitive development include exposure to toxins, extreme stress, trauma, abuse, and addiction issues, such as alcoholism in the family.

Nutrition and Health

Maintaining good nutrition and health is vital for a child’s cognitive development. Adequate nutrition is essential for the proper growth and functioning of the brain . Key micronutrients that contribute to cognitive development include iron, zinc, and vitamins A, C, D, and B-complex vitamins. Additionally, a child’s overall health, including physical fitness and immunity, ensures they have the energy and resources to engage in learning activities and achieve cognitive milestones effectively .

Emotional and Social Factors

Emotional well-being and social relationships can also greatly impact a child’s cognitive development. A supportive, nurturing, and emotionally healthy environment allows children to focus on learning and building cognitive skills. Children’s emotions and stress levels can impact their ability to learn and process new information. Additionally, positive social interactions help children develop important cognitive skills such as empathy, communication, and collaboration.

In summary, cognitive development in children is influenced by various factors, including genetics, environmental influences, nutrition, health, and emotional and social factors. Considering these factors can help parents, educators, and policymakers create suitable environments and interventions for promoting optimal child development.

Language and Communication Development

Language skills and milestones.

Children’s language development is a crucial aspect of their cognitive growth. They begin to acquire language skills by listening and imitating sounds they hear from their environment. As they grow, they start to understand words and form simple sentences.

• Infants (0-12 months): Babbling, cooing, and imitating sounds are common during this stage. They can also identify their name by the end of their first year. Facial expressions play a vital role during this period, as babies learn to respond to emotions.
• Toddlers (1-3 years): They rapidly learn new words and form simple sentences. They engage more in spoken communication, constantly exploring their language environment.
• Preschoolers (3-5 years): Children expand their vocabulary, improve grammar, and begin participating in more complex conversations.

It’s essential to monitor children’s language development and inform their pediatrician if any delays or concerns arise.

Nonverbal Communication

Nonverbal communication contributes significantly to children’s cognitive development. They learn to interpret body language, facial expressions, and gestures long before they can speak. Examples of nonverbal communication in children include:

• Eye contact: Maintaining eye contact while interacting helps children understand emotions and enhances communication.
• Gestures: Pointing, waving goodbye, or using hand signs provide alternative ways for children to communicate their needs and feelings.
• Body language: Posture, body orientation, and movement give clues about a child’s emotions and intentions.

Teaching children to understand and use nonverbal communication supports their cognitive and social development.

Parent and Caregiver Interaction

Supportive interaction from parents and caregivers plays a crucial role in children’s language and communication development. These interactions can improve children’s language skills and overall cognitive abilities . Some ways parents and caregivers can foster language development are:

• Reading together: From an early age, reading books to children enhance their vocabulary and listening skills.
• Encouraging communication: Ask open-ended questions and engage them in conversations to build their speaking skills.
• Using rich vocabulary: Expose children to a variety of words and phrases, promoting language growth and understanding.

By actively engaging in children’s language and communication development, parents and caregivers can nurture cognitive, emotional, and social growth.

Cognitive Abilities and Skills

Cognitive abilities are the mental skills that children develop as they grow. These skills are essential for learning, adapting, and thriving in modern society. In this section, we will discuss various aspects of cognitive development, including reasoning and problem-solving, attention and memory, decision-making and executive function, as well as academic and cognitive milestones.

Reasoning and Problem Solving

Reasoning is the ability to think logically and make sense of the world around us. It’s essential for a child’s cognitive development, as it enables them to understand the concept of object permanence , recognize patterns, and classify objects. Problem-solving skills involve using these reasoning abilities to find solutions to challenges they encounter in daily life .

Children develop essential skills like:

• Logical reasoning : The ability to deduce conclusions from available information.
• Perception: Understanding how objects relate to one another in their environment.
• Schemes: Organizing thoughts and experiences into mental categories.

Attention and Memory

Attention refers to a child’s ability to focus on specific tasks, objects, or information, while memory involves retaining and recalling information. These cognitive abilities play a critical role in children’s learning and academic performance . Working memory is a vital component of learning, as it allows children to hold and manipulate information in their minds while solving problems and engaging with new tasks.

• Attention: Focuses on relevant tasks and information while ignoring distractions.
• Memory: Retains and retrieves information when needed.

Decision-Making and Executive Function

Decision-making is the process of making choices among various alternatives, while executive function refers to the higher-order cognitive processes that enable children to plan, organize, and adapt in complex situations. Executive function encompasses components such as:

• Inhibition: Self-control and the ability to resist impulses.
• Cognitive flexibility: Adapting to new information or changing circumstances.
• Planning: Setting goals and devising strategies to achieve them.

Academic and Cognitive Milestones

Children’s cognitive development is closely linked to their academic achievement. As they grow, they achieve milestones in various cognitive domains that form the foundation for their future learning. Some of these milestones include:

• Language skills: Developing vocabulary, grammar, and sentence structure.
• Reading and mathematics: Acquiring the ability to read and comprehend text, as well as understanding basic mathematical concepts and operations.
• Scientific thinking: Developing an understanding of cause-and-effect relationships and forming hypotheses.

Healthy cognitive development is essential for a child’s success in school and life. By understanding and supporting the development of their cognitive abilities, we can help children unlock their full potential and prepare them for a lifetime of learning and growth.

Developmental Delays and Early Intervention

Identifying developmental delays.

Developmental delays in children can be identified by monitoring their progress in reaching cognitive, linguistic, physical, and social milestones. Parents and caregivers should be aware of developmental milestones that are generally expected to be achieved by children at different ages, such as 2 months, 4 months, 6 months, 9 months, 18 months, 1 year, 2 years, 3 years, 4 years, and 5 years. Utilizing resources such as the “Learn the Signs. Act Early.” program can help parents and caregivers recognize signs of delay early in a child’s life.

Resources and Support for Parents

There are numerous resources available for parents and caregivers to find information on developmental milestones and to learn about potential developmental delays, including:

• Learn the Signs. Act Early : A CDC initiative that provides pdf checklists of milestones and resources for identifying delays.
• Parental support groups : Local and online communities dedicated to providing resources and fostering connections between families experiencing similar challenges.

Professional Evaluations and Intervention Strategies

If parents or caregivers suspect a developmental delay, it is crucial to consult with healthcare professionals or specialists who can conduct validated assessments of the child’s cognitive and developmental abilities. Early intervention strategies, such as the ones used in broad-based early intervention programs , have shown significant positive impacts on children with developmental delays to improve cognitive development and outcomes.

Professional evaluations may include:

• Pediatricians : Primary healthcare providers who can monitor a child’s development and recommend further assessments when needed.
• Speech and language therapists : Professionals who assist children with language and communication deficits.
• Occupational therapists : Experts in helping children develop or improve on physical and motor skills, as well as social and cognitive abilities.

Depending on the severity and nature of the delays, interventions may involve:

• Individualized support : Tailored programs or therapy sessions specifically developed for the child’s needs.
• Group sessions : Opportunities for children to learn from and interact with other children experiencing similar challenges.
• Family involvement : Parents and caregivers learning support strategies to help the child in their daily life.

Fostering Healthy Cognitive Development

Play and learning opportunities.

Encouraging play is crucial for fostering healthy cognitive development in children . Provide a variety of age-appropriate games, puzzles, and creative activities that engage their senses and stimulate curiosity. For example, introduce building blocks and math games for problem-solving skills, and crossword puzzles to improve vocabulary and reasoning abilities.

Playing with others also helps children develop social skills and better understand facial expressions and emotions. Provide opportunities for cooperative play, where kids can work together to achieve a common goal, and open-ended play with no specific rules to boost creativity.

Supportive Home Environment

A nurturing and secure home environment encourages healthy cognitive growth. Be responsive to your child’s needs and interests, involving them in everyday activities and providing positive reinforcement. Pay attention to their emotional well-being and create a space where they feel safe to ask questions and explore their surroundings.

Promoting Independence and Decision-Making

Support independence by allowing children to make decisions about their playtime, activities, and daily routines. Encourage them to take age-appropriate responsibilities and make choices that contribute to self-confidence and autonomy. Model problem-solving strategies and give them opportunities to practice these skills during play, while also guiding them when necessary.

Healthy Lifestyle Habits

Promote a well-rounded lifestyle, including:

• Sleep : Ensure children get adequate and quality sleep by establishing a consistent bedtime routine.
• Hydration : Teach the importance of staying hydrated by offering water frequently, especially during play and physical activities.
• Screen time : Limit exposure to electronic devices and promote alternative activities for toddlers and older kids.
• Physical activity : Encourage children to engage in active play and exercise to support neural development and overall health .

Frequently Asked Questions

What are the key stages of child cognitive development.

Child cognitive development can be divided into several key stages based on Piaget’s theory of cognitive development . These stages include the sensorimotor stage (birth to 2 years), preoperational stage (2-7 years), concrete operational stage (7-11 years), and formal operational stage (11 years and beyond). Every stage represents a unique period of cognitive growth, marked by the development of new skills, thought processes, and understanding of the world.

What factors influence cognitive development in children?

Several factors contribute to individual differences in child cognitive development, such as genetic and environmental factors. Socioeconomic status, access to quality education, early home environment, and parental involvement all play a significant role in determining cognitive growth. In addition, children’s exposure to diverse learning experiences, adequate nutrition, and mental health also influence overall cognitive performance .

How do cognitive skills vary during early childhood?

Cognitive skills in early childhood evolve as children progress through various stages . During the sensorimotor stage, infants develop fundamental skills such as object permanence. The preoperational stage is characterized by the development of symbolic thought, language, and imaginative play. Children then enter the concrete operational stage, acquiring the ability to think logically and solve problems. Finally, in the formal operational stage, children develop abstract reasoning abilities, complex problem-solving skills and metacognitive awareness.

What are common examples of cognitive development?

Examples of cognitive development include the acquisition of language and vocabulary, the development of problem-solving skills, and the ability to engage in logical reasoning. Additionally, memory, attention, and spatial awareness are essential aspects of cognitive development. Children may demonstrate these skills through activities like puzzle-solving, reading, and mathematics.

How do cognitive development theories explain children’s learning?

Piaget’s cognitive development theory suggests that children learn through active exploration, constructing knowledge based on their experiences and interactions with the world. In contrast, Vygotsky’s sociocultural theory emphasizes the role of social interaction and cultural context in learning. Both theories imply that cognitive development is a dynamic and evolving process, influenced by various environmental and psychological factors.

Why is it essential to support cognitive development in early childhood?

Supporting cognitive development in early childhood is critical because it lays a strong foundation for future academic achievement, social-emotional development, and lifelong learning. By providing children with diverse and enriching experiences, caregivers and educators can optimize cognitive growth and prepare children to face the challenges of today’s complex world. Fostering cognitive development early on helps children develop resilience, adaptability, and critical thinking skills essential for personal and professional success.

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9 cognitive skill examples and how to improve them

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What are cognitive skills?

Cognitive development

Types of cognitive skills.

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How to improve your cognitive skills

Stay focused.

Out of the blue, your team leader drops a curveball: the team is adopting a new project management app and updating work performance standards. 

Such an abrupt shift pushes your most basic cognitive skills into action. You diligently listen to your manager's instructions, process the influx of new information, and use logic to understand it all.

Normally, your thinking skills operate in the background, quietly supporting your daily work. But moments like this emphasize the incredible potential of your brain and the importance of honing your cognitive abilities.

Of course, some abilities — such as reasoning, visual learning, and listening — may come more naturally than others. Don’t worry: like any skill, you can grow and develop your brain power.

Prepare to unlock the full potential of your mind . Let's explore examples of cognitive skills and discover practical ways to elevate them in the workplace.

What are cognitive skills? 

The definition of cognitive skills encompasses your brain's remarkable capacity to process, store, and utilize information . These include abilities such as concentration , memory , and problem-solving.

Your cognitive skills operate subtly yet significantly, shaping your social interactions, learning processes, and ability to complete tasks successfully.

Say you meet a potential client at a networking event. Your brain effortlessly processes various pieces of information, from nonverbal social cues (like gestures ) to your elevator pitch . In this scenario, your adaptability is the defining factor between a successful and unsuccessful connection.

Cognitive development begins in infancy and early childhood and continues throughout your life. Your brain learns and grows as you age — a process called neuroplasticity . The more you train your mind through goal-setting and skill learning, the sharper your brain becomes. 

Research suggests the greater your cognitive ability, the better your performance . But there’s a caveat: your cognitive skills don’t operate in a vacuum. Self-discipline and planning also play a strong role in your ability to access and improve these abilities.

Although you may lean toward certain skills — perhaps your auditory processing is stronger than your visual learning — you can improve in any area with thoughtful practice and goal-setting .

Remember: your cognitive skills define your capacity for processing incoming information, building memories, and interpreting stimuli. Before jumping into cognitive skills to fine-tune, let’s explore eight different types of cognitive skills and their daily applications:  

Attention abilities

The world is full of stimuli. With so many distractions, it’s important to build up your ability to keep your focus. 

Your attention span is divided into three categories: 

• Sustained attention: This is your ability to focus and concentrate your thought processes over an extended period of time. You’ve likely been in a meeting or call where your mind started to wander — that was your sustained attention clocking off. But when you let distractions get the best of you, you might procrastinate , take exc essive time to complete tasks, or lose out on important information. 
• Selective attention : When various stimuli battle for your attention, your selective attention helps you suppress distractions and stay on task. Giving into distractions pushes your workflow off course and disrupts your productivity.  
• Divided attention : When you’re working on a project, you often have constructive feedback from your manager, requests from your client, and the scope of work to consider. Your divided attention allows you to take in all this information and find the right path forward. Without it, you might become overwhelmed and struggle to chart a course of action.

Memory skills

At work, building your memory helps ensure that information doesn't go in one ear and out the other. These are the two types of memories to polish: 

• Working memory : Sometimes referred to as your short-term memory, working memory allows you to hold on to information while you use it. Imagine a virtual onboarding with a new project management app: your working memory allows you to process instructions as you work through the platform. Weak working memory can cost you time. You might re-read directions, forget what someone just told you, or have difficulty following step-by-step instructions.
• Long-term memory : Long-term memories are the procedures, facts, and experiences you use to interact with your environment and learn new skills . Your long-term memory guides your professional development as you build upon your knowledge and expertise. Without a sharp long-term memory, you may struggle to fine-tune important technical skills or build relationships impo rtant to your career. 

Information processing skills

Pings on your phone, numbers on a chart, and the inflection of a coworker's voice all signal different messages. Here are three ways your brain processes information: 

• Auditory processing: Noise is identified, analyzed, and separated by your auditory processing abilities. Auditory processing disorder is a common cognitive disorder that impacts your ability to listen to speech with background noise, follow spoken instructions, or learn new languages. 
• Visual processing: This is your ability to perceive, analyze, and synthesize visual patterns — as well as form visual imagery and memory. It’s not uncommon to struggle with visual pro cessing, which can make pattern recognition in math and written instructions difficult. Fortunately, this can often be improved with a vision therapist . 
• Processing speed: This is the time required to respond to and process information from your environment. Low processing speeds can cause you to take longer to complete tasks — especially under pressure — which throws off your efficiency and workflow.

What are examples of cognitive skills at work? 

Ready to level up your performance? Here are nine examples of cognitive skills to work on to strengthen your professional development:

1. Logic and reasoning 

The ability to draw specific conclusions based on varied facts or data is your deductive reasoning. Even mundane tasks, like organizing your calendar, require strong logic and problem-solving skills. Deductive reasoning also helps you gauge importance, estimate work times, and set realistic goals. Without these logical thinking skills, you would struggle to work productively. 

2. Language

Language is divided into four skills: reading, writing, listening, and speaking. Every person is different — you may be an excellent writer but struggle with verbally expressing your ideas. However, clearly communicating your ideas is valuable in just about any role. Strong language skills can help you overcome miscommunications, resolve conflict, and encourage teamwork.  

3. Critical thinking

Critical thinking is a union of several soft skills , including attention to detail, intellectual curiosity , and open-mindedness. These traits are integral to problem-solving because they help you work through biases and arrive at independent, out-of-the-box solutions . That’s likely why critical thinking is considered one of the most durable skills in the workplace . 

4. Planning

Your day-to-day is full of short-term tasks and long-term objectives. Without proper planning, you could become disorganized or miss important deadlines. Planning requires logic and memory recall — these skills allow you to estimate a task's relevance and how long it should take to complete. Learning to organize and prioritize your tasks empowers you to be efficient, responsible, and proactive.

5. Quantitative skills

An understanding of statistics and math helps you turn ideas into data and eliminate emotional biases from important decisions. Data analysis is an increasingly important hard skill to have on your resume .

And as artificial intelligence and big data can contribute to businesses project growth and calculate risk, learning quantitative tools might help you stay competitive in the job market. Similarly, if you’re a freelancer building a personal brand , being able to read analytics allows you to engage wider audiences and find opportunities in your market. 

6. Networking

Making the right first impression is a science. It requires you to pay attention to social cues and process several visual and auditory stimuli from the person you’re networking with. Practicing active listening trains your brain to sustain its focus and pick up on information that will lead to positive and productive professional interactions. 

In the digital age, we work with more emails, project management tools, and messenger apps than ever before. While you don’t have to aspire to be a copywriting master, learning to organize your thoughts and contextualize them for your readers can reduce miscommunications. And when someone understands a message immediately, it saves you and your colleagues time that you can dedicate to more important tasks. 

8. Reading comprehension

Reading requires you to connect ideas, sustain your focus, and recall past experiences or know-how to de-code information. Similar to writing, analyzing and contextualizing information can help you avoid misunderstandings and improve your productivity. Reading comprehension is important in any job, particularly remote jobs that depend heavily on written communication. 

9. Collaboration

While collaboration may sound more like a social skill than a cognitive function, efficient teamwork requires abstract thinking. These skills help you break a project down into different tasks, leverage everyone’s strengths, and keep on top of all your team members’ deliverables. 

Inspired to level up your cognitive capacities? Here are four ways to take care of your brain: 

1. Stay healthy

Your physical and mental health are intimately connected to one another. Besides working up a sweat, physical exercise builds new neurons and stimulates memory by increasing blood flow to the brain. 

Consider developing a routine to get your 150 minutes of recommended weekly exercise , like an after work swim, joining a jogging club, or hiring a personal trainer. Similarly, a firm sleep schedule , staying hydrated , and good nutrition are complimentary habits that contribute to better brain health. 

2. Practice focusing

Repetition leads to success, which also applies to strengthening your focus. Methods like the Pomodoro Technique and concentration-based apps are great ways to build self-awareness and discover how you can stay on track.

Learning task management methods (like the Eisenhower Matrix) , adopting work productivity tools, or occasional digital detoxes are more ways to prioritize your focus. Find what works for you and practice until it becomes a habit. This prolonged ability to concentrate will strengthen your overall cognitive abilities.  

3. Reduce your stress

Worry activates your fight or flight response , which can cause mental fatigue and poor sleep. Acute stress or anxiety can often be improved by developing regular self-care practices, such as meditation , yoga, and deep breathing. 

Chronic stress is a more serious mental health risk with serious implications on your short term wellness and long-term cognitive health. Mental health professionals can help you identify the root cause of your stress and provide you with the tools and resources to ease your mind.

4. Train your brain

Your brain is like any other muscle in your body — to keep it in peak condition, you need to work it out. Incorporate some mental activities into your free time , such as reading before bed, playing chess on your lunch break, or following a serial podcast during your daily commute. You ca n also try memory or reasoning games to sharpen your cognitive skills in fun and practical ways. Even two minutes a day dedicated to self-improvement can grow your skills. 

Your brain is working even when you aren’t. But even though many of your cognitive skills are firing off in the background, you can still work to actively sharpen your abilities. 

The next time you’re tackling a new task, pay close attention to your focus. How easily do you succumb to distractions? Do you respond better to visual or auditory learning? Once you understand your strengths and acknowledge your weaknesses, you can incorporate techniques to improve. 

Eventually, you won’t have to focus so much on focusing. And the next time your coworker comes at you with a curveball, you’ll have the resources and know-how to take the change in stride. 

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Make meaningful changes and become the best version of yourself. BetterUp's professional Coaches are here to support your personal growth journey.

Elizabeth Perry, ACC

Elizabeth Perry is a Coach Community Manager at BetterUp. She uses strategic engagement strategies to cultivate a learning community across a global network of Coaches through in-person and virtual experiences, technology-enabled platforms, and strategic coaching industry partnerships. With over 3 years of coaching experience and a certification in transformative leadership and life coaching from Sofia University, Elizabeth leverages transpersonal psychology expertise to help coaches and clients gain awareness of their behavioral and thought patterns, discover their purpose and passions, and elevate their potential. She is a lifelong student of psychology, personal growth, and human potential as well as an ICF-certified ACC transpersonal life and leadership Coach.

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Chapter 7: Thinking and Intelligence

Learning objectives.

• Distinguish between concepts and prototypes
• Explain the difference between natural and artificial concepts
• Describe problem-solving strategies, including algorithms and heuristics
• Explain some common roadblocks to effective problem solving, including choice blindness
• Explain the triarchic theory of intelligence
• Explain the multiple intelligences theory
• Define creativity, divergent, and convergent thinking
• Describe the development of IQ tests, their purposes, and benefits
• Explain the bell curve, and how IQ is measured
• Describe how genetics and the environment affect intelligence
• Describe common learning disabilities

Figure 1 . Thinking is an important part of our human experience, and one that has captivated people for centuries. Today, it is one area of psychological study. The 19th-century Girl with a Book by José Ferraz de Almeida Júnior, the 20th-century sculpture The Thinker by August Rodin, and Shi Ke’s 10th-century painting Huike Thinking all reflect the fascination with the process of human thought. (credit “middle”: modification of work by Jason Rogers; credit “right”: modification of work by Tang Zu-Ming)

Why is it so difficult to break habits—like reaching for your ringing phone even when you shouldn’t, such as when you’re driving? How does a person who has never seen or touched snow in real life develop an understanding of the concept of snow? How do young children acquire the ability to learn language with no formal instruction? Psychologists who study thinking explore questions like these.

Cognitive psychologists also study intelligence. What is intelligence, and how does it vary from person to person? Are “street smarts” a kind of intelligence, and if so, how do they relate to other types of intelligence? What does an IQ test really measure? These questions and more will be explored in this module as you study thinking and intelligence.

As a part of this discussion, we will consider thinking and briefly explore the development and use of language. We will also discuss problem solving and creativity, intelligence testing, and how our biology and environments interact to affect intelligence. After finishing this module, you will have a greater appreciation of the higher-level cognitive processes that contribute to our distinctiveness as a species.

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National Research Council (US) Committee on the Assessment of 21st Century Skills. Assessing 21st Century Skills: Summary of a Workshop. Washington (DC): National Academies Press (US); 2011.

Assessing 21st Century Skills: Summary of a Workshop.

• Hardcopy Version at National Academies Press

2 Assessing Cognitive Skills

As described in Chapter 1 , the steering committee grouped the five skills identified by previous efforts ( National Research Council, 2008 , 2010 ) into the broad clusters of cognitive skills, interpersonal skills, and intrapersonal skills. Based on this grouping, two of the identified skills fell within the cognitive cluster: nonroutine problem solving and systems thinking. The definition of each, as provided in the previous report ( National Research Council, 2010 , p. 3), appears below:

• Nonroutine problem solving: A skilled problem solver uses expert thinking to examine a broad span of information, recognize patterns, and narrow the information to reach a diagnosis of the problem. Moving beyond diagnosis to a solution requires knowledge of how the information is linked conceptually and involves metacognition—the ability to reflect on whether a problem-solving strategy is working and to switch to another strategy if it is not working ( Levy and Murnane, 2004 ). It includes creativity to generate new and innovative solutions, integrating seemingly unrelated information, and entertaining possibilities that others may miss ( Houston, 2007 ).
• Systems thinking: The ability to understand how an entire system works; how an action, change, or malfunction in one part of the system affects the rest of the system; adopting a “big picture” perspective on work ( Houston, 2007 ). It includes judgment and decision making, systems analysis, and systems evaluation as well as abstract reasoning about how the different elements of a work process interact ( Peterson et al., 1999 ).

After considering these definitions, the committee decided a third cognitive skill, critical thinking, was not fully represented. The committee added critical thinking to the list of cognitive skills, since competence in critical thinking is usually judged to be an important component of both skills ( Mayer, 1990 ). Thus, this chapter focuses on assessments of three cognitive skills: problem solving, critical thinking, and systems thinking.

• DEFINING THE CONSTRUCT

One of the first steps in developing an assessment is to define the construct and operationalize it in a way that supports the development of assessment tasks. Defining some of the constructs included within the scope of 21st century skills is significantly more challenging than defining more traditional constructs, such as reading comprehension or mathematics computational skills. One of the challenges is that the definitions tend to be both broad and general. To be useful for test development, the definition needs to be specific so that there can be a shared conception of the construct for use by those writing the assessment questions or preparing the assessment tasks.

This set of skills also generates debate about whether they are domain general or domain specific. A predominant view in the past has been that critical thinking and problem-solving skills are domain general: that is, that they can be learned without reference to any specific domain and, further, once they are learned, can be applied in any domain. More recently, psychologists and learning theorists have argued for a domain-specific conception of these skills, maintaining that when students think critically or solve problems, they do not do it in the absence of subject matter: instead, they think about or solve a problem in relation to some topic. Under a domain-specific conception, the learner may acquire these skills in one domain as he or she acquires expertise in that domain, but acquiring them in one domain does not necessarily mean the learner can apply them in another.

At the workshop, Nathan Kuncel, professor of psychology with University of Minnesota, and Eric Anderman, professor of educational psychology with Ohio State University, discussed these issues. The sections below summarize their presentations and include excerpts from their papers, 1 dealing first with the domain-general and domain-specific conceptions of critical thinking and problem solving and then with the issue of transferring skills from one domain to another.

Critical Thinking: Domain-Specific or Domain-General

It is well established, Kuncel stated, that foundational cognitive skills in math, reading, and writing are of central importance and that students need to be as proficient as possible in these areas. Foundational cognitive abilities, such as verbal comprehension and reasoning, mathematical knowledge and skill, and writing skills, are clearly important for success in learning in college as well as in many aspects of life. A recent study documents this. Kuncel and Hezlett (2007) examined the body of research on the relationships between traditional measures of verbal and quantitative skills and a variety of outcomes. The measures of verbal and quantitative skills included scores on six standardized tests—the GRE, MCAT, LSAT, GMAT, MAT, and PCAT. 2 The outcomes included performance in graduate school settings ranging from Ph.D. programs to law school, medical school, business school, and pharmacy programs. Figure 2-1 shows the correlations between scores on the standardized tests and the various outcome measures, including (from bottom to top) first-year graduate GPA (1st GGPA), cumulative graduate GPA (GGPA), qualifying or comprehensive examination scores, completion of the degree, estimate of research productivity, research citation counts, faculty ratings, and performance on the licensing exam for the profession. For instance, the top bar shows a correlation between performance on the MCAT and performance on the licensing exam for physicians of roughly .65, the highest of the correlations reported in this figure. The next bar indicates the correlation between performance on the LSAT and performance on the licensing exam for lawyers is roughly .35. Of the 34 correlations shown in the figure, all but 11 are over .30. Kuncel characterized this information as demonstrating that verbal and quantitative skills are important predictors of success based on a variety of outcome measures, including performance on standardized tests, whether or not people finish their degree program, how their performance is evaluated by faculty, and their contribution to the field.

Correlations between scores on standardized tests and academic and job outcome measures. SOURCE: Kuncel and Hezlett (2007). Reprinted with permission of American Association for the Advancement of Science.

Kuncel has also studied the role that broader abilities have in predicting future outcomes. A more recent review ( Kuncel and Hezlett, 2010 ) examined the body of research on the relationships between measures of general cognitive ability (historically referred to as IQ) and job outcomes, including performance in high, medium, and low complexity jobs; training success in civilian and military settings; how well leaders perform on objective measures; and evaluations of the creativity of people’s work. Figure 2-2 shows the correlations between performance on a measure of general cognitive ability and these outcomes. All of the correlations are above .30, which Kuncel characterized as demonstrating a strong relationship between general cognitive ability and job performance across a variety of performance measures. Together, Kuncel said, these two reviews present a body of evidence documenting that verbal and quantitative skills along with general cognitive ability are predictive of college and career performance.

Correlations between measures of cognitive ability and job performance. SOURCE: Kuncel and Hezlett (2011). Copyright 2010 by Sage Publications. Reprinted with permission of Sage Publications.

Kuncel noted that other broader skills, such as critical thinking or analytical reasoning, may also be important predictors of performance, but he characterizes this evidence as inconclusive. In his view, the problems lie both with the conceptualization of the constructs as domain-general (as opposed to domain-specific) as well as with the specific definition of the construct. He finds the constructs are not well defined and have not been properly validated. For instance, a domain-general concept of the construct of “critical thinking” is often indistinguishable from general cognitive ability or general reasoning and learning skills. To demonstrate, Kuncel presented three definitions of critical thinking that commonly appear in the literature:

• “[Critical thinking involves] cognitive skills or strategies that increase the probability of a desirable outcome—in the long run, critical thinkers will have more desirable outcomes than ‘noncritical’ thinkers. . . . Critical thinking is purposeful, reasoned, and goal-directed. It is the kind of thinking involved in solving problems, formulating inferences, calculating likelihoods, and making decisions” ( Halpern, 1998 , pp. 450–451).
• “Critical thinking is reflective and reasonable thinking that is focused on deciding what to believe or do” ( Ennis, 1985 , p. 45).
• “Critical thinking [is] the ability and willingness to test the validity of propositions” ( Bangert-Drowns and Bankert, 1990 , p. 3).

He characterizied these definitions both very general and very broad. For instance, Halpern’s definition essentially encompasses all of problem solving, judgment, and cognition, he said. Others are more specific and focus on a particular class of tasks (e.g., Bangert-Drowns and Bankert, 1990 ). He questioned the extent to which critical thinking so conceived is distinct from general cognitive ability (or general intelligence).

Kuncel conducted a review of the literature for empirical evidence of the validity of the construct of critical thinking. The studies in the review examined the relationships between various measures of critical thinking and measures of general intelligence and expert performance. He looked for two types of evidence—convergent validity evidence 3 and discriminant validity 4 evidence.

Kuncel found several analyses of the relationships among different measures of critical thinking (see Bondy et al., 2001 ; Facione, 1990 ; and Watson and Glaser, 1994 ). The assessments that were studied included the Watson-Glaser Critical Thinking Appraisal (WGCTA), the Cornell Critical Thinking Test (CCTT), the California Critical Thinking Skills Test (CCTST), and the California Critical Thinking Disposition Inventory (CCTDI). The average correlation among the measures was .41. Considering that all of these tests purport to be measures of the same construct, Kuncel judged this correlation to be low. For comparison, he noted a correlation of .71 between two subtests of the SAT intended to measure critical thinking (the SAT-critical reading test and the SAT-writing test).

With regard to discriminant validity, Kuncel conducted a literature search that yielded 19 correlations between critical-thinking skills and traditional measures of cognitive abilities, such as the Miller Analogies Test and the SAT ( Adams et al., 1999 ; Bauer and Liang, 2003 ; Bondy et al., 2001 ; Cano and Martinez, 1991 ; Edwards, 1950 ; Facione et al., 1995 , 1998 ; Spector et al., 2000 ; Watson and Glaser, 1994 ). He separated the studies into those that measured critical-thinking skills and those that measured critical-thinking dispositions (i.e., interest and willingness to use one’s critical-thinking skills). The average correlation between general cognitive ability measures and critical-thinking skills was .48, and the average correlation between general cognitive ability measures and critical-thinking dispositions was .21.

Kuncel summarized these results as demonstrating that different measures of critical thinking show lower correlations with each other (i.e., average of .41) than they do with traditional measures of general cognitive ability (i.e., average of .48). Kuncel judges that these findings provide little support for critical thinking as a domain-general construct distinct from general cognitive ability. Given this relatively weak evidence of convergent and discriminant validity, Kuncel argued, it is important to determine if critical thinking is correlated differently than cognitive ability with important outcome variables like grades or job performance. That is, do measures of critical-thinking skills show incremental validity beyond the information provided by measures of general cognitive ability?

Kuncel looked at two outcome measures: grades in higher education and job performance. With regard to higher education, he examined data from 12 independent samples with 2,876 subjects ( Behrens, 1996 ; Gadzella et al., 2002 , 2004 ; Kowalski and Taylor, 2004 ; Taube, 1997 ; Williams, 2003 ). Across these studies, the average correlation between critical-thinking skills and grades was .27 and between critical-thinking dispositions and grades was .24. To put these correlations in context, the SAT has an average correlation with 1st year college GPA between .26 to .33 for the individual scales and .35 when the SAT scales are combined ( Kobrin et al., 2008 ). 5

There are very limited data that quantify the relationship between critical-thinking measures and subsequent job performance. Kuncel located three studies with the Watson-Glaser Appraisal ( Facione and Facione, 1996 , 1997 ; Giancarlo, 1996 ). They yielded an average correlation of .32 with supervisory ratings of job performance (N = 293).

Kuncel described these results as “mixed” but not supporting a conclusion that assessments of critical thinking are better predictors of college and job performance than other available measures. Taken together with the convergent and discriminant validity results, the evidence to support critical thinking as an independent construct distinct from general cognitive ability is weak.

Kuncel believes these correlational results do not tell the whole story, however. First, he noted, a number of artifactual issues may have contributed to the relatively low correlation among different assessments of critical thinking, such as low reliability of the measures themselves, restriction in range, different underlying definitions of critical thinking, overly broad definitions that are operationalized in different ways, different kinds of assessment tasks, and different levels of motivation in test takers.

Second, he pointed out, even though two tests correlate highly with each other, they may not measure the same thing. That is, although the critical-thinking tests correlate .48, on average, with cognitive ability measures, it does not mean that they measure the same thing. For example, a recent study ( Kuncel and Grossbach, 2007 ) showed that ACT and SAT scores are highly predictive of nursing knowledge. But, obviously, individuals who score highly on a college admissions test do not have all the knowledge needed to be a nurse. The constructs may be related but not overlap entirely.

Kuncel explained that one issue with these studies is they all conceived of critical thinking in its broadest sense and as a domain-general construct. He said this conception is not useful, and he summarized his meta-analysis findings as demonstrating little evidence that critical thinking exists as a domain-general construct distinct from general cognitive ability. He highlighted the fact that some may view critical thinking as a specific skill that, once learned, can be applied in many situations. For instance, many in his field of psychology mention the following as specific critical-thinking skills that students should acquire: understanding the law of large numbers, understanding what it means to affirm the consequent, being able to make judgments about sample bias, understanding control groups, and understanding Type I versus Type II errors. However, Kuncel said many tasks that require critical thinking would not make use of any of these skills.

In his view, the stronger argument is for critical thinking as a domain-specific construct that evolves as the person acquires domain-specific knowledge. For example, imagine teaching general critical-thinking skills that can be applied across all reasoning situations to students. Is it reasonable, he asked, to think a person can think critically about arguments for different national economic policies without understanding macro-economics or even the current economic state of the country? At one extreme, he argued, it seems clear that people cannot think critically about topics for which they have no knowledge, and their reasoning skills are intimately tied to the knowledge domain. For instance, most people have no basis for making judgments about how to conduct or even prioritize different experiments for CERN’s Large Hadron Collider. Few people understand the topic of particle physics sufficiently to make more than trivial arguments or decisions. On the other hand, perhaps most people could try to make a good decision about which among a few medical treatments would best meet their needs.

Kuncel also talked about the kinds of statistical and methodological reasoning skills learned in different disciplines. For instance, chemists, engineers, and physical scientists learn to use these types of skills in thinking about the laws of thermodynamics that deal with equilibrium, temperature, work, energy, and entropy. On the other hand, psychologists learn to use these skills in thinking about topics such as sample bias and self-selection in evaluating research findings. Psychologists who are adept at thinking critically in their own discipline would have difficulty thinking critically about problems in the hard sciences, unless they have specific subject matter knowledge in the discipline. Likewise, it is difficult to imagine that a scientist highly trained in chemistry could solve a complex problem in psychology without knowing some subject matter in psychology.

Kuncel said it is possible to train specific skills that aid in making good judgments in some situations, but the literature does not demonstrate that it is possible to train universally effective critical thinking skills. He noted, “I think you can give people a nice toolbox with all sorts of tools they can apply to a variety of tasks, problems, issues, decisions, citizenship questions, and learning those things will be very valuable, but I dissent on them being global and trainable as a global skill.”

Transfer from One Context to Another

There is a commonplace assumption, Eric Anderman noted in his presentation, that learners readily transfer the skills they have learned in one course or context to situations and problems that arise in another. Anderman argued research on human learning does not support this assumption. Research suggests such transfer seldom occurs naturally, particularly when learners need to transfer complex cognitive strategies from one domain to another ( Salomon and Perkins, 1989 ). Transfer is only likely to occur when care is taken to facilitate that transfer: that is, when students are specifically taught strategies that facilitate the transfer of skills learned in one domain to another domain ( Gick and Holyoak, 1983 ).

For example, Anderman explained, students in a mathematics class might be taught how to solve a problem involving the multiplication of percentages (e.g., 4.79% × 0.25%). The students then might encounter a problem in their social studies courses that involves calculating compounded interest (such as to solve a problem related to economics or banking). Although the same basic process of multiplying percentages might be necessary to solve both problems, it is unlikely that students will naturally, on their own, transfer the skills learned in the math class to the problem encountered in the social studies class.

In the past, Anderman said, there had been some notion that critical-thinking and problem-solving skills could be taught independent of context. For example, teaching students a complex language such as Latin, a computer programming language such as LOGO, or other topics that require complex thinking might result in an overall increase in their ability to think critically and problem solve.

Both Kuncel and Anderman maintained that the research does not support this idea. Instead, the literature better supports a narrower definition in which critical thinking is considered a finite set of specific skills. These skills are useful for effective decision making for many, but by no means all, tasks or situations. Their utility is further curtailed by task-specific knowledge demands. That is, a decision maker often has to have specific knowledge to make more than trivial progress with a problem or decision.

Anderman highlighted four important messages emerging from recent research. First, research documents that it is critical that students learn basic skills (such as basic arithmetic skills like times tables) so the skills become automatic. Mastery of these skills is required for the successful learning of more complex cognitive skills. Second, the use of general practices intended to improve students’ thinking are not usually successful as a means of improving their overall cognitive abilities. The research suggests students may become more adept in the specific skill taught, but this does not transfer to an overall increase in cognitive ability. Third, when general problem-solving strategies are taught, they should be taught within meaningful contexts and not as simply rote algorithms to be memorized. Finally, educators need to actively teach students to transfer skills from one context to another by helping students to recognize that the solution to one type of problem may be useful in solving a problem with similar structural features ( Mayer and Wittrock, 1996 ).

He noted that instructing students in general problem-solving skills can be useful but more elaborate scaffolding and domain-specific applications of these skills are often necessary. Whereas general problem-solving and critical-thinking strategies can be taught, research indicates these skills will not automatically or naturally transfer to other domains. Anderman stressed that educators and trainers must recognize that 21st century skills should be taught within specific domains; if they are taught as general skills, he cautioned, then extreme care must be taken to facilitate the transfer of these skills from one domain to another.

• ASSESSMENT EXAMPLES

The workshop included examples of four different types of assessments of critical-thinking and problem-solving skills—one that will be used to make international comparisons of achievement, one used to license lawyers, and two used for formative purposes (i.e., intended to support instructional decision making). The first example was the computerized problem-solving component of the Programme for International Student Assessment (PISA). This assessment is still under development but is scheduled for operational administration in 2012. 6 Joachim Funke, professor of cognitive, experimental, and theoretical psychology with the Heidelberg University in Germany, discussed this assessment.

The second example was the Multistate Bar Exam, a paper-and-pencil test that consists of both multiple-choice and extended-response components. This test is used to qualify law students for practice in the legal profession. Susan Case, director of testing with the National Conference of Bar Exams, made this presentation.

The two formative assessments both make use of intelligent tutors, with assessments embedded into instruction modules. The “Auto Tutor” described by Art Graesser, professor of psychology with the University of Memphis, is used in instructing high school and higher education students in critical thinking skills in science. The Auto Tutor is part of a system Graesser has developed called Operation ARIES! (Acquiring Research Investigative and Evaluative Skills). The “Packet Tracer,” described by John Beherns, director of networking academy learning systems development with Cisco, is intended for individuals learning computer networking skills.

Problem Solving on PISA

For the workshop, Joachim Funke supplied the committee with the draft framework for PISA (see Organisation for Economic Co-operation and Development, 2010 7 ) and summarized this information in his presentation. 8 The summary below is based on both documents.

PISA, Funke explained, defines problem solving as an individual’s capacity to engage in cognitive processing to understand and resolve problem situations where a solution is not immediately obvious. The definition includes the willingness to engage with such situations in order to achieve one’s potential as a constructive and reflective citizen ( Organisation for Co-operation and Development, 2010 , p. 12). Further, the PISA 2012 assessment of problem-solving competency will not test simple reproduction of domain-based knowledge, but will focus on the cognitive skills required to solve unfamiliar problems encountered in life and lying outside traditional curricular domains. While prior knowledge is important in solving problems, problem-solving competency involves the ability to acquire and use new knowledge or to use old knowledge in a new way to solve novel problems. The assessment is concerned with nonroutine problems, rather than routine ones (i.e., problems for which a previously learned solution procedure is clearly applicable). The problem solver must actively explore and understand the problem and either devise a new strategy or apply a strategy learned in a different context to work toward a solution. Assessment tasks center on everyday situations, with a wide range of contexts employed as a means of controlling for prior knowledge in general.

The key domain elements for PISA 2012 are as follows:

• The problem context: whether it involves a technological device or not, and whether the focus of the problem is personal or social
• The nature of the problem situation: whether it is interactive or static (defined below)
• The problem-solving processes: the cognitive processes involved in solving the problem

The PISA 2012 framework (pp. 18–19) defines four processes that are components of problem solving. The first involves information retrieval. This process requires the test taker to quickly explore a given system to find out how the relevant variables are related to each other. The test taker must explore the situation, interact with it, consider the limitations or obstacles, and demonstrate an understanding of the given information. The objective is for the test taker to develop a mental representation of each piece of information presented in the problem. In the PISA framework, this process is referred to as exploring and understanding.

The second process is model building, which requires the test taker to make connections between the given variables. To accomplish this, the examinee must sift through the information, select the information that is relevant, mentally organize it, and integrate it with relevant prior knowledge. This requires the test taker to represent the problem in some way and formulate hypotheses about the relevant factors and their interrelationships. In the PISA framework, this dimension is called representing and formulating.

The third process is called forecasting and requires the active control of a given system. The framework defines this process as setting goals, devising a strategy to carry them out, and executing the plan. In the PISA framework, this dimension is called planning and executing.

The fourth process is monitoring and reflecting. The framework defines this process as checking the goal at each stage, detecting unexpected events, taking remedial action if necessary, and reflecting on solutions from different perspectives by critically evaluating assumptions and alternative solutions.

Each of these processes requires the use of reasoning skills, which the framework describes as follows ( Organisation for Economic Co-operation and Development, 2010 , p. 19):

In understanding a problem situation, the problem solver may need to distinguish between facts and opinion, in formulating a solution, the problem solver may need to identify relationship between variables, in selecting a strategy, the problem solver may need to consider cause and effect, and in communicating the results, the problem solver may need to organize information in a logical manner. The reasoning skills associated with these processes are embedded within problem solving. They are important in the PISA context since they can be taught and modeled in classroom instruction (e.g., Adey et al., 2007 ; Klauer and Phye, 2008 ).

For any given test taker, the test lasts for 40 minutes. PISA is a survey-based assessment that uses a balanced rotation design. A total of 80 minutes of material is organized into four 20-minute clusters, with each student taking two clusters.

The items are grouped into units around a common stimulus that describes the problem. Reading and numeracy demands are kept to a minimum. The tasks all consist of authentic stimulus items, such as refueling a moped, playing on a handball team, mixing a perfume, feeding cats, mixing elements in a chemistry lab, taking care of a pet, and so on. Funke noted that the different contexts for the stimuli are important because test takers might be motivated differentially and might be differentially interested depending on the context. The difficulty of the items is manipulated by increasing the number of variables or the number of relations that the test taker has to deal with.

PISA 2012 is a computer-based test in which items are presented by computer and test takers respond on the computer. Approximately three-quarters of the items are in a format that the computer can score (simple or complex multiple-choice items). The remaining items are constructed-response, and test takers enter their responses into text boxes.

Scoring of the items is based on the processes that the test taker uses to solve the problem and involves awarding points for the use of certain processes. For information retrieval, the focus is on identifying the need to collect baseline data (referred to in PISA terminology as identifying the “zero round”) and the method of manipulating one variable at a time (referred to in PISA terminology as “varying one thing at a time” or VOTAT). Full credit is awarded if the subject uses VOTAT strategy and makes use of zero rounds. Partial credit is given if the subject uses VOTAT but does not make use of zero rounds.

For model building, full credit is awarded if the generated model is correct. If one or two errors are present in the model, partial credit is given. If more than two errors are present, then no credit is awarded.

For forecasting, full credit is given if the target goals are reached. Partial credit is given if some progress toward the target goals can be registered, and no credit is given if there is no progress toward target goals at all.

PISA items are classified as static versus interactive. In static problems, all the information the test taker needs to solve the problem is presented at the outset. In contrast, interactive problems require the test taker to explore the problem to uncover important relevant information ( Organisation for Economic Co-operation and Development, 2010 , p. 15). Two sample PISA items appear in Box 2-1 .

Sample Problem-Solving Items for PISA 2012. Digital Watch–interactive: A simulation of a digital watch is presented. The watch is controlled by four buttons, the functions of which are unknown to the student at the outset of the problems. The (more...)

Funke and his colleagues have conducted analyses to evaluate the construct validity of the assessment. They have examined the internal structure of the assessment using structural equation modeling, which evaluates the extent to which the items measure the dimensions they are intended to measure. The results indicate the three dimensions are correlated with each other. Model Building and Forecasting correlate at .77; Forecasting and Information Retrieval correlate at .71; and Information Retrieval and Model Building correlate at .75. Funke said that the results also document that the items “load on” the three dimensions in the way the test developers hypothesized. He indicated some misfit related to the items that measure Forecasting, and he attributes this to the fact that the Forecasting items have a skewed distribution. However, the fit of the model does not change when these items are removed.

Funke reported results from studies of the relationship between test performance and other variables, including school achievement and two measures of problem solving on the PISA German National Extension on Complex Problem Solving. The latter assessment, called HEIFI, measures knowledge about a system and the control of the system separately. Scores on the PISA Model Building dimension are statistically significant (p < .05) related to school achievement (r = .64) and to scores on the HEIFI knowledge component (r = .48). Forecasting is statistically significant (p < .05) related to both of the HEIFI scores (r = .48 for HEIFI knowledge and r = .36 for HEIFI control). Information Retrieval is statistically significant (p < .05) related to HEIFI control (r = .38). The studies also show that HEIFI scores are not related to school achievement.

Funke closed by discussing the costs associated with the assessment. He noted it is not easy to specify the costs because in a German university setting, many costs are absorbed by the department and its equipment. Funke estimates that development costs run about $13 per unit, 9 plus $6.5 for the Cognitive Labs used to pilot test and refine the items. 10 The license for the Computer Based Assessment (CBA) Item-builder and the execution environment is given for free for scientific use from DIPF 11 Frankfurt.

The Bar Examination for Lawyers 12

The Bar examination is administered by each jurisdiction in the United States as one step in the process to license lawyers. The National Council of Bar Examiners (NCBE) develops a series of three exams for use by the jurisdictions. Jurisdictions may use any or all of these three exams or may administer locally developed exam components if they wish. The three major components developed by the NCBE include the Multi-state Bar Exam (MBE), the Multi-state Essay Exam (MEE), and the Multi-state Performance Test (MPT). All are paper-and-pencil tests. Examinees pay to take the test, and the costs are $54 for the MBE, $20 for the MEE, and $20 for the MPT.

Susan Case, who has spent her career working on licensing exams—first the medical licensing exam for physicians and then the bar exam for lawyers—noted the Bar examination is like other tests used to award professional licensure. The focus of the test is on the extent to which the test taker has the knowledge and skills necessary to be licensed in the profession on the day of the test. The test is intended to ensure the newly licensed professional knows what he/she needs to know to practice law. The test is not designed to measure the curriculum taught in law schools, but what licensed professionals need to know. When they receive the credential, lawyers are licensed to practice in all fields of law. This is analogous to medical licensing in which the licensed professional is eligible to practice any kind of medicine.

The Bar exam includes both multiple-choice and constructed-response components. Both require examinees to be able to gather and synthesize information and apply their knowledge to the given situation. The questions generally follow a vignette that describes a case or problem and asks the examinee to determine the issues to resolve before advising the client or to determine other information needed in order to proceed. For instance, what questions should be asked next? What is the best strategy to implement? What is the best defense? What is the biggest obstacle to relief? The questions may require the examinee to synthesize the law and the facts to predict outcomes. For instance, is the ordinance constitutional? Should a conviction be overturned?

The purpose of the MBE is to assess the extent to which an examinee can apply fundamental legal principles and legal reasoning to analyze a given pattern of facts. The questions focus on the understanding of legal principles rather than memorization of local case or statutory law. The MBE consists of 60 multiple-choice questions and lasts a full day.

A sample question follows:

A woman was told by her neighbor that he planned to build a new fence on his land near the property line between their properties. The woman said that, although she had little money, she would contribute something toward the cost. The neighbor spent $2,000 in materials and a day of his time to construct the fence. The neighbor now wants her to pay half the cost of the materials. Is she liable for this amount?

The purpose of the MEE is to assess the examinee’s ability to (1) identify legal issues raised by a hypothetical factual situation; (2) separate material that is relevant from that which is not; (3) present a reasoned analysis of the relevant issues in a clear, concise, and well-organized composition; and (4) demonstrate an understanding of the fundamental legal principles relevant to the probable resolution of the issues raised by the factual situation.

The MEE lasts for 6 hours and consists of nine 30-minute questions. An excerpt from a sample question follows:

The CEO/chairman of the 12-member board of directors (the Board) of a company plus three other members of the Board are senior officers of the company. The remaining eight members of the Board are wholly independent directors. Recently, the Board decided to hire a consulting firm to market a new product . . . The CEO disclosed to the Board that he had a 25% partnership interest in the consulting firm. The CEO stated that he would not be involved in any work to be performed by the consulting firm. He knew but did not disclose to the Board that the consulting firm’s proposed fee for this consulting assignment was substantially higher than it normally charged for comparable work . . . The Board discussed the relative merits of the two proposals for 10 minutes. The Board then voted unanimously (CEO abstaining) to hire the consulting firm . . . Did the CEO violate his duty of loyalty to his company? Explain. Assuming the CEO breached his duty of loyalty to his company, does he have any defense to liability? Explain. Did the other directors violate their duty of care? Explain.

The purpose of the MPT is to assess fundamental lawyering skills in realistic situations by asking the candidate to complete a task that a beginning lawyer should be able to accomplish. The MPT requires applicants to sort detailed factual materials; separate relevant from irrelevant facts; analyze statutory, case, and administrative materials for relevant principles of law; apply relevant law to the facts in a manner likely to resolve a client’s problem; identify and resolve ethical dilemmas; communicate effectively in writing; and complete a lawyering task within time constraints.

Each task is completely self-contained and includes a file, a library, and a task to complete. The task might deal with a car accident, for example, and therefore might include a file with pictures of the accident scene and depositions from the various witnesses, as well as a library with relevant case law. Examinees are given 90 minutes to complete each task.

For example, in a case involving a slip and fall in a store, the task might be to prepare an initial draft of an early dispute resolution for a judge. The draft should candidly discuss the strengths and weaknesses of the client’s case. The file would contain the instructional memo from the supervising attorney, the local rule, the complaint, an investigator’s report, and excerpts of the depositions of the plaintiff and a store employee. The library would include a jury instruction concerning the premises liability with commentary on contributory negligence.

The MBE is a multiple-choice test and thus scored by machine. However, the other two components require human scoring. The NCBE produces the questions and the grading guidelines for the MEE and MPT, but the essays and performance tests are scored by the jurisdictions themselves. The scorers are typically lawyers who are trained during grading seminars held at the NCBE offices, after the exam is administered. At this time, they review sample papers and receive training on how to apply the scoring guidelines in a consistent fashion.

Each component of the Bar examination (MBE, MEE, MPT) is intended to assess different skills. The MBE focuses on breadth of knowledge, the MEE focuses on depth of knowledge, and the MPT focuses on the ability to demonstrate practical skills. Together, the three formats cover the different types of tasks that a new lawyer needs to do.

Determinations about weighting the three components are left to the jurisdictions; however, the NCBE urges them to weight the MBE score by 50 percent and the MEE and MPT by 25 percent each. The recommendation is an attempt to balance a number of concerns, including authenticity, psychometric considerations, logistical issues, and economic concerns. The recommendation is to award the highest weight to the MBE because it is the most psychometrically sound. The reliability of scores on the MBE is generally over .90, much higher than scores on the other portions, and the MBE is scaled and equated across time. The recommended weighting helps to ensure high decision consistency and comparability of pass/fail decisions across administrations.

Currently the MBE is used by all but three jurisdictions (Louisiana, Washington, and Puerto Rico). The essay exam is used by 27 jurisdictions, and the performance test is used by 34 jurisdictions.

Test Development

Standing test development committees that include practicing lawyers, judges, and lawyers on staff with law schools write the test questions. The questions are reviewed by outside experts, pretested on appropriate populations, analyzed and revised, and professionally edited before operational use. Case said the test development procedures for the Bar exam are analogous to those used for the medical licensure exams.

Operation ARIES! (Acquiring Research Investigative and Evaluative Skills)

The summary below is based on materials provided by Art Graesser, including his presentation 13 and two background papers he supplied to the committee ( Graesser et al., 2010 ; Millis et al., in press ).

Operation ARIES! is a tutorial system with a formative assessment component intended for high school and higher education students, Graesser explained. It is designed to teach and assess critical thinking about science. The program operates in a game environment intended to be engaging to students. The system includes an “Auto Tutor,” which makes use of animated characters that converse with students. The Auto Tutor is able to hold conversations with students in natural language, interpret the student’s response, and respond in a way that is adaptive to the student’s response. The designers have created a science fiction setting in which the game and exercises operate. In the game, alien creatures called “Fuaths” are disguised as humans. The Fuaths disseminate bad science through various media outlets in an attempt to confuse humans about the appropriate use of the scientific method. The goal for the student is to become a “special agent of the Federal Bureau of Science (FBS), an agency with a mission to identify the Fuaths and save the planet” ( Graesser et al., 2010 , p. 328).

The system addresses scientific inquiry skills, developing research ideas, independent and dependent variables, experimental control, the sample, experimenter bias, and relation of data to theory. The focus is on use of these skills in the domains of biology, chemistry, and psychology. The system helps students to learn to evaluate evidence intended to support claims. Some examples of the kinds of research questions/claims that are evaluated include the following:

From Biology

• Do chemical and organic pesticides have different effects on food quality?
• Does milk consumption increase bone density?

From Chemistry

• Does a new product for winter roads prevent water from freezing?
• Does eating fish increase blood mercury levels?

From Psychology

• Does using cell phones hurt driving?
• Is a new cure for autism effective?

The system includes items in real-life formats, such as articles, advertisements, blogs, and letters to the editor, and makes use of different types of media where it is common to see faulty claims.

Through the system, the student encounters a story told by video, combined with communications received by e-mail, text message, and updates. The student is engaged through the Auto Tutor, which involves a “tutor agent” that serves as a narrator, and a “student agent” that serves in different roles, depending on the skill level of the student.

The system makes use of three kinds of modules—interactive training, case studies, and interrogations. The interactive training exchanges begin with the student reading an e-book, which provides the requisite information used in later modules. After each chapter, the student responds to a set of multiple-choice questions intended to assess the targeted skills. The text is interactive in that it involves “trialogs” (three-way conversations) between the primary agent, the student agent, and the actual (human) student. It is adaptive in that the strategy used is geared to the student’s performance. If the student is doing poorly, the two auto-tutor agents carry on a conversation that promotes vicarious learning: that is, the tutor agent and the student agent interact with each other, and the human student observes. If the student is performing at an intermediate level, normal tutoring occurs in which the student carries on a conversational exchange with the tutor agent. If the student is doing very well, he or she may be asked to teach the student agent, under the notion that the act of teaching can help to perfect one’s skills.

In the case study modules, the student is expected to apply what he or she has learned. The case study modules involve some type of flawed science, and the student is to identify the flaws by applying information learned from the interactive text in the first module. The student responds by verbally articulating the flaws, and the system makes use of advances in computational linguistics to analyze the meaning of the response. The researchers adopted the case study approach because it “allows learners to encode and discover the rich source of constraints and interdependencies underlying the target elements (flaws) within the cases. [Prior] cases provide a knowledge base for assessing new cases and help guide reasoning, problem solving, interpretation and other cognitive processes” ( Millis et al., in press , p. 17).

In the interrogation modules, insufficient information is provided, so students must ask questions. Research is presented in an abbreviated fashion, such as through headlines, advertisements, or abstracts. The student is expected to identify the relevant questions to ask and to learn to discriminate good research from flawed research. The storyline is advanced by e-mails, dialogues, and videos that are interspersed among the learning activities.

Through the three kinds of modules, the system interweaves a variety of key principles of learning that Graesser said have been shown to increase learning. These include

• Self-explanation (where the learner explains the material to another student, such as the automated student)
• Immediate feedback (through the tutoring system)
• Multimedia effects (which tend to engage the student)
• Active learning (in which students actually participate in solving a problem)
• Dialog interactivity (in which students learn by engaging in conversations and tutorial dialogs)
• Multiple, real-life examples (intended to help students transfer what they learn in one context to another context and to real world situations)

Graesser closed by saying that he and his colleagues are beginning to collect data from evaluation studies to examine the effects of the Auto Tutor. Research has focused on estimating changes in achievement before and after use of the system, and, to date, the results are promising.

Packet Tracer

The summary below is based on materials provided by John Behrens, including his presentation 14 and a background paper he forwarded in preparation for the workshop ( Behrens et al., in press ).

To help countries around the world train their populations in networking skills, Cisco created the Networking Academy. The academy is a public/private partnership through which Cisco provides free online curricula and assessments. Behrens pointed out that in order to become adept with networking, students need both a conceptual understanding of networking and the skills to apply this knowledge to real situations. Thus, hands-on practice and assessment on real equipment are important components of the academy’s instructional program. Cisco also wants to provide students with time for out-of-class practice and opportunities to explore on their own using online equipment that is not typically available in the average classroom setting. In the Networking Academy, students work with an online instructor, and they proceed through an established curriculum that incorporates numerous interactive activities.

Behrens talked specifically about a new program Cisco has developed called “Packet Tracer,” a computer package that uses simulations to provide instruction and includes an interactive and adaptable assessment component. Cisco has incorporated Packet Tracer activities into the curricula for training networking professionals. Through this program, instructors and students can construct their own activities, and students can explore problems on their own. In Cisco’s Networking Academy, assessments can be student-initiated or instructor-initiated. Student-initiated assessments are primarily embedded in the curriculum and include quizzes, interactive activities, and “challenge labs,” which are a feature of Packet Tracer. The student-initiated assessments are designed to provide feedback to the student to help his or her learning. They use a variety of technologies ranging from multiple-choice questions (in the quizzes) to complex simulations (in the challenge labs). Before the development of Packet Tracer, the instructor-initiated assessments consisted either of hands-on exams with real networking equipment or multiple-choice exams in the online assessment system. Packet Tracer provides more simulation-based options, and also includes detailed reporting and grade-book integration features.

Each assessment consists of one extensive network configuration or troubleshooting activity that may require up to 90 minutes to complete. Access to the assessment is associated with a particular curricular unit, and it may be re-accessed repeatedly based on instructor authorization. The system provides simulations of a broad range of networking devices and networking protocols, including features set around the Cisco IOS (Internet Operating System). Instructions for tasks can be presented through HTML-formatted text boxes that can be preauthored, stored, and made accessible by the instructor at the appropriate time.

Behrens presented an example of a simulated networking problem in which the student needs to obtain the appropriate cable. To complete this task, the student must determine what kind of cable is needed, where on the computer to plug it in, and how to connect it. The student’s performance is scored, and his or her interactions with the problem are tracked in a log. The goal is not to simply assign a score to the student’s performance but to provide detailed feedback to enhance learning and to correct any misinterpretations. The instructors can receive and view the log in order to evaluate how well the student understands the tasks and what needs to be done.

Packet Tracer can simulate a broad range of devices and networking protocols, including a wide range of PC facilities covering communication cards, power functionality, web browsers, and operating system configurations. The particular devices, configurations, and problem states are determined by the author of the task (e.g., the instructor) in order to address whatever proficiencies the chapter, course, or instruction targets. When icons of the devices are touched in the simulator, more detailed pictures are presented with which the student can interact. The task author can program scoring rules into the system. Students can be observed trying and discarding potential solutions based on feedback from the game resulting in new understandings. The game encourages students to engage in problem-solving steps (such as problem identification, solution generation, and solution testing). Common incorrect strategies can be seen across recordings.

For Kuncel’s presentation, see http://www7 ​.national-academies ​.org/bota/21st ​_Century_Workshop_Kuncel.pdf . For Kuncel’s paper, see http://www7 ​.national-academies ​.org/bota/21st ​_Century_Workshop_Kuncel_Paper.pdf . For Anderman’s presentation, see http://www7 ​.national-academies ​.org/bota/21st ​_Century_Workshop_Anderman.pdf . For Anderman’s paper, see http: ​//nrc51/xpedio/groups ​/dbasse/documents ​/webpage/060387~1.pdf [August 2011].

Respectively, the Graduate Record Exam, Medical College Admission Test, Law School Admission Test, Graduate Management Admission Test, Miller Analogies Test, and Pharmacy College Admission Test.

Convergent validilty indicates the degree to which an operationalized construct is similar to other operationalized constructs that it theoretically should also be similar to. For instance, to show the convergent validity of a test of critical thinking, the scores on the test can be correlated with scores on other tests that are also designed to measure critical thinking. High correlations between the test scores would be evidence of convergent validity.

Discriminant validity evaluates the extent to which a measure of an operationalized construct differs from measures of other operationalized constructs that it should differ from. In the present context, the interest is in verifying that critical thinking is a construct distinct from general intelligence and expert performance. Thus, discriminant validity would be examined by evaluating the patterns of correlations between and among scores on tests of critical thinking and scores on tests of the other two constructs (general intelligence and expert performance).

It is important to note that when corrected for restriction in range, these coefficients increase to .47 to .51 for individual scores and .51 for the combined score.

For a full description of the PISA program, see http://www ​.oecd.org/pages ​/0,3417,en_32252351 ​_32235731_1_1_1_1_1,00.html [August 2011].

Available at http://www ​.oecd.org/dataoecd ​/8/42/46962005.pdf [August 2011].

Available at http://www7 ​.national-academies ​.org/bota/21st ​_Century_Workshop_Funke.pdf [August 2011].

A unit consists of stimulus materials, instructions, and the associated questions.

Costs are in American dollars.

DIPF stands for the Deutsches Institut für Internationale Pädagogische Forschung, which translates to the German Institute for Educational Research and Educational Information.

The summary is based on a presentation by Susan Case, see http://www7 ​.nationalacademies ​.org/bota/21st ​_Century_Workshop_Case.pdf [August 2011].

For Graesser’s presentation, see http: ​//nrc51/xpedio/groups ​/dbasse/documents ​/webpage/060267~1.pdf [August 2011].

For Behrens’ presentation, see http://www7 ​.national-academies ​.org/bota/21st ​_Century_Workshop_Behrens.pdf [August 2011].

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Levels of problem-solving competency identified through Bebras Computing Challenge

• Published: 23 April 2021
• Volume 26 , pages 5477–5498, ( 2021 )

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• Kyungbin Kwon   ORCID: orcid.org/0000-0001-8646-0144 1 ,
• Jongpil Cheon 2 &
• Hyunchang Moon 3  

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As computational thinking (CT) gains more attention in K-16 education, problem-solving has been more emphasized as a core competency that can be found across various domains. To develop an evaluation framework that reveals students’ problem-solving competency, this study examined solutions for the Bebras Computing Challenge which requires students to utilize problem-solving skills in a CT domain. A total of 246 solutions of three Bebras tasks were analyzed based on a qualitative content analysis method and four levels of solutions were identified. The solution levels revealed how students (1) failed to understand a problem (No solution), (2) solved the problem but failed to identify the pattern (Premature level), (3) identified principles embedded in the problem but failed to apply them to devise an automized solution (Intermediate level), and (4) identified principles and solved the problem by applying them (Advanced level). This study presented solution levels across Bebras tasks and discussed how task difficulty affected student solutions differently. Implications for teaching problem-solving skills were discussed.

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Assessing Computational Thinking Across the Curriculum

It’s Computational Thinking! Bebras Tasks in the Curriculum

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Appendix: Bebras tasks

1.1 cipher wheel.

A secret message was left on a beaver’s gravestone by using a cipher wheel and we want to find out what it means.

The wheel works such that only the inner wheel (with small letters) can be rotated. The outer wheel is for the actual message.

The second image shows that when the key is 17 (because the inner wheel has been rotated by 17 positions counter-clockwise) 'A' is encoded as 'r'.

With the key equal to 17, we can encode the message “WHO ARE YOU” as “nyf riv pfl” The message “mgvw ny twao” is received. We know that this was encrypted in a clever way: For the first letter the key was 1, for the second letter the key was 2, the key for the third letter was 3, etc. For instance, “sgg” with the same encryption method would be “RED”.

Decipher the encrypted message and choose the original message. (PLEASE attach a picture of problem-solving procedure/your work. It could be handwritten or digital).

LOVE IS HERE

LIFE IS GOOD

LOVE IS MINE

LESS IS MORE

1.2 Red Raider School

Red Raider School encourages its teachers to include games in their lessons.

One teacher invented the following game and he asks his students to play this game. The winner will leave school before dismissal.

Rules of the game:

The school has one hallway with four doors in a row. The students form a queue and take turns to walk down the hallway. When they get to an open door, they must close it and move to the next door. When they get to a closed door, they must open it, go into the classroom, leave the door open and wait there until the teacher dismisses them.

At the start of the game all the doors are closed. For example,

If a student finds all the doors open, he or she will shut all of them, and leave school early!

If the students are numbered 1 to 20, which student gets to leave school first? (PLEASE attach a picture of problem-solving procedure/your work. It could be handwritten or digital) *HINT: Use a binary system.

15th student

16th student

17th student

18th student

1.3 Ballroom dance partners

Andy, Bert, Chris, David, and Eric are professional male ballroom dancers that take part in a TV show. Amy, Brenda, Carol, Dianna, and Emma are female participants that will learn to dance during this show. Each dancer will be assigned a single participant to teach.

Before the show, the producer organizes a party where everybody meets. After the party, the professionals and participants fill out a questionnaire:

each professional dancer ranks the participants in the order that he thinks they can be successful

each participant ranks the professionals in the order of how fast she can learn from him (1 = 1st choice, 2 = 2nd choice, etc.)

Here are the results of these choices:

The producers want to match the professionals with their ideal participants so that every participant is satisfied with his/her choice. You are asked to match the professionals with the contestants so that everyone has a perfect partner. You must also make sure that all unmatched pairs would still be happy with their partners.

When you have finished assigning partners, who is Eric's partner? (PLEASE attach a picture of problem-solving procedure/your work. It could be handwritten or digital).

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About this article

Kwon, K., Cheon, J. & Moon, H. Levels of problem-solving competency identified through Bebras Computing Challenge. Educ Inf Technol 26 , 5477–5498 (2021). https://doi.org/10.1007/s10639-021-10553-9

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Received : 10 February 2021

Accepted : 18 April 2021

Published : 23 April 2021

Issue Date : September 2021

DOI : https://doi.org/10.1007/s10639-021-10553-9

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