Chapter 7: Thinking, Problem Solving and Creativity
In the 1860s a German chemist named August Kekulé was stuck. He was trying to work out the structure of benzene, a common chemical whose atoms simply would not fit into any arrangement he could draw. He read, he calculated, he sketched chains of atoms for weeks, and got nowhere. Tired one evening, he dozed off by the fire. In a half-dream, he later recalled, he saw atoms dancing and linking into long chains that twisted and coiled like snakes. Then one of the snakes seized its own tail and spun in a circle before his eyes. Kekulé woke with a jolt. What if the atoms of benzene were not a straight chain at all, but a closed ring?
That single image solved a problem that careful, step-by-step effort had failed to crack. His story captures something strange about the way we think. Some of our best ideas do not arrive while we are staring hard at a problem. They surface quietly, sometimes after we have stopped trying, drawing on pictures and knowledge stored deep in the mind. Thinking is not one single activity. It can be logical and orderly, or free and imaginative. It can reason carefully, weigh a decision, hunt for a solution, or leap to something entirely new.
Diagram
Fig. 7.0-1 – a thinker at a desk with tangled chains of scribbled thought looping upward into a glowing ring above the head, with faint branching paths labelled ‘reasoning’, ‘problem solving’ and ‘creativity’
By the end of this chapter you will be able to answer each of these questions, and understand a little more about the remarkable machinery of your own mind.
7.1 Thinking
You are reading these words, picturing what they mean, perhaps disagreeing with one of them. All of that is thinking. Thinking is the mental activity of working with images, ideas and concepts in the mind, so as to understand a situation, reach a goal or solve a problem. It is the ability that most clearly sets human beings apart from other animals.
Thinking is usually silent and hidden. Nobody can watch you think the way they can watch you walk. Psychologists therefore call it an implicit activity, meaning it goes on inside, without any obvious outward sign. Its results, however, are everywhere, from a farmer planning a crop to an engineer designing a bridge.
Thinking belongs to a family of mental activities called cognition — all the processes by which we gain, store and use knowledge. It serves three broad purposes. It helps us understand the situations we meet, identify the goals that will satisfy our needs, and work out ideas to reach those goals.
7.1.1 The Nature of Thinking
Thinking is a basic cognitive process — the foundation on which other mental abilities such as remembering, imagining, reasoning, decision making and problem solving are built. Take thinking away, and these fall with it.
At its core, thinking is the analysis, organisation and re-organisation of information. We take in information through our senses, store it, and later rearrange it to evaluate a situation or find a solution.
We rarely work with raw sensations directly. Instead the mind uses symbols — signs, words and mental images — that stand in for objects and events. When you hear the word “elephant”, you do not need an elephant in the room; a stored image appears, and you think using that.
Thinking ranges from the concrete to the abstract. At its simplest it works on concrete objects we can see and touch. At a higher level it becomes imaginative and abstract, handling ideas such as justice or infinity that have no physical shape.
Much of our thinking is purposive, aimed at a goal. But thinking can also drift without any goal, and it can shift from one subject to another in a fraction of a second.
7.1.2 The Elements of Thinking
To understand how thinking works, psychologists break it into three basic building blocks: the image, the concept and language. As shown in Fig. 7.1-1, these three feed into every act of thought. We examine image and language here; the concept is important enough to deserve a section of its own, which follows.
Diagram
Fig. 7.1-1 – three linked circles labelled ‘Image’, ‘Concept’ and ‘Language’ feeding arrows into a central circle labelled ‘Thinking’
7.1.2.1 Image
An image is a mental picture of a person, object or event, built up from past sensory experience. When you close your eyes and “see” your mother’s face, you are using a visual image.
Images are not limited to sight. We also form auditory images (the tune of a song), olfactory images (the smell of rain on dry earth) and kinesthetic images (the feel of riding a bicycle). Each is stored from the sense that first recorded it.
An image is more than a replay of the past. It also carries imagination and creative additions. While we think, we manipulate and re-organise these images to understand new information and to search for solutions.
Before rearranging your room, for instance, you may “see” the bed against the far wall in your mind without moving a thing. That mental trial run uses images to test an idea in advance.
7.1.2.2 Language
Language is a system of symbols — spoken, written or gestured — that lets us represent ideas and share them with others. It is one of the highest achievements of the human mind, and it is bound up closely with thought.
Language has two parts. The first is articulated vocal speech: the words, phrases and sentences we say aloud. The second is an unarticulated, sub-vocal part: sign language, gestures, posture, body language and eye contact. Writing, diagrams and pictures also carry language.
The spoken part of language can be analysed into a few elements.
Sound
Every language is built from a set of basic speech sounds. The smallest unit of sound in a language is called a phoneme — the /a/ in “ant” is one example. Sounds are produced as air from the lungs vibrates the vocal cords and is shaped by the tongue, lips and mouth.
Syntax
Syntax is the set of rules for arranging words into meaningful sentences. If a friend messages “What you doing are?”, the words are familiar but the syntax is wrong, and the meaning is muddled. The smallest unit of meaning in a language, meanwhile, is called a morpheme — for instance the “un” in “unfair”, or the “s” that turns a word plural.
Semantics
Semantics is the aspect of language that deals with meaning — what words and sentences actually stand for. The same string of sounds can carry more than one meaning, and semantics is what lets a listener recover the meaning the speaker intended.
Signs and Symbols
Signs, signals and symbols stand in for real objects and actions, and steer our thinking. The signs +, −, × and ÷ instantly direct a calculation. A red traffic light tells us to stop before we have put the thought into words. Because a symbol can replace a long idea with a short code, it saves time and effort while we think. Signs and symbols are, in this way, a further building block of thought.
7.1.3 Language and Thought
Are language and thought simply two names for one process? They are not. They arise from different origins, yet they are tightly linked, and psychologists have long argued about exactly how.
One strong view comes from Benjamin Lee Whorf. He proposed the linguistic relativity hypothesis — the idea that the language we speak shapes the way we think. The strongest form of the claim, that thought is entirely controlled by language, is called linguistic determinism. Because Whorf built on the work of his teacher Edward Sapir, this is often called the Sapir–Whorf hypothesis.
Not everyone agrees that language comes first. Jean Piaget, the Swiss psychologist, argued that thinking begins in a child before language does. A baby watches and imitates others long before it can speak, and imitation already involves thought. For Piaget, thinking is not built on language at the start; language later supplies thought with its symbols.
Lev Vygotsky, the Russian psychologist, offered a middle path. He suggested that around the age of two, as the child begins to speak, thought and language start to merge. From then on much of our thinking takes the form of inner speech — a silent conversation with ourselves. Some thoughts still need no language, and some speech needs little thought, but higher, conceptual thinking grows with the quality of that inner speech.
7.1.4 Types of Thinking
Thinking is not one uniform activity. Depending on our purpose, it takes different forms. The psychologist J.P. Guilford drew a famous distinction between two of them, convergent and divergent thinking. Adding the everyday varieties, we can describe five kinds of thinking: convergent, divergent, creative, goal-oriented and aimless, as set out in Fig. 7.1-2. Creative thinking is so central to this chapter that it has a full section of its own later; the other four are described here.
Diagram
Fig. 7.1-2 – a central hub labelled ‘Types of Thinking’ with five branches to circles labelled ‘Convergent’, ‘Divergent’, ‘Creative’, ‘Goal-oriented’ and ‘Aimless’
7.1.4.1 Convergent Thinking
Convergent thinking is logical, focused thinking that moves step by step towards a single, correct answer. The word “converge” means to come together; here many pieces of information come together to point at one solution.
Convergent thinking works best when a problem has one right answer that can be reached by analysing what we already know. Most school examinations, arithmetic sums and multiple-choice questions call for it. Its outcome is conventional and expected, and it needs little creativity.
7.1.4.2 Divergent Thinking
Divergent thinking moves in the opposite direction. Instead of narrowing towards one answer, it spreads outwards, generating many possible solutions to a single problem. The word “diverge” means to branch apart.
Divergent thinking is loosely organised, spontaneous and free-flowing. It suits open-ended problems that allow many answers, such as thinking up new uses for an old bottle, or designing a poster. It is the engine of creativity, and it depends on brain-storming — producing a flood of ideas without judging them too soon.
Convergent and divergent thinking are not rivals; a skilled thinker uses both. We often spread out first to generate many options, then narrow down to choose the best one.
7.1.4.3 Goal-oriented Thinking
Goal-oriented thinking is directed thinking that fixes a target, measures progress towards it, and evaluates the outcome. It is the disposition to aim at a useful, realistic result and to keep adjusting one’s actions until that result is reached.
The goal must be realistic, matched to the person’s ability and situation. An unrealistic goal pushes thinking into frustration rather than progress.
Goal-oriented thinking carries a hidden danger called tunnel vision — the tendency to fix so tightly on one goal that all other possibilities are ignored. A student pushed towards a single career by anxious parents may develop a tunnel vision that keeps other talents from blooming.
7.1.4.4 Aimless Thinking
Aimless thinking is thinking that has no goal and no fixed subject. The mind drifts from one idea to the next, as it often does in a dull class or on a long bus ride.
Because it is undirected, aimless thinking usually does little to solve problems. If it hardens into a habit, it can waste time and feed low moods. Goal-oriented thinking, by contrast, tends to carry a person forward.
Yet undirected thinking is not entirely useless. When the mind wanders freely, it sometimes makes fresh connections that focused effort had missed — which is part of how a ring first appeared to Kekulé by the fire. The skill lies in giving most of our thinking a direction, while allowing a little free wandering that can quietly feed new ideas.
7.2 Concepts and Concept Formation
Look at a hatchback, a sedan and a large SUV. They differ in size, shape and price, yet you have no trouble calling all three a “car”. You are using a concept — a general idea that groups together objects or events sharing common features.
A concept is a way of sorting the world into categories on the basis of shared features, so that we can understand and remember things easily. However different the individual cars may look, they all have wheels, a steering system, an engine and brakes, and that common core is the concept of a car.
Concepts are not formed only about concrete objects that we can see and touch. They also cover abstract ideas, qualities and feelings, such as love, honesty or independence. A child builds the concept of independence not from any single object but from many stories, examples and experiences.
Once formed, concepts are stored in memory as mental representations of a whole group of similar things. They are the products of understanding and logical thinking, and they become the foundation for all higher-order thinking that follows.
The importance of concepts in thinking is hard to overstate. They organise a crowded world into a manageable number of categories, they let us recognise new things quickly, and they make communication and reasoning possible. Without them, every object would have to be understood from scratch.
7.2.1 Types of Concepts
Concepts differ in how they are built and how much thinking they demand. Four types are usually recognised: simple, complex, natural and artificial concepts.
Simple Concepts
A simple concept is formed on the basis of a single common feature of an object or event. Because it needs only one feature, it develops early, and can even be seen in animals and very young children.
Simple concepts are built by ordinary observation of similar features in the surroundings. When a child forms the concept of the colour “yellow”, they begin to pick out all yellow things — flowers, fruits, clothes, buses — regardless of any other differences between them.
Complex Concepts
A complex concept goes further, involving the identification, understanding, generalisation and differentiation of the common features of many items. It is built through repeated interaction with the environment and demands intelligence and logical thinking, so it is mostly seen in older children and adults.
Complex concepts come in two forms. A conjunctive concept is defined by two or more features that must be present together. A landlady, for example, must be both a woman and an owner of property; either feature alone is not enough.
A disjunctive concept is defined by any one of two or more features, or by all of them. An ice-cream counts as an ice-cream whether it contains chocolate, or fruit, or a mix of both. The features do not have to occur together.
Natural Concepts
A natural concept develops naturally through direct and indirect experience with the world. Concepts of man, woman, food, animals and day and night are natural concepts that we use in everyday thinking without effort.
Direct experience is not always necessary. A person living in Kashmir has direct experience of snowfall, while someone who has never left the Andaman Islands may know snow only through photographs and films. Both, however, form a natural concept of snow.
Natural concepts grow around a prototype — the original, most typical pattern to which new examples are compared. New items that match the prototype closely are quickly folded into the concept. Ask most Indians to think of “food” and rice, dal, chapatti and vegetables come to mind first; noodles or pizza are added later, after enough exposure.
Artificial Concepts
An artificial concept, also called a logical concept, is built from a unique and precise set of rules and features. Grammar rules, mathematical formulae and geometric figures are all artificial concepts, and they are clear and unambiguous by design.
A circle, for instance, is defined by fixed rules: it must have a centre, and every point on it must lie at a fixed distance from that centre. Anything meeting those rules is a circle, and anything failing them is not.
It is worth noticing that our everyday concepts are often harder to pin down than artificial ones. If several people are asked to define love, each will describe it in their own way, because everyday concepts are a rich blend of many ideas and personal convictions.
7.2.2 The Formation of Concepts
How does a concept take shape in the first place? Much of what we know about this comes from the work of the cognitive psychologists Jerome Bruner, Jacqueline Goodnow and George Austin.
Concept formation is the process by which the common features of certain things are identified and then used to classify those things into a group. It asks the thinker to compare items, keeping the features that matter and setting aside those that do not.
The process of concept formation moves through three stages, each leading into the next, as shown in Fig. 7.2-1.
Diagram
Fig. 7.2-1 – flow: Perception (forming mental images of things) → Abstraction (picking out the common features and ignoring the uncommon) → Generalisation (forming one general idea, which is the concept)
The first stage is perception. Experiences and learning give us mental images of the objects and events around us, and these images are the starting point for building any concept.
The second stage is abstraction. The mind examines the perceived images and picks out what is common to them all, while ignoring what is uncommon or exceptional. This act of noticing shared features is abstraction.
The third stage is generalisation. After abstracting the common features many times over, the thinker forms one general idea about the shared properties of a group of things. That general idea is the concept.
Consider a small child who sees a black cow for the first time and is told it is a cow. At first the idea is very particular. Later, meeting a white cow and a red cow, the child hesitates, then works out that these too are cows, by comparing them with the stored image of the first. Across many such comparisons the differences of colour and size fall away, and the common features remain to form the concept “cow”.
Because natural concepts are formed early and stored in memory, they shape how we handle everything new. When we meet an unfamiliar object, its features are automatically compared and matched with the features we have already stored. The better the match, the faster the new object is absorbed into an existing concept.
Concepts are also supported by our schema — a bundle of organised knowledge we hold about an object, idea or event. A rich schema for mothers and motherhood may, over time, help a child build wider concepts such as patience, compassion and forgiveness.
7.3 Reasoning
Suppose your bus does not arrive and you work out, from the time, the traffic and the distance, that walking will get you to school faster than waiting. You have just used reasoning — the logical part of thinking that moves us from what we know to a sensible conclusion.
Reasoning is the ability to think in a logical way towards a goal, in order to serve our adjustive demands — that is, our everyday need to cope with and adjust to situations. It draws on all the information stored in the mind to find solutions, and it is especially valuable for complex problems.
At heart, reasoning is about recognising links between cause and effect, and using them to predict or explain events. It is a higher, more advanced form of thinking, and it is organised, systematic, careful and purposive.
7.3.1 Types of Reasoning
Reasoning is usually classified into two broad kinds, which run in opposite directions, as Fig. 7.3-1 shows.
Diagram
Fig. 7.3-1 – two horizontal arrows: the upper labelled ‘Deductive reasoning’ pointing from ‘General principle’ to ‘Specific conclusion’; the lower labelled ‘Inductive reasoning’ pointing from ‘Specific observations’ to ‘General rule’
Inductive Reasoning
Inductive reasoning moves from particular cases to a general rule. If a fact is found true in one, two or several similar cases, we combine these instances and conclude that it holds for all such cases.
Suppose Seema feels tense before an examination, and so do Shekhar and Bobby. From these separate cases we reason to a general conclusion: examinations are stressful. Each observed case is a premise, a starting statement on which the reasoning rests.
A conclusion reached by induction is only probable, never guaranteed. It rests on the cases seen so far, and a single genuine exception can overturn it.
Deductive Reasoning
Deductive reasoning works the other way, from a general principle to a specific conclusion. We begin with a rule we accept and apply it to a particular case.
Starting from “examinations are stressful”, and knowing that Seema has an examination tomorrow, we conclude that Seema is stressed. Deduction is also used to test or verify established principles and hypotheses.
When the starting rule is true and the logic is valid, a deductive conclusion is certain. Its strength therefore depends entirely on the truth of the principle we begin with.
7.3.2 Common Errors in Reasoning
Reasoning does not always run true. Several kinds of error can slip in and pull a conclusion off course. Five of the most common are described below, and the first and last of them are worth watching especially closely.
Faulty Premises
The clearest way for reasoning to fail is to begin from a false assumption. Since reasoning verifies an idea by building on it, a wrong starting concept guarantees a wrong conclusion. This error is called faulty premises.
For example, from “this car is expensive” a person may reason that “its engine must be very powerful”. The premise is shaky: a car may be costly for its safety features, its electronics or its looks, not its engine. Believing something about a person only because of the group they belong to — their caste or religion, say — is also reasoning from a faulty premise, because the assumption about the group is itself false.
Biases
A bias is a preconceived idea that produces systematic errors in reasoning. Such cognitive biases are habitual tendencies to think in certain ways, and they quietly pull our judgement away from a fair, rational conclusion.
Because a bias operates as a fixed leaning rather than a one-off slip, its errors repeat in the same direction again and again, which makes them harder to notice.
Hindsight Error
The hindsight error is the tendency to believe, after an event has happened, that we had predicted it all along. It is also called the knew-it-all-along effect.
Once we know how something turned out, the outcome feels obvious and even inevitable, though we may have had no real basis for predicting it. The effect works at a few levels: we misremember our earlier opinion, we come to feel the event was bound to happen, and we believe we could have foreseen it.
Rationalisation
Rationalisation is reasoning used to defend ourselves. When a person has done something inappropriate, they may cook up a comfortable but false justification for it, rather than accept the real reason.
A student who has done badly in a test, for instance, may decide the teacher marked unfairly, instead of admitting to poor preparation. Rationalisation distorts the actual facts, and so it deforms the reasoning built on them.
Fallacy of Single Cause
The fallacy of single cause is the mistaken belief that an event has one simple root cause, when in reality several hidden causes act together. It is also known as the reduction fallacy.
Publishers, for example, often blame falling book sales entirely on television. But video games, smartphones, social media and e-books all play a part as well. Picking out one cause and ignoring the rest leads to faulty reasoning.
7.3.3 Improving Reasoning
Good reasoning helps in school, at work and in relationships, yet it does not always run in the right direction or reach the right answer. The good news is that reasoning can be trained and improved. Three broad strategies are useful.
Avoiding Errors
The first step is to become aware of one’s own errors and biases, which differ from person to person, and then to correct them. A biased person tends to see a situation from a single point of view, so it helps to gather ideas and look at the matter from several perspectives.
It also helps to resist over-generalising, since each event may have its own particular cause; to avoid making abrupt assumptions or predictions; and to steer clear of catastrophic thinking, where a stressful situation is imagined at its extreme worst. Keeping an open mind and weeding out logical fallacies through practice make reasoning steadier over time.
When errors in reasoning begin to cause real damage in daily life, a person may seek help from a trained expert, who can assess where the thinking goes wrong and suggest ways to correct it.
Examining Assumptions and Ambiguities
Before drawing any conclusion, it is wise to examine the concepts and assumptions the reasoning rests on. If they are faulty or unclear, no sound conclusion can follow; if they are true, the conclusion is far more likely to be true as well.
This calls for questioning one’s own decisions and beliefs repeatedly before acting on them, and for clearing away any ambiguity about the goal. Turning vague ideas into clear, testable form is one of the first and most important steps in reasoning well.
Guarding against Emotion
Strong emotions interfere with rational thinking. The Greek philosopher Plato pictured emotion and reason as two horses pulling a person in opposite directions, because emotional activity tends to be sudden and uncontrolled, while reasoning is meant to be calm and orderly.
During intense anxiety, anger or stress, the mind struggles to reason clearly and may make hasty, regrettable decisions. Guarding against this does not mean bottling emotions up. It means processing and releasing them in a healthy way that restores clear thinking.
Useful techniques include avoiding situations that trigger unwanted emotions, staying hopeful, changing the beliefs that fuel intense feelings, consciously shifting attention away from the flashpoint, and using physical methods such as exercise, yoga and progressive muscle relaxation to calm the body.
It is worth remembering that not all emotion is the enemy of reason. Positive emotions can sharpen our focus and strengthen our drive towards a goal.
7.4 Decision Making
Every day brings choices, from what to eat to which course to take after school. Decision making is the act of selecting a logical option from among several that are available.
More fully, it is the process of making a choice by identifying and gathering relevant information and then assessing the alternative solutions to a problem. Along the way a person weighs the positive and negative outcomes of each choice and selects the one best suited to the situation.
Breaking the process into small steps helps a person decide more thoughtfully, by organising the relevant information and searching out alternatives in an orderly way. A good decision-maker looks ahead, foresees the likely outcome of each option and picks the most appropriate one within a reasonable time.
A decision always involves two or more alternatives. If there is only one possible course of action, no decision has to be made. Choosing, then, is a matter of applying some criteria to select one alternative from among several.
Decision making does not always run smoothly. Common challenges include having too much or too little information, misidentifying the real problem in the first place, and being overconfident about how an option will turn out.
7.4.1 Characteristics of Decision Making
Decision making has a few defining features. It is, first of all, a process of selecting one option from among several alternatives, and it is carried out through deliberation and reasoning rather than on impulse.
The decision itself is the end product of this process. A good decision then helps a person plan suitable strategies for reaching their objectives through a clear, purposeful course of action.
Rational thinking and reasoning lie at the centre of it all. In the end, decision making aims at determining the exact solution to a problem.
7.4.2 Steps in Decision Making
Sound decisions tend to follow an orderly sequence of steps. Understanding these steps helps a person decide with more care and less confusion.
Identifying the particular problem. The first step is to recognise the problem clearly and decide to address it. It helps to ask what exactly the problem is, why it needs to be solved, and whether there is a deadline.
Gathering relevant information. Good decisions rest on facts, so the next step is to collect the information that matters. Some comes from within, such as one’s own perceptions and past experience, and some from outside, such as the advice of others, books and media.
Identifying the options. Once the problem is clear, the person lists the various possible courses of action. This is an internal preparation for judging each alternative against what is needed.
Weighing the options. Each option is now evaluated for its feasibility, acceptability and desirability. Weighing the pros and cons in this way shows which alternative has the best chance of success.
Choosing among the alternatives. The person becomes ready to select the best alternative, or a combination of alternatives, while staying aware of the risks attached to the chosen option.
Committing to the decision. Once the choice is made, the person stays with it and takes positive action to put it into effect.
Reviewing the decision and its consequences. Finally, the outcome is evaluated, to judge how correct the decision was and how far it achieved the target.
7.5 Problem Solving
Sometimes the way to a goal is not open. Imagine a boy who cannot find his school diary. The obvious path — reaching into his bag — has failed, and he must now think his way around the obstacle.
Situations like this call for problem solving. It is a higher form of cognitive functioning, used when the direct route to a goal is blocked and we must find some new or unusual way through.
Problem solving draws on facts, logic and stored knowledge to reach a goal that ordinary, habitual methods cannot. The boy searching for his diary must generate and test many possibilities — asking his teacher, checking the classroom, asking the bus attendant — until one works.
7.5.1 Steps in Problem Solving
Effective problem solving tends to move through an orderly set of steps, from spotting the problem to putting a solution into action.
Identifying or formulating the problem. We must first pin down exactly what the problem is, and identify the obstacles that are blocking the way to the goal.
Analysing the elements of the problem. The parts of the problem are understood and reorganised. We usually perceive a problem in one habitual way, and only by restructuring its elements do we often find a fresh, workable approach.
Searching for potential solutions. Using all the information available, the person generates as many possible solutions as they can. The more solutions are formulated, the better the chances of finding a good one.
Evaluating the possibilities of each solution. Each hypothetical solution is weighed carefully, to see which brings the person closest to the goal while making economical use of time and resources.
Attempting the appropriate solution. Finally the chosen solution is put into practice in the best possible way, and the result is evaluated. Without proper implementation, even a good plan fails.
7.5.2 Strategies of Problem Solving
People solve problems in different ways, ranging from crude guesswork to careful, systematic methods. Six common strategies are described below.
Trial and Error
Trial and error is the simplest and most primitive strategy. It amounts to guessing and then checking the result, trying one response after another until one works.
It is used when a person lacks the information for a more systematic approach. It gives no guarantee of a dependable solution, yet it has its place: new medicines such as antibiotics were once found largely by testing chemicals on laboratory animals until a useful effect appeared.
Algorithm
An algorithm is a set of step-by-step procedures that, if followed correctly, provides the correct answer to a problem. It is systematic, thorough and complete, working through every possibility in order.
Suppose the task is to make a real word from the letters O, N and S. An algorithm arranges them in every possible order — ONS, NOS, SNO, NSO — until it arrives at SON. The method is reliable but can be slow, and for large problems it becomes tedious.
Heuristics
A heuristic is a mental shortcut, or rule of thumb, drawn from past experience. Heuristics are fast and efficient, and they help us meet immediate goals, but they do not guarantee the best or perfect solution. The psychologists Amos Tversky and Daniel Kahneman identified two important kinds.
The availability heuristic judges how likely or common something is by how easily examples come to mind. The idea is roughly “if you can think of it easily, it must be important”. Because the media report dramatic, rare events heavily, such events spring to mind readily, and we overestimate their frequency.
The representativeness heuristic judges whether something belongs to a category by how closely it resembles the typical member of that category. Seeing a well-dressed man with an expensive watch, we may quickly decide he is wealthy, simply because he matches our image of a rich person.
Means-end Analysis
Means-end analysis arose from the problem-solving work of Allen Newell and Herbert Simon. It compares the present situation with the situation the person wants to reach, identifies the biggest difference between the two, and sets up sub-goals to close that difference.
The problem is thus broken into smaller sub-problems, and solving each one brings the person closer to the final goal. A man who wants to drive from home to work in the least time analyses the routes, compares their distances, checks the traffic and the road conditions on each, and finally selects the route that is shortest and clearest.
Analogy
Analogy solves a new problem by drawing on a similar situation from the past. We compare the two situations, find where they are alike, and apply the earlier solution to the new case.
We use analogy very often, usually without noticing it. When a familiar approach worked for one problem, we reach for it again the next time a similar problem appears.
Optimum Expected Utility
Optimum expected utility is the strategy of choosing the best option when the outcome is uncertain, as in gambling or buying insurance. Instead of ordinary reasoning, the person weighs each possible outcome by both its probability and its utility — the value that outcome holds for them.
In symbols, the expected utility of an option is \(EU = \sum (p \times u)\), where \(p\) is the probability of each outcome and \(u\) is its utility to the person. If a game offers ₹100 with a probability of \(0.5\), its expected value is \(0.5 \times 100 = ₹50\). A doctor uses the same logic in choosing, from several possible treatments, the one most likely to cure the patient.
7.5.3 Psychological Factors Affecting Problem Solving
Two psychological factors have a strong influence on how well a person solves problems.
Mental Set
Mental set is the psychological state a person brings to a problem. It is a tendency to approach new problems with the same methods that worked in the past.
A mental set can help, letting us solve a familiar problem quickly. But it can also hinder, blinding us to a better method simply because it is unfamiliar. When a person is under stress, this state also weakens attention and clear thinking, which further reduces their problem-solving power.
Functional Fixedness
Functional fixedness is a rigidity of thinking in which we see an object only in terms of its usual function, and fail to imagine a new use for it. It stops us from thinking differently or finding an alternative means to a solution.
The person in Fig. 7.5-1 feels too hot and wonders what to do, without realising that the thick book in their hands could be used as a fan. Functional fixedness of this kind often springs from stereotyped, habitual thinking, and it can quietly reduce a person’s ability to solve problems.
Diagram
Fig. 7.5-1 – a person fanning themselves with one hand while holding a thick book in the other, a thought-bubble showing the same book being waved as a fan; the point being an object seen only for its usual purpose
7.6 Creative Thinking
Most thinking looks for the standard answer. Creative thinking looks for a fresh one. It is the process that produces a novel, unusual solution rather than a conventional or general one.
Creative thinking calls for a high level of imagination. It combines and connects different, seemingly unrelated ideas and events, and out of them generates something new. Turning such imaginative ideas into reality is what we mean by creativity.
Creativity leans heavily on the divergent thinking you met earlier in this chapter — the free, outward-fanning search for many possible ideas. It then uses convergent judgement to pick out and refine the best of them.
Psychologists describe creative thinking as having three components. The intellectual component is the ability to produce ideas. The motivational component is the readiness to think of something new and to put it forward. The emotional component is the courage to think differently, resist the pressure to conform, and take a risk.
7.6.1 Characteristics of Creative Thinking
Creative thinking is exceptional and rare. Across the whole history of human civilisation, only a handful of truly great creative thinkers stand out. Those who are highly creative tend to share a set of characteristics.
- Curiosity — a strong desire to explore and to know the most unusual aspects of things.
- Generating ideas — the ability to produce a large number of unique and unusual solutions to a problem.
- Tenacity — the persistence to stay with their own ideas even when progress is slow.
- Risk-taking — being energetic and adventurous, and willing to take risks.
- Higher emotional sensitivity — feeling and responding to experiences, relationships and beauty more deeply than most.
- Self-expression — communicating their ideas clearly and well.
- Constructive self-criticism — being honestly critical of their own work, always looking to improve it.
- Independence — a strong need for autonomy, and a dislike of domineering, authoritarian control.
- High aspiration — setting high goals and striving to use their full potential.
Beyond these traits, creative thinking involves imagination, flows in many diverging directions at once, and encompasses open-mindedness, flexibility and a playful frame of mind. It is also an important element of critical thinking. Encouragingly, research suggests creativity is not entirely fixed and can be enhanced through training.
7.6.2 Stages of Creative Thinking
The whole process of creative thinking can be divided into four stages, first set out by the psychologist Graham Wallas. You saw all four at work in the story of Kekulé and the benzene ring at the start of this chapter. The sequence is shown in Fig. 7.6-1.
Diagram
Fig. 7.6-1 – flow: Preparation (gather facts, define the problem) → Incubation (set the problem aside; the mind works unconsciously) → Illumination (the solution appears in a sudden flash) → Verification (test the idea); with a loop-back arrow from Verification to Preparation if the solution fails
The first stage is preparation. Here the thinker sets a target, identifies the problem they are likely to meet, and gathers the facts, materials and past experience needed to solve it. They may read, brainstorm and collaborate with others. Often the problem resists solution even after days or weeks of effort, and the thinker eventually turns away from it.
The second stage is incubation. This stage shows no obvious outward activity. The thinker seems busy with other things, and the problem is set aside, yet thinking continues quietly in the unconscious mind. Ideas that were leading nowhere fade, while useful connections keep forming out of sight.
The third stage is illumination. Now the solution suddenly appears, often while the mind is busy with something quite different. This flash of insight is the Aha experience, also called the Eureka experience. It arrives all at once and is usually accompanied by a strong feeling of certainty and joy.
The fourth stage is verification. The illuminated idea is now evaluated and tested to check whether it actually works. If it does, it is accepted, with or without minor changes. If it does not, the thinker reverts to the beginning and starts the process again.
7.6.3 Fostering Creativity
Creativity is, to a large extent, a natural and spontaneous ability of the mind. But it can also be nurtured, and schools, workplaces and families can do a great deal to bring it out. Four broad approaches help.
Enriching Knowledge and Experience
Creativity grows on a wide base of knowledge and varied experience. Exposure to many situations, and to the creative work of others, gives the mind more raw material to combine. Active learning, hands-on work, and exploring different cultures and languages all enrich this base.
Encouraging Independence and Autonomy
Creativity flourishes in a friendly, fear-free environment where people can express their thoughts and act on their ideas. Ideas should be supported enthusiastically rather than mocked or penalised. And because creative ideas are only worthwhile when they can be tried out, people must be allowed to act on them.
Enhancing Curiosity
Curiosity is the drive that pushes a person to explore and learn. A person’s own interest is the starting point from which creative thinking takes off, which is why creativity is intrinsic — arising from within, for its own sake. Engaging people in activities around their interests, and fostering a spirit of inquiry, keeps curiosity alive.
Promoting Positive Mood
Research shows that creativity works best in a positive mood. Although some creative work does emerge from difficult moods, strong positive emotions are the surest encouragement. Feelings of love and affection, in particular, have long inspired art, music and literature. Whatever lifts our mood and generates positive feeling also tends to encourage creative thinking.
7.7 Summary
Thinking is an implicit cognitive process in which we manipulate mental representations — images, concepts and language — to understand situations, solve problems and reach goals. It underlies all other higher processes such as remembering, reasoning and deciding.
The three basic elements of thinking are image, concept and language. An image is a mental picture drawn from past sensory experience, and it can be visual, auditory, olfactory or kinesthetic.
Language is a rule-governed system of symbols. Its structural elements are sound (built from phonemes), syntax (the rules for arranging words, using morphemes) and semantics (meaning). Signs and symbols also stand in for objects and speed up thought.
Language and thought are distinct but interdependent. Whorf’s linguistic relativity hypothesis holds that language shapes thought; Piaget argued thought comes first; Vygotsky argued the two converge around age two. Thinking can occur without language, yet the two are deeply linked.
There are several kinds of thinking: convergent (logical, aimed at one best answer), divergent (free, generating many answers), goal-oriented (directed at a target, with the risk of tunnel vision), aimless (without a target) and creative (producing novel solutions).
A concept is a general category that groups objects or events by their common features. Concepts organise a crowded world into manageable groups and make higher thinking possible.
Concepts are of four types: simple (a single feature), complex (conjunctive, needing features together, or disjunctive, needing any one), natural (formed from experience and built around prototypes) and artificial or logical (built from precise defined rules).
Concept formation moves through three stages in order — perception, then abstraction (picking out common features while ignoring the uncommon), then generalisation (forming one general idea, which is the concept). Schemas of organised knowledge support this process.
Reasoning is organised, purposeful thinking that draws logical conclusions from available information to meet the demands of a situation.
Reasoning is of two kinds. Inductive reasoning moves from specific cases to a general rule, giving a conclusion that is only probable. Deductive reasoning moves from a general rule to a specific case, giving a conclusion that is certain if the premises are true.
Common errors in reasoning include faulty premises (false starting assumptions), biases, the hindsight error (the knew-it-all-along effect), rationalisation (self-justifying excuses) and the fallacy of single cause (blaming one cause where many act together).
Reasoning can be improved by avoiding errors, examining one’s assumptions and ambiguities, and guarding against strong emotion, which disrupts clear thinking.
Decision making is the process of choosing one option from two or more alternatives by gathering and assessing information. It follows seven steps: identifying the problem, gathering information, identifying options, weighing them, choosing, committing, and reviewing the outcome.
Problem solving is higher cognitive functioning used when the direct path to a goal is blocked by obstacles. Its steps are identifying the problem, analysing its elements, searching for solutions, evaluating them, and attempting the best one.
Strategies of problem solving include trial and error, algorithms (which guarantee the correct answer but can be slow), heuristics (fast mental shortcuts such as the availability and representativeness heuristics), means-end analysis (breaking a problem into sub-goals), analogy, and optimum expected utility (weighing each outcome by its probability and its value).
Two psychological factors can hinder problem solving: mental set (approaching new problems with old, fixed methods) and functional fixedness (seeing an object only in terms of its usual function).
Creative thinking produces novel, unconventional solutions rather than standard ones. It draws on divergent thinking and has intellectual, motivational and emotional components.
Creative thinkers tend to be curious, idea-rich, persistent, risk-taking, emotionally sensitive, self-expressive, independent and high in aspiration. Creativity is rare but can be strengthened through training.
Creative thinking passes through four stages, as described by Wallas: preparation (gathering material), incubation (setting the problem aside while the mind works unconsciously), illumination (the sudden Aha experience) and verification (testing the idea).
Creativity can be fostered by enriching knowledge and experience, encouraging independence and autonomy, enhancing curiosity, and promoting a positive mood.
7.8 Key Terms
- Abstraction — The stage of concept formation in which the mind picks out the features common to several things and ignores the uncommon ones.
- Aha experience — The sudden flash of insight during illumination, when a solution appears all at once with a feeling of certainty and joy; also called the Eureka experience.
- Aimless thinking — Thinking that drifts from one idea to another without any goal or settled subject.
- Algorithm — A step-by-step procedure that, if followed correctly, is guaranteed to give the correct answer, though it may be slow.
- Analogy — A problem-solving strategy that solves a new problem by applying the solution from a similar situation in the past.
- Artificial concept — A concept built from a precise, fixed set of defined rules, such as a circle or a grammar rule; also called a logical concept.
- Availability heuristic — A mental shortcut that judges how likely something is by how easily examples of it come to mind.
- Bias — A preconceived leaning that produces systematic errors in reasoning.
- Cognition — All the mental processes by which we gain, store and use knowledge.
- Complex concept — A concept that groups items by several features and requires intelligence and logic; it may be conjunctive or disjunctive.
- Concept — A general category that groups objects, events or ideas together by their common features.
- Concept formation — The process of identifying the common features of things and using them to sort those things into a category.
- Conjunctive concept — A concept defined by two or more features that must all be present together.
- Convergent thinking — Focused, logical thinking directed at finding a single correct or best answer.
- Creative thinking — The mental ability to produce a new, unconventional and useful solution rather than the usual one.
- Decision making — The process of choosing one option from two or more alternatives after gathering and weighing information.
- Deductive reasoning — Reasoning that moves from a general rule to a specific conclusion, which is certain if the premises are true.
- Disjunctive concept — A concept defined by any one of several features, which need not occur together.
- Divergent thinking — Free, spontaneous thinking that generates many possible solutions to an open-ended problem.
- Escalation of commitment — Sticking with a failing decision because of the effort already invested in it.
- Fallacy of single cause — The error of assuming one simple cause for an event that really has many causes; also called the reduction fallacy.
- Faulty premises — A reasoning error that begins from false or inaccurate assumptions.
- Functional fixedness — A rigidity of thinking in which an object is seen only in terms of its usual function.
- Generalisation — The stage of concept formation in which repeated abstraction leads to one general idea about a group of things.
- Goal-oriented thinking — Thinking directed at a chosen target, involving setting a goal, tracking progress and evaluating outcomes.
- Heuristic — A mental shortcut or rule of thumb that gives a quick but not guaranteed solution.
- Hindsight error — The tendency to believe, after an event, that one had predicted it all along; also called the knew-it-all-along effect.
- Illumination — The stage of creative thinking in which the solution suddenly appears in the mind.
- Image — A mental picture of a person, object or event, drawn from past sensory experience.
- Incubation — The stage of creative thinking in which the problem is set aside and the mind works on it unconsciously.
- Inductive reasoning — Reasoning that moves from specific cases to a general rule, giving a conclusion that is only probable.
- Language — A rule-governed system of symbols used for communication and for carrying on thought.
- Linguistic determinism — The strong claim that thought is entirely dependent on, and determined by, language.
- Linguistic relativity hypothesis — Whorf’s idea that the language a person speaks shapes how they perceive and think about the world.
- Means-end analysis — A problem-solving strategy that reduces the gap between the present state and the goal by setting sub-goals.
- Mental set — The tendency to approach new problems with methods that worked in the past.
- Morpheme — The smallest unit of meaning in a language, such as the “un-” in “unkind”.
- Natural concept — A concept formed through everyday experience and built around a typical example, or prototype.
- Optimum expected utility — Choosing, under uncertainty, the option with the highest value once each outcome is weighed by its probability and its worth.
- Phoneme — The smallest unit of sound in a language, such as the /a/ in “apple”.
- Post-traumatic growth — Positive personal change, including creative growth, that can follow a difficult or painful experience.
- Premise — A starting statement or assumption on which a piece of reasoning is built.
- Preparation — The first stage of creative thinking, in which the problem is defined and information is gathered.
- Prototype — The most typical example around which a natural concept is organised.
- Rationalisation — A reasoning error in which a person invents an acceptable but false justification to defend their behaviour.
- Reasoning — Organised, purposeful thinking that draws logical conclusions from available information.
- Representativeness heuristic — A mental shortcut that judges whether something fits a category by how much it resembles the typical member.
- Schema — An organised bundle of knowledge a person holds about an object, idea or event.
- Semantics — The aspect of language that deals with meaning.
- Simple concept — A concept formed on the basis of a single common feature.
- Sub-goals — Smaller intermediate targets set while breaking a problem down towards its final goal.
- Syntax — The set of rules for arranging words in an order that makes sense.
- Thinking — An implicit cognitive process in which mental representations are manipulated to understand situations, solve problems and reach goals.
- Trial and error — A simple problem-solving strategy of trying one response after another until one works.
- Tunnel vision — Focusing so narrowly on one goal that other options and possibilities go unnoticed.
- Verification — The final stage of creative thinking, in which the new idea is tested and evaluated.
7.9 Points to Ponder
If a concept exists in your mind for which your language has no single word, does that concept become harder to think about — and what does your answer suggest about how far Whorf was right?
Schools reward convergent thinking far more often than divergent thinking. What might be gained, and what might be lost, if that balance were reversed?
Heuristics are described as errors waiting to happen, yet the mind uses them constantly. If they are so unreliable, why has evolution left us leaning on them so heavily?
Incubation suggests that stepping away from a problem can help solve it. Where, then, is the line between a productive pause and simple avoidance?
Creativity is said to flourish in freedom, yet classical Indian music and dance produce startling originality within tight inherited rules. Does structure limit creativity, or does it make creativity possible?
If a person’s reasoning is shaped by biases they cannot see in themselves, how could they ever know their own thinking is faulty?
7.10 Objective Questions
The smallest unit of sound in a language is called a —
- morpheme
- phoneme
- prototype
- schema [1 mark]
Thinking that generates many possible solutions to an open-ended problem is called —
- convergent thinking
- goal-oriented thinking
- divergent thinking
- aimless thinking [1 mark]
The concept of a landlady, defined as a woman who is also an owner of property, is an example of a —
- disjunctive concept
- conjunctive concept
- simple concept
- natural concept [1 mark]
Reasoning that begins with several specific observations and arrives at a general rule is —
- deductive reasoning
- inductive reasoning
- analogical reasoning
- rationalisation [1 mark]
A problem-solving strategy that is guaranteed to produce the correct answer if followed correctly, but may be slow, is —
- a heuristic
- trial and error
- an algorithm
- analogy [1 mark]
The stage of creative thinking in which the problem is set aside while the mind continues to work on it unconsciously is —
- preparation
- incubation
- illumination
- verification [1 mark]
Judging an event to be more likely simply because vivid examples of it come to mind easily is called the —
- representativeness heuristic
- availability heuristic
- hindsight error
- fallacy of single cause [1 mark]
Fill in the blank: The stage of concept formation in which the mind picks out what is common to several perceived images, while ignoring what is uncommon, is called _______. [1 mark]
Fill in the blank: Seeing an object only in terms of the function it is normally used for, and failing to imagine a new use for it, is known as _______. [1 mark]
Fill in the blank: According to Whorf’s _______ hypothesis, the language a person speaks shapes the way that person perceives and thinks about the world. [1 mark]
State whether True or False: A conclusion reached by deductive reasoning is only probable and can be overturned by a single exception. [1 mark]
State whether True or False: Natural concepts are organised around prototypes formed through everyday direct and indirect experience. [1 mark]
State whether True or False: Rationalisation involves accepting the true reason for one’s poor performance rather than offering an excuse. [1 mark]
Match the following:
| Column A | Column B |
|---|---|
| a. Joy Paul Guilford | i. Linguistic relativity hypothesis |
| b. Benjamin Lee Whorf | ii. Four stages of creative thinking |
| c. Graham Wallas | iii. Convergent and divergent thinking |
| d. Newell and Simon | iv. Means-end analysis |
[1 mark]
- Match the term with its definition:
| Column A | Column B |
|---|---|
| a. Morpheme | i. Rules for arranging words meaningfully |
| b. Syntax | ii. The branch of language dealing with meaning |
| c. Semantics | iii. The smallest unit of meaning in a language |
| d. Prototype | iv. The most typical example of a natural concept |
[1 mark]
- Assertion (A): A conclusion reached through inductive reasoning is only probable, never certain. Reason (R): Inductive reasoning generalises from a limited number of observed cases, so a single exception can overturn it.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true and R is false.
- A is false and R is true. [1 mark]
- Assertion (A): Goal-oriented thinking can sometimes reduce a person’s flexibility. Reason (R): Rigid fixation on a single goal produces tunnel vision, so other useful options are never noticed.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true and R is false.
- A is false and R is true. [1 mark]
- Assertion (A): Heuristics always give a better solution than algorithms. Reason (R): Heuristics are mental shortcuts that save time by drawing on past experience.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true and R is false.
- A is false and R is true. [1 mark]
- Assertion (A): Artificial concepts are clearer and less ambiguous than everyday concepts such as love. Reason (R): Artificial concepts are built from a precise, fixed set of defining rules.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true and R is false.
- A is false and R is true. [1 mark]
- Assertion (A): A person may sincerely believe they predicted an election result that in fact surprised them. Reason (R): Once an outcome is known, the hindsight error makes it feel obvious and reshapes the memory of one’s earlier confidence.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true and R is false.
- A is false and R is true. [1 mark]
7.11 Check Your Concepts
Define thinking. [2 marks]
State the three basic elements of thinking. [2 marks]
What is a prototype? Give one example. [2 marks]
Define reasoning, and state any one of its characteristics. [2 marks]
List the three stages of concept formation in the correct order. [2 marks]
What is meant by tunnel vision in goal-oriented thinking? [2 marks]
Differentiate between a phoneme and a morpheme, with one example of each. [3 marks]
Differentiate between conjunctive and disjunctive concepts, with an example of each. [3 marks]
Give reasons why a conclusion reached by inductive reasoning can never be fully certain. [3 marks]
Briefly explain the fallacy of single cause, with one example. [3 marks]
Explain the availability heuristic with an example, and state one way it can mislead judgement. [4 marks]
Describe the process of means-end analysis, using an everyday example to show how sub-goals are used. [4 marks]
Explain functional fixedness with an example, and say why it acts as a barrier to problem solving. [4 marks]
Why does emotion interfere with reasoning? Describe any three techniques for guarding reasoning against strong emotion. [4 marks]
What would happen to a person’s everyday thinking if they were unable to form concepts? Explain with reference to the functions concepts serve. [4 marks]
Describe the four stages of creative thinking in order, explaining why the stage of incubation must follow preparation rather than precede it. [6 marks]
Compare convergent and divergent thinking in terms of their direction, the kinds of problem each suits, and the nature of the solutions each produces. Give one example of each. [6 marks]
Discuss the views of Whorf, Piaget and Vygotsky on the relationship between language and thought. How far does the evidence support the claim that thinking is impossible without language? [6 marks]
Describe the steps involved in decision making, and explain why the step of reviewing the decision matters even after the choice has been carried out. [6 marks]
Describe the main strategies of problem solving, and evaluate the claim that algorithms are always superior to heuristics. [6 marks]
Explain any four ways in which creativity may be encouraged, and discuss why a positive and autonomous environment matters for creative work. [6 marks]
7.12 Case-Based Questions
- Ananya’s school bus does not arrive one morning. She first works out exactly what her difficulty is — reaching school before the first bell. She then gathers information about the auto stand, the metro timing and the walking distance. She lists three possible routes, weighs each for cost and time, chooses the metro, and later that evening thinks about whether her choice worked well.
- Identify the cognitive process Ananya is using, and define it. [2 marks]
- Name and explain any three steps of this process visible in the situation. [3 marks]
- Why is the final step she carries out in the evening a useful one? [1 mark] [6 marks]
- During a family gathering, an uncle argues that children today read fewer books entirely because of television. A cousin adds that she “knew all along” that the local cricket team would lose yesterday’s match, though she had confidently predicted a win that morning. A third relative insists he failed his driving test only because the examiner disliked him.
- Identify the reasoning error shown by each of the three speakers. [3 marks]
- Explain why each of these errors distorts the conclusion reached. [3 marks] [6 marks]
- Kabir has been struggling for two weeks with the design of a working model for his school science exhibition. Nothing he tries fits together. Frustrated, he abandons it and spends the weekend helping his grandmother in the garden. While watering the plants, the arrangement he needs suddenly comes to him, complete and clear. He rushes back, sketches it, and then spends the next two days testing whether it actually holds together.
- Name the four stages of creative thinking, and identify which part of Kabir’s experience corresponds to each. [4 marks]
- Explain what is meant by the “Aha experience” as it occurred in the garden. [2 marks] [6 marks]
- A group of students is asked to find a way to carry a stack of heavy books across the school grounds. Ishaan repeatedly tries the same method that worked last year, though it is clearly failing this time. Nearby, Tara needs something to prop open a door and searches everywhere, walking past a large flat stone several times without noticing it could serve the purpose.
- Identify the psychological factor affecting problem solving in Ishaan’s case, and explain it. [2 marks]
- Identify the factor affecting Tara, and explain how it operates. [2 marks]
- Suggest one way each student could overcome the difficulty. [2 marks] [6 marks]