K Y T M Thinking Skills h C in k Critical Thinking and Problem Solving i John Butterworth and Geoff Thwaites n r e g v Second edition o S C John Butterworth and Geoff Thwaites k Thinking Skills n This lively coursebook encourages students to develop more ills o ti sophisticated and mature thinking processes by learning specifi c, di transferable skills independent of subject content which assist confi dent e d engagement in argument and reasoning. Critical Thinking and Problem Solving n As well as giving a thorough grounding in critical thinking and problem o c solving, the book discusses how to analyse and evaluate arguments, e manipulate numerical and graphical information and develop a range of S skills including data handling, logic and reasoning. The second edition of the book has been substantially updated with new s l and revised content throughout. The only endorsed coursebook offering l ki complete coverage of the Cambridge AS and A Level Thinking Skills S syllabus, this resource also contains extensive extra material to cover a g wide range of related awards. n i k Features include: n i • clearly focused and differentiated critical thinking and problem solving h T units that provide complete coverage of the Thinking Skills syllabus and beyond t • a range of stimulating student activities with commentaries to develop ai analytical skills w h • summary of key concepts at the end of each chapter to review learning B T u f • end-of-chapter assignments to reinforce knowledge and skills, with t f t o answers at the back for self-assessment e e r G • a mapping grid to demonstrate the applicability of each unit to awards w o d including Critical Thinking, BMAT and TSA. r n t h a Thinking Skills is written by two experienced examiners, who have a h produced a lively and accessible text which all students of Thinking Skills n t or will fi nd invaluable. d w T r Visit education.cambridge.org/cie for information on our full range of h e w Cambridge International A Level titles including e-book versions and t t a u mobile apps. i B te n s h o J 2 0 3 6 0 6 7 ISBN 978-1-107-66996-3 0 1 1 8 Second edition 7 9 John Butterworth and Geoff Thwaites Thinking Skills Critical Thinking and Problem Solving Second edition cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK www.cambridge.org Information on this title: www.cambridge.org/9781107606302 © Cambridge University Press 2005, 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2005 Second edition 2013 Printed in Italy by L.E.G.O. S.p.A. A catalogue record for this publication is available from the British Library ISBN 978-1-107-60630-2 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. Contents Unit 1 Thinking and reasoning 1.1 Thinking as a skill 1 1.2 An introduction to critical thinking 7 1.3 Solutions not problems 13 Unit 2 Critical thinking: the basics 2.1 Claims, assertions, statements 16 2.2 Judging claims 21 2.3 Argument 28 2.4 Identifying arguments 33 2.5 Analysing arguments 38 2.6 Complex arguments 43 2.7 Conclusions 50 2.8 Reasons 58 2.9 Assumptions 63 2.10 Flaws and fallacies 70 Unit 3 Problem solving: basic skills 3.1 What do we mean by a ‘problem’? 79 3.2 How do we solve problems? 82 3.3 Selecting and using information 86 3.4 Processing data 90 3.5 Finding methods of solution 93 3.6 Solving problems by searching 98 3.7 Recognising patterns 102 3.8 Hypotheses, reasons, explanations and inference 106 3.9 Spatial reasoning 112 3.10 Necessity and sufficiency 116 3.11 Choosing and using models 119 3.12 Making choices and decisions 123 Unit 4 Applied critical thinking 4.1 Inference 126 4.2 Explanation 137 4.3 Evidence 144 4.4 Credibility 150 4.5 Two case studies 156 4.6 Critical thinking and science 163 Contents iii 4.7 Introducing longer arguments 170 4.8 Applying analysis skills 177 4.9 Critical evaluation 183 4.10 Responding with further argument 191 4.11 A self-assessment 195 Unit 5 Advanced problem solving 5.1 Combining skills – using imagination 205 5.2 Developing models 211 5.3 Carrying out investigations 220 5.4 Data analysis and inference 225 Unit 6 Problem solving: further techniques 6.1 Using other mathematical methods 231 6.2 Graphical methods of solution 235 6.3 Probability, tree diagrams and decision trees 240 6.4 Have you solved it? 246 Unit 7 Critical reasoning: Advanced Level 7.1 Conditions and conditionals 249 7.2 Soundness and validity: a taste of logic 254 7.3 Non-deductive reasoning 262 7.4 Reasoning with statistics 269 7.5 Decision making 279 7.6 Principles 287 7.7 An argument under the microscope 295 7.8 Critical writing 301 Answers to assignments 311 Appendix 342 Acknowledgements 344 Index 345 iv Contents Unit 1 Thinking and reasoning 1.1 Thinking as a skill This book is about thinking. But it is not about there are advanced skills like gymnastics or any thinking. It is about those kinds of woodwork or piano playing. It doesn’t make thinking that take conscious effort, and which much sense to talk about jumping ‘well’ can be done well or badly. Most of our unless you mean jumping a significant thinking takes little or no conscious effort. We distance, or clearing a high bar, or just do it. You could almost say that we think somersaulting in mid-air and landing on without thinking! If I am asked whether I your feet. There has to be a degree of would like coffee or tea, I don’t have to challenge in the task. But even when the exercise skill to reply appropriately. Similarly if challenge is met, there is still more to be said I am asked a factual question, and I know the about the quality of the performance. One answer, it takes no skill to give it. Expressing a gymnast may look clumsy and untidy, preference or stating a fact are not in another perfectly controlled and balanced. themselves thinking skills. There are language Both have performed the somersault, but one and communication skills involved, of course, has done it better than the other: with more and these are very considerable skills in their economy of effort, and more skilfully. own right. But they are contributory skills to The first of these two criteria also applies to the activities which we are calling ‘thinking’. thinking. Once we have learned to count and This distinction is often made by assigning add, tell the time, read and understand a text, some skills a ‘higher order’ than others. Much recognise shapes, and so on, we do these work has been done by psychologists, things without further thought, and we don’t educationalists, philosophers and others to really regard them as skilled. You don’t have classify and even rank different kinds of to think ‘hard’ unless there is a hard problem thinking. Most would agree that activities to solve, a decision to make, or a difficult such as analysis, evaluation, problem solving concept to understand. So, as with physical and decision making present a higher order of performance, we judge thinking partly by the challenge than simply knowing or recalling or degree of challenge posed by the task. If a understanding facts. What distinguishes student can solve a difficult problem, within higher orders of thinking is that they apply a set time, that is usually judged as a sign of knowledge, and adapt it to different purposes. greater skill than solving an easier one. They require initiative and independence on However, when it comes to assessing the the part of the thinker. It is skills of this order quality of someone’s thinking, matters are that form the content of this book. more complicated. Mental performance is Skills are acquired, improved, and judged largely hidden inside a person’s head, unlike by performance. In judging any skill, there physical performance which is very visible. If are two key criteria: (1) the expertise with two students give the same right answer to a which a task is carried out; (2) the difficulty of question, there is no telling from the answer the task. We are very familiar with this in the alone how it was reached. One of the two case of physical skills. There are basic skills may simply have known the answer, or have like walking and running and jumping; and learned a mechanical way to obtain it – or 1.1 Thinking as a skill 1 even just guessed it. The other may have to suggest that there are two distinct ways of worked it out independently, by reasoning thinking: cold hard reason on one hand and and persistence and imagination. Although free-ranging creativity on the other. In fact, the difference may not show from the answer there is so much overlap and interdependence given, the second student scores over the first between the two that it is very difficult to say in the long term, because he or she has the where one begins and the other ends. Clearly ability to adapt to different challenges. The there are times when a seemingly insoluble first is limited to what he or she knew and problem has been cracked by an imaginative could recall, or simply guessed correctly. leap rather than a methodical process. Some of the greatest advances in science have been the Reasoning result of creative thinking that appeared to Reasoning is the ability most closely conflict with reason when first put forward. associated with human advancement. It is Yet it is just as clear that many apparent often cited as the faculty which marks the flashes of genius, which seem to come ‘out of difference between humans and other the blue’, actually come on the back of a lot of animals. The famous apes studied by the careful and methodical work. Likewise, new psychologist Wolfgang Köhler learned ways to and creative ideas have to be understood and overcome problems, such as using a stick to explained to be of any practical value. get at food that was beyond their reach; but Reasoning is required both to enable and to they discovered the solution by trial and error, apply creative thinking, just as creative and then remembered it for the next time. thinking is needed to give a spark to This is evidence of animal intelligence, and reasoning. certainly of skill; but it is not evidence that Reflection apes can ‘reason’. As far as we can tell, no animal ever draws conclusions on the basis of Another quality that is evidently exclusive to observable facts. None of Köhler’s apes human thinking is reflection. Reflecting thought anything like, ‘That banana is further means giving deep or serious or concentrated from the bars than the length of my arm. thought to something, beyond the immediate Therefore I need to find a stick’; or ‘If this response to stimuli. When we are engaged in stick is too short, I will need a longer one.’ reflection we don’t just make up our minds on Reasoning is the process by which we impulse, but carefully consider alternatives, advance from what we know already to new think about consequences, weigh up available knowledge and understanding. Being rational evidence, draw conclusions, test hypotheses is recognising that from some facts or beliefs and so on. Critical thinking, problem solving others follow, and using that understanding and decision making are all forms of reflective to make decisions or form judgements with thinking. confidence. If there is one overriding aim of Moreover, the reflective thinker does not this book it is to improve students’ focus only on the problem to be solved, the confidence in reasoning. decision to be made, or the argument to be won, but also on the reasoning processes that Creative thinking go into those activities. Reflecting on the way Reasoning is not the only higher thinking we think – or thinking about thinking – helps skill, nor the only kind of rationality. us to evaluate how effective our thinking is, Imaginative and creative activities are no less what its strengths are, where it sometimes important in the history of human goes wrong and, most importantly, how it development and achievement. But that is not can be improved. 2 Unit 1 Thinking and reasoning Using this book examination are covered, though not Throughout the book there are activities and necessarily in the same order as they appear in discussion topics to prompt and encourage the specification. The book does not follow reflection on thinking and reasoning the syllabus step by step or confine itself to themselves. At regular intervals in the chapters just one examination. If it did it would not you will find ‘Activity’ panels. You can use help you either to think more effectively or to these as opportunities to close the book, or do well in your exam. Critical thinking and cover up the rest of the page, and think or talk problem solving are very broad skills, not – or both – about the question or task. Each bodies of knowledge to be learned and activity is followed by a commentary offering repeated. A competent thinker is one who is an appropriate answer, or some guidance on able to deal with the unexpected as well as the the task, before returning to the chapter. By expected. This book therefore takes you well comparing the discussion or solution in the beyond the content of one particular exam commentary with your own reflections and and equips you with a deeper understanding responses, you can judge whether to go back of the processes involved, as well as a flexible, and look at a section again, or whether to adaptive approach to the tasks you are set. move on to the next one. Because thinking skills are general and Although it is not essential to do all of these transferable, the topics and concepts dealt activities, you are strongly urged to give some with in the coming units will also prepare time to them, as they will help greatly with you for many other awards that involve your understanding of the concepts and critical thinking and/or problem solving. The procedures that make up the Thinking Skills table on pages 342–43 shows a range of syllabus. The tasks also act as opportunities for public examinations and admissions tests self-assessment, both of your own personal whose content is covered by some or all of responses, and of those of your colleagues if the chapters. These include A Level Critical you are working in groups. Small-group Thinking (OCR and AQA); the BioMedical discussion of the tasks is particularly valuable Admissions Test (BMAT); Cambridge because it gives you insight into other ways to Thinking Skills Assessment (TSA); Singapore think and reason besides your own. You have H2 Knowledge and Inquiry; and Theory the opportunity to compare your responses of Knowledge in the International with those of others, as well as with the Baccalaureate (IB). responses suggested in the commentary. The Other subjects activities and commentaries are like a dialogue Finally, the value of developing your thinking between you and the authors of the book. skills extends far beyond passing exams called The book can be used either for a school or ‘Thinking Skills’! It has been shown, college course in thinking skills, or by the student unsurprisingly, that confidence and aptitude for individual study. It is divided into seven units in critical thinking and problem solving will with varying numbers of chapters within them. assist students to achieve higher grades across Although it is not a straight-line progression, all the subjects that they study. Accordingly there is an overall advance from basic skills to you will find critical thinking, problem applied skills and to higher levels of challenge. solving and presenting well-reasoned Preparing for examinations argument among the learning and assessment The backbone of this book is the Cambridge objectives of just about every senior-school or syllabus for A and AS Level Thinking Skills. All university course, whether in the sciences or of the assessment objectives for that the arts and humanities. 1.1 Thinking as a skill 3 Beyond that, too, these are sought-after No more than one of the statements on each qualities in a great many professions and envelope is false. occupations. Hardly surprisingly, employers want staff who can think for themselves, On envelope X it says: solve problems, make decisions and A The jailhouse key is solid brass. construct arguments. B The jailhouse key is not in this What to expect envelope. To give a taste of the structure and style of the On envelope Y it says: book, this chapter ends with an activity similar to those which appear at regular C The jailhouse key is not in this intervals in all of the coming units. You can envelope either. think of it as a trial run. The task is to solve a D The jailhouse key is in envelope Z. puzzle entitled ‘The Jailhouse Key’. It is a On envelope Z it says: simple puzzle, but it introduces some of the reasoning skills you will encounter in future E The jailhouse key is solid silver. chapters, giving a foretaste of all of three F The jailhouse key is not in envelope X. disciplines: problem solving, critical thinking The prisoners may look inside the envelopes and decision making. if they wish, before deciding. They have five minutes to make up their minds. Activity Decide which envelope the prisoners should choose in order to escape from the cell. Two prisoners are held in a dungeon. One The best way to do this activity is to night a mysterious visitor appears in their cell discuss it with a partner, just as the two and offers them a chance to escape. It is prisoners would do in the story. As well as only a chance because they must first reason deciding which envelope to choose, answer to a decision which will determine whether or this further question: not they actually do go free. Why is the envelope you have chosen the Their cell is at the bottom of a long flight right one; and why can it not be either of the of steps. At the top is the outer door. Three others? envelopes, marked X, Y and Z, are placed on the table in the prisoners’ cell. One of them, they are told, contains the key to the outer Commentary door, but they may take only one envelope Throughout this book you will be given when they attempt to leave the cell. If they questions to answer, problems to solve, ideas choose the wrong one, they will stay locked to think about or discuss, followed, as we have up forever, and the chance will not come said, by commentaries. The commentaries will again. It is an all-or-nothing decision. vary: some will provide the correct answer, if There are six clues, A to F, to help them – there is one. Some will suggest various possible or puzzle them, depending on how you look at answers, or different directions you could have it. Two are printed on each envelope. There is taken in your thinking. The purpose of the also a general instruction, on a separate activities and commentaries is to allow you to card, which stipulates: assess your own progress and to give you useful advice for tackling future tasks. 4 Unit 1 Thinking and reasoning Sometimes you may question or disagree It also tells you that: with the commentary, especially later on when [1b] The statements on any one envelope you have gained experience. On other cannot both be false. occasions you will see from the commentary where you went wrong, or missed an Although [1a] says exactly the same as the important point, or reasoned ineffectively. card, it states it in a positive way rather than a Don’t be disheartened if you do find you have negative one. Negative statements can be taken the wrong tack. It is part of the learning confusing to work with. A positive statement process. Very often we learn more from making may express the information more practically. mistakes than we do from easy successes. [1b] also says the same as the card, and In the present example there is only one although it is negative it restates it in a plainer answer to the question: the key is in envelope way. Just rewording statements in this kind of Z. The clues, although they seem confusing way draws useful information from them, and and contradictory, do give you all the helps you to organise your thoughts. information you need to make the correct Now let’s look at the envelopes and ask decision. Nonetheless, there are any number what more we can learn from the clues on of different ways to get to the solution, and them. Here are some suggestions: you may have found a quicker, clearer or [2] Statements B and F are both true or more satisfying procedure than the one you both false (because they say the same are about to see. You may even have taken thing). one look at the puzzle and ‘seen’ the solution [3] A and E cannot both be true. (You only straight away. Occasionally this happens. have to look at them to see why.) However, you still have to explain and/or justify your decision. That is the reflective part Taking these two points together, we can apply of the task. a useful technique known as ‘suppositional reasoning’. Don’t be alarmed by the name. You Procedures and strategies do this all the time. It just means asking Procedures and strategies can help with questions that begin: ‘What if . . .?’ For puzzles and problems. These may be quite example: ‘What if B and F were both false?’ obvious; or you may find it hard even to know Well, it would mean A and E would both have where to begin. One useful opening move is to to be true, because (as we know from [1a]) at look at the information and identify the parts least one statement on each envelope has to be that seem most relevant. At the same time you true. But, as we know from [3], A and E cannot can write down other facts which emerge from both be true (because no key can be solid silver them. Selecting and interpreting information and solid brass). in this way are two basic critical thinking and Therefore: problem solving skills. Start with the general claim, on the card, [4] B and F have to be true: the key is not in that: envelope X: it is in either Y or Z. [1] N o more than one of the statements on This is a breakthrough. Now all the clues we each envelope is false. need are on envelope Y. Using suppositional reasoning again we ask: What if the key were in This also tells you that: Y? Well, then C and D would both be false. But [1a] At least one of the statements on each we know (from [1b]) that they can’t both be envelope must be true. false. Therefore the key must be in envelope Z. 1.1 Thinking as a skill 5