factors reasoning and problem solving

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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

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Problem-Solving Strategies and Obstacles

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• Application
• Improvement

From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

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

7.3 Problem-Solving

Learning objectives.

By the end of this section, you will be able to:

• Describe problem solving strategies
• Define algorithm and heuristic
• Explain some common roadblocks to effective problem solving

   People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy for solving the problem.

The study of human and animal problem solving processes has provided much insight toward the understanding of our conscious experience and led to advancements in computer science and artificial intelligence. Essentially much of cognitive science today represents studies of how we consciously and unconsciously make decisions and solve problems. For instance, when encountered with a large amount of information, how do we go about making decisions about the most efficient way of sorting and analyzing all the information in order to find what you are looking for as in visual search paradigms in cognitive psychology. Or in a situation where a piece of machinery is not working properly, how do we go about organizing how to address the issue and understand what the cause of the problem might be. How do we sort the procedures that will be needed and focus attention on what is important in order to solve problems efficiently. Within this section we will discuss some of these issues and examine processes related to human, animal and computer problem solving.

PROBLEM-SOLVING STRATEGIES

   When people are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution.

Problems themselves can be classified into two different categories known as ill-defined and well-defined problems (Schacter, 2009). Ill-defined problems represent issues that do not have clear goals, solution paths, or expected solutions whereas well-defined problems have specific goals, clearly defined solutions, and clear expected solutions. Problem solving often incorporates pragmatics (logical reasoning) and semantics (interpretation of meanings behind the problem), and also in many cases require abstract thinking and creativity in order to find novel solutions. Within psychology, problem solving refers to a motivational drive for reading a definite “goal” from a present situation or condition that is either not moving toward that goal, is distant from it, or requires more complex logical analysis for finding a missing description of conditions or steps toward that goal. Processes relating to problem solving include problem finding also known as problem analysis, problem shaping where the organization of the problem occurs, generating alternative strategies, implementation of attempted solutions, and verification of the selected solution. Various methods of studying problem solving exist within the field of psychology including introspection, behavior analysis and behaviorism, simulation, computer modeling, and experimentation.

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them (table below). For example, a well-known strategy is trial and error. The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

   Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

• When one is faced with too much information
• When the time to make a decision is limited
• When the decision to be made is unimportant
• When there is access to very little information to use in making the decision
• When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

Further problem solving strategies have been identified (listed below) that incorporate flexible and creative thinking in order to reach solutions efficiently.

Additional Problem Solving Strategies :

• Abstraction – refers to solving the problem within a model of the situation before applying it to reality.
• Analogy – is using a solution that solves a similar problem.
• Brainstorming – refers to collecting an analyzing a large amount of solutions, especially within a group of people, to combine the solutions and developing them until an optimal solution is reached.
• Divide and conquer – breaking down large complex problems into smaller more manageable problems.
• Hypothesis testing – method used in experimentation where an assumption about what would happen in response to manipulating an independent variable is made, and analysis of the affects of the manipulation are made and compared to the original hypothesis.
• Lateral thinking – approaching problems indirectly and creatively by viewing the problem in a new and unusual light.
• Means-ends analysis – choosing and analyzing an action at a series of smaller steps to move closer to the goal.
• Method of focal objects – putting seemingly non-matching characteristics of different procedures together to make something new that will get you closer to the goal.
• Morphological analysis – analyzing the outputs of and interactions of many pieces that together make up a whole system.
• Proof – trying to prove that a problem cannot be solved. Where the proof fails becomes the starting point or solving the problem.
• Reduction – adapting the problem to be as similar problems where a solution exists.
• Research – using existing knowledge or solutions to similar problems to solve the problem.
• Root cause analysis – trying to identify the cause of the problem.

The strategies listed above outline a short summary of methods we use in working toward solutions and also demonstrate how the mind works when being faced with barriers preventing goals to be reached.

One example of means-end analysis can be found by using the Tower of Hanoi paradigm . This paradigm can be modeled as a word problems as demonstrated by the Missionary-Cannibal Problem :

Missionary-Cannibal Problem

Three missionaries and three cannibals are on one side of a river and need to cross to the other side. The only means of crossing is a boat, and the boat can only hold two people at a time. Your goal is to devise a set of moves that will transport all six of the people across the river, being in mind the following constraint: The number of cannibals can never exceed the number of missionaries in any location. Remember that someone will have to also row that boat back across each time.

Hint : At one point in your solution, you will have to send more people back to the original side than you just sent to the destination.

The actual Tower of Hanoi problem consists of three rods sitting vertically on a base with a number of disks of different sizes that can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top making a conical shape. The objective of the puzzle is to move the entire stack to another rod obeying the following rules:

• 1. Only one disk can be moved at a time.
• 2. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack or on an empty rod.
• 3. No disc may be placed on top of a smaller disk.

  Figure 7.02. Steps for solving the Tower of Hanoi in the minimum number of moves when there are 3 disks.

Figure 7.03. Graphical representation of nodes (circles) and moves (lines) of Tower of Hanoi.

The Tower of Hanoi is a frequently used psychological technique to study problem solving and procedure analysis. A variation of the Tower of Hanoi known as the Tower of London has been developed which has been an important tool in the neuropsychological diagnosis of executive function disorders and their treatment.

GESTALT PSYCHOLOGY AND PROBLEM SOLVING

As you may recall from the sensation and perception chapter, Gestalt psychology describes whole patterns, forms and configurations of perception and cognition such as closure, good continuation, and figure-ground. In addition to patterns of perception, Wolfgang Kohler, a German Gestalt psychologist traveled to the Spanish island of Tenerife in order to study animals behavior and problem solving in the anthropoid ape.

As an interesting side note to Kohler’s studies of chimp problem solving, Dr. Ronald Ley, professor of psychology at State University of New York provides evidence in his book A Whisper of Espionage  (1990) suggesting that while collecting data for what would later be his book  The Mentality of Apes (1925) on Tenerife in the Canary Islands between 1914 and 1920, Kohler was additionally an active spy for the German government alerting Germany to ships that were sailing around the Canary Islands. Ley suggests his investigations in England, Germany and elsewhere in Europe confirm that Kohler had served in the German military by building, maintaining and operating a concealed radio that contributed to Germany’s war effort acting as a strategic outpost in the Canary Islands that could monitor naval military activity approaching the north African coast.

While trapped on the island over the course of World War 1, Kohler applied Gestalt principles to animal perception in order to understand how they solve problems. He recognized that the apes on the islands also perceive relations between stimuli and the environment in Gestalt patterns and understand these patterns as wholes as opposed to pieces that make up a whole. Kohler based his theories of animal intelligence on the ability to understand relations between stimuli, and spent much of his time while trapped on the island investigation what he described as  insight , the sudden perception of useful or proper relations. In order to study insight in animals, Kohler would present problems to chimpanzee’s by hanging some banana’s or some kind of food so it was suspended higher than the apes could reach. Within the room, Kohler would arrange a variety of boxes, sticks or other tools the chimpanzees could use by combining in patterns or organizing in a way that would allow them to obtain the food (Kohler & Winter, 1925).

While viewing the chimpanzee’s, Kohler noticed one chimp that was more efficient at solving problems than some of the others. The chimp, named Sultan, was able to use long poles to reach through bars and organize objects in specific patterns to obtain food or other desirables that were originally out of reach. In order to study insight within these chimps, Kohler would remove objects from the room to systematically make the food more difficult to obtain. As the story goes, after removing many of the objects Sultan was used to using to obtain the food, he sat down ad sulked for a while, and then suddenly got up going over to two poles lying on the ground. Without hesitation Sultan put one pole inside the end of the other creating a longer pole that he could use to obtain the food demonstrating an ideal example of what Kohler described as insight. In another situation, Sultan discovered how to stand on a box to reach a banana that was suspended from the rafters illustrating Sultan’s perception of relations and the importance of insight in problem solving.

Grande (another chimp in the group studied by Kohler) builds a three-box structure to reach the bananas, while Sultan watches from the ground.  Insight , sometimes referred to as an “Ah-ha” experience, was the term Kohler used for the sudden perception of useful relations among objects during problem solving (Kohler, 1927; Radvansky & Ashcraft, 2013).

Solving puzzles.

   Problem-solving abilities can improve with practice. Many people challenge themselves every day with puzzles and other mental exercises to sharpen their problem-solving skills. Sudoku puzzles appear daily in most newspapers. Typically, a sudoku puzzle is a 9×9 grid. The simple sudoku below (see figure) is a 4×4 grid. To solve the puzzle, fill in the empty boxes with a single digit: 1, 2, 3, or 4. Here are the rules: The numbers must total 10 in each bolded box, each row, and each column; however, each digit can only appear once in a bolded box, row, and column. Time yourself as you solve this puzzle and compare your time with a classmate.

How long did it take you to solve this sudoku puzzle? (You can see the answer at the end of this section.)

   Here is another popular type of puzzle (figure below) that challenges your spatial reasoning skills. Connect all nine dots with four connecting straight lines without lifting your pencil from the paper:

Did you figure it out? (The answer is at the end of this section.) Once you understand how to crack this puzzle, you won’t forget.

   Take a look at the “Puzzling Scales” logic puzzle below (figure below). Sam Loyd, a well-known puzzle master, created and refined countless puzzles throughout his lifetime (Cyclopedia of Puzzles, n.d.).

What steps did you take to solve this puzzle? You can read the solution at the end of this section.

Pitfalls to problem solving.

   Not all problems are successfully solved, however. What challenges stop us from successfully solving a problem? Albert Einstein once said, “Insanity is doing the same thing over and over again and expecting a different result.” Imagine a person in a room that has four doorways. One doorway that has always been open in the past is now locked. The person, accustomed to exiting the room by that particular doorway, keeps trying to get out through the same doorway even though the other three doorways are open. The person is stuck—but she just needs to go to another doorway, instead of trying to get out through the locked doorway. A mental set is where you persist in approaching a problem in a way that has worked in the past but is clearly not working now.

Functional fixedness is a type of mental set where you cannot perceive an object being used for something other than what it was designed for. During the Apollo 13 mission to the moon, NASA engineers at Mission Control had to overcome functional fixedness to save the lives of the astronauts aboard the spacecraft. An explosion in a module of the spacecraft damaged multiple systems. The astronauts were in danger of being poisoned by rising levels of carbon dioxide because of problems with the carbon dioxide filters. The engineers found a way for the astronauts to use spare plastic bags, tape, and air hoses to create a makeshift air filter, which saved the lives of the astronauts.

   Researchers have investigated whether functional fixedness is affected by culture. In one experiment, individuals from the Shuar group in Ecuador were asked to use an object for a purpose other than that for which the object was originally intended. For example, the participants were told a story about a bear and a rabbit that were separated by a river and asked to select among various objects, including a spoon, a cup, erasers, and so on, to help the animals. The spoon was the only object long enough to span the imaginary river, but if the spoon was presented in a way that reflected its normal usage, it took participants longer to choose the spoon to solve the problem. (German & Barrett, 2005). The researchers wanted to know if exposure to highly specialized tools, as occurs with individuals in industrialized nations, affects their ability to transcend functional fixedness. It was determined that functional fixedness is experienced in both industrialized and nonindustrialized cultures (German & Barrett, 2005).

In order to make good decisions, we use our knowledge and our reasoning. Often, this knowledge and reasoning is sound and solid. Sometimes, however, we are swayed by biases or by others manipulating a situation. For example, let’s say you and three friends wanted to rent a house and had a combined target budget of $1,600. The realtor shows you only very run-down houses for $1,600 and then shows you a very nice house for $2,000. Might you ask each person to pay more in rent to get the $2,000 home? Why would the realtor show you the run-down houses and the nice house? The realtor may be challenging your anchoring bias. An anchoring bias occurs when you focus on one piece of information when making a decision or solving a problem. In this case, you’re so focused on the amount of money you are willing to spend that you may not recognize what kinds of houses are available at that price point.

The confirmation bias is the tendency to focus on information that confirms your existing beliefs. For example, if you think that your professor is not very nice, you notice all of the instances of rude behavior exhibited by the professor while ignoring the countless pleasant interactions he is involved in on a daily basis. Hindsight bias leads you to believe that the event you just experienced was predictable, even though it really wasn’t. In other words, you knew all along that things would turn out the way they did. Representative bias describes a faulty way of thinking, in which you unintentionally stereotype someone or something; for example, you may assume that your professors spend their free time reading books and engaging in intellectual conversation, because the idea of them spending their time playing volleyball or visiting an amusement park does not fit in with your stereotypes of professors.

Finally, the availability heuristic is a heuristic in which you make a decision based on an example, information, or recent experience that is that readily available to you, even though it may not be the best example to inform your decision . Biases tend to “preserve that which is already established—to maintain our preexisting knowledge, beliefs, attitudes, and hypotheses” (Aronson, 1995; Kahneman, 2011). These biases are summarized in the table below.

Were you able to determine how many marbles are needed to balance the scales in the figure below? You need nine. Were you able to solve the problems in the figures above? Here are the answers.

   Many different strategies exist for solving problems. Typical strategies include trial and error, applying algorithms, and using heuristics. To solve a large, complicated problem, it often helps to break the problem into smaller steps that can be accomplished individually, leading to an overall solution. Roadblocks to problem solving include a mental set, functional fixedness, and various biases that can cloud decision making skills.

References:

Openstax Psychology text by Kathryn Dumper, William Jenkins, Arlene Lacombe, Marilyn Lovett and Marion Perlmutter licensed under CC BY v4.0. https://openstax.org/details/books/psychology

Review Questions:

1. A specific formula for solving a problem is called ________.

a. an algorithm

b. a heuristic

c. a mental set

d. trial and error

2. Solving the Tower of Hanoi problem tends to utilize a  ________ strategy of problem solving.

a. divide and conquer

b. means-end analysis

d. experiment

3. A mental shortcut in the form of a general problem-solving framework is called ________.

4. Which type of bias involves becoming fixated on a single trait of a problem?

a. anchoring bias

b. confirmation bias

c. representative bias

d. availability bias

5. Which type of bias involves relying on a false stereotype to make a decision?

6. Wolfgang Kohler analyzed behavior of chimpanzees by applying Gestalt principles to describe ________.

a. social adjustment

b. student load payment options

c. emotional learning

d. insight learning

7. ________ is a type of mental set where you cannot perceive an object being used for something other than what it was designed for.

a. functional fixedness

c. working memory

Critical Thinking Questions:

1. What is functional fixedness and how can overcoming it help you solve problems?

2. How does an algorithm save you time and energy when solving a problem?

Personal Application Question:

1. Which type of bias do you recognize in your own decision making processes? How has this bias affected how you’ve made decisions in the past and how can you use your awareness of it to improve your decisions making skills in the future?

anchoring bias

availability heuristic

confirmation bias

functional fixedness

hindsight bias

problem-solving strategy

representative bias

trial and error

working backwards

Answers to Exercises

algorithm:  problem-solving strategy characterized by a specific set of instructions

anchoring bias:  faulty heuristic in which you fixate on a single aspect of a problem to find a solution

availability heuristic:  faulty heuristic in which you make a decision based on information readily available to you

confirmation bias:  faulty heuristic in which you focus on information that confirms your beliefs

functional fixedness:  inability to see an object as useful for any other use other than the one for which it was intended

heuristic:  mental shortcut that saves time when solving a problem

hindsight bias:  belief that the event just experienced was predictable, even though it really wasn’t

mental set:  continually using an old solution to a problem without results

problem-solving strategy:  method for solving problems

representative bias:  faulty heuristic in which you stereotype someone or something without a valid basis for your judgment

trial and error:  problem-solving strategy in which multiple solutions are attempted until the correct one is found

working backwards:  heuristic in which you begin to solve a problem by focusing on the end result

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What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
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Critical thinking and problem-solving, jump to: , what is critical thinking, characteristics of critical thinking, why teach critical thinking.

• Teaching Strategies to Help Promote Critical Thinking Skills

References and Resources

When examining the vast literature on critical thinking, various definitions of critical thinking emerge. Here are some samples:

• "Critical thinking is the intellectually disciplined process of actively and skillfully conceptualizing, applying, analyzing, synthesizing, and/or evaluating information gathered from, or generated by, observation, experience, reflection, reasoning, or communication, as a guide to belief and action" (Scriven, 1996).
• "Most formal definitions characterize critical thinking as the intentional application of rational, higher order thinking skills, such as analysis, synthesis, problem recognition and problem solving, inference, and evaluation" (Angelo, 1995, p. 6).
• "Critical thinking is thinking that assesses itself" (Center for Critical Thinking, 1996b).
• "Critical thinking is the ability to think about one's thinking in such a way as 1. To recognize its strengths and weaknesses and, as a result, 2. To recast the thinking in improved form" (Center for Critical Thinking, 1996c).

Perhaps the simplest definition is offered by Beyer (1995) : "Critical thinking... means making reasoned judgments" (p. 8). Basically, Beyer sees critical thinking as using criteria to judge the quality of something, from cooking to a conclusion of a research paper. In essence, critical thinking is a disciplined manner of thought that a person uses to assess the validity of something (statements, news stories, arguments, research, etc.).

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Wade (1995) identifies eight characteristics of critical thinking. Critical thinking involves asking questions, defining a problem, examining evidence, analyzing assumptions and biases, avoiding emotional reasoning, avoiding oversimplification, considering other interpretations, and tolerating ambiguity. Dealing with ambiguity is also seen by Strohm & Baukus (1995) as an essential part of critical thinking, "Ambiguity and doubt serve a critical-thinking function and are a necessary and even a productive part of the process" (p. 56).

Another characteristic of critical thinking identified by many sources is metacognition. Metacognition is thinking about one's own thinking. More specifically, "metacognition is being aware of one's thinking as one performs specific tasks and then using this awareness to control what one is doing" (Jones & Ratcliff, 1993, p. 10 ).

In the book, Critical Thinking, Beyer elaborately explains what he sees as essential aspects of critical thinking. These are:

• Dispositions: Critical thinkers are skeptical, open-minded, value fair-mindedness, respect evidence and reasoning, respect clarity and precision, look at different points of view, and will change positions when reason leads them to do so.
• Criteria: To think critically, must apply criteria. Need to have conditions that must be met for something to be judged as believable. Although the argument can be made that each subject area has different criteria, some standards apply to all subjects. "... an assertion must... be based on relevant, accurate facts; based on credible sources; precise; unbiased; free from logical fallacies; logically consistent; and strongly reasoned" (p. 12).
• Argument: Is a statement or proposition with supporting evidence. Critical thinking involves identifying, evaluating, and constructing arguments.
• Reasoning: The ability to infer a conclusion from one or multiple premises. To do so requires examining logical relationships among statements or data.
• Point of View: The way one views the world, which shapes one's construction of meaning. In a search for understanding, critical thinkers view phenomena from many different points of view.
• Procedures for Applying Criteria: Other types of thinking use a general procedure. Critical thinking makes use of many procedures. These procedures include asking questions, making judgments, and identifying assumptions.

Oliver & Utermohlen (1995) see students as too often being passive receptors of information. Through technology, the amount of information available today is massive. This information explosion is likely to continue in the future. Students need a guide to weed through the information and not just passively accept it. Students need to "develop and effectively apply critical thinking skills to their academic studies, to the complex problems that they will face, and to the critical choices they will be forced to make as a result of the information explosion and other rapid technological changes" (Oliver & Utermohlen, p. 1 ).

As mentioned in the section, Characteristics of Critical Thinking , critical thinking involves questioning. It is important to teach students how to ask good questions, to think critically, in order to continue the advancement of the very fields we are teaching. "Every field stays alive only to the extent that fresh questions are generated and taken seriously" (Center for Critical Thinking, 1996a ).

Beyer sees the teaching of critical thinking as important to the very state of our nation. He argues that to live successfully in a democracy, people must be able to think critically in order to make sound decisions about personal and civic affairs. If students learn to think critically, then they can use good thinking as the guide by which they live their lives.

Teaching Strategies to Help Promote Critical Thinking

The 1995, Volume 22, issue 1, of the journal, Teaching of Psychology , is devoted to the teaching critical thinking. Most of the strategies included in this section come from the various articles that compose this issue.

• CATS (Classroom Assessment Techniques): Angelo stresses the use of ongoing classroom assessment as a way to monitor and facilitate students' critical thinking. An example of a CAT is to ask students to write a "Minute Paper" responding to questions such as "What was the most important thing you learned in today's class? What question related to this session remains uppermost in your mind?" The teacher selects some of the papers and prepares responses for the next class meeting.
• Cooperative Learning Strategies: Cooper (1995) argues that putting students in group learning situations is the best way to foster critical thinking. "In properly structured cooperative learning environments, students perform more of the active, critical thinking with continuous support and feedback from other students and the teacher" (p. 8).
• Case Study /Discussion Method: McDade (1995) describes this method as the teacher presenting a case (or story) to the class without a conclusion. Using prepared questions, the teacher then leads students through a discussion, allowing students to construct a conclusion for the case.
• Using Questions: King (1995) identifies ways of using questions in the classroom:
• Reciprocal Peer Questioning: Following lecture, the teacher displays a list of question stems (such as, "What are the strengths and weaknesses of...). Students must write questions about the lecture material. In small groups, the students ask each other the questions. Then, the whole class discusses some of the questions from each small group.
• Reader's Questions: Require students to write questions on assigned reading and turn them in at the beginning of class. Select a few of the questions as the impetus for class discussion.
• Conference Style Learning: The teacher does not "teach" the class in the sense of lecturing. The teacher is a facilitator of a conference. Students must thoroughly read all required material before class. Assigned readings should be in the zone of proximal development. That is, readings should be able to be understood by students, but also challenging. The class consists of the students asking questions of each other and discussing these questions. The teacher does not remain passive, but rather, helps "direct and mold discussions by posing strategic questions and helping students build on each others' ideas" (Underwood & Wald, 1995, p. 18 ).
• Use Writing Assignments: Wade sees the use of writing as fundamental to developing critical thinking skills. "With written assignments, an instructor can encourage the development of dialectic reasoning by requiring students to argue both [or more] sides of an issue" (p. 24).
• Written dialogues: Give students written dialogues to analyze. In small groups, students must identify the different viewpoints of each participant in the dialogue. Must look for biases, presence or exclusion of important evidence, alternative interpretations, misstatement of facts, and errors in reasoning. Each group must decide which view is the most reasonable. After coming to a conclusion, each group acts out their dialogue and explains their analysis of it.
• Spontaneous Group Dialogue: One group of students are assigned roles to play in a discussion (such as leader, information giver, opinion seeker, and disagreer). Four observer groups are formed with the functions of determining what roles are being played by whom, identifying biases and errors in thinking, evaluating reasoning skills, and examining ethical implications of the content.
• Ambiguity: Strohm & Baukus advocate producing much ambiguity in the classroom. Don't give students clear cut material. Give them conflicting information that they must think their way through.
• Angelo, T. A. (1995). Beginning the dialogue: Thoughts on promoting critical thinking: Classroom assessment for critical thinking. Teaching of Psychology, 22(1), 6-7.
• Beyer, B. K. (1995). Critical thinking. Bloomington, IN: Phi Delta Kappa Educational Foundation.
• Center for Critical Thinking (1996a). The role of questions in thinking, teaching, and learning. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
• Center for Critical Thinking (1996b). Structures for student self-assessment. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univclass/trc.nclk
• Center for Critical Thinking (1996c). Three definitions of critical thinking [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
• Cooper, J. L. (1995). Cooperative learning and critical thinking. Teaching of Psychology, 22(1), 7-8.
• Jones, E. A. & Ratcliff, G. (1993). Critical thinking skills for college students. National Center on Postsecondary Teaching, Learning, and Assessment, University Park, PA. (Eric Document Reproduction Services No. ED 358 772)
• King, A. (1995). Designing the instructional process to enhance critical thinking across the curriculum: Inquiring minds really do want to know: Using questioning to teach critical thinking. Teaching of Psychology, 22 (1) , 13-17.
• McDade, S. A. (1995). Case study pedagogy to advance critical thinking. Teaching Psychology, 22(1), 9-10.
• Oliver, H. & Utermohlen, R. (1995). An innovative teaching strategy: Using critical thinking to give students a guide to the future.(Eric Document Reproduction Services No. 389 702)
• Robertson, J. F. & Rane-Szostak, D. (1996). Using dialogues to develop critical thinking skills: A practical approach. Journal of Adolescent & Adult Literacy, 39(7), 552-556.
• Scriven, M. & Paul, R. (1996). Defining critical thinking: A draft statement for the National Council for Excellence in Critical Thinking. [On-line]. Available HTTP: http://www.criticalthinking.org/University/univlibrary/library.nclk
• Strohm, S. M., & Baukus, R. A. (1995). Strategies for fostering critical thinking skills. Journalism and Mass Communication Educator, 50 (1), 55-62.
• Underwood, M. K., & Wald, R. L. (1995). Conference-style learning: A method for fostering critical thinking with heart. Teaching Psychology, 22(1), 17-21.
• Wade, C. (1995). Using writing to develop and assess critical thinking. Teaching of Psychology, 22(1), 24-28.

Other Reading

• Bean, J. C. (1996). Engaging ideas: The professor's guide to integrating writing, critical thinking, & active learning in the classroom. Jossey-Bass.
• Bernstein, D. A. (1995). A negotiation model for teaching critical thinking. Teaching of Psychology, 22(1), 22-24.
• Carlson, E. R. (1995). Evaluating the credibility of sources. A missing link in the teaching of critical thinking. Teaching of Psychology, 22(1), 39-41.
• Facione, P. A., Sanchez, C. A., Facione, N. C., & Gainen, J. (1995). The disposition toward critical thinking. The Journal of General Education, 44(1), 1-25.
• Halpern, D. F., & Nummedal, S. G. (1995). Closing thoughts about helping students improve how they think. Teaching of Psychology, 22(1), 82-83.
• Isbell, D. (1995). Teaching writing and research as inseparable: A faculty-librarian teaching team. Reference Services Review, 23(4), 51-62.
• Jones, J. M. & Safrit, R. D. (1994). Developing critical thinking skills in adult learners through innovative distance learning. Paper presented at the International Conference on the practice of adult education and social development. Jinan, China. (Eric Document Reproduction Services No. ED 373 159)
• Sanchez, M. A. (1995). Using critical-thinking principles as a guide to college-level instruction. Teaching of Psychology, 22(1), 72-74.
• Spicer, K. L. & Hanks, W. E. (1995). Multiple measures of critical thinking skills and predisposition in assessment of critical thinking. Paper presented at the annual meeting of the Speech Communication Association, San Antonio, TX. (Eric Document Reproduction Services No. ED 391 185)
• Terenzini, P. T., Springer, L., Pascarella, E. T., & Nora, A. (1995). Influences affecting the development of students' critical thinking skills. Research in Higher Education, 36(1), 23-39.

On the Internet

• Carr, K. S. (1990). How can we teach critical thinking. Eric Digest. [On-line]. Available HTTP: http://ericps.ed.uiuc.edu/eece/pubs/digests/1990/carr90.html
• The Center for Critical Thinking (1996). Home Page. Available HTTP: http://www.criticalthinking.org/University/
• Ennis, Bob (No date). Critical thinking. [On-line], April 4, 1997. Available HTTP: http://www.cof.orst.edu/cof/teach/for442/ct.htm
• Montclair State University (1995). Curriculum resource center. Critical thinking resources: An annotated bibliography. [On-line]. Available HTTP: http://www.montclair.edu/Pages/CRC/Bibliographies/CriticalThinking.html
• No author, No date. Critical Thinking is ... [On-line], April 4, 1997. Available HTTP: http://library.usask.ca/ustudy/critical/
• Sheridan, Marcia (No date). Internet education topics hotlink page. [On-line], April 4, 1997. Available HTTP: http://sun1.iusb.edu/~msherida/topics/critical.html

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

Neil Almond

Fluency, reasoning and problem solving are central strands of mathematical competency, as recognized by the National Council of Teachers of Mathematics (NCTM) and the National Research Council’s report ‘Adding It Up’.

They are key components to the Standards of Mathematical Practice, standards that are interwoven into every mathematics lesson. Here we look at how these three approaches or elements of math can be interwoven in a child’s math education through elementary and middle school.

We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help elementary and middle school teachers think carefully about their practice and the pedagogical choices they make around the teaching of what the common core refers to as ‘mathematical practices’, and reasoning and problem solving in particular.

Before we can think about what this would look like in Common Core math examples and other state-specific math frameworks, we need to understand the background to these terms.

What is fluency in math?

What is reasoning in math, what is problem solving in math, mathematical problem solving is a learned skill, performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving.

• What IS ‘performance vs learning’?
• Teaching to “cover the curriculum” hinders development of strong problem solving skills.
• Fluency and reasoning – Best practice in a lesson, a unit, and a semester

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

Fluency, reasoning and problem solving should not be taught by rote .

The Ultimate Guide to Problem Solving Techniques

Develop problem solving skills in the classroom with this free, downloadable worksheet

Fluency in math is a fairly broad concept. The basics of mathematical fluency – as defined by the Common Core State Standards for math – involve knowing key mathematical skills and being able to carry them out flexibly, accurately and efficiently.

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

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

Read more: How the best schools develop math fluency

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

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

Read more: Developing math reasoning: the mathematical skills required and how to teach them .

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

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

We are all problem solvers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In that paper he produces this pyramid:

This is important for two reasons:

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

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

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

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

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

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

At some point, schools become obsessed with the ridiculous notion of moving students through content at an accelerated rate. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘accelerated progress in math’ in this lesson,’ ‘School officials will be looking for ‘accelerated progress’ etc.

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

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

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

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

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

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

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

What IS ‘performance vs learning’?

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

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

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

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

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers (a process referred to as “number plucking”) – not exactly high level problem solving skills.

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

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

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content being missed in the classroom.

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

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

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

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

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

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

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

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

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

This blog on mental math strategies students should learn at each grade level is a good place to start when thinking about the core aspects of fluency that students should achieve.

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

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

Read more: How to teach multiplication for instant recall

What about best practice over a longer period?

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

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

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

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

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

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

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

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

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

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

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

Practice with facts that are secure

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

You could get students in 3rd grade to solve complicated place value problems with the numbers they should know from 1st or 2nd grade. This would lessen the cognitive load , freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Increase complexity gradually

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

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

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

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

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

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

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

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

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

Read more: 

• Math Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Problem Solving and Reasoning Questions and Answers
• Get to Grips with Math Problem Solving For Elementary Students
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Math Challenges To Really Stretch Your More Able Students
• Why You Should Be Incorporating Stem Sentences Into Your Elementary Math Teaching

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by primary school lead teacher Neil Almond and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

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20 Effective Math Strategies To Approach Problem-Solving 

Why Student Centered Learning Is Important: A Guide For Educators

13 Effective Learning Strategies: A Guide to Using them in your Math Classroom

Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

Ultimate Guide to Metacognition [FREE]

Looking for a summary on metacognition in relation to math teaching and learning?

Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

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Critical Thinking and Decision-Making  - What is Critical Thinking?

Critical thinking and decision-making  -, what is critical thinking, critical thinking and decision-making what is critical thinking.

Critical Thinking and Decision-Making: What is Critical Thinking?

Lesson 1: what is critical thinking, what is critical thinking.

Critical thinking is a term that gets thrown around a lot. You've probably heard it used often throughout the years whether it was in school, at work, or in everyday conversation. But when you stop to think about it, what exactly is critical thinking and how do you do it ?

Watch the video below to learn more about critical thinking.

Simply put, critical thinking is the act of deliberately analyzing information so that you can make better judgements and decisions . It involves using things like logic, reasoning, and creativity, to draw conclusions and generally understand things better.

This may sound like a pretty broad definition, and that's because critical thinking is a broad skill that can be applied to so many different situations. You can use it to prepare for a job interview, manage your time better, make decisions about purchasing things, and so much more.

The process

As humans, we are constantly thinking . It's something we can't turn off. But not all of it is critical thinking. No one thinks critically 100% of the time... that would be pretty exhausting! Instead, it's an intentional process , something that we consciously use when we're presented with difficult problems or important decisions.

Improving your critical thinking

In order to become a better critical thinker, it's important to ask questions when you're presented with a problem or decision, before jumping to any conclusions. You can start with simple ones like What do I currently know? and How do I know this? These can help to give you a better idea of what you're working with and, in some cases, simplify more complex issues.  

Real-world applications

Let's take a look at how we can use critical thinking to evaluate online information . Say a friend of yours posts a news article on social media and you're drawn to its headline. If you were to use your everyday automatic thinking, you might accept it as fact and move on. But if you were thinking critically, you would first analyze the available information and ask some questions :

• What's the source of this article?
• Is the headline potentially misleading?
• What are my friend's general beliefs?
• Do their beliefs inform why they might have shared this?

After analyzing all of this information, you can draw a conclusion about whether or not you think the article is trustworthy.

Critical thinking has a wide range of real-world applications . It can help you to make better decisions, become more hireable, and generally better understand the world around you.

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STEM Problem Solving: Inquiry, Concepts, and Reasoning

• Published: 29 January 2022
• Volume 32 , pages 381–397, ( 2023 )

Cite this article

• Aik-Ling Tan   ORCID: orcid.org/0000-0002-4627-4977 1 ,
• Yann Shiou Ong   ORCID: orcid.org/0000-0002-6092-2803 1 ,
• Yong Sim Ng   ORCID: orcid.org/0000-0002-8400-2040 1 &
• Jared Hong Jie Tan 1  

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Balancing disciplinary knowledge and practical reasoning in problem solving is needed for meaningful learning. In STEM problem solving, science subject matter with associated practices often appears distant to learners due to its abstract nature. Consequently, learners experience difficulties making meaningful connections between science and their daily experiences. Applying Dewey’s idea of practical and science inquiry and Bereiter’s idea of referent-centred and problem-centred knowledge, we examine how integrated STEM problem solving offers opportunities for learners to shuttle between practical and science inquiry and the kinds of knowledge that result from each form of inquiry. We hypothesize that connecting science inquiry with practical inquiry narrows the gap between science and everyday experiences to overcome isolation and fragmentation of science learning. In this study, we examine classroom talk as students engage in problem solving to increase crop yield. Qualitative content analysis of the utterances of six classes of 113 eighth graders and their teachers were conducted for 3 hours of video recordings. Analysis showed an almost equal amount of science and practical inquiry talk. Teachers and students applied their everyday experiences to generate solutions. Science talk was at the basic level of facts and was used to explain reasons for specific design considerations. There was little evidence of higher-level scientific conceptual knowledge being applied. Our observations suggest opportunities for more intentional connections of science to practical problem solving, if we intend to apply higher-order scientific knowledge in problem solving. Deliberate application and reference to scientific knowledge could improve the quality of solutions generated.

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1 Introduction

As we enter to second quarter of the twenty-first century, it is timely to take stock of both the changes and demands that continue to weigh on our education system. A recent report by World Economic Forum highlighted the need to continuously re-position and re-invent education to meet the challenges presented by the disruptions brought upon by the fourth industrial revolution (World Economic Forum, 2020 ). There is increasing pressure for education to equip children with the necessary, relevant, and meaningful knowledge, skills, and attitudes to create a “more inclusive, cohesive and productive world” (World Economic Forum, 2020 , p. 4). Further, the shift in emphasis towards twenty-first century competencies over mere acquisition of disciplinary content knowledge is more urgent since we are preparing students for “jobs that do not yet exist, technology that has not yet been invented, and problems that has yet exist” (OECD, 2018 , p. 2). Tan ( 2020 ) concurred with the urgent need to extend the focus of education, particularly in science education, such that learners can learn to think differently about possibilities in this world. Amidst this rhetoric for change, the questions that remained to be answered include how can science education transform itself to be more relevant; what is the role that science education play in integrated STEM learning; how can scientific knowledge, skills and epistemic practices of science be infused in integrated STEM learning; what kinds of STEM problems should we expose students to for them to learn disciplinary knowledge and skills; and what is the relationship between learning disciplinary content knowledge and problem solving skills?

In seeking to understand the extent of science learning that took place within integrated STEM learning, we dissected the STEM problems that were presented to students and examined in detail the sense making processes that students utilized when they worked on the problems. We adopted Dewey’s ( 1938 ) theoretical idea of scientific and practical/common-sense inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge building process to interpret teacher-students’ interactions during problem solving. There are two primary reasons for choosing these two theoretical frameworks. Firstly, Dewey’s ideas about the relationship between science inquiry and every day practical problem-solving is important in helping us understand the role of science subject matter knowledge and science inquiry in solving practical real-world problems that are commonly used in STEM learning. Secondly, Bereiter’s ideas of referent-centred and problem-centred knowledge augment our understanding of the types of knowledge that students can learn when they engage in solving practical real-world problems.

Taken together, Dewey’s and Bereiter’s ideas enable us to better understand the types of problems used in STEM learning and their corresponding knowledge that is privileged during the problem-solving process. As such, the two theoretical lenses offered an alternative and convincing way to understand the actual types of knowledge that are used within the context of integrated STEM and help to move our understanding of STEM learning beyond current focus on examining how engineering can be used as an integrative mechanism (Bryan et al., 2016 ) or applying the argument of the strengths of trans-, multi-, or inter-disciplinary activities (Bybee, 2013 ; Park et al., 2020 ) or mapping problems by the content and context as pure STEM problems, STEM-related problems or non-STEM problems (Pleasants, 2020 ). Further, existing research (for example, Gale et al., 2000 ) around STEM education focussed largely on description of students’ learning experiences with insufficient attention given to the connections between disciplinary conceptual knowledge and inquiry processes that students use to arrive at solutions to problems. Clarity in the role of disciplinary knowledge and the related inquiry will allow for more intentional design of STEM problems for students to learn higher-order knowledge. Applying Dewey’s idea of practical and scientific inquiry and Bereiter’s ideas of referent-centred and problem-centred knowledge, we analysed six lessons where students engaged with integrated STEM problem solving to propose answers to the following research questions: What is the extent of practical and scientific inquiry in integrated STEM problem solving? and What conceptual knowledge and problem-solving skills are learnt through practical and science inquiry during integrated STEM problem solving?

2 Inquiry in Problem Solving

Inquiry, according to Dewey ( 1938 ), involves the direct control of unknown situations to change them into a coherent and unified one. Inquiry usually encompasses two interrelated activities—(1) thinking about ideas related to conceptual subject-matter and (2) engaging in activities involving our senses or using specific observational techniques. The National Science Education Standards released by the National Research Council in the US in 1996 defined inquiry as “…a multifaceted activity that involves making observations; posing questions; examining books and other sources of information to see what is already known; planning investigations; reviewing what is already known in light of experimental evidence; using tools to gather, analyze, and interpret data; proposing answers, explanations, and predictions; and communicating the results. Inquiry requires identification of assumptions, use of critical and logical thinking, and consideration of alternative explanations” (p. 23). Planning investigation; collecting empirical evidence; using tools to gather, analyse and interpret data; and reasoning are common processes shared in the field of science and engineering and hence are highly relevant to apply to integrated STEM education.

In STEM education, establishing the connection between general inquiry and its application helps to link disciplinary understanding to epistemic knowledge. For instance, methods of science inquiry are popular in STEM education due to the familiarity that teachers have with scientific methods. Science inquiry, a specific form of inquiry, has appeared in many science curriculum (e.g. NRC, 2000 ) since Dewey proposed in 1910 that learning of science should be perceived as both subject-matter and a method of learning science (Dewey, 1910a , 1910b ). Science inquiry which involved ways of doing science should also encompass the ways in which students learn the scientific knowledge and investigative methods that enable scientific knowledge to be constructed. Asking scientifically orientated questions, collecting empirical evidence, crafting explanations, proposing models and reasoning based on available evidence are affordances of scientific inquiry. As such, science should be pursued as a way of knowing rather than merely acquisition of scientific knowledge.

Building on these affordances of science inquiry, Duschl and Bybee ( 2014 ) advocated the 5D model that focused on the practice of planning and carrying out investigations in science and engineering, representing two of the four disciplines in STEM. The 5D model includes science inquiry aspects such as (1) deciding on what and how to measure, observe and sample; (2) developing and selecting appropriate tools to measure and collect data; (3) recording the results and observations in a systematic manner; (4) creating ways to represent the data and patterns that are observed; and (5) determining the validity and the representativeness of the data collected. The focus on planning and carrying out investigations in the 5D model is used to help teachers bridge the gap between the practices of building and refining models and explanation in science and engineering. Indeed, a common approach to incorporating science inquiry in integrated STEM curriculum involves student planning and carrying out scientific investigations and making sense of the data collected to inform engineering design solution (Cunningham & Lachapelle, 2016 ; Roehrig et al., 2021 ). Duschl and Bybee ( 2014 ) argued that it is needful to design experiences for learners to appreciate that struggles are part of problem solving in science and engineering. They argued that “when the struggles of doing science is eliminated or simplified, learners get the wrong perceptions of what is involved when obtaining scientific knowledge and evidence” (Duschl & Bybee, 2014 , p. 2). While we concur with Duschl and Bybee about the need for struggles, in STEM learning, these struggles must be purposeful and grade appropriate so that students will also be able to experience success amidst failure.

The peculiar nature of science inquiry was scrutinized by Dewey ( 1938 ) when he cross-examined the relationship between science inquiry and other forms of inquiry, particularly common-sense inquiry. He positioned science inquiry along a continuum with general or common-sense inquiry that he termed as “logic”. Dewey argued that common-sense inquiry serves a practical purpose and exhibits features of science inquiry such as asking questions and a reliance on evidence although the focus of common-sense inquiry tends to be different. Common-sense inquiry deals with issues or problems that are in the immediate environment where people live, whereas the objects of science inquiry are more likely to be distant (e.g. spintronics) from familiar experiences in people’s daily lives. While we acknowledge the fundamental differences (such as novel discovery compared with re-discovering science, ‘messy’ science compared with ‘sanitised’ science) between school science and science that is practiced by scientists, the subject of interest in science (understanding the world around us) remains the same.

The unfamiliarity between the functionality and purpose of science inquiry to improve the daily lives of learners does little to motivate learners to learn science (Aikenhead, 2006 ; Lee & Luykx, 2006 ) since learners may not appreciate the connections of science inquiry in their day-to-day needs and wants. Bereiter ( 1992 ) has also distinguished knowledge into two forms—referent-centred and problem-centred. Referent-centred knowledge refers to subject-matter that is organised around topics such as that in textbooks. Problem-centred knowledge is knowledge that is organised around problems, whether they are transient problems, practical problems or problems of explanations. Bereiter argued that referent-centred knowledge that is commonly taught in schools is limited in their applications and meaningfulness to the lives of students. This lack of familiarity and affinity to referent-centred knowledge is likened to the science subject-matter knowledge that was mentioned by Dewey. Rather, it is problem-centred knowledge that would be useful when students encounter problems. Learning problem-centred knowledge will allow learners to readily harness the relevant knowledge base that is useful to understand and solve specific problems. This suggests a need to help learners make the meaningful connections between science and their daily lives.

Further, Dewey opined that while the contexts in which scientific knowledge arise could be different from our daily common-sense world, careful consideration of scientific activities and applying the resultant knowledge to daily situations for use and enjoyment is possible. Similarly, in arguing for problem-centred knowledge, Bereiter ( 1992 ) questioned the value of inert knowledge that plays no role in helping us understand or deal with the world around us. Referent-centred knowledge has a higher tendency to be inert due to the way that the knowledge is organised and the way that the knowledge is encountered by learners. For instance, learning about the equation and conditions for photosynthesis is not going to help learners appreciate how plants are adapted for photosynthesis and how these adaptations can allow plants to survive changes in climate and for farmers to grow plants better by creating the best growing conditions. Rather, students could be exposed to problems of explanations where they are asked to unravel the possible reasons for low crop yield and suggest possible ways to overcome the problem. Hence, we argue here that the value of the referent knowledge is that they form the basis and foundation for the students to be able to discuss or suggest ways to overcome real life problems. Referent-centred knowledge serves as part of the relevant knowledge base that can be harnessed to solve specific problems or as foundational knowledge students need to progress to learn higher-order conceptual knowledge that typically forms the foundations or pillars within a discipline. This notion of referent-centred knowledge serving as foundational knowledge that can be and should be activated for application in problem-solving situation is shown by Delahunty et al. ( 2020 ). They found that students show high reliance on memory when they are conceptualising convergent problem-solving tasks.

While Bereiter argues for problem-centred knowledge, he cautioned that engagement should be with problems of explanation rather than transient or practical problems. He opined that if learners only engage in transient or practical problem alone, they will only learn basic-category types of knowledge and fail to understand higher-order conceptual knowledge. For example, for photosynthesis, basic-level types of knowledge included facts about the conditions required for photosynthesis, listing the products formed from the process of photosynthesis and knowing that green leaves reflect green light. These basic-level knowledges should intentionally help learners learn higher-level conceptual knowledge that include learners being able to draw on the conditions for photosynthesis when they encounter that a plant is not growing well or is exhibiting discoloration of leaves.

Transient problems disappear once a solution becomes available and there is a high likelihood that we will not remember the problem after that. Practical problems, according to Bereiter are “stuck-door” problems that could be solved with or without basic-level knowledge and often have solutions that lacks precise definition. There are usually a handful of practical strategies, such as pulling or pushing the door harder, kicking the door, etc. that will work for the problems. All these solutions lack a well-defined approach related to general scientific principles that are reproducible. Problems of explanations are the most desirable types of problems for learners since these are problems that persist and recur such that they can become organising points for knowledge. Problems of explanations consist of the conceptual representations of (1) a text base that serves to represent the text content and (2) a situation model that shows the portion of the world in which the text is relevant. The idea of text base to represent text content in solving problems of explanations is like the idea of domain knowledge and structural knowledge (refers to knowledge of how concepts within a domain are connected) proposed by Jonassen ( 2000 ). He argued that both types of knowledges are required to solve a range of problems from well-structured problems to ill-structured problems with a simulated context, to simple ill-structured problems and to complex ill-structured problems.

Jonassen indicated that complex ill-structured problems are typically design problems and are likely to be the most useful forms of problems for learners to be engaged in inquiry. Complex ill-structured design problems are the “wicked” problems that Buchanan ( 1992 ) discussed. Buchanan’s idea is that design aims to incorporate knowledge from different fields of specialised inquiry to become whole. Complex or wicked problems are akin to the work of scientists who navigate multiple factors and evidence to offer models that are typically oversimplified, but they apply them to propose possible first approximation explanations or solutions and iteratively relax constraints or assumptions to refine the model. The connections between the subject matter of science and the design process to engineer a solution are delicate. While it is important to ensure that practical concerns and questions are taken into consideration in designing solutions (particularly a material artefact) to a practical problem, the challenge here lies in ensuring that creativity in design is encouraged even if students initially lack or neglect the scientific conceptual understanding to explain/justify their design. In his articulation of wicked problems and the role of design thinking, Buchanan ( 1992 ) highlighted the need to pay attention to category and placement. Categories “have fixed meanings that are accepted within the framework of a theory or a philosophy and serve as the basis for analyzing what already exist” (Buchanan, 1992 , p. 12). Placements, on the other hand, “have boundaries to shape and constrain meaning, but are not rigidly fixed and determinate” (p. 12).

The difference in the ideas presented by Dewey and Bereiter lies in the problem design. For Dewey, scientific knowledge could be learnt from inquiring into practical problems that learners are familiar with. After all, Dewey viewed “modern science as continuous with, and to some degree an outgrowth and refinement of, practical or ‘common-sense’ inquiry” (Brown, 2012 ). For Bereiter, he acknowledged the importance of familiar experiences, but instead of using them as starting points for learning science, he argued that practical problems are limiting in helping learners acquire higher-order knowledge. Instead, he advocated for learners to organize their knowledge around problems that are complex, persistent and extended and requiring explanations to better understand the problems. Learners are to have a sense of the kinds of problems to which the specific concept is relevant before they can be said to have grasp the concept in a functionally useful way.

To connect between problem solving, scientific knowledge and everyday experiences, we need to examine ways to re-negotiate the disciplinary boundaries (such as epistemic understanding, object of inquiry, degree of precision) of science and make relevant connections to common-sense inquiry and to the problem at hand. Integrated STEM appears to be one way in which the disciplinary boundaries of science can be re-negotiated to include practices from the fields of technology, engineering and mathematics. In integrated STEM learning, inquiry is seen more holistically as a fluid process in which the outcomes are not absolute but are tentative. The fluidity of the inquiry process is reflected in the non-deterministic inquiry approach. This means that students can use science inquiry, engineering design, design process or any other inquiry approaches that fit to arrive at the solution. This hybridity of inquiry between science, common-sense and problems allows for some familiar aspects of the science inquiry process to be applied to understand and generate solutions to familiar everyday problems. In attempting to infuse elements of common-sense inquiry with science inquiry in problem-solving, logic plays an important role to help learners make connections. Hypothetically, we argue that with increasing exposure to less familiar ways of thinking such as those associated with science inquiry, students’ familiarity with scientific reasoning increases, and hence such ways of thinking gradually become part of their common-sense, which students could employ to solve future relevant problems. The theoretical ideas related to complexities of problems, the different forms of inquiry afforded by different problems and the arguments for engaging in problem solving motivated us to examine empirically how learners engage with ill-structured problems to generate problem-centred knowledge. Of particular interest to us is how learners and teachers weave between practical and scientific reasoning as they inquire to integrate the components in the original problem into a unified whole.

3.1 Context

The integrated STEM activity in our study was planned using the S-T-E-M quartet instructional framework (Tan et al., 2019 ). The S-T-E-M quartet instructional framework positions complex, persistent and extended problems at its core and focusses on the vertical disciplinary knowledge and understanding of the horizontal connections between the disciplines that could be gained by learners through solving the problem (Tan et al., 2019 ). Figure  1 depicts the disciplinary aspects of the problem that was presented to the students. The activity has science and engineering as the two lead disciplines. It spanned three 1-h lessons and required students to both learn and apply relevant scientific conceptual knowledge to solve a complex, real-world problem through processes that resemble the engineering design process (Wheeler et al., 2019 ).

Connections across disciplines in integrate STEM activity

Frequency of different types of reasoning

In the first session (1 h), students were introduced to the problem and its context. The problem pertains to the issue of limited farmland in a land scarce country that imports 90% of food (Singapore Food Agency [SFA], 2020 ). The students were required to devise a solution by applying knowledge of the conditions required for photosynthesis and plant growth to design and build a vertical farming system to help farmers increase crop yield with limited farmland. This context was motivated by the government’s effort to generate interests and knowledge in farming to achieve the 30 by 30 goal—supplying 30% of country’s nutritional needs by 2030. The scenario was a fictitious one where they were asked to produce 120 tonnes of Kailan (a type of leafy vegetable) with two hectares of land instead of the usual six hectares over a specific period. In addition to the abovementioned constraints, the teacher also discussed relevant success criteria for evaluating the solution with the students. Students then researched about existing urban farming approaches. They were given reading materials pertaining to urban farming to help them understand the affordances and constraints of existing solutions. In the second session (6 h), students engaged in ideation to generate potential solutions. They then designed, built and tested their solution and had opportunities to iteratively refine their solution. Students were given a list of materials (e.g. mounting board, straws, ice-cream stick, glue, etc.) that they could use to design their solutions. In the final session (1 h), students presented their solution and reflected on how well their solution met the success criteria. The prior scientific conceptual knowledge that students require to make sense of the problem include knowledge related to plant nutrition, namely, conditions for photosynthesis, nutritional requirements of Kailin and growth cycle of Kailin. The problem resembles a real-world problem that requires students to engage in some level of explanation of their design solution.

A total of 113 eighth graders (62 boys and 51 girls), 14-year-olds, from six classes and their teachers participated in the study. The students and their teachers were recruited as part of a larger study that examined the learning experiences of students when they work on integrated STEM activities that either begin with a problem, a solution or are focused on the content. Invitations were sent to schools across the country and interested schools opted in for the study. For the study reported here, all students and teachers were from six classes within a school. The teachers had all undergone 3 h of professional development with one of the authors on ways of implementing the integrated STEM activity used in this study. During the professional development session, the teachers learnt about the rationale of the activity, familiarize themselves with the materials and clarified the intentions and goals of the activity. The students were mostly grouped in groups of three, although a handful of students chose to work independently. The group size of students was not critical for the analysis of talk in this study as the analytic focus was on the kinds of knowledge applied rather than collaborative or group think. We assumed that the types of inquiry adopted by teachers and students were largely dependent on the nature of problem. Eighth graders were chosen for this study since lower secondary science offered at this grade level is thematic and integrated across biology, chemistry and physics. Furthermore, the topic of photosynthesis is taught under the theme of Interactions at eighth grade (CPDD, 2021 ). This thematic and integrated nature of science at eighth grade offered an ideal context and platform for integrated STEM activities to be trialled.

The final lessons in a series of three lessons in each of the six classes was analysed and reported in this study. Lessons where students worked on their solutions were not analysed because the recordings had poor audibility due to masking and physical distancing requirements as per COVID-19 regulations. At the start of the first lesson, the instructions given by the teacher were:

You are going to present your models. Remember the scenario that you were given at the beginning that you were tasked to solve using your model. …. In your presentation, you have to present your prototype and its features, what is so good about your prototype, how it addresses the problem and how it saves costs and space. So, this is what you can talk about during your presentation. ….. pay attention to the presentation and write down questions you like to ask the groups after the presentation… you can also critique their model, you can evaluate, critique and ask questions…. Some examples of questions you can ask the groups are? Do you think your prototype can achieve optimal plant growth? You can also ask questions specific to their models.

3.2 Data collection

Parental consent was sought a month before the start of data collection. The informed consent adhered to confidentiality and ethics guidelines as described by the Institutional Review Board. The data collection took place over a period of one month with weekly video recording. Two video cameras, one at the front and one at the back of the science laboratory were set up. The front camera captured the students seated at the front while the back video camera recorded the teacher as well as the groups of students at the back of the laboratory. The video recordings were synchronized so that the events captured from each camera can be interpreted from different angles. After transcription of the raw video files, the identities of students were substituted with pseudonyms.

3.3 Data analysis

The video recordings were analysed using the qualitative content analysis approach. Qualitative content analysis allows for patterns or themes and meanings to emerge from the process of systematic classification (Hsieh & Shannon, 2005 ). Qualitative content analysis is an appropriate analytic method for this study as it allows us to systematically identify episodes of practical inquiry and science inquiry to map them to the purposes and outcomes of these episodes as each lesson unfolds.

In total, six h of video recordings where students presented their ideas while the teachers served as facilitator and mentor were analysed. The video recordings were transcribed, and the transcripts were analysed using the NVivo software. Our unit of analysis is a single turn of talk (one utterance). We have chosen to use utterances as proxy indicators of reasoning practices based on the assumption that an utterance relates to both grammar and context. An utterance is a speech act that reveals both meaning and intentions of the speaker within specific contexts (Li, 2008 ).

Our research analytical lens is also interpretative in nature and the validity of our interpretation is through inter-rater discussion and agreement. Each utterance at the speaker level in transcripts was examined and coded either as relevant to practical reasoning or scientific reasoning based on the content. The utterances could be a comment by the teacher, a question by a student or a response by another student. Deductive coding is deployed with the two codes, practical reasoning and scientific reasoning derived from the theoretical ideas of Dewey and Bereiter as described earlier. Practical reasoning refers to utterances that reflect commonsensical knowledge or application of everyday understanding. Scientific reasoning refers to utterances that consist of scientifically oriented questions, scientific terms, or the use of empirical evidence to explain. Examples of each type of reasoning are highlighted in the following section. Each coded utterance is then reviewed for detailed description of the events that took place that led to that specific utterance. The description of the context leading to the utterance is considered an episode. The episodes and codes were discussed and agreed upon by two of the authors. Two coders simultaneously watched the videos to identify and code the episodes. The coders interpreted the content of each utterance, examine the context where the utterance was made and deduced the purpose of the utterance. Once each coder has established the sense-making aspect of the utterance in relation to the context, a code of either practical reasoning or scientific reasoning is assigned. Once that was completed, the two coders compared their coding for similarities and differences. They discussed the differences until an agreement was reached. Through this process, an agreement of 85% was reached between the coders. Where disagreement persisted, codes of the more experienced coder were adopted.

4 Results and Discussion

The specific STEM lessons analysed were taken from the lessons whereby students presented the model of their solutions to the class for peer evaluation. Every group of students stood in front of the class and placed their model on the bench as they presented. There was also a board where they could sketch or write their explanations should they want to. The instructions given by the teacher to the students were to explain their models and state reasons for their design.

4.1 Prevalence of Reasoning

The 6h of videos consists of 1422 turns of talk. Three hundred four turns of talk (21%) were identified as talk related to reasoning, either practical reasoning or scientific reasoning. Practical reasoning made up 62% of the reasoning turns while 38% were scientific reasoning (Fig. 2 ).

The two types of reasoning differ in the justifications that are used to substantiate the claims or decisions made. Table 1 describes the differences between the two categories of reasoning.

4.2 Applications of Scientific Reasoning

Instances of engagement with scientific reasoning (for instance, using scientific concepts to justify, raising scientifically oriented questions, or providing scientific explanations) revolved around the conditions for photosynthesis and the concept of energy conversion when students were presenting their ideas or when they were questioned by their peers. For example, in explaining the reason for including fish in their plant system, one group of students made connection to cyclical energy transfer: “…so as the roots of the plants submerged in the water, faeces from the fish will be used as fertilizers so that the plant can grow”. The students considered how organic matter that is still trapped within waste materials can be released and taken up by plants to enhance the growth. The application of scientific reasoning made their design one that is innovative and sustainable as evaluated by the teacher. Some students attempted more ecofriendly designs by considering energy efficiencies through incorporating water turbines in their farming systems. They applied the concept of different forms of energy and energy conversion when their peers inquired about their design. The same scientific concepts were explained at different levels of details by different students. At one level, the students explained in a purely descriptive manner of what happens to the different entities in their prototypes, with implied changes to the forms of energy─ “…spins then generates electricity. So right, when the water falls down, then it will spin. The water will fall on the fan blade thing, then it will spin and then it generates electricity. So, it saves electricity, and also saves water”. At another level, students defended their design through an explanation of energy conversion─ “…because when the water flows right, it will convert gravitational potential energy so, when it reaches the bottom, there is not really much gravitational potential energy”. While these instances of applying scientific reasoning indicated that students have knowledge about the scientific phenomena and can apply them to assist in the problem-solving process, we are not able to establish if students understood the science behind how the dynamo works to generate electricity. Students in eighth grade only need to know how a generator works at a descriptive level and the specialized understanding how a dynamo works is beyond the intended learning outcomes at this grade level.

The application of scientific concepts for justification may not always be accurate. For instance, the naïve conception that students have about plants only respiring at night and not in the day surfaced when one group of students tried to justify the growth rates of Kailan─ “…I mean, they cannot be making food 24/7 and growing 24/7. They have nighttime for a reason. They need to respire”. These students do not appreciate that plants respire in the day as well, and hence respiration occurs 24/7. This naïve conception that plants only respire at night is one that is common among learners of biology (e.g. Svandova, 2014 ) since students learn that plant gives off oxygen in the day and takes in oxygen at night. The hasty conclusion to that observation is that plants carry out photosynthesis in the day and respire at night. The relative rates of photosynthesis and respiration were not considered by many students.

Besides naïve conceptions, engagement with scientific ideas to solve a practical problem offers opportunities for unusual and alternative ideas about science to surface. For instance, another group of students explained that they lined up their plants so that “they can take turns to absorb sunlight for photosynthesis”. These students appear to be explaining that the sun will move and depending on the position of the sun, some plants may be under shade, and hence rates of photosynthesis are dependent on the position of the sun. However, this idea could also be interpreted as (1) the students failed to appreciate that sunlight is everywhere, and (2) plants, unlike animals, particularly humans, do not have the concept of turn-taking. These diverse ideas held by students surfaced when students were given opportunities to apply their knowledge of photosynthesis to solve a problem.

4.3 Applications of Practical Reasoning

Teachers and students used more practical reasoning during an integrated STEM activity requiring both science and engineering practices as seen from 62% occurrence of practical reasoning compared with 38% for scientific reasoning. The intention of the activity to integrate students’ scientific knowledge related to plant nutrition to engineering practice of building a model of vertical farming system could be the reason for the prevalence of practical reasoning. The practical reasoning used related to structural design considerations of the farming system such as how water, light and harvesting can be carried out in the most efficient manner. Students defended the strengths of designs using logic based on their everyday experiences. In the excerpt below (transcribed verbatim), we see students applied their everyday experiences when something is “thinner” (likely to mean narrower), logically it would save space. Further, to reach a higher level, you use a machine to climb up.

Excerpt 1. “Thinner, more space” Because it is more thinner, so like in terms of space, it’s very convenient. So right, because there is – because it rotates right, so there is this button where you can stop it. Then I also installed steps, so that – because there are certain places you can’t reach even if you stop the – if you stop the machine, so when you stop it and you climb up, and then you see the condition of the plants, even though it costs a lot of labour, there is a need to have an experienced person who can grow plants. Then also, when like – when water reach the plants, cos the plants I want to use is soil-based, so as the water reach the soil, the soil will xxx, so like the water will be used, and then we got like – and then there’s like this filter that will filter like the dirt.

In the examples of practical reasoning, we were not able to identify instances where students and teachers engaged with discussion around trade-off and optimisation. Understanding constraints, trade-offs and optimisations are important ideas in informed design matrix for engineering as suggested by Crismond and Adams ( 2012 ). For instance, utterances such as “everything will be reused”, “we will be saving space”, “it looks very flimsy” or “so that it can contains [sic] the plants” were used. These utterances were made both by students while justifying their own prototypes and also by peers who challenged the design of others. Longer responses involving practical reasoning were made based on common-sense, everyday logic─ “…the product does not require much manpower, so other than one or two supervisors like I said just now, to harvest the Kailan, hence, not too many people need to be used, need to be hired to help supervise the equipment and to supervise the growth”. We infer that the higher instances of utterances related to practical reasoning could be due to the presence of more concrete artefacts that is shown, and the students and teachers were more focused on questioning the structure at hand. This inference was made as instructions given by the teacher at the start of students’ presentation focus largely on the model rather than the scientific concepts or reasoning behind the model.

4.4 Intersection Between Scientific and Practical Reasoning

Comparing science subject matter knowledge and problem-solving to the idea of categories and placement (Buchanan, 1992 ), subject matter is analogous to categories where meanings are fixed with well-established epistemic practices and norms. The problem-solving process and design of solutions are likened to placements where boundaries are less rigid, hence opening opportunities for students’ personal experiences and ideas to be presented. Placements allow students to apply their knowledge from daily experiences and common-sense logic to justify decisions. Common-sense knowledge and logic are more accessible, and hence we observe higher frequency of usage. Comparatively, while science subject matter (categories) is also used, it is observed less frequently. This could possibly be due either to less familiarity with the subject matter or lack of appropriate opportunity to apply in practical problem solving. The challenge for teachers during implementation of a STEM problem-solving activity, therefore, lies in the balance of the application of scientific and practical reasoning to deepen understanding of disciplinary knowledge in the context of solving a problem in a meaningful manner.

Our observations suggest that engaging students with practical inquiry tasks with some engineering demands such as the design of modern farm systems offers opportunities for them to convert their personal lived experiences into feasible concrete ideas that they can share in a public space for critique. The peer critique following the sharing of their practical ideas allows for both practical and scientific questions to be asked and for students to defend their ideas. For instance, after one group of students presented their prototype that has silvered surfaces, a student asked a question: “what is the function of the silver panels?”, to which his peers replied : “Makes the light bounce. Bounce the sunlight away and then to other parts of the tray.” This question indicated that students applied their knowledge that shiny silvered surfaces reflect light, and they used this knowledge to disperse the light to other trays where the crops were growing. An example of a practical question asked was “what is the purpose of the ladder?”, to which the students replied: “To take the plants – to refill the plants, the workers must climb up”. While the process of presentation and peer critique mimic peer review in the science inquiry process, the conceptual knowledge of science may not always be evident as students paid more attention to the design constraints such as lighting, watering, and space that was set in the activity. Given the context of growing plants, engagement with the science behind nutritional requirements of plants, the process of photosynthesis, and the adaptations of plants could be more deliberately explored.

5 Conclusion

The goal of our work lies in applying the theoretical ideas of Dewey and Bereiter to better understand reasoning practices in integrate STEM problem solving. We argue that this is a worthy pursue to better understand the roles of scientific reasoning in practical problem solving. One of the goals of integrated STEM education in schools is to enculture students into the practices of science, engineering and mathematics that include disciplinary conceptual knowledge, epistemic practices, and social norms (Kelly & Licona, 2018 ). In the integrated form, the boundaries and approaches to STEM learning are more diverse compared with monodisciplinary ways of problem solving. For instance, in integrated STEM problem solving, besides scientific investigations and explanations, students are also required to understand constraints, design optimal solutions within specific parameters and even to construct prototypes. For students to learn the ways of speaking, doing and being as they participate in integrated STEM problem solving in schools in a meaningful manner, students could benefit from these experiences.

With reference to the first research question of What is the extent of practical and scientific reasoning in integrated STEM problem solving, our analysis suggests that there are fewer instances of scientific reasoning compared with practical reasoning. Considering the intention of integrated STEM learning and adopting Bereiter’s idea that students should learn higher-order conceptual knowledge through engagement with problem solving, we argue for a need for scientific reasoning to be featured more strongly in integrated STEM lessons so that students can gain higher order scientific conceptual knowledge. While the lessons observed were strong in design and building, what was missing in generating solutions was the engagement in investigations, where learners collected or are presented with data and make decisions about the data to allow them to assess how viable the solutions are. Integrated STEM problems can be designed so that science inquiry can be infused, such as carrying out investigations to figure out relationships between variables. Duschl and Bybee ( 2014 ) have argued for the need to engage students in problematising science inquiry and making choices about what works and what does not.

With reference to the second research question , What is achieved through practical and scientific reasoning during integrated STEM problem solving? , our analyses suggest that utterance for practical reasoning are typically used to justify the physical design of the prototype. These utterances rely largely on what is observable and are associated with basic-level knowledge and experiences. The higher frequency of utterances related to practical reasoning and the nature of the utterances suggests that engagement with practical reasoning is more accessible since they relate more to students’ lived experiences and common-sense. Bereiter ( 1992 ) has urged educators to engage learners in learning that is beyond basic-level knowledge since accumulation of basic-level knowledge does not lead to higher-level conceptual learning. Students should be encouraged to use scientific knowledge also to justify their prototype design and to apply scientific evidence and logic to support their ideas. Engagement with scientific reasoning is preferred as conceptual knowledge, epistemic practices and social norms of science are more widely recognised compared with practical reasoning that are likely to be more varied since they rely on personal experiences and common-sense. This leads us to assert that both context and content are important in integrated STEM learning. Understanding the context or the solution without understanding the scientific principles that makes it work makes the learning less meaningful since we “…cannot strip learning of its context, nor study it in a ‘neutral’ context. It is always situated, always relayed to some ongoing enterprise”. (Bruner, 2004 , p. 20).

To further this discussion on how integrated STEM learning experiences harness the ideas of practical and scientific reasoning to move learners from basic-level knowledge to higher-order conceptual knowledge, we propose the need for further studies that involve working with teachers to identify and create relevant problems-of-explanations that focuses on feasible, worthy inquiry ideas such as those related to specific aspects of transportation, alternative energy sources and clean water that have impact on the local community. The design of these problems can incorporate opportunities for systematic scientific investigations and scaffolded such that there are opportunities to engage in epistemic practices of the constitute disciplines of STEM. Researchers could then examine the impact of problems-of-explanations on students’ learning of higher order scientific concepts. During the problem-solving process, more attention can be given to elicit students’ initial and unfolding ideas (practical) and use them as a basis to start the science inquiry process. Researchers can examine how to encourage discussions that focus on making meaning of scientific phenomena that are embedded within specific problems. This will help students to appreciate how data can be used as evidence to support scientific explanations as well as justifications for the solutions to problems. With evidence, learners can be guided to work on reasoning the phenomena with explanatory models. These aspects should move engagement in integrated STEM problem solving from being purely practice to one that is explanatory.

6 Limitations

There are four key limitations of our study. Firstly, the degree of generalisation of our observations is limited. This study sets out to illustrate what how Dewey and Bereiter’s ideas can be used as lens to examine knowledge used in problem-solving. As such, the findings that we report here is limited in its ability to generalise across different contexts and problems. Secondly, the lessons that were analysed came from teacher-frontal teaching and group presentation of solution and excluded students’ group discussions. We acknowledge that there could potentially be talk that could involve practical and scientific reasonings within group work. There are two practical consideration for choosing to analyse the first and presentation segments of the suite of lesson. Firstly, these two lessons involved participation from everyone in class and we wanted to survey the use of practical and scientific reasoning by the students as a class. Secondly, methodologically, clarity of utterances is important for accurate analysis and as students were wearing face masks during the data collection, their utterances during group discussions lack the clarity for accurate transcription and analysis. Thirdly, insights from this study were gleaned from a small sample of six classes of students. Further work could involve more classes of students although that could require more resources devoted to analysis of the videos. Finally, the number of students varied across groups and this could potentially affect the reasoning practices during discussions.

Data Availability

The datasets used and analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors would like to acknowledge the contributions of the other members of the research team who gave their comment and feedback in the conceptualization stage.

This study is funded by Office of Education Research grant OER 24/19 TAL.

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Tan, AL., Ong, Y.S., Ng, Y.S. et al. STEM Problem Solving: Inquiry, Concepts, and Reasoning. Sci & Educ 32 , 381–397 (2023). https://doi.org/10.1007/s11191-021-00310-2

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STEM Problem Solving: Inquiry, Concepts, and Reasoning

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• 1 Natural Sciences and Science Education, meriSTEM@NIE, National Institute of Education, Nanyang Technological University, Singapore, Singapore.
• PMID: 35125658
• PMCID: PMC8799441
• DOI: 10.1007/s11191-021-00310-2

Balancing disciplinary knowledge and practical reasoning in problem solving is needed for meaningful learning. In STEM problem solving, science subject matter with associated practices often appears distant to learners due to its abstract nature. Consequently, learners experience difficulties making meaningful connections between science and their daily experiences. Applying Dewey's idea of practical and science inquiry and Bereiter's idea of referent-centred and problem-centred knowledge, we examine how integrated STEM problem solving offers opportunities for learners to shuttle between practical and science inquiry and the kinds of knowledge that result from each form of inquiry. We hypothesize that connecting science inquiry with practical inquiry narrows the gap between science and everyday experiences to overcome isolation and fragmentation of science learning. In this study, we examine classroom talk as students engage in problem solving to increase crop yield. Qualitative content analysis of the utterances of six classes of 113 eighth graders and their teachers were conducted for 3 hours of video recordings. Analysis showed an almost equal amount of science and practical inquiry talk. Teachers and students applied their everyday experiences to generate solutions. Science talk was at the basic level of facts and was used to explain reasons for specific design considerations. There was little evidence of higher-level scientific conceptual knowledge being applied. Our observations suggest opportunities for more intentional connections of science to practical problem solving, if we intend to apply higher-order scientific knowledge in problem solving. Deliberate application and reference to scientific knowledge could improve the quality of solutions generated.

Keywords: Dewey; Practical Inquiry; Problem-centered knowledge; Referent-centered knowledge; Science Inquiry.

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Connections across disciplines in integrate…

Connections across disciplines in integrate STEM activity

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Frequency of different types of reasoning

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How to Improve Problem Solving Skills

Quick Summary

• Problem-solving skills are essential for personal growth, career advancement, and tackling life’s challenges. 62% of recruiters seek people who can solve complex problems.
• Learning how to improve problem solving skills will help you applying these skills to various situations from debugging to organizing schedules.
• Core techniques include breaking problems down, analyzing systematically, and applying creative thinking are essential strategies in different fields. Continuous practice, embracing challenges, and learning from mistakes helps to improve problem-solving abilities.

Table of Contents

Picture this situation

“You’re faced with a tricky situation at work, a challenging coding problem, or a complex personal decision. Your heart races, your palms get sweaty, and your mind goes completely blank. Sound familiar?”

That’s where solid problem-solving skills come in handy. They’re not just for math whzzes or tech gurus – they’re essential tools for everyone, every day.

According to Abraham Lincoln “ Give me six hours to chop down a tree and I will spend the first four sharpening the axe. “  

This famous quote depicts the importance of problem solving skills. 

How to improve problem solving skills isn’t just about acing tests or impressing your boss (though those are nice perks). It’s about growing as a person, boosting your confidence, and opening doors to new opportunities you might never have imagined. Currently, 62% of recruiters are seeking people who can solve complex things. 

Whether you’re debugging code that refuses to cooperate, engineering the next big thing that could change the world, or just figuring out how to organise your chaotic schedule, these skills are your trusty sidekick.

So, let’s roll up our sleeves and get into the nitty-gritty of becoming a problem-solving pro.

Core Problem Solving Techniques

At the heart of how to improve problem solving skills lies a set of core techniques. These are your go-to strategies, applicable across various fields and situations. Think of them as the Swiss Army knife in your mental toolkit – versatile, reliable, and always ready when you need them.

• Break it down

When faced with a big, scary problem, your first instinct might be to run for the hills. Instead, take a deep breath and start slicing that monster into smaller, more manageable chunks. It’s like eating an elephant – one bite at a time. This approach makes even the most daunting tasks seem doable.

Here’s where you let your imagination run wild. Let your ideas flow freely, no matter how crasy they might seem. No judgment, just pure creativity. You never know which wild idea might lead to the perfect solution. Remember, some of the world’s greatest inventions started as “crasy” ideas.

Once you have a list of potential solutions, it’s time to put on your critic’s hat. Weigh the pros and cons of each option. Consider factors like feasibility, resources required, potential outcomes, and possible obstacles. This step helps you separate the wheat from the chaff.

Choose the best solution and put it into action. Remember, a good plan today is better than a perfect plan tomorrow. Don’t get stuck in analysis paralysis – sometimes, you need to take the plunge and learn as you go.

After implementing your solution, take a step back and assess the results. What worked? What didn’t? This reflection is crucial for continuous improvement. It’s not just about solving the current problem, but also about becoming better at problem-solving in general.

Problem-solving is rarely a one-and-done deal. Use what you’ve learned to refine your approach and tackle similar problems more effectively in the future. Each problem you solve is a stepping stone to becoming a better problem solver.

Improving Problem Solving Skills in Different Fields

Now, let’s explore how to improve problem solving skills in specific areas. Whether you’re a budding programmer dreaming of creating the next big app, an aspiring engineer with visions of innovative designs, or a student preparing for competitive exams, we’ve got you covered.

Programming Problem Solving Skills

In the fast-paced world of technology, knowing how to improve problem solving skills in programming is like having a superpower. Here’s how you can level up your coding game:

• Code regularly: Practice makes perfect, and coding is no exception. Set aside time each day to write code, even if it’s just for fun. The more you code, the more natural it becomes.
• Take on challenges: Platforms like LeetCode, HackerRank, and CodeWars offer coding pussles that will put your skills to the test. Start with easier problems and gradually work your way up to more complex ones.
• Learn algorithms: Understanding different algorithms and data structures is like adding new tools to your programming toolkit. They help you solve problems more efficiently and elegantly.
• Pair program: Two heads are better than one. Collaborate with fellow coders to tackle problems together. You’ll learn new approaches and perspectives while improving your communication skills .
• Review and refactor: Look back at your old code. Can you make it more efficient? Cleaner? This process will sharpen your skills over time and help you develop a keen eye for quality code.

Engineering Problem Solving Skills

For those wondering how to improve problem solving skills in engineering , here are some targeted strategies:

• Think analytically: Break down complex engineering problems into smaller, solvable components. This approach helps you tackle even the most daunting projects step by step.
• Use simulations: Leverage software tools to model and test your solutions before implementation. This can save time, resources, and prevent costly mistakes.
• Stay updated: Engineering practices evolve rapidly. Keep learning to stay ahead of the curve. Attend workshops, read journals, and engage with the engineering community.
• Cross-disciplinary approach: Don’t limit yourself to one field. Often, the best engineering solutions come from combining knowledge from different areas. Biology might inspire a mechanical design, or psychology could inform a user interface.

Tips to Improve General Problem-Solving Skills

Wondering how to improve solving problem skills in general? Here are some universal tips that apply across all fields:

• Identify and define the problem clearly: Start by pinpointing the exact issue at hand. Ask yourself, “What’s the real problem here?” Often, what seems to be the problem is just a symptom of a deeper issue. Take time to articulate the problem in clear, specific terms. This clarity will guide your entire problem-solving process.
• Gather all relevant information and data: Before jumping to solutions, collect as much pertinent information as possible. This might involve research, asking questions, or analysing data. The more informed you are, the better equipped you’ll be to find an effective solution.
• Brainstorm multiple solutions without judgment: Let your creativity flow freely. Generate as many potential solutions as you can, no matter how outlandish they might seem at first. This divergent thinking can lead to innovative approaches you might not have considered otherwise.
• Evaluate and compare potential solutions: Once you have a list of possible solutions, critically assess each one. Consider factors such as feasibility, resources required, potential outcomes, and possible obstacles. This analytical approach helps you narrow down your options to the most promising ones.
• Break the problem down into smaller, manageable steps: Large, complex problems can be overwhelming. By breaking them down into smaller components, you make them more approachable and easier to tackle. This method also helps you identify specific areas that might need more attention or resources.
• Develop a step-by-step action plan: Once you’ve chosen a solution, create a detailed plan for implementation. Outline the specific steps you’ll take, set deadlines, and allocate resources. This roadmap will keep you focused and organised throughout the problem-solving process.
• Implement the chosen solution with confidence: With your plan in place, it’s time to take action. Move forward decisively, trusting in the thought process that led you to this solution. Remember, even if things don’t go perfectly, you can always adjust your approach.
• Monitor progress and make adjustments as needed: Regularly assess how well your solution is working. Be prepared to make tweaks or even significant changes if you encounter unexpected challenges. Flexibility is key in effective problem-solving.
• Reflect on the outcome to learn from the experience: Once you’ve resolved the problem, take time to review the entire process. What worked well? What could have been done differently? This reflection helps you refine your problem-solving skills for future challenges.
• Practice problem-solving regularly to build skills and confidence: Like any skill, problem-solving improves with practice. Seek out opportunities to solve problems in your daily life, work, or even through pussles and brain teasers. The more you practice, the more natural and effective your problem-solving abilities will become.

Specific Techniques for Enhancing Problem Solving Skills

Let’s dive deeper into how to improve analytical and problem solving skills, how to improve complex problem solving skills, and more.

Analytical and Logical Reasoning

To learn how to improve logical reasoning and problem solving skills, and boost your analytical prowess, follow the tips below:

• Play strategy games: Chess, Sudoku, and similar games can sharpen your analytical skills. They force you to think several steps ahead and consider multiple possibilities.
• Practice logical pussles: Engage in logic problems regularly to strengthen your reasoning abilities. Crosswords, riddles, and brain teasers are great for this.
• Study mathematics: Math is the language of logic. Improving your math skills will naturally enhance your analytical thinking. Even if you’re not a “math person,” basic mathematical concepts can significantly boost your problem-solving abilities.

Creative Problem Solving

Wondering how to improve creative problem solving skills? Try these techniques:

• Brainstorm without limits: Let your imagination run wild. The crasiest ideas often lead to innovative solutions. Use techniques like mind mapping or free writing to get your creative juices flowing.
• Use mind mapping: Visualise problems and potential solutions to spark creativity. This technique helps you see connections you might have missed otherwise.
• Take breaks: Sometimes, stepping away from a problem allows your subconscious to work its magic. Ever noticed how great ideas often come to you in the shower or while taking a walk? That’s your subconscious at work.

Critical Thinking and Decision Making

For those pondering how to improve critical thinking and problem solving skills or how to improve decision making and problem solving skills, consider these strategies:

• Question assumptions: Don’t take things at face value. Always ask “why?” Challenging assumptions can lead to breakthrough solutions.
• Consider multiple perspectives: Look at problems from different angles to develop a well-rounded view. Try to put yourself in others’ shoes to gain new insights.
• Use decision-making frameworks: Tools like SWOT analysis, decision matrices, or the Eisenhower Box can help structure your thinking and lead to better decisions.

Enhancing Problem Solving Skills for Specific Exams

Preparing for exams requires a targeted approach. Here’s how to fine-tune your skills for specific tests:

If you’re wondering how to improve problem solving skills for JEE, try these tips:

• Understand the syllabus: Know what topics are covered and focus your efforts accordingly. This will help you prioritise your study time effectively.
• Practice time management: JEE is as much about speed as it is about accuracy. Learn to pace yourself and know when to move on from a difficult question.
• Join study groups: Collaborative learning can expose you to different problem-solving approaches. Explaining concepts to others can also reinforce your own understanding.

For those wondering how to improve problem solving skills in physics:

• Master the fundamentals: A strong grasp of basic principles will help you tackle complex problems. Make sure you have a solid foundation before moving on to advanced topics.
• Use mnemonics: Create memory aids to recall important formulas and concepts quickly. This can be a lifesaver during exams when time is of the essence.
• Solve problems daily: Consistent practice is key to improving your physics problem-solving skills. Set aside time each day to work on physics problems, gradually increasing the difficulty level.

Mastering Problem-Solving Skills: A Lifelong Journey

Mastering how to improve problem solving skills is a lifelong journey. It’s not just about acing exams or excelling at work – it’s about equipping yourself with the tools to navigate life’s challenges with confidence and creativity.

Remember, every problem you face is an opportunity to grow. Whether you’re debugging stubborn code, tackling a tough engineering problem, or just figuring out your daily schedule, each challenge helps you build your problem-solving muscles.

So, keep practicing, stay curious, and don’t be afraid to make mistakes. Embrace challenges as opportunities to learn and grow. After all, some of the world’s greatest discoveries came from problem-solving gone “wrong.” Who knows? Your next “failed” solution might just lead to an incredible breakthrough that changes the world.

As you continue on your journey to become a master problem solver, remember that the skills you’re developing are invaluable in every aspect of life. They’ll help you in your career, in your personal relationships, and in achieving your goals. So keep pushing yourself, keep learning, and never stop asking “How can I solve this?”

Frequently Added Questions (FAQs)

What are the key techniques to improve problem-solving skills.

The core techniques include breaking down problems into manageable parts, brainstorming a wide range of solutions, carefully evaluating options, implementing the best solution, reviewing the outcomes, and iterating based on what you’ve learned. Regular practice and exposure to diverse problems also play a crucial role.

How can I enhance my problem-solving skills in programming?

To improve your programming problem-solving skills, practice coding regularly, tackle coding challenges on platforms like LeetCode or HackerRank, learn and apply various algorithms and data structures, engage in pair programming, and regularly review and refactor your code. Additionally, working on personal projects can provide real-world problem-solving experience.

What role do problem-solving skills play in the workplace?

Problem-solving skills are crucial in the workplace for handling daily tasks, managing projects, resolving conflicts, and driving innovation. They help employees navigate challenges, make informed decisions, and contribute to the overall success of the organisation. Strong problem-solving skills can also lead to career advancement opportunities.

How can I improve my analytical and logical reasoning abilities?

To boost analytical and logical reasoning skills, engage in activities like solving pussles (e.g., Sudoku, crosswords), playing strategy games (e.g., chess), practicing logical reasoning problems, and studying mathematics. Reading books on logic and critical thinking can also be beneficial. Regular practice and challenging yourself with increasingly difficult problems is key.

What are some ways to boost creative problem-solving skills?

To enhance creative problem-solving, engage in open-ended brainstorming sessions, use mind mapping techniques to visualise problems and solutions, practice lateral thinking exercises, and allow time for ideas to incubate. Exposing yourself to diverse experiences and perspectives can also stimulate creativity. Remember, sometimes the most innovative solutions come from combining ideas from different fields.

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Simply 'having a go' is a great way to make a start on a mathematical problem.  Whatever happens, you will have learnt more about the situation and can then tweak your approach.  These activities all lend themselves to this 'trial and improvement' way of working.

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What is Subitising? A hidden gem in developing mental arithmetic skills

When they practise subitising, children develop their ability to recognize and quickly comprehend the quantities of objects, a skill that can help them with mental calculations, estimation and problem-solving.

In a world where calculators and smartphones have made mental arithmetic seem less important, it's crucial to remember the value of building strong foundational maths skills in children. One often-overlooked method that can greatly improve mental arithmetic is subitising. In this blog post, we'll explore the concept of subitising, its connection to mental arithmetic and how you can help your child develop these essential skills.

Subitising plays an important role in how young children learn and is a key part within the requirements of the National Curriculum. It is formally introduced in the Early Years Foundation Stage ( EYFS ), where children begin to develop this skill as part of learning number sense. By understanding and nurturing subitising skills, educators and parents can lay a strong foundation for a child's mathematical journey.

Understanding subitising

When they practise subitising, children develop their ability to recognise and quickly comprehend the quantities of objects through number patterns, a skill that can help them with mental calculations, estimation and problem-solving.

What is subitising?

Subitising — from the Latin word meaning ‘suddenly’ — is the skill of instantly seeing how many items are in a group . For example, if I hold up my hand, you normally wouldn’t have to count my fingers and thumb to know it’s five. Most people can subitise as many as six or seven randomly arranged items.

In the EYFS, children typically start by subitising up to 5, gradually progressing to understanding numbers up to 10. This progression aligns with the development of their overall number sense and prepares them for more advanced mathematical concepts in later years.

Editor’s note: There are some exceptions. Read our ‘ parents and teachers guide to identifying specific learning difficulties ’ to learn more.

Difference between perceptual and conceptual subitising

There are two primary types of subitising: perceptual and conceptual. According to the website Early Math Counts , perceptual subitising is instantly knowing how many are in a given set, while conceptual subitising is the ability to see sets of numbers within larger sets, such as seeing two fours in the eight of a domino. If the number of items is too large to see all at once our brains ‘decompose’ the group into smaller chunks before adding them back together.

Here are some examples to illustrate these types:

• Perceptual Subitising : Quickly recognizing that there are four dots on a dice without counting
• Conceptual Subitising : Seeing a domino with 6 dots and instantly recognizing it as two groups of three
• Advanced Conceptual Subitising : Looking at a 10-frame with 7 counters and immediately seeing it as a full five and two more

The role of subitising in early years maths education

Sue Gifford at the University of Roehampton explains that subitising helps children build images for numbers, visualise and learn number facts . Very young children are usually able to recognise the number of items in a group without having to count them. For instance, most four-year-olds recognise five dots on a dice, which helps them understand the cardinal value — or 'howmanyness' — of five. They can then link to the word ‘five’ and the numeric symbol ‘5.’

The Connection Between Subitising and Mental Arithmetic

How subitising develops number sense.

Subitising is important because it helps children gain a deep understanding of quantity. Without an understanding of quantity, they may have no option but to count everything when they start to face more complex problems.

John Van de Walle, a prominent mathematics educator and author , emphasised the importance of subitising in developing number sense and mathematical fluency in young children. He said subitising lays the foundation for more advanced mathematical understanding by allowing children to grasp the concept of number as a quantity, rather than just a symbol or a sequence. By being able to recognize small groups of objects without counting, children can develop a strong sense of numbers and their relationships.

As children progress through Key Stage 2, their ability to subitise can significantly enhance their understanding and application of column addition and subtraction.

For example:

• Quick recognition of number bonds within 10 (developed through subitising) can speed up mental calculations in column addition.
• Understanding of number composition (e.g., seeing 8 as 5 and 3) helps in regrouping during written subtraction.
• The ability to visualise quantities supports estimation skills, which are crucial for checking the reasonableness of answers in formal calculations.

The role of subitising in fostering mental arithmetic skills

There are several ways in which the skill of subitising contributes to the development of mental arithmetic abilities:

• Rapid recognition of small quantities : by instantly recognising quantities, children can perform mental calculations more rapidly, which leads to improved mental arithmetic skills
• Number combinations : subitising helps children develop an understanding of number combinations or number bonds. For example, when children are presented with the dots on a pair of dice, subitising allows them to see the numbers as parts that make up a whole, which supports mental arithmetic skills such as addition and subtraction
• Number decomposition : subitising helps in number decomposition, or breaking a number down into its component parts. This allows children to quickly recognise different smaller groups within the larger set, which is crucial for mental arithmetic skills such as multiplication
• Mental estimation : by quickly perceiving quantities without counting, children can estimate the total number of objects or the result of an operation more effectively, which is a valuable skill in various mental arithmetic tasks that require approximations and quick calculations
• Building number sense : subitising supports the development of an intuitive understanding of numbers and their relationships

The link between subitising and other cognitive abilities

A strong understanding of numbers continues to develop over time through the exploration of numbers, visualising numbers and relating to numbers in different ways. Judy Hornigold provides this checklist for tracking how children are progressing with number sense :

• An awareness of the relationship between number and quantity
• An understanding of number symbols, vocabulary, and meaning
• The ability to engage in systematic counting , including notions of cardinality — knowing the symbol 4 denotes a group of four objects — and ordinality — knowing that 4 comes before 5 and after 3.
• An awareness of magnitude, or the ability to compare groups
• An understanding of different representations of number
• Competence with simple mathematical operations

Subitising is also linked to the following cognitive skills:

• Visual perception : the ability to discriminate and differentiate small groups of objects quickly
• Working memory : the ability to hold a mental representation of the quantity and rapidly process it
• Attention : the ability to focus attention on the visual stimuli and selectively attend to relevant information
• Numerical cognition : understanding and manipulating numerical quantities
• Executive functioning : a capacity for mental flexibility, inhibition of irrelevant information and cognitive control, which contribute to the development of executive functions involved in goal-directed behaviour, self-regulation, and problem-solving, since children must quickly process the quantity and inhibit the impulse to count

Activities to Enhance Subitising and Mental Arithmetic Skills

There are a variety of ways to practise, but one fun way is through introducing subitising games in the classroom. Let’s take a look at a few that support early years learning.

Dot card games

Dot card games present a variety of dot arrangements that help children develop their mental image of specific quantities. Try this downloadable dot card game from Young Mathematicians called ‘ Which One Doesn’t Belong .’ As children practise, explain how chunking dots into groups can make it easier to subitise. For example, chunking two dots and two dots helps you see that there are four dots.

For a deeper dive into dot patterns and cards using early number sense, see this article by Jennie Back .

Ten-frame activities

Ten-frames not only help children learn to subitise, but also to develop their understanding of ten as one of the most significant numbers in our number system. The National Council of Teachers of Mathematics has created a cool applet that helps develop addition and counting skills through playing games with ten-frames.

These informative blogs take a closer look at ten-frames and number sense:

• ‘ Early years maths mastery: introducing the CPA approach ,’ by  Sabrina Pinnock
• ‘ Number Sense Series: A Sense of 'Ten' and Place Value ,’ by Jenni Way

An example of ten frames built inside the Maths — No Problem! Visualiser app

Number line games

Teachers have used number lines for decades to teach number sense and arithmetic. Try these number line activities to help children strengthen their subitising skills:

• Use sticky notes to line up numbers
• Build a Lego number line
• Create a life-size number line (see video below)

Dice games are useful for helping children build images for numbers, visualise and learn number facts. Here are some good dice games from The Teacher Next Door , including ‘Build it Big’ and ‘Roll and Round.’ We Are Teachers also has some fun ideas for dice games including ‘Tenzi’ and ‘Dice Wars.’

Using hands and fingers in games

Hold up a number of fingers quickly and ask children to say how many they see. Gradually increase the difficulty by using both hands or asking children to show numbers in different ways.

You can use this video to quickly set up the exercise as you teach subitising.

Tips for Parents and Educators

Encourage number talks and discussions.

Talking with their peers, asking questions, and debating best approaches to problem-solving help students develop more complex thinking and reasoning skills, says Nina Parrish in Edutopia . Psychologists such as Jean Piaget and Lev Vygotsky also advocated the idea of metacognitive talk because children benefit from being active participants in the construction of knowledge. Conversations can create productive conflict that helps students develop multiple perspectives, leading to deeper learning.

There are many sources for exploring maths talks:

• Catherine Reed on her website The Brown Bag Teacher offers some unique methods for encouraging number talks
• The National Association for the Education of Young Children gives several tips for supporting maths readiness through maths talks
• Ruth Parker’s Number talks are a great way to develop number flexibility with your children or pupils
• Here’s Jo Boaler from Stanford University teaching a visual dot card number talk .

In addition, there are several strategies for encouraging metacognitive talk:

• Limit teacher talk time: most of the thinking and reasoning work should be left to the students
• Use open-ended questions and encourage problem-solving: try to ask more questions that are open-ended and encourage students to find the answer on their own or explore the process to find the answer collaboratively
• Explain the steps or outline the process: encourage students to explain their thinking process and how to arrive at an answer. Journalling is an excellent way to do this
• Generate knowledge and new examples: have students create their own unique examples and then collaborate with a partner to compare and contrast what they came up with and check each other’s work
• Take on a specific role in the critical thinking process: talk can be used as a scaffold that allows students to engage in assignments with increased rigour. Suggest strategies such as reciprocal teaching , where students work in groups to analyse a complex fiction or nonfiction text or sort through a maths word problem, each taking on a specific thinking role in order to practise making predictions, asking questions, clarifying and summarising

Incorporating subitising activities into daily routines

There are many ways to blend subitising activities into your daily routine. Some examples of these for the classroom from Rhody Girl Resource are brain breaks and time fillers. For some tips on how to encourage early maths skills at home, check out this blog post from The Preschool Toolbox , which advises parents to use mittens, socks, play dough pieces and dice to create small groups of items. Once your child can count a group correctly, progress to visual scanning for mental maths. Learning how to do quick visual scans will help your children strengthen their subitising skills.

From Jump Math , here are some examples of how to develop numeracy skills at home, including ‘The Bedtime Maths Challenge’ and ‘Bathroom Tiles.’

Creating a supportive learning environment

“The single most important factor in a child’s schooling is how they feel about themselves as individuals and as learners within the classroom,” says Katie Bowles, who looks at ‘ the psychologically safe classroom and restorative practice ' through the work of Becky Carlzon, co-author of The Learning Powered Approach and Mark Finnis, who wrote Restorative Practice , in which he explores the value of relationships within a school setting.

According to Teresa Duncan in Math for All , the saying ‘Maslow before Bloom’ sums up the decades of research that show how a sense of physical and emotional safety is the foundation for the development of social, emotional, and academic competencies. In other words, safe, supportive, classroom learning environments are important in supporting mathematics achievement.

This is demonstrated in a study by Rebekah Berlin and Julie Cohen , in which they analyse 400 lessons from 49 elementary schools. They discovered that the highest degree of mathematical engagement was found only when the learning environment was emotionally supportive.

To emphasise this point, a 2022 study in Frontiers of Psychology found that among several factors, “classroom climate provides the most critical microsystems directly affecting student learning process and outcomes.” The study further explains classroom climate includes at least three essential components — teacher-student interaction, instructional support and social-emotional support.

By understanding the concept of subitising and incorporating it into your child's learning, you can help them build a strong foundation in maths. With consistent practice and encouragement, your child will soon be able to make quick mental calculations, setting them up for success by helping them build maths skills for the future.

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A research team from the Beijing Academy of Artificial Intelligence and China University of Mining & Technology has proposed a scalable dataset for programmatic mathematical reasoning called InfinityMath. According to the authors, InfinityMath is supposed to decouple numeric values from problems stated in mathematics. This way, creating a huge, diverse dataset will require a manageable amount of computational resources. The dataset was created from seven high-quality math sources. It has over 101,380 data points. This makes it quite a comprehensive tool for enhancing the reasoning ability of artificial intelligence models.

The methodology of InfinityMath is multistep for maximum scalability and logical consistency. Masking numerical values of math problems creates generic templates that provide a base for generating problem-solving programs. These are then taken as general templates for developing programs that do not refer to specific numbers, logically following the same reasoning procedure for all possible numerical variations. It can efficiently scale data and improve the resiliency of AI models across different mathematical challenges. Such programs could be generated with sophisticated language models like GPT-4 to reduce potential errors and improve overall quality.

The models fine-tuned with the InfinityMath dataset performed quite well across several benchmarks. For example, aided by the InfinityMath dataset, the Llama2 model showed sensational accuracy improvements in the GSM8K dataset at 316.44% and in the MATH dataset at 1067.6%. Another model fine-tuned on this dataset was CodeLlama, which also showed huge improvements: 120.58% in SVAMP and 1118.09% in SimulEq. These results show that, at the very least, InfinityMath can increase AI models’ accuracy and robustness and improve their reliability in solving various mathematical problems. This consistency was also ahead regarding logical outcomes due to numerical variations; traditional datasets often lack performance.

Therefore, The InfinityMath effect extends beyond mere numerical accuracy to strike at perhaps the most fundamental feature of mathematical reasoning. The authors performed strict, improved evaluations with existing test sets, such as GSM8K+ and MATH+, differing only in the numerical values. Models trained on InfinityMath showed higher performance in logical consistency than any other dataset in accuracy and model efficacy. This success underlines the role played by InfinityMath in further pushing the frontiers of mathematical reasoning and scaling and making an effective solution available to a very large class of AI models.

In other words, InfinityMath is a major improvement in mathematical reasoning, solving two major challenges: scalability and logical consistency. The dataset was curated by a dedicated research team from the Beijing Academy of Artificial Intelligence and the China University of Mining & Technology to ensure that a robust and highly extensible solution could ultimately allow AI models to solve extremely complex mathematical problems. In this case, the InfinityMath process not only separates numerical values from solving processes but also makes constructing a large, highly diversified dataset more efficient to enhance the accuracy and reliability of the AI models. These results thus enable gains in improvement to be witnessed with multiple benchmark-related performances. Therefore, this dataset could further improve AI and its applications in various fields.

Check out the Paper and Dataset . All credit for this research goes to the researchers of this project. Also, don’t forget to follow us on  Twitter and join our  Telegram Channel and  LinkedIn Gr oup . If you like our work, you will love our  newsletter..

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Nikhil is an intern consultant at Marktechpost. He is pursuing an integrated dual degree in Materials at the Indian Institute of Technology, Kharagpur. Nikhil is an AI/ML enthusiast who is always researching applications in fields like biomaterials and biomedical science. With a strong background in Material Science, he is exploring new advancements and creating opportunities to contribute.

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