Sbfs1103 thinking skills and prob solving

SBFS1103
THINKING SKILLS
AND PROBLEM
SOLVING
Dr Haresh Kumar Kantilal
Dr Anton Espira
Copyright © Open University Malaysia (OUM)

First Edition, August 2008
Second Edition, December 2011
Third Edition, December 2013 (rs)
Copyright © University Malaysia (OUM), December 2013, SBFS1103
All rights reserved. No part of this work may be reproduced in any form or by any means
without the written permission of the President, Open University Malaysia (OUM).
Project Directors: Prof Dato’ Dr Mansor Fadzil
Assoc Prof Dr Norlia T. Goolamally
Open University Malaysia
Module Writers: Dr Haresh Kumar Kantilal
MAHSA University College Malaysia
Dr Anton Espira
Moderators: Assoc Prof Dr Arifin Hj Zainal
P Rajesh Kumar
Developed by: Centre for Instructional Design and Technology
Printed by: Meteor Doc Sdn Bhd
Lot 47-48, Jalan SR 1/9, Seksyen 9,
Jalan Serdang Raya, Taman Serdang Raya,
43300 Seri Kembangan, Selangor Darul Ehsan

Course Guide xi–xviii
Topic 1 Problem Solving 1
1.1 What is a Problem? 2
1.2 What is Problem Solving? 4
1.2.1 Stages in Problem Solving 5
1.3 Importance of Goals in Problem Solving 8
1.4 General Guidelines for Problem Solving 9
1.4.1 Explore the Problem 11
1.4.2 Establish Your Goals 14
1.4.3 Generate Ideas 15
1.4.4 Select Your Ideas 16
1.4.5 Implement Your Action Plan 17
1.4.6 Evaluate and Follow up 17
1.5 Root Cause Analysis 18
1.6 Constraints 19
1.7 What is a Solution? 19
1.7.1 The „Stop It‰ Approach 20
1.7.2 The „Mop It‰ Approach 21
1.8 Identifying Possible Solutions 22
1.9 Planning the Solution 23
1.10 Taking Responsibility 24
Summary 24
Key Terms 25
Self-Assessment 26
References 27
Topic 2 Research on Thinking and Problem Solving 28
2.1 Major Antecedents of the Science of Psychology 28
2.1.1 Associationism 29
2.1.2 Faculties of the Mind 31
2.2 The Psychology of Thinking 32
2.2.1 William James 32
2.2.2 Functionalism 32
2.2.3 Later Functionalism 33
2.2.4 Behaviourism 34
2.2.5 Gestalt Psychology 35
2.3 Research on Concepts 37
2.3.1 Common-Element Concepts 38
2.3.2 Mediational Processes and Hypotheses 38
Table of Contents

TABLE OF CONTENTSiv
2.3.3 Strategies and Kinds of Concepts 39
2.4 Research on Problem Solving 41
2.4.1 Trial and Error 41
2.4.2 Insight 42
2.4.3 Problem-Solving Set 43
2.4.4 Transfer of Principles 46
2.5 Research on Reasoning 46
2.5.1 Atmosphere Effect 46
2.5.2 Content Effects 48
2.6 Transition to the New Cognitive Era 49
Summary 50
Key Terms 51
Self-Assessment 52
References 52
Topic 3 Identifying Reasons and Conclusions 55
3.1 Deciding when Reasoning is Present 56
3.2 Examples of Reasoning 58
3.3 Language of Reasoning 60
3.4 Thinking Map for Understanding and Evaluating Reasoning 62
3.5 Acceptability of Reasons 65
3.6 Judging Acceptability of a Claim 67
3.7 Judging Credibility of Sources 68
3.8 Evaluating Inferences 70
Summary 74
Key Terms 76
Self-Assessment 76
References 76
Topic 4 Development of Problem Solving 77
4.1 The Developement of Problem Solving 78
4.1.1 Strategic Development 79
4.1.2 Representational Development 82
4.1.3 Self-Regulation 87
Summary 91
Key Terms 92
Self-Assessment 92
References 93

TABLE OF CONTENTS v
Topic 5 Thinking Skills 99
5.1 Definition of Thinking 101
5.2 Importance of Thinking 103
5.3 Increasing Thinking Power 105
5.3.1 Production Phase 106
5.3.2 Judgement Phase 106
5.4 Roles and Functions of the Right and Left Brain 108
Summary 111
Key Terms 112
Self-Assessment 112
References 113
Topic 6 Critical and Creative Thinking 114
6.1 Critical Thinking 115
6.1.1 Eight Characteristics of Critical Thinking 116
6.1.2 Classification Games in Teaching Critical Thinking 119
6.1.3 Barriers to Critical Thinking 120
6.1.4 Importance of Critical Thinking 121
6.2 Creative Thinking 121
6.2.1 Five Characteristics of Creative Thinking 123
6.2.2 Applying Creativity to Problems and Issues 126
6.3 Lateral Thinking 128
6.4 Right and Left Brain Crossover 130
Summary 132
Key Terms 132
Self-Assessment 133
References 134
Topic 7 Problem Solving Strategies 136
7.1 Conceptualisation in Problem Solving 137
7.2 Problem Solving Strategies 138
7.2.1 Decomposition or Subgoaling 139
7.2.2 Working Backwards 140
7.2.3 Hill Climbing 141
7.2.4 Means-end Analysis 142
7.2.5 Forward Chaining 143
7.3 Other Problem Solving Strategies 144
7.3.1 Analogy in Problem Solving 144
7.3.2 Specialisation and Generalisation 145
7.3.3 Use of Extreme Cases 146
Summary 148
Key Terms 149

TABLE OF CONTENTSvi
Self-Assessment 149
References 150
Topic 8 Problem-Solving Tools 151
8.1 Brainstorming 152
8.2 Benchmarking 154
8.3 Cause and Effect Diagrams 155
8.4 Flow Charts 158
8.5 SWOT Analysis 161
8.6 Prioritisation Matrix 162
Summary 164
Key Terms 165
Self-Assessment 166
References 166
Topic 9 Decision Making 167
9.1 What is Decision Making? 168
9.2 Types of Decisions 169
9.3 Decision Making is a Recursive Process 170
9.4 Factors Influencing Decision Making Processes 170
9.5 Four Quadrants of the Brain and Our Different
Thinking Styles 172
9.6 Decision Making in Everyday Life 174
9.7 Evidence-Driven Decision Making Cycle 175
9.8 Effects of Quantity on Decision Making 180
9.9 Decision Streams 180
9.10 Decision Quality 181
9.11 Risk 182
Summary 183
Key Terms 184
Self-Assessment 184
References 185
Topic 10 Strategies and Tools for Making the Final Decision 187
" 10.1 Approaches to Decision Making 188
10.2 Decision Making Strategies 189
10.3 Group Decision Making: Methods 190
10.4 Decision Making Tools 191
10.4.1 Grid Analysis: Making a Choice Whereby
Many Factors Must be Considered 192
10.4.2 Decision Tree Analysis: Choosing Among
Options by Projecting Likely Outcomes 193
10.4.3 PMI: Weighing the Pros and Cons of a Decision 194

TABLE OF CONTENTS vii
10.5 Some Final Decision Making Tips 196
Summary 197
Key Terms 199
Self-Assessment 199
References 199
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TABLE OF CONTENTSviii
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COURSEGUIDE
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COURSE GUIDE DESCRIPTION
You must read this Course Guide carefully from the beginning to the end. It tells
you briefly what the course is about and how you can work your way through
the course material. It also suggests the amount of time you are likely to spend in
order to complete the course successfully. Please keep on referring to Course
Guide as you go through the course material as it will help you to clarify
important study components or points that you might miss or overlook.
INTRODUCTION
SBFS1103 Thinking Skills and Problem Solving is one of the courses offered by
Faculty of Science and Technology at Open University Malaysia (OUM). This
course is worth three credit hours and should be covered over 8 to 15 weeks.
COURSE AUDIENCE
This course is designed for students taking Bachelor programmes in OUM.
As an open and distance learner, you should be acquainted with learning
independently and being able to optimise the learning modes and environment
available to you. Before you begin this course, please confirm the course material,
the course requirements and how the course is conducted.
STUDY SCHEDULE
It is a standard OUM practice that learners accumulate 40 study hours for every
credit hour. As such, for a three-credit hour course, you are expected to spend
120 study hours. Table 1 gives an estimation of how the 120 study hours could be
accumulated.

COURSE GUIDE
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xii
Table 1: Estimation of Time Accumulation of Study Hours
Study Activities Study Hours
Briefly go through the course content and participate in initial
discussion
2
Study the module 60
Attend 4 tutorial sessions 8
Online Participation 12
Revision 18
Assignment(s), Test(s) and Examination(s) 20
TOTAL STUDY HOURS 120
COURSE OUTCOMES
By the end of this course, learners should be able to:
1." Explain the process of problem solving and the different approaches to
problem solving;
2." Describe the major antecedents and research in the science of psychology;
3." Discuss the reasoning process, the acceptability of reasons and evaluate
inferences;
4." Describe the development of problem solving;
5." Explain the theories, models and conceptual framework involved in
problem solving and the stepwise conceptions of problem solving;
6." Explain the thinking process, its importance and its application;
7." Describe critical thinking, creative thinking and lateral thinking processes;
8." Discuss the various strategies employed in problem solving including the
common tools and techniques;
9." Describe decision making, the factors that influence the process and our
ability to make decisions; and
10." Identify decision-making strategies, risk and tools.

COURSE GUIDE xiii
COURSE SYPNOPSIS
This course will provide knowledge on thinking skills and problem solving,
focusing mainly on the thinking process, its importance and the phases involved,
the roles and functions of the left and right brain, the characteristics and
applications of both critical thinking and creative thinking, identifying reasoning,
the acceptability of reasons and claims, and evaluating inferences. In addition,
there is focus on problem-solving strategies, common tools and techniques. The
final section of this course covers decision making process and strategies.
Topic 1 presents alternative conceptual approaches to define a problem and
introduces both the process and stages involved in problem solving. The
importance of goal setting and the generation of the right questions to ask when
faced with a problem is also discussed. It also introduces root cause analysis and
constraints faced in problem solving. The topic concludes by defining solution
and introducing the stop it and mop it approach in problem solving.
Topic 2 introduces the major antecedents in the science of psychology and
psychology of thinking. It also elaborates on the research carried out on concepts,
problem solving and reasoning. The topic concludes by describing the transition
to the cognitive era.
Topic 3 discusses how to identify reasons and conclusions, examples of
reasoning, language of reasoning, the thinking map of understanding and
evaluating reasoning, the acceptability of reasons, judging acceptability of claims,
judging the credibility of sources, evaluating inferences and deductive validity.
Topic 4 introduces the development of problem solving that includes strategic
development, representational development and self-regulation development. It
also presents the various theories, models and conceptual framework in problem
solving. The topic concludes by describing the stepwise conception of problem
solving.
Topic 5 discusses the definition and importance of thinking, ways to increase
thinking power, the two phases of the thinking process and the roles and
functions of the right and left parts of the brain.
Topic 6 discusses the critical thinking process which includes its characteristics,
examples, barriers and importance. It also describes the creative thinking process,
its characteristics and how to apply creativity to problems and issues. The topic
concludes by describing lateral thinking and the crossover between the right and
left brain.

COURSE GUIDE
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xiv
Topic 7 presents both the employment of different strategies on its own and
combination of strategies in problems solving. It also introduces other concepts
such as consideration of analogous problems, specialisation and generalisation
and consideration of extreme cases.
Topic 8 introduces common tools and techniques used in the process of problem
solving.
Topic 9 introduces the decision making process and describes different kinds of
decisions and the various factors that affect decision making. It introduces the
Four Quadrants of the Brain Theory and explains how this relates to our ability
to make decisions. The topic includes the Evidence-Driven Decision-Making
Cycle and the effects of quantity on decision making. The topic concludes by
describing decision streams, decision quality and risking.
Topic 10 introduces the different approaches to decision making and identifies
some decision-making strategies. It introduces the Group Decision-Making
Process and concludes by describing some basic tools and techniques that can aid
the decision-making process.
TEXT ARRANGEMENT GUIDE
Before you go through this module, it is important that you note the text
arrangement. Understanding the text arrangement will help you to organise your
study of this course in a more objective and effective way. Generally, the text
arrangement for each topic is as follows:
Learning Outcomes: This section refers to what you should achieve after you
have completely covered a topic. As you go through each topic, you should
frequently refer to these learning outcomes. By doing this, you can continuously
gauge your understanding of the topic.
Self-Check: This component of the module is inserted at strategic locations
throughout the module. It may be inserted after one sub-section or a few sub-
sections. It usually comes in the form of a question. When you come across this
component, try to reflect on what you have already learnt thus far. By attempting
to answer the question, you should be able to gauge how well you have
understood the sub-section(s). Most of the time, the answers to the questions can
be found directly from the module itself.
Activity: Like Self-Check, the Activity component is also placed at various
locations or junctures throughout the module. This component may require you

COURSE GUIDE xv
to solve questions, explore short case studies, or conduct an observation or
research. It may even require you to evaluate a given scenario. When you come
across an Activity, you should try to reflect on what you have gathered from the
module and apply it to real situations. You should, at the same time, engage
yourself in higher order thinking where you might be required to analyse,
synthesise and evaluate instead of only having to recall and define.
Summary: You will find this component at the end of each topic. This component
helps you to recap the whole topic. By going through the summary, you should
be able to gauge your knowledge retention level. Should you find points in the
summary that you do not fully understand, it would be a good idea for you to
revisit the details in the module.
Key Terms: This component can be found at the end of each topic. You should go
through this component to remind yourself of important terms or jargon used
throughout the module. Should you find terms here that you are not able to
explain, you should look for the terms in the module.
References: The References section is where a list of relevant and useful
textbooks, journals, articles, electronic contents or sources can be found. The list
can appear in a few locations such as in the Course Guide (at the References
section), at the end of every topic or at the back of the module. You are
encouraged to read or refer to the suggested sources to obtain the additional
information needed and to enhance your overall understanding of the course.
PRIOR KNOWLEDGE
There are no pre-requisites for this course. Students are required to understand
this course guide well before starting with the topics in this module.
ASSESSMENT METHOD
Please refer to myVLE.
REFERENCES
Adair, J. (2010). Decision making and problem solving strategies: Learn key
problem solving strategies; Sharpen your creative thinking strategies; Make
effective decisions (2nd ed.). London, UK: Kogan Page.

COURSE GUIDE
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Bedell, J. R., & Lennox, S. S. (1997). Handbook for communication and problem-
solving skills training: A cognitive-behavioural approach (1st ed). New York:
John Wiley.
Butterworth, J., & Thwaites, G. (2005). Thinking skills. Cambridge International
Examinations (Paperback).
De Bono, E. (2005). Thinking course. (Revised ed.). Barnes & Noble Books.
Elder, L. (2007). Critical thinking: Tools for taking charge of your professional
and personal life. New Jersey: Pearson Education.
Fisher, A. (2001). Critical thinking: An introduction. Cambridge, UK: Cambridge
University Press.
Harrison, A. F., & Bramson, R. M. (2002). The art of thinking. New York, NY: The
Berkley Publishing Group (Paperback).
Hurson, T. (2007). Think better: An innovatorÊs guide to productive thinking
(1st ed.). New York, USA: McGraw-Hill.
John, C. (2006). Thinking critically. Houghton Mifflin Co.
Lumsdaine, E., & Lumsdaine, M. (1995). Creative problem solving. McGraw-
Hill, Inc.
Meador, K. S. (1997). Creative thinking and problem solving for young learners.
Englewood, CO: Teacher Ideas Press.
Moore, B. N., & Parker, R. (2005). Critical thinking (8th ed.). McGraw-Hill.
Paul, R., & Linda, E. (2002). Critical thinking. Pearson Education Inc.
Sternberg, R. J. (1998). Thinking and problem solving: Handbook of perception
and cognition (2nd ed.). London: Academic Press.
Vaughn, L. (2004). The power of critical thinking: Effective reasoning about
ordinary and extraordinary claims. USA: Oxford University Press.
Wright L. (2001). Critical thinking: An introduction to analytical reading and
reasoning. New York: Oxford University Press.

COURSE GUIDE xvii
Vaughn, L. (2004). The power of critical thinking: Effective reasoning about
ordinary and extraordinary claims. (Paperback).
Wright L. (2001). Critical thinking: An Introduction to analytical reading and
reasoning. New York: Oxford University Press."
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TAN SRI DR ABDULLAH SANUSI (TSDAS) DIGITAL
LIBRARY
The TSDAS Digital Library has a wide range of print and online resources for the
use of its learners. This comprehensive digital library, which is accessible
through the OUM portal, provides access to more than 30 online databases
comprising e-journals, e-theses, e-books and more. Examples of databases
available are EBSCOhost, ProQuest, SpringerLink, Books24x7, InfoSci Books,
Emerald Management Plus and Ebrary Electronic Books. As an OUM learner,
you are encouraged to make full use of the resources available through this
library.
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COURSE GUIDE
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xviii
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" INTRODUCTION
Regardless of what we do for a living or where we are, we spend most of our
waking hours constantly trying to solve problems. For many students, problem
solving is a procedure they carry out on a task assigned by a teacher. They view
the problem as something to be completed within a set time so as to avoid
punitive consequences. Unfortunately, in their daily lives, students, and people
in general, have a tendency to overanalyse every action or event deemed
problematic. The problems we face can be small or complex, but they need to be
solved in a constructive manner. Hence, problem solving is an important skill.
Whether we arrive at an effective solution or not depends on how we confront
the problem (Figure 1.1).
TTooppiicc
11
Problem Solving
LEARNING OUTCOMES
By the end of this topic, you should be able to:
1." Define what a problem is;
2." Describe how a problem can present an opportunity for
improvement;
3." Explain the four stages involved in the process of problem solving;
4." Explain the importance of goal setting in the problem-solving
process;
5." Establish the general guidelines to effective problem solving;
6." Discuss the role of Root Cause Analysis (RCA) technique in the
problem-solving process;
7." Examine the common constraints in the problem-solving process;
and
8." Compare and contrast the different approaches to problem solving.

TOPIC 1 PROBLEM SOLVING
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Figure 1.1: How we view a problem is the key to determining its outcome
Source: http://beautybrainsnbeyond.blogspot.com/2010/05/thought-of- day_20.html
WHAT IS A PROBLEM?
Based on some dictionaries, a problem is defined as:
(a)" A question raised for consideration or solution (WordNet® 2.0, Princeton
University, 2003).
(b)" A question to be considered, solved or answered (The American Heritage®
Dictionary of the English Language, Fourth Edition, Updated in 2009).
(c)" Anything, matter, person, etc, that is difficult to deal with, solve or
overcome (Collins English Dictionary: Complete and Unabridged 6th
Edition, 2003).
(d)" Any question or matter involving doubt, uncertainty or difficulty
(Dictionary.com, Unabridged).
1.1

TOPIC 1 PROBLEM SOLVING 3
Figure 1.2: A negative approach to a problem: "If the problem is too big, just blame it on
someone else!"
Depending on the context, some problems may be minor, personal or caused by
external factors. Some problems may be too big or too complex and need to be
passed on to people who are more capable of dealing with them. Figure 1.2
illustrates how NOT to go about it!
Different authors define a problem differently. Generally, we can define a
problem using three approaches:
(a)" A problem is an opportunity for improvement.
(b)" A problem is the difference between your current state and your goal state.
(c)" A problem results from the recognition of a present imperfection and the
belief in the possibility of a better future.
The definition that a problem is an opportunity for improvement will be the
focus of our discussion here. Thinking out of the box enables one to view a
problem as an opportunity – a chance to open up new ways of looking at, and
responding to, the difficulties being faced. Seen from this perspective, problems
are not necessarily the result of external factors or to be perceived as negative or
"bad" events. This shift in perspective creates a new viewpoint or awareness of
the problem. New opportunities for improvement will present themselves to
you, and as they do, they bring about new challenges that demand a solution
from you.

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Such a situation requires flexibility in approaching the problem at hand. This
requirement for flexibility offers a distinct advantage to creative people in the
course of problem solving, since changes in the problem necessitate innovative
and inventive adjustments in how it is solved. Perceiving problems as
opportunities for improvement nurtures a positive mindset, which can transform
you into a more confident person with more control of your life.
At times, a problem can manifest itself in different ways. An example is a
situation where we see ourselves in conflict with where we are at present and
where we want to be. When we can identify the difference between the situation
we are in at present and the situation we desire, we can define the problem,
which we can thereafter attempt to resolve in order to achieve our goal.
WHAT IS PROBLEM SOLVING?
Problem solving and the techniques associated with it are domain-independent
reasoning components which specify patterns of behaviour that can be reused
across applications (Fensel and Motta, 1997). This simply means that problem
solving is a series of techniques, both cognitive and behavioural, that can be
applied in various fields or areas to generate solutions.
As Woods (1975) and colleagues suggested, "Problem solving is the process of
obtaining a satisfactory solution to a novel problem, or at least a problem which
the problem solver has not seen before."
1.2
"Each problem has hidden in it an opportunity so powerful that it literally
dwarfs the problem. The greatest success stories were created by people
who recognised a problem and turned it into an opportunity."
Joseph Sugarman (Inglish, P. 2011)
ACTIVITY 1.1
"There is so far no limit to the rise in food and oil prices, a situation that
is leading many consumers and producers to a point where they are
carrying an unbearable burden." (The Star, 23 June 2008)
Identify one consumer-related problem that can result from the situation
reported above. Can you approach this problem as an opportunity?
Explain your answer.

As a whole, problem solving involves a series of steps, from the identification of
a problem leading to the development of a solution. Problem solving can be seen
as a structured, directional-oriented process. However, in determining the
direction for the solution, we need to define the problem first, followed by the
desired goal or the changes we want.
1.2.1 Stages in Problem Solving
As highlighted in the previous subtopic, to better understand problem solving
and to improve our personal problem-solving capabilities, we should view
problem solving as a process that leads to a positive result. The importance of
this approach is that it focuses on the process of attaining the solution rather than
the solution itself. This means more emphasis is placed on the steps in reaching
the solution than the end result. Thus, it is vital that we understand the various
stages involved.
The four stages involved are presented in Figure 1.3.
Figure 1.3: Stages in problem solving
(a)" Recognising and defining a problem
A problem can only be solved when its existence is recognised. However, in
certain situations, a problem is only recognised when it becomes a serious
matter. After recognising a problem, the next action will be to thoroughly
define it. A problem can be referred to as a closed-ended problem or an
open-ended problem. A closed-ended problem involves all circumstances
surrounding the deviation from the norm that need to be defined. On the
other hand, an open-ended problem involves both identifying and defining
your objectives and any hindrance that can stop you from attaining them.

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The main differences between closed-ended and open-ended problems are
summarised in Table 1.1.
Analysing a problem thoroughly is important in order for you to fully
understand it and subsequently come up with various solutions that can
help in achieving your goals. Analysing a problem involves identifying and
bringing together relevant information, and presenting it in a meaningful
way. It eventually helps you to decide what the best solution would be.
Table 1.1: Differences between Closed- and Open-ended Problems
Closed-ended Problems Open-ended Problems
Have one or a limited number of
possible solutions.
Can be solved using many different
methods.
Problem analysis involves identifying all
the possible causes before looking for a
solution.
Problem analysis involves looking for
information which will help to
suggest a range of possible ways to
solve the problem.
(b)" Finding possible solutions
Generally, coming up with various actions to solve a problem involves
finding out what appropriate actions to take, what the hindrances are and
finally working towards achieving your goals. It may be necessary to
modify and combine ideas using any creative technique available in order
to devise a workable solution. In truth, the more ideas you have to choose
from, the higher the probability of finding an effective solution.
(c)" Selecting the best solution
This is a decision-making process that is based on a comparison of the
potential outcomes of alternative solutions. This stage involves:
(i)" Identifying the characteristics of the possible solutions, including
any limitations that might be encountered;
(ii)"Eliminating the solutions that cannot overcome the limitations;
(iii)"Evaluating the remaining solutions;
(iv)"Evaluating the risks linked to arriving at the best solution; and
(v)"Coming up with a decision to implement the chosen solution.
It is believed that a problem is considered solved when a solution has been
implemented. However, in certain circumstances, the selected solution has

to be accepted or authorised by other individuals in order for it to be
implemented, and this may involve several strategies of persuasion.
(d)" Implementing the solution
Three steps are involved in this stage, as presented in Table 1.2.
Table 1.2: Steps Involved in the Implementation of a Solution
Step Description
Planning and preparing to
implement the solution.
Involves the conclusion of all efforts and thorough
planning that describes the sequence of actions
needed to fulfil the objective, timescale and
resources required at each stage.
Taking the appropriate
action and monitoring its
effect.
Problems or potential problems have to be dealt
with immediately in order to ensure that things run
smoothly.
Reviewing the success of
the action.
This is important to estimate the efficacy of the
action in solving future similar problems and to
make sure that the problem has been solved.
To sum up, the emphasis of a closed-ended problem is on defining and
analysing the problem to indicate possible causes, whereas the focus of an
open-ended problem is on the idea generation stage to come up with a
broader range of possible solutions. It may be necessary to recalibrate the
work done earlier, during any stage in problem solving.
In all the four stages discussed, we need to use our reasoning and thinking
skills. These will be discussed in later topics.
1." Explain the four stages involved in problem solving.
40" Explain briefly the differences between open-ended and closed-
ended problems."
SELF-CHECK 1.1

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IMPORTANCE OF GOALS IN PROBLEM
SOLVING
Problem solving centres on thinking about goals and ideal states that one would
like to achieve. Usually, when a goal is reached, the problem is considered
settled. However, this will only happen if the goal originally set is appropriate.
Defining a goal is essential to determine how much of a problem exists or
whether or not there is a problem in the first place. There are, of course,
innumerable problems that need solutions. Do not get carried away by trying to
solve every problem you identify and do not let others set goals for you that you
will not be able to achieve (as illustrated in Figure 1.4).
Figure 1.4: Set yourself realistic goals; do not try to solve all problems and do not let
others set goals for you that you know you cannot meet
1.3

Robert Harris (1998), in the following excerpt, gives a very good example to
illustrate the need for a goal to identify whether a problem exists:
Let's say you have just brought a pizza home from the pizza parlour and
it is beginning to cool. If your ideal state is to eat very hot pizza, then you
have a problem – you may define it as how to keep the pizza from
cooling, how to heat it back up or how to eat it quickly. On the other
hand, if you like moderately warm pizza, then you do not have a
problem. Similarly, if your friend comes over an hour later and you offer
him a piece of leftover pizza only to discover that your oven is on the
blink, you have a problem: how to heat up the pizza. But if your friend
says, "I really like cold pizza better than hot", you do not have a problem.
Source: http://www.virtualsalt.com/crebook3.htm
As the above example illustrates, in order to determine whether or not a problem
exists, and to what extent it does, you must first consider:
(a)" Your goal(s); and
(b)" Your current state.
Your goals must be achievable within the boundaries of your capabilities and the
tools you possess. Take time to think about your goals and plan your strategy.
Remember, failing to plan is planning to fail.
Goals are never static. Your priorities will change over time, and so will your
goals. The changes may involve a major rethinking of your lifeÊs purpose, or they
may be small refinements to your holiday budget. Whatever the changes may be,
it is essential that you keep your problems and solutions in tandem with your
goals.
GENERAL GUIDELINES FOR PROBLEM
SOLVING
These guidelines are meant to help you analyse, define and solve problems in an
orderly way, and to give some structure to the various problem-solving activities
you encounter. When using these guidelines, it is important to keep in mind that
problem solving does not proceed through a predetermined sequence of steps at
all times, nor is it necessarily linear. Rather, problem solving is a recursive
process whereby you must continually go back and forth between steps, and
repeat some steps. This can best be viewed as a problem-solving cycle (illustrated
in Figure 1.5) whereby each process can feed into another process through
feedback and correction.
1.4

"
10
Figure 1.5: The problem-solving cycle
Whenever we are faced with a problem, an effective solution to the problem can
be developed using the six steps, as shown in Figure 1.6.
Figure 1.6: The six steps in problem solving and decision making
These steps can each be broken down into smaller steps to better enable you to
carry out the problem-solving process. We will look further into each step in the
following subsections.

1.4.1 Explore the Problem
This is the first step in the process of effective problem solving. The problem is
investigated and broken down into sub-problems; the terms are defined and the
nature of the problem is determined. Some research should be done on whether
the problem has been encountered in the past, and if so, how it was resolved, if at
all.
Obviously, the nature of the problem will determine how much time is spent on
problem exploration. Simple problems may not need extensive exploration while
complex problems may require extensive exploration. Compare, for example, the
amount of exploration needed to determine why a little bulb is not working in
your house and why a car is not working properly. Problem exploration may
involve some steps which include:
(a)" Problem statement
(i)" What is the problem?
•" Is the problem multi-faceted? If so, what are its various facets or
aspects?
•" If the problem was passed on to you from someone else (another
individual or another department, for example), it might be useful
to have a statement stating the problem as was given to you and
another statement stating the problem in your own words.
•" In any problem definition or statement, it is useful to state what
the problem is not. By doing this, you will have a clearer idea of
what it is.
(ii)" Restate it differently.
•" Restate the problem in a completely different way.
•" Do this several different times.
•" This helps to separate the real problem from our verbal
representations of it.
(iii)" Make a broad statement about the problem.
•" State the problem in abstract terms, perhaps even with a
philosophical slant. Often, how a problem is worded reflects a
specific instance of a broader issue. By restating the problem in
more general terms, one is able to determine if this is so.

"
12
•" Using general statements allows the issues surrounding the
problem to be argued from different perspectives; thereby
suggesting solutions that would not otherwise be considered.
(b)" Problem clarification
(i)" What are the key terms of the problem?
•" Use words with similar meanings.
•" Define the terms recursively, making them more general or more
specific, until the problem is articulated as best as can be.
•" With this technique, the problem is broken down into its
attributes, components and general features, so that possible
solutions can be more readily apparent.
•" Anything that is ambiguous or uncertain about the problem must
be clarified. It is not uncommon that problem statements
originally written are vague and fuzzy, and need to be fine-tuned.
(ii)" What are the assumptions made about the problem? Under what
circumstances would a solution work?
•" This step involves the conscious listing of all assumptions that can
be identified. The listing must be impartial, without prejudice or
judgement.
•" It is important to list the obvious assumptions because often it is
those that later turn out to be alterable.
•" Many assumptions are necessary while some are not.
(iii)" Obtain further information about the problem.
•" Get ideas for solving the problem by researching how similar
problems were approached or solved in the past.
•" Often, you may discover alternative solutions used previously to
solve very similar problems faced at present.
(c)" Explanation of the problem
(i)" Brainstorm with others the issues surrounding the problem.
•" Articulate the problem carefully and listen to your own
explanation; then, try to visualise how someone else may see the
problem.

•" Discussion may help you find a solution from someone else who
has encountered a similar problem or has a different perspective.
•" Generally, when discussing a problem or idea, listen to yourself as
well as to others you talk to.
(ii)" View the problem from different perspectives.
•" Would your problem look different to different people? How
would the issue you consider a problem look like when viewed
from a different perspective?
•" To get the best perspective, the problem should be looked at from
the points of view of those who cause it, those who suffer from it,
those who have to fix it, those who have to pay for it, etc.
(iii)" Ask a series of clarifying "Whys?"
•" Asking "Why?" can serve a purpose similar to that of broadening
the definition of the problem, which can lead to new ways of
looking at the problem and at possible solutions.
•" You can also ask yourself if the problem is a symptom or result of
another problem. For example, at first glance, the production of
low-quality cars appears to be a problem of quality control. A
deeper analysis, however, may reveal it to be a symptom of a
bigger underlying problem, such as poor management, sloppy
work habits, unreliable parts and materials, old machinery, bad
maintenance and so on.
(d)" Put the problem in context
(i)" What is the history and historical context of the problem?
•" It helps to know the history of a problem. By knowing what has
worked and not worked before, you can concentrate your efforts
on solutions that are more likely to succeed.
•" Even if the problem appears to be the same, the context in which
the solution worked (or not worked) previously will put a
different spin to the solution now.
(ii)" What is the problem environment?
•" What are the contexts surrounding the problem?

"
14
•" What factors contribute to the cause of the problem and cause it to
continue being a problem?
•" By understanding the contributing or perpetuating factors, you
will be able to take steps to prevent a problem from recurring.
(iii)" What are the circumstances affecting the problem?
•" Is the problem affected by specific conditions?
•" Would different conditions or circumstances worsen the problem
or reduce it?
•" Would removing the conditions or circumstances eliminate the
problem?
(iv)" What are the constraints?
•" Constraints are the limitations imposed or the requirements that
must be observed in solving the problem. They are part of the
solution framework that must be kept in mind. It is therefore
important to identify the constraints.
•" Examples of constraints are legal obligations, contractual
requirements, environmental concerns, physical circumstances,
financial and budgetary considerations, factors beyond your
control and so on.
1.4.2 Establish Your Goals
Now, let us look at the second step of problem solving – goal setting.
(a)" What goals would you consider ideal?
(i)" Very often, instead of considering what an ideal solution to a problem
would be, we simply settle for a quick solution.
ACTIVITY 1.2
Problems constrained by budgetary or legal realities are constantly
encountered by governments and large industries. Can you think of a
few examples from real life? Can you find one or two examples in the
newspapers?"

(ii)" We may reduce our expectation for practical reasons like the
constraints discussed above but we need not do this until we have
envisioned the ideal scenario.
(iii)" Sometimes the ideal scenario, after discussion and analysis, may seem
more achievable than previously thought.
(b)" Establish realistic and practical goals
(i)" In order for goals to be measured, they must be clear, definite and
precise. A list of such goals can be used to monitor the progress of
problem-solving attempts.
(ii)" Setting up goals gives direction to problem-solving tasks and gives
the problem-solver an objective to aim for.
(iii)" The goals must be realistic, otherwise achieving the goals will seem
an insurmountable task.
(iv)" Try to be as specific as possible in describing your goal; do not
hesitate if your goals remain vague or unquantifiable.
1.4.3 Generate Ideas
For problems to be solved, we need to generate ideas to look for possible
solutions. The following will give you ideas on how to find possible solutions.
Generate ideas for possible solutions
(i)" Read, research, think, ask questions and discuss.
•" Start by gathering information about your problem, and this includes
reading, discussing, brainstorming, asking questions and seeking
opinions. Be curious, look for ideas and learn as much as you can about
the problem.
(ii)" Use idea generation techniques such as brainstorming, forced relationships,
random stimulation and so on (these will be discussed in detail in the next
topic).
•" Generate as many ideas of all kinds as possible. This will give you a
vast pool of ideas from which you can choose, adapt or stimulate other
ideas.
(iii)" Allow time for ideas to develop during various stages of idea generation.
•" The major cycle of creativity has several stages, namely:
−" Preparation: Initial thought, research, study, work;

"
16
−" Incubation: Time to let the unconscious work;
−" Insight: The flash of recognition of a solution path, also called the
eureka experience or the eureka moment;
−" Implementation: Working out the solution; and
−" Evaluation.
1.4.4 Select Your Ideas
Now, based on the ideas you have found, evaluate and choose the best solution.
(a)" Evaluate the possible solutions for likelihood of success
(i)" Evaluate the ideas and possible solutions that you have collected so
far.
(ii)" Set up another possible solution that can be implemented („Plan B‰)
in case your main plan does not work.
(iii)" Your evaluation will involve finding the solution that will be the most
effective (works best), efficient (costs the least, in terms of money,
time, etc), and has the fewest side effects.
(iv)" Establish a ranking order for the possible solutions.
(b)" Choose the solution(s)
(i)" Select one or more solutions to try.
•" From the solutions ranked near the top of your evaluation list,
choose one or more to try. (The solution chosen for
implementation is not always, or even necessarily, the very top-
ranked one).
(ii)" Allow others to see and criticise your selected solution and to suggest
improvements or alternatives.
•" It is important to let others criticise and evaluate your solution
because you will then have a chance to improve it.
•" Do not let criticisms affect your confidence in ideas you believe
are good, but also do not be blind to criticisms that legitimately
point out your weaknesses. Always be willing to incorporate new
ideas.

1.4.5 Implement Your Action Plan
Once you have selected a solution, you can implement the action plan. However,
it is best for you to try out the solution and make necessary changes before the
actual implementation.
(a)" Try out the solution(s)
Experiment with the idea; test it on something closely resembling the real-
life scenario you are facing. This is especially vital in solutions that may
affect people's health or life.
(b)" Make adjustments or changes as needed during implementation
(i)" Always remain flexible in this application phase. Practically every
solution needs some modification when put into effect.
(ii)" Remember that the goal is to solve the problem, not to implement the
solution; don't expect that your solution will be exactly as you
originally proposed.
1.4.6 Evaluate and Follow Up
Now, after you have completed all the earlier steps in problem solving, it is time
for you to evaluate whether or not the steps that you have taken are effective.
(a)" Investigate whether the solution(s) worked and to what extent
(i)" Do you require modifications? Do other solutions need to be tested? Is
a different approach needed?
(ii)" One of the most frequent failures of problem solving is the lack of
evaluation of the implemented solution. The solution may not have
worked or not worked completely or it may have caused other
problems in the process.
(b)" Do not deal in absolutes (success or failure only)
(i)" Remember that many solutions are better described as partially
successful or unsuccessful, rather than a total success or failure.
(ii)" In many cases, an incomplete remedy is better than none at all.
Next, let us learn about a popular mechanism in exploring a problem.

"
18
ROOT CAUSE ANALYSIS
Root Cause Analysis (RCA) is a popular and frequently employed technique that
assists individuals in finding out why the particular problem happened in the
first place. Basically, its main aim is to investigate the origin of a problem which
is carried out using a specific set of steps together with associated tools. By
identifying the main cause of the problem, you can determine what happened,
why it happened and also find out the steps that should be taken to avoid its
recurrence in the future.
In general, RCA works on the principle that systems and events are interrelated.
According to this technique, a symptom can be traced back right to the initial
action that had actually triggered the cascade of events leading to the current
problem. The three basics types of causes that are usually looked at include:
(a)" Physical causes – tangible, material items that failed in certain ways.
(b)" Human causes – a mistake committed by an individual, or a certain task
that was not carried out accordingly by an individual.
(c)" Organisational causes – a faulty system, process, or policy that people
employ to make decisions.
RCA has been found to be effective in investigating the patterns of negative
effects, identifying hidden flaws in a particular system, and also uncovering
specific actions that have contributed to the problem. Thus RCA can be applied
to practically all types of problems as long as one is determined to go far in his or
her investigation with a good judgment and common sense. The steps involved
in the RCA process include:
(a)" Defining the problem – in this step you are to observe what is happening
and at the same time look out for specific symptoms.
(b)" Collection of data – this step is important to prove that the problem exists,
or that the problem has been there for a period of time and has created a
particular impact. In this step, assistance from individuals who understand
the situation and are most familiar with the problem will help you in
tackling the problem better. CATWOE is a useful tool that can be employed
at this stage in order for you to look at the problem from different
perspectives [the Customers, the people (Actors) who implement the
solutions, the Transformation process affected, the World view, the process
Owner, and Environmental constraints].
(c)" Identification of possible causal factors – this stage allows you to recognise
the sequence of events that led to the problem, the conditions that
permitted the problem to occur and the other problems that surrounded the
1.5

occurrence of the main problem. The most important fact is that one should
always look for as many causal factors as possible.
(d)" Identification of the root cause(s) – this step is carried out to find out the
reason behind the existence of the causal factor and the occurrence of the
problem.
(e)" Recommendation and implementation of solutions – in this final step, you
are supposed to come up with ideas to prevent the problem from occurring
again, planning the implementation of a solution by the assigned
individuals, and identifying the risks that may arise with the
implementation of the chosen solution.
You are encouraged to plan ahead to predict the effects of your solution in order
for potential failures to be spotted even before they occur using tools such as
Failure Mode and Effects Analysis (FMEA), Impact Analysis and Kaizen.
CONSTRAINTS
According to Dörner & Wearing (1995) and Funke (1991) some of the constraints
faced in solving difficult problems include:
•" Lack of clarity of the situation – in this case, one may face difficulty in either
initiating or continuing in the problem solving.
•" Polytely – this involves complex problem-solving characterised by the
presence of several goals or endings.
•" Complexity – problems with a large number of items, interrelations and
decisions.
•" Dynamics – problems that cannot be solved due the insufficient time frame
provided. In certain cases, the exact time required to solve the problem is
unpredictable.
Finally, we will explore the term solution again and relook at planning of the
solution.
WHAT IS A SOLUTION?
In everyday life, our understanding of solving a problem is by eliminating the
problem. This is just one approach to solving problems. There are problems,
however, that cannot simply be eliminated. These include things like waste, wear
and tear, or the occurrence of illness. Often, such "problems" are related to the
laws of nature and are therefore an integral part of our existence. When faced
1.6
1.7

"
20
with such problems, our main task is to make them less disruptive to the process
of reaching our goals. For this purpose, Harris, R. (1998) defines „solution‰ as the
management of a problem that successfully meets the goals established for
treating that problem. Harris goes on to clarify that "sometimes the goal will be to
eliminate the problem entirely; sometimes the goal will be only to treat the effects
of the problem. As such, it is vital to realise that your goal as a problem solver
will be shaped by various factors that include your ambitiousness, resources,
values and expectations, as well as the possibilities inherent in the problem itself.
Problems can be solved by applying two basic approaches:
(a)" Addressing the cause or source of the problem; and
(b)" Addressing the effects or symptoms of the problem.
Harris conveniently calls these the „stop it‰ and the „mop it‰ approaches
respectively. Each of these approaches has three basic forms.
1.7.1 The “Stop It” Approach
When we use the stop-it approach, we tackle the problem in such a way that, for
all intents and purposes, the problem no longer exists. There are three ways in
which the problem may be tackled:
•" Prevent the problem
•" Eliminate the problem
•" Reduce the problem
(a)" Prevent the Problem
(i)" Preventing a problem from occurring (or recurring) in the first place is
probably the ideal solution.
(ii)" This is often done by approaching any task with the desire for utmost
quality, regardless of cost or time.
(iii)" The prevention approach is often difficult to apply because it requires
predictive foresight (the ability to foresee a problem before it occurs).
(iv)" However, most problems appear unexpectedly.
(v)" Similarly, by preventing misunderstandings, the need for lots of
damage control and emotional healing can be avoided.

(b)" Eliminate the Problem
(i)" An excellent way of dealing with a problem is to eliminate it.
(ii)" There is no recurrence of the problem if elimination is properly done.
(iii)" You should always consider elimination solutions as possible
solutions.
(iv)" Elimination solutions, however may not always be feasible. For
example, the AIDS problem might be solved by eliminating unwanted
social behaviours, such as sexual practices and drug use. Parties with
vested interests would expectedly resist the implementation of such a
solution. Therefore, the mop-it approach is the usual approach taken
to solve the AIDS problem.
"
(c)" Reduce the Problem
(i)" There are problems, such as garbage production, that cannot be
eliminated entirely. When faced with such a situation, reducing the
problem can prove to be highly effective.
(ii)" We can lessen the impact of practically any problem by reducing its
size. A real-life example is the current approach to the flow of illegal
drugs into many countries. Most governments use reduction
strategies to approach the illegal drug problem. The flow of drugs
cannot be eliminated as long as demand continues, so law
enforcement agencies work to reduce as much as possible the routes
of entry, the middlemen dealing the drugs on the street and the
number of end-users.
1.7.2 The “Mop It” Approach
A mop it approach focuses on the effects of a problem rather than on the problem
itself. There are three forms of the mop it approach:
•" Treat the problem
•" Redirect the problem
•" Tolerate the problem
(a)" Treat the Problem
(i)" In this form, the damage caused by the problem is repaired or treated.
(ii)" Note, however, that by itself, a „treat it‰ solution is not going to be
nearly as effective as some form of „stop it‰ solution.

"
22
(b)" Redirect the Problem
(i)" In this approach, the problem is deflected, either physically by
moving it to another locality (a common solution in garbage
problems) or by postponing its solution by redefining it as not a
problem.
(ii)" Remember that a problem is a problem only when someone defines it
as such.
(c)" Tolerate the Problem
In this form, the effects of the problem are tolerated. It is debatable whether
this can be considered a form of "problem solving" but for the sake of
completion, we will leave it in the list. Basically, in this scenario, the effects
are taken for granted and measures are taken to endure them.
The best approach is, of course, the stop-it approach. However, this is not always
possible in all cases, and it may be that the only solution to a problem at hand is
the mop-it one. Nevertheless, an effective problem-solver will always consider
stop-it solutions before exploring mop-it alternatives. It is also important that
problems be solved by treating the root (or underlying) causes rather than merely
addressing the symptoms.
IDENTIFYING POSSIBLE SOLUTIONS
It is believed that „the best way to have a good idea is to have a lot of ideas‰.
Thus, identifying possible solutions to a particular problem is also known as
looking out for „optional solution‰ due to the fact that the main objective is to
bring to an end the list of all possible alternatives to the particular problem. A
wide range of conceivable solutions can be generated using a variety of creative
techniques.
This works best where a team is concerned. Every team member should
contribute one or more suggestions in order for all ideas to be taken into account.
This is important because when all viewpoints are thoroughly considered by the
team, members have a sense of direct ownership in the decision making process.
Eventually, this can assist the team in coming up with a consensus later in the
decision making process.
1.8

PLANNING THE SOLUTION
It is often vital to plan the solution before you can proceed to solve it or choose
the most suitable tool for it.
Planning your solution can be approached in the following way:
(a)" Defining four or five key causes of the problem, followed by identifying the
intervention needed to resolve them.
(b)" There may be several alternatives, with the best one depending on the cost,
effectiveness and forces that favour or impede it.
(c)" You can determine the best approach in the following manner:
(i)" Write down clearly what you expect your results to be.
(ii)" Evaluate possible alternatives. Are the alternatives doable, practical
and feasible? The following techniques are helpful:
•" Generate ideas by brainstorming.
•" Identify how forces can affect the viability of alternatives by using
the force field analysis.
•" Identify and explore the Strengths, Weaknesses, Opportunities
and Threats of the alternative solutions using the SWOT analysis.
•" Benchmark your processes with another similar organisation and
look for learning opportunities.
(iii)" Follow the analysis process to resolve the most important causes of
the problem:
•" Compare the effectiveness and viability of the alternative
solutions by using the prioritization matrix.
•" For the strategies selected, use the hierarchical flow chart to
determine which activities are the major focus, and what the
chronological order of their execution is.
(iv)" As part of your implementation plan, use the programme matrix to
define your goals and allocate the resources (human, material,
financial, time) that you need.
These techniques are explored again in later topics.
1.9

"
24
TAKING RESPONSIBILITY
Finally, it is crucial to take responsibility of a problem should we want to bring
about constructive change to the situation. When we deflect ownership and
blame only external circumstances for the problems we face, it can lead to a
feeling of helplessness which will undermine our capability to find a way out of
the problem.
„External locus of control‰ refers to a person who believes that his problems are
caused and determined by external stimulus, which are beyond their own
personal control and hence there is nothing he or she can do to change a
situation. This mindset can be unhealthy in the context of positive thinking.
We should not succumb to defeat with the belief that forces outside of ourselves
affect our ability to succeed.
•" A problem exists when there is a mismatch between what actually happens
with what we want or expect to have happened.
•" A problem can be defined as an opportunity for improvement.
•" Problem solving is the process of obtaining a satisfactory solution, preferably
in a novel way.
•" Problem solving can be seen as a structured or directional. The stages in
problem solving are:
−" Recognising and defining a problem;
−" Finding possible solutions;
−" Selecting the best solution; and
−" Implementing the solution.
•" Defining a goal is essential in determining how much of a problem exists or
whether or not there is a problem in the first place. It is vital to realise that
goals change with time and so do the problems you face.
•" Whenever faced with a problem, an effective solution can be developed by
following these steps:
−" Problem Exploration;
1.10

−" Goal Establishment;
−" Idea Generation;
−" Idea Selection;
−" Implementation; and
−" Evaluation or Follow Up.
•" Root Cause Analysis (RCA) involves the following processes;
−" Defining the problem;
−" Collection of data;
−" Identification of possible causal factors or root causes; and
−" Recommendation and implementation of solutions.
•" Constraints faced in problem solving include lack of clarity, polytely,
complexity and dynamics.
•" Two basic approaches to solving problems can be tackling it at the source, or
tackling its symptoms or effects.
•" It is important to plan the solution for the problem before proceeding to the
actual process of solving it.
"
Closed-ended problems
External Locus of Control
"Mop It" Approach
Open-ended problems
Problem-solving cycle
Root Cause Analysis (RCA)
"Stop It" Approach
SWOT Analysis
Thinking out of the box

"
26
1." Based on your understanding, explain what constitutes a problem.
2." A problem can be seen as both an opportunity for improvement and a
yardstick showing the difference between your current state and your ideal
state. Can you elaborate on both of these states?
3." Elaborate on the process of problem solving.
4." Explain briefly the steps involved in implementing the solution to a
problem.
5." Problem solving centres on thinking about ideal states that one would like
to achieve. How is goal setting useful in enhancing the problem-solving
process?
6." Whenever faced with a problem, an effective solution to the problem can be
developed by following these steps:
•" Problem Exploration
•" Goal Establishment
•" Idea Generation
•" Idea Selection
•" Implementation
•" Evaluation or Follow Up
Briefly explain the processes involved in each of these steps.
7." What are the three types of causes that are usually looked at in the Root
Cause Analysis (RCA) technique?
8." List the common constraints faced in difficult problems.
9." There are two basic approaches to solving problems:
•" Where the cause or source of the problem is attacked;
•" Where the effects or symptoms of the problem are attacked.
Can you elaborate on these approaches?
10." Describe briefly how to use the SWOT Analysis approach in solution
planning.

Adair, J. (2010). Decision making and problem solving strategies: Learn key
problem solving strategies; Sharpen your creative thinking strategies; Make
effective decisions (2nd ed.). London: Kogan Page.
Bedell, J. R., & Lennox, S. S. (1997). Handbook for communication and problem
solving skills training: A cognitive-behavioural approach (1st ed.). Canada:
John Wiley and Sons.
Dörner, D., & Wearing, A. (1995). Complex problem solving: Toward a
(computer-simulated) theory. In Frensch, P. A., & Funke, J. (Eds.), Complex
problem solving: The European Perspective (pp. 65-99). Hillsdale, NJ:
Lawrence Erlbaum Associates.
Fensel, D., & Motta, E. (2001). Structured development of problem solving
methods. IEEE Transactions on Knowledge and Data Engineering. (913-932).
Retrieved from http://people.kmi.open.ac.uk/motta
Funke, J. (1991). Solving complex problems: Human identification and control of
complex systems. In Sternberg, R. J., & Frensch P. A., (Eds.), Complex
problem solving: Principles and mechanisms (pp. 185-222). Hillsdale, NJ:
Lawrence Erlbaum Associates.
Harris, R. (1998). Introduction to problem solving. Retrieved from
http://www.virtualsalt.com/crebook3.htm
Sternberg, R. J. (1998). Thinking and problem solving (Handbook of perception
and cognition) (2nd ed.). San Diego, California: Academic Press.
Woods, D. R. (1975). Teaching problem-solving skills. Engineering Education,
66(3), 238-243.
Woods, D. R. (1980). Problem solving workshop. Annual Conference of American
Society for Engineering Education. University of Massachusetts, Amherst.

" INTRODUCTION
Mental representations and mental computations are the two main issues that
characterise modern psychology. Most individuals feel that thinking is solely
related to something internal or cognitive rather than behavioural. In other
words, thinking is something that takes place almost invisibly and, therefore, we
have no direct observational access to it. Thoughts about thinking date back to
early philosophers like Aristotle. We will discover some of the modern theories
about thinking in this topic. And we will explore a little further on the research
about the thinking process related to problem solving and reasoning later.
MAJOR ANTECEDENTS OF THE SCIENCE
OF PSYCHOLOGY
Thinking is carried out on versions of the external world called representations
existing in the human brain. In addition, thinking also involves processes that
2.1
TTooppiicc
22
Research on
Thinking and
Problem Solving
LEARNING OUTCOMES
By the end of this topic, you should be able to:
1." Describe the major antecedents of the science of psychology;
2." Explain the psychology of thinking;
3." Identify and describe the various concepts of problem solving;
4." Discuss the research carried out in problem solving and reasoning;
and
5." Explain the transition to the new cognitive era.

TOPIC 2 RESEARCH ON THINKING AND PROBLEM SOLVING 29
work to modify or enhance these representations, which are the functional or
computational aspects of thought. According to Dellarosa (1988), even the
earliest efforts to understand human thought, in philosophy and other domains
of inquiry, conversed to either or both of these issues of representation and
computation.
2.1.1 Associationism
Associationism, the first movement within science and philosophy to describe in
a formal sense the problems of representation and computation that extends as
far back to Aristotle, states that the world is represented within ourselves, and
our behaviour is guided by associations formed among these ideas. According to
John Locke (Figure 2.1), an English philosopher who characterised psychology as
the study of mental life, exerts that the mind represents the world as a network of
ideas, interrelated to another through experiences. An idea can be defined as the
irreducible representational unit of human thought that represents meaning,
knowledge and abstraction. Whatever the human mind can comprehend is
represented by a set of complex ideas that is analysable into its component
elemental ideas. Locke went on by insisting that ideas are not inborn or innate,
but rather developed from experiences (Boring, 1950). Locke, states that the
human mind without experience is like a blank slate; thus experience both
engraves its message on this slate and leaves behind signs of itself which
translates into ideas.
ACTIVITY 2.1
The thinking routine is a helpful tool in making thinking a visible
process. One thinking routine that we have found to be useful in many
settings involves two key questions: "What's going on here?" and "What
do you see that makes you say so?" Discuss these two key questions in a
classroom setting."

TOPIC 2 RESEARCH ON THINKING AND PROBLEM SOLVING
"
30
Figure 2.1: John Locke
Source: http://www.philosophypages.com/ph/lock.htm
Complex ideas are the result of the binding and association of simple ideas. The
level of complexity and connectedness among ideas develop from childhood into
adulthood through experiences. The two experimental sources of simple ideas
are our senses that notice changes in our surroundings and the mind itself,
through reflection upon its own activity. Hence, the mind is able of being
conscious of its own matter and processes; a phenomenon known as
introspection. Due to the fact that the mind is fundamentally independent of
obvious behaviour, an individualÊs capability to explain it seems to be the only
way its activity can be known. This again assumes a certain degree of awareness
of mental representation and/or computation.
The principles that ruled the association of ideas and have stood the test of time
and data include:
(i)" Contiguity: Two ideas that happen together in time or in space are likely to
be associated or linked;
(ii)" Similarity: The higher the degree of similarity between two ideas, the
greater their probability of being associated; and
(iii)" Repetition: The more frequent two ideas come about together, the greater
their strength of association.
Subsequently, British philosophers, James Mill and his son John Stuart Mill,
differentiated between sensations and ideas as basic classes of elements in the
mind. According to them, sensations are the primary elements of consciousness
from which ideas, which are pure abstractions, are drawn from. They developed
the criteria of strength which included:
(i)" Permanence: The more persistent an association as time passes by, the
greater its initial strength;

(ii)" Certainty: The more confident the individual of an association, the greater
the strength; and
(iii)" Facility: The greater the speed and effortlessness with which an association
comes by, the greater its strength.
2.1.2 Faculties of the Mind
According to the Scottish School of Philosophy (Boring, 1950), the mind is not a
set of ideas or representations of the world but a set of faculties, powers, or
computational capacities that operate on incoming sensations. Their main
disagreement to Associationism was its importance on elementary elements,
which among other things appeared to go against the conventional religious
notions of the unity of the mind and soul prominent at the time.
However, as time passed, psychology drew closer to Associationism. Thomas
Brown proposed that it was necessary to invoke the principle of association or
representation, along with faculties, to account for the succession of thoughts that
seem to be typical of mental life. Brown proposed a series of laws that guided
associational processes and accounted for the finer differentiations among ideas
such as relative frequency, recency and vividness.
Aristotle introduced another principle to govern the association of ideas
called the „The Law of Contrast‰. What do you understand from this
theory?
SELF-CHECK 2.1
ACTIVITY 2.2
Brown extended the principles of Associationism by being the first to
identify the secondary laws of Association. Primary laws accounted for
the transformation of simple ideas into complex ideas. The secondary
laws, however, explained why one association would be stronger than
another. Carry out a search on the internet on these six secondary laws of
association."

"
32
THE PSYCHOLOGY OF THINKING
In general, early American psychologists were more concerned with the „whys‰
and „hows‰ of consciousness and behaviour than with the „whats‰. Importance
was placed on the functional significance of the mind, on its role in the
adaptation of the organism to the environment, and on its properties as a
mechanism that controls human behaviour.
2.2.1 William James
A theorist and a systematiser whose fundamental position on psychology was set
forth in a classic book entitled Principles of Psychology (1890). According to
James, the human mind can be described as an array of functions and
consciousness as a dynamic stream of interacting events. He characterised
psychology as the study of an adaptive process. Being a person who believed in
practicality, he attempted to describe both behaviour and consciousness through
the concept of the mind as a quasi-biological organ whose function(s) can be used
for environmental adaptation. Besides its greater intricacy, abstractness and
complexity, the mind is almost similar with any other bodily structure such as
the heart or the lungs. In addition, according to him, the mind has also evolved in
human beings to a point where its functions are more versatile and pervasive
than those of any other single organ.
2.2.2 Functionalism
Functionalism mainly focused on consciousness and Elementism, and was led by
William James and John Dewey. JamesÊs principles were the foundation of
Functionalism, however, they were enhanced by the insights and observations of
Dewey and others.
(a)" The Significance of Consciousness
Although the importance of consciousness was not disagreed by
functionalists, its position as the primary problem for psychology was.
Consciousness was treated by the functionalists much as mediators and
symbolic processes.
(b)" DeweyÊs Theory of Thinking
Dewey (1910) defined thinking as a multi-staged, goal-oriented process that
possesses the following characteristics:
(i)" Recognition of a problem, or a „felt difficulty‰;
2.2

(ii)" Location and definition of the problem and the isolation of its relevant
features;
(iii)" Formulation of possible alternative solutions;
(iv)" Thinking over or reasoning through the various possibilities to
determine the most likely candidate solution; and
(v)" Testing the selected solution possibilities.
A stage-like analysis of thought was later refined by Graham Wallas based
on a survey of anecdotal evidence on the symbolic processes of creative
scientists. The four stages of creative acts include:
(i)" Preparation: The collection and assembly of problem-relevant
information;
(ii)" Incubation: The pondering period;
(iii)" Illumination or insight: The conception of a solution; and
(iv)" Verification: The process ensuring that the solution is effective.
2.2.3 Later Functionalism
According to studies carried out by Thorndike (1898), learning and problem
solving are, in general, gradual processes based on the increasing strength of the
connection between the stimulus situation and certain response possibilities. The
law of repetition or exercise and The Law of Effect strengthens the connection
between stimulus and response.
The Law of Effect which is rather similar to the modern-day principle of
reinforcement states that an act or response that is followed by a pleasurable state
of affairs tends to become associated with the stimulus situation effective at the
time of its occurrence. On the other hand, the connection between a response that
1." Define the term „pragmatism‰. In what way was William James a
pragmatist?
40" You are having tough time thinking about your current job due to
the heavy workload and the difficult attitude of your boss. Explain
how you would apply DeweyÊs theory of thinking in handling the
above scenario."
SELF-CHECK 2.2

"
34
is accompanied by a displeasing state of affairs and the stimulus to which it
occurs tends to be weakened. Due to the fact that experimental data of both
human and animal subjects appeared similar, Thorndike stated that responses
are linked directly to the stimulus situation as it is sensed by the subject.
2.2.4 Behaviourism
The system introduced by Watson (1924), called Behaviourism, placed behaviour
or performance as the focus of attention; making it the fundamental element in
psychology. According to behaviourists, it is of no use trying to understand the
inaccessible and possibly non-existing consciousness when the real data of
psychology are open, observable and available to direct measurement. The Law
of Effect: Pain avoidance vs. Pleasure theory was formalised by Edward L.
Thorndike (Figure 2.2) who suggested that behaviour, which results in pleasing
outcomes, tends to be repeated; while behaviour that leads to distasteful
consequences, does not.
Figure 2.2: Edward L. Thorndike
Source: http://www-distance.syr.edu/pvitaelt.html
Watson is of the opinion that inferred conscious states and processes are excess
baggage, while knowledge of the regularities of performance is all that a
psychologist really requires to be interested in.
Watson further added that we should eventually find out that much of what is
considered mentalistic is in reality unspoken or miniaturised motor activity,
mainly in the voice mechanism. Words, also known as symbols, are responses
that we have learned to apply to objects and events in the environment.

One can think of these objects or events in terms of their verbal counterparts.
When we „think to ourselves‰ about them, we are basically suppressing the
obvious verbal response of naming to a point where it becomes difficult or
impossible for others to find out. However, with suitable sensitive recording and
measuring equipments, we should find proof of them in tiny laryngeal
movements. Hence, thinking is actually sub-audible speech or what is known as
the motor theory of thinking, which places the emphasis on muscular or
glandular changes rather than on central or ideational changes.
Results carried out by Jacobson (1930) revealed that specific patterns of muscular
activity accompany and correlate with the content of thought process. In
addition, Freeman (1931) reported that elementary mental activities could be
facilitated by inducing an increase in generalised muscular tone in the subject.
This leads to the conclusion that there is a correlation between muscular
(especially vocal) movements and thinking. In other words, it may be that mental
activity is in some sense dependent on certain specific muscular tones.
Nevertheless, the evidence has been inadequate to convince all psychologists that
thought and action are identical or that thought can be represented completely
by patterns of peripheral activity.
2.2.5 Gestalt Psychology
Gestalt is a German word which means „organised whole‰ or „configuration‰.
The believers of this psychology feel that psychological experience is anti-
elementaristic or in other words, not made up of discrete representational
elements, but an organised, dynamic field of events that interact or mutually
affect one another. An organism reacts to the whole configuration when it
experiences its environment. Properties of the whole psychological field are
different from the sum of its individual parts, and thus no analysis into parts can
be entirely successful. It also states that sensations, perceptions, images,
associations, reflexes and the like are not accepted as meaningful elemental
psychological units. In order to understand psychological phenomena, one must
consider a system of stimulation in which the change in any part can affect all the
other parts.
(a)" Gestalt Description of Thinking
In the Gestalt point of view, a problem is considered to exist when there are
unresolved tensions or stresses in the psychological field, resulting from
some interaction of perceptual and/or memory factors. Thinking takes
place as the stresses work themselves out, which in turn force the thinking
organism into action. It emphasises the computational process that follows
the principles of field theory from physics. Past experience with related
problems is no guarantee for solution, however, solutions arise from the

"
36
problem as it is perceived. Thinking is a process of resolving field stresses.
Looking at the problem from many different angles is vital before the
interaction of events leads to a clearer picture of a solution. In other words,
the moment one analyses the problem correctly, the solution appears
almost immediately. The Gestalt theory believes that there is a clear and
strong relation between thinking and perception where both are controlled
by practically the same principle of field theory. The main difference
between the two is that thinking occurs in a symbolic manner and is less
under the control of external factors.
(b)" Evaluation
The emphasis of Gestalt psychologists on the fundamental and basic
importance of insight as a principle of learning has got them involved in a
long-standing controversy with individuals who regard learning as a
gradual process. The arrival of Gestalt psychology was during the times
when experimentalism was strongly in the grip of Behaviourism that
pictured the organism as a machine, subject to the many different sources of
external stimulation and responding to them in an indispensible automatic
fashion. Overall, Gestalt theory assisted psychology to attain a more
balanced and practical view of complex human behaviour that would have
been difficult to be described solely on the basis of Behaviourism.

RESEARCH ON CONCEPTS
The two important questions that are required to understand human conceptual
behaviour are:
(i)" How are concepts defined?
(ii)" How are concepts acquired?
2.3
Similarity, continuation, closure, proximity and figure & ground are
among the Gestalt principles. In the following examples, identify the
principle that is being applied:
(a)" Hint: Do the 11 objects that appear as a single unit look alike?
(b)" Hint: Although the panda is not complete, enough is present for
the eye to complete the shape; the viewer's perception completes a
shape.
SELF-CHECK 2.3

"
38
2.3.1 Common-Element Concepts
Hull (1920) defined concepts as the abstraction of elements common to a category
of stimuli and the attachment of a single response to those elements. He added
that new stimuli sharing those elements would bring out the same response,
producing a phenomenon called stimulus generalisation. According to Hull, the
learning process was relatively unconscious and automatic. However, the two
ideas, that concepts are defined by common elements and that concept learning is
a gradual, automatic associative process, remained popular for a long time. In
contrast, Osgood (1953) felt that the common-elements approach was too
restrictive and did not exactly capture the meaning of concept formation.
Furthermore, many concepts not based on common elements that require the
learner to show only a consistent response did not differentiate between simple
labelling and a more complete understanding of a concept.
2.3.2 Mediational Processes and Hypotheses
In contrast to the common-element concept, Smoke (1932) studied the learning of
concepts based not on common physical elements but on common perceptual
relations or immediate sensory experiences. Osgood (1953) stated that common
perceptual relations were insufficient to describe many concepts. A good
example here would be the common perception that Singapore, Malaysia and
Myanmar are all nations. However, Osgood further added that the key notion
was that concepts reflect the mediational processes generated by the concept
learner and not by external factors.
Heidbreder (1946) explained the idea that concepts can be based on different
kinds of criteria. Nevertheless, none of the concepts she demonstrated was based
on common physical elements; in fact, the categories were based on either
concrete objects (e.g., trees), spatial forms (e.g., anything circular) or numbers
(e.g., five objects). According to HeidbrederÊs research, members of a concept
always had „something in common‰, however, that „something‰ may be
complex and abstract. As for Osgood, coming up with a common symbolic
response to physically different stimuli was essentially different from responding
to common physical elements.
Both Hull (1920) and Heidbreder (1924) had contradicting views on the concept-
learning process. In HullÊs approach, learning was passive where stimulus
elements get conditioned to responses in an automatic fashion. On the other
hand, Heidbreder reported two types of behaviour, namely participant and
spectator behaviour. As for participants, they formulated and actively tested
hypotheses on the basis of the concepts they were learning. In contrast,

spectators, normally after a series of wrong responses, responded „randomly‰ in
categorising stimuli, waiting for a new hypothesis to suggest itself.
The two important conclusions that we can arrive at this points are that:
(i)" Concept learners at times actively pursue their ideas about what a
particular concept might be; and
(ii)" A concept-learning task may be approached in more than one method.
2.3.3 Strategies and Kinds of Concepts
Great emphasis was focused on the use of strategies for obtaining concepts.
Several different strategies were explained, with each strategy considered in
terms of its logical, information-gaining properties as well as its cognitive strain.
Bruner and colleagues (1956) described cognitive strain by the load on memory
and inference. These researchers also realised that there could be conflict between
maximising information gain and minimising cognitive strain, in order that a
concept learner may use a less-than-optimal strategy to keep cognitive strain to a
tolerable level.
A strategy called conservative focusing applicable to learning common-element
concepts under the selection procedure, both maximises information gain and at
the same time keeps cognitive strain to a low level. This strategy consists of
choosing an example that differs in one and only one characteristic from the
initial positive example, taking note whether the chosen example is positive or
negative. This strategy is an excellent method to identify a conjunctive
(connective) concept (Figure 2.3).

"
40
Figure 2.3: A concept formation study (Bruner et al. (1956))
Source: http://www.palgrave.com/psychology/malim/pdfs/chap_15.pdf
Successive scanning is a another strategy introduced by Bruner and colleagues
(1956) which consists of making a guess on what the target concept may be and
simply holding on to it until it is proven wrong, followed by making another
guess, and so on. It was found that as the concept-learning task becomes more
„stressful‰, the probability increases that learners will employ simpler and less
effective strategies.
In logic, correct predictions usually lead to maintaining hypotheses while wrong
predictions ought to be followed by employing new hypotheses that are in

harmony with the information presented. The type of instance (positive or
negative) does not matter; however, it can be expected to affect performance due
to the greater processing demands normally linked with negative instances.
These researchers further added that learners are least likely to produce correct
inferences from trials involving wrong predictions about negative instances.
They also pointed out that different strategies have unequal chances of coming
across the various types of trials, hence providing a trial-by-trial processing
explanation of overall differences in strategy success rates.
RESEARCH ON PROBLEM SOLVING
There has been rational agreement that the essential characteristics of a problem
are that an individual has a goal, but lacks a clear or well-studied path to the
goal. Hence the importance in research on problem solving has been on response
discovery or in other words, how the individual arrives at effective, goal-
achieving behaviour.
2.4.1 Trial and Error
Due to the fact that a problem solver has to find a suitable solution, it is expected
that a crucial activity is trying different approaches and making errors until the
right approach is found. Trial and error is linked with the view that obtaining a
solution is a gradual, undirected process that does not involve perception or
comprehension of problem requirements or structure. In his studies on animal
behaviour (Figure 2.4), Thorndike (1898) placed a hungry cat in a cage where the
food could not be reached from inside the cage unless the cat triggered some
mechanism by pulling a string or pushing a pole somewhere in the cage.
2.4
ACTIVITY 2.3
Link up with a friend and simulate Bruner's procedure in the following
way: Take a pack of playing cards which has attributes of suits (hearts,
diamonds, spades and clubs), colours (black and red) and numbers (ace
to ten, Jack, Queen and King). Ask your friend to select a concept and go
through Bruner's procedure. First, get your friend to point out a positive
example of the concept selected, then select further cards and for each
ask, „Is that one?‰ Take both of the strategies described above in turn
and decide for yourself which is the most effective one."

"
42
Figure 2.4: ThorndikeÊs puzzle box
Source: http://www.nickyhayes.co.uk/nicky/OHP/ch18ohp.html
Thorndike observed that over repeated trials, the catsÊ behaviour gradually
changed, and after many trials the animals would make relatively direct efforts to
activate the door-opening mechanism. He observed in this behavioural pattern,
no proof of the cat „seeing through‰ the problem. In fact, problem solving was
regarded as a process in which unsuccessful responses were gradually
eliminated and successful responses „stamped in‰. An important characteristic of
the puzzle-box problems was that the solution mechanism was concealed and the
critical elements of the situation were not readily available for inspection.
Woodworth and Schlosberg (1954) stated clearly that trial and error can at times
be necessary, as a strategy of exploring the environment and obtaining
information necessary. In addition, even hypothesis testing could be regarded as
a form of trial and error.
2.4.2 Insight
Thorndike concluded that no perception of critical relations was involved. On the
other hand, Yerkes (1916) reported a sudden shift from trial and error to solution
in chimpanzees, who exhibited good retention of solutions and some transfer to
new situations. Kohler (1925; 1976) realised that when presented with a problem,
the apes tried several different methods such as stretching, jumping, reaching out
for or even bashing any objects that were handy enough. After a certain amount
of time, these chimpanzees would go into a period of calmness, perhaps thinking
thoroughly on its mission. Eventually, with a sudden and planned movement,

the ape would utilise the critical object(s) to obtain the particular fruit
(Figure 2.5).
Figure 2.5: KohlerÊs most well-known work on chimp cognition was in the use of tools to
gain access to food
Source: http://wkprc.eva.mpg.de/english/files/wolfgang_koehler.htm
Kohler indicated that there was a significant variation in the time required to
arrive at a solution where some animals solved the task in less than 15 minutes
while some required hours before being able to find a solution. Furthermore,
Kohler also observed that the positioning of the objective and the implement in
the animalÊs visual field was vital. The positioning of the fruit and the critical
implement at opposite ends of the cage drastically reduced the attainment of the
solution due to the fact that the animal was not able to look at both these objects
simultaneously. Hence, the effect of visual separation was particularly crucial on
the first exposure to a problem situation, a finding that is in harmony with the
notion that solving a problem involves a change in perception of the situation.
2.4.3 Problem-Solving Set
The phenomenon of set happens in many situations, reflecting a state of
preparation for a particular input and response. In most situations where a quick
but accurate reaction is desired, sets serve as positive purposes. Thus, to be
prepared actually means to detect and react more quickly than would otherwise
be the case. The positive effects of set mentioned above are best understood in
terms of general preparedness. Another aspect of set is to prepare an individual
for a specific type of input or response.
As stated by Woodworth and Schlosberg (1954), the term „direction‰ proposed
by Maier (1930) which guides problem-solving efforts, was a kind of set.
Directions are not habits but indicate the influence of the problem situation. In

"
44
other words, the structure of the problem situation was maintained to affect the
direction that would be followed. Maier (1940) differentiated between habitual
directions and new directions as summarised in Table 2.1.
Table 2.1: Habitual and New Directions
Habitual directions New directions
Are overcome when problem solving
requires doing something new.
Are created when problem solving
requires doing something new.
Results in reproducing old solutions. Give rise to a new combination of
experiences.
Accounted for by memory and associative
processes.
Require a direction that is established in
the problem situation and controls
memory integration.
Functional fixedness, an example of negative set, is a term proposed by Duncker
(1945) that refers to the fact that an object with a strong expected function cannot
be observed as playing a different role. In a classic experiment explaining
functional fixedness, Duncker (1945) provided experimental subjects a candle, a
box of nails, and several other objects, and asked them to attach the candle to the
wall so that it did not drip onto the table below (Figure2.6).
Figure 2.6: Classic candle box experiment that illustrates functional fixedness

He found that the most of the subjects either tried to nail the candle directly to
the wall or to glue it onto the wall by melting it. In fact, very few of them came
up with the idea of using the inside of the nail box as a candle-holder and nailing
this to the wall. In DunckerÊs terms the participants were „fixated‰ on the boxÊs
usual function of holding nails and could not re-conceptualise it in an approach
that allowed them to solve the problem.
Research on functional fixedness stressed on uses for particular objects. A rather
different approach to the problem-solving set concerned the dominance of a
particular type of response to situations with a high degree of surface similarity.
The usual procedure was to provide individuals with solving a series of similar
problems in „the same way‰, followed by testing them with similar problems
that may or must be solved using a different approach. A suitable example for
this was demonstrated by the „water jar‰ problem (Luchin, 1942). In this
experiment, individuals were provided with three different jars with specified
capacities, however, without any gradations. They were required to show how
they would obtain a particular volume of water. For instance, a volume of 100
quarts can be obtained from three separate jars holding 21 quarts (jar A), 127
quarts (jar B) and 3 quarts (jar C) by firstly filling the largest jar B, followed by
filling the second largest jar A once from B, and finally filling the smallest jar C
twice from B. This would finally leave a total volume of 100 quarts in the largest
jar B as demonstrated by the equation B – A – 2C.
Imagine that you have three jars, A, B and C. For each of the seven
problems below, the capacity of the three jars is listed. You must use
these jars in order to obtain the amount of liquid specified in the Goal
column. You may obtain the goal amount by adding or subtracting the
quantities listed in A, B and C.
Problem A B C Goal
1 24 130 3 100
2 9 44 7 21
3 21 58 4 29
4 12 160 25 98
5 19 75 5 46
6 23 49 3 20
7 18 48 4 22
SELF-CHECK 2.4

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