Learning to Solve Problems

A Handbook for Designing Problem-Solving Learning Environments

David H. Jonassen

• Publisher: Routledge
• Year: 2010
• ISBN:
  9781136941887
• Edition: 1
• Topic: Education
• Subtopic: Education General

This book provides a comprehensive, up-to-date look at problem solving research and practice over the last fifteen years. The first chapter describes differences in types of problems, individual differences among problem-solvers, as well as the domain and context within which a problem is being solved. Part one describes six kinds of problems and the methods required to solve them. Part two goes beyond traditional discussions of case design and introduces six different purposes or functions of cases, the building blocks of problem-solving learning environments. It also describes methods for constructing cases to support problem solving. Part three introduces a number of cognitive skills required for studying cases and solving problems. Finally, Part four describes several methods for assessing problem solving. Key features includes:

• Teaching Focus – The book is not merely a review of research. It also provides specific research-based advice on how to design problem-solving learning environments.
• Illustrative Cases – A rich array of cases illustrates how to build problem-solving learning environments. Part two introduces six different functions of cases and also describes the parameters of a case.
• Chapter Integration – Key theories and concepts are addressed across chapters and links to other chapters are made explicit. The idea is to show how different kinds of problems, cases, skills, and assessments are integrated.
• Author expertise – A prolific researcher and writer, the author has been researching and publishing books and articles on learning to solve problems for the past fifteen years.

This book is appropriate for advanced courses in instructional design and technology, science education, applied cognitive psychology, thinking and reasoning, and educational psychology. Instructional designers, especially those involved in designing problem-based learning, as well as curriculum designers who seek new ways of structuring curriculum will find it an invaluable reference tool.

1
HOW DOES PROBLEM SOLVING VARY?

WHAT IS A PROBLEM?

This book is about learning to solve problems, so first I shall describe what a problem is, that is, what is being solved. There are many conceptions of a problem. The word “problem” derives from the Greek problema, meaning obstacle. The word “problem,” as used in this book, refers to a question or issue that is uncertain and so must be examined and solved. Everyday life and work are filled with uncertain situations for which no resolution is immediately known. What route should I take to work to minimize traffic congestion? How can we afford an addition to the school building? How can we accelerate the collection of receivables? What will be the most effective method for marketing our new product to the target group? How can we increase fatigue strength to this material without increasing cost significantly? Which medical-insurance program should I select? These are all questions about situations that are currently unknown and therefore need resolution. Those problem situations vary from algorithmic math calculations to vexing and complex social problems, such as mitigating violence in the schools. For me, finding or solving the problem must have some social, cultural, or intellectual value. That is, someone believes that the problem is worth solving. “Problems become problems when there is a ‘felt need’ or difficulty that propels one toward resolution” (Arlin, 1989, p. 230). That is, someone believes that the question is worth answering. If no one perceives a need to answer the question, there is no problem. This latter attribute may eliminate most formal, in-school problems from the category of real problems because students often do not perceive a need to find the unknowns to the types of problems posed in schools. However, because their teachers do perceive such a need, they are normally regarded as problems.

For many people, the concept problem has a more affective conotation. For them, a problem is a situation or matter that presents a perceived difficulty. They may have a problem child, or I am having a problem with my boss. Synonyms for problems include “dilemma,” “quandary,” “obstacle,” “predicament,” and “difficulty,” all of which have heavy affective connotations. Indeed, problems often do represent predicaments, and problems are often difficult. However, for purposes of this book, I regard problem solving as a primarily cognitive activity. Although many problems engage affect, I will not deal explicitly with those issues. The cognitive perspective on problems considers a problem as “a question to be resolved.” That is the spirit in which this book addresses problems. Why? Because we are constantly solving problems in our everyday and professional lives, so educators ought to help students learn to solve the problems they will face in their professional lives and perhaps those that plague their personal lives.

Psychologists have examined problems and problem solving fairly extensively, beginning with information-processing theorists. A problem, from an information-processing perspective, consists of sets of initial states, goals states, and path constraints (Wood, 1983). Solving a problem means finding a path through the problem space that starts with initial states passing along paths that satisfy the path constraints and ends in the goal state. According to Davidson, Deuser, and Sternberg (1994), problems consist of givens (the elements, relations, and conditions that define the initial state), goal (desired solution), and obstacles (characteristics of the problem solver or the problem situation that make it difficult to transform initial state into goal state). Unfortunately this information-processing conception has been used largely to describe well-structured problems (to be described later in this chapter and more extensively in Chapter 2). For most everyday, ill-structured problems (also described later in this chapter), the goal states and path constraints are often unknown or are open to negotiation, and so there are no established routes through path constraints toward the goal state. Information-processing models of problem solving are inadequate for representing the many kinds of problem solving, especially those that engage situated, distributed, and social aspects of problem solving. As problems become more ill defined, their solutions become more socially and culturally mediated (Kramer, 1986; Roth & McGinn, 1997). What becomes a problem arises from the interaction of participants, activity, and context.

WHAT IS PROBLEM SOLVING?

My assumption in this book is that problem solving is primarily a cognitive process. In the Introduction and earlier in this chapter, I recognize the importance of affect and motivation on problem solving; however, in this book, I focus on the cognition of problem solving.

There have been many cognitive conceptions of problem solving. As alluded to before, a number of information-processing models of problem solving, such as the classic General Problem Solver (Newell & Simon, 1972), have explained problem-solving processes. The General Problem Solver specifies two sets of thinking processes associated with the problem-solving processes, understanding processes and search processes. Another popular problem-solving model, the IDEAL problem solver (Bransford & Stein, 1984) describes problem solving as a uniform process of identifying potential problems, defining and representing the problem, exploring possible strategies, acting on those strategies, and looking back and evaluating the effects of those activities.

Gick (1986) synthesized these and other problem-solving models (Greeno, 1980) into a simplified model of the problem-solving process, including the processes of constructing a problem representation, searching for solutions, and implementing and monitoring solutions. Although descriptively useful, these problem-solving models assume that all problems are solved in pretty much the same way and that these generalizable processes can be applied in different contexts with different types of problems in order to yield similar results. A serious weakness of general problem-solving approaches is their underestimation of the role of domain knowledge and thus pattern recognition (analogical reasoning) which has resulted in the misrepresentation of knowledge, thereby inhibiting far transfer, which is the true purpose of education and training. Treating problem solving as a reproducible, algorithmic process has failed to focus on the highest-value learning outcomes, which is certainly part of the reason that school learning and corporate training are often perceived as irrelevant and boring.

Among the most commonly referenced models of problem solving is that proposed by Polya (1957). In How to Solve It, Polya (1957) addressed some of the limitations of information-processing models of problem solving in his general problem-solving approach, even before they were conceived. He recommended four steps to solving mathematical problems:

1. understand the problem (what is being asked for; is there enough information);

2. make a plan (look for patterns; organize information);

3. carry out the plan;

4. evaluate its effectiveness.

Polya recommended numerous heuristics to improve problem solving, such as analogies (Can you find a similar problem?), induction (generalizing from examples of problems), and pattern matching (Have you solved a similar problem?). Many of his recommendations are described throughout this book. Analogies are described in Chapters 11 and 16. Induction is described in Chapter 9, and pattern recognition is described in Chapter 12.

For me, problem solving as a process also has two critical attributes. First, problem solving requires the mental representation of the problem, known as the problem space, problem schema (Chapter 15), or mental model of the problem. The problem space consists of a set of symbolic structures (the states of space) and the set of operators over the space (Newell, 1980; Newell & Simon, 1972). Once again, those states of space are easily identifiable in well-structured problems (see discussion below); however, they are much more difficult to identify for ill-structured problems. Problem spaces may be externalized as formal models (see Chapter 19 of this volume for descriptions of methods used to externalize problem representations). However represented, the construction of a mental model of the problem is one of the most critical problem-solving processes. In this book, I emphasize the importance of constructing mental models of the problem in order to understand the elements of the problems and how they interact as well as the procedures for solving a problem. Until the problem solver constructs a model of the problem in its context, a viable solution is only probable, while understanding and transfer are improbable. Second, problem solving requires some manipulation and testing of the mental model of the problem in order to generate a solution. Problem solvers act on the problem space in order to generate and test hypotheses and solutions.

Schema-theoretic conceptions of problem solving opened the door for different problem types by arguing that problem-solving skill is dependent on a schema for solving particular types of problems. The construction of those problem schemas results from the extraction and application of domain knowledge. If the learner possesses a complete schema for any problem type, then constructing the problem representation is simply a matter of mapping an existing problem schema onto a problem. Existing problem schemas are the result of previous experiences in solving particular types of problems, enabling the learner to proceed directly to the implementation stage of problem solving (Gick, 1986) and try out the activated solution. Experts are better problem solvers because they recognize different problem states that invoke certain solutions (Sweller, 1988). If the type of problem is recognized, then little searching through the problem space is required. Novices, who do not possess well-developed problem schemas, are not able to recognize problem types, so they must rely on weaker problem-solving strategies, such as means–ends analysis.

My theory of problem solving diverges from traditional approaches to problem solving that articulate single approaches to solving all kinds of problems. In the remainder of this chapter, I argue that problems and therefore problem-solving processes vary. The ways that physicians diagnose medical maladies is different from the ways that mechanical engineers design a new part for an automobile or the ways that we make personal decisions about our needs. Next, I describe how problems and problem solving vary.

HOW DO PROBLEMS VARY?

Problems and the methods and strategies used by individuals and groups to solve them, both in the everyday and classroom worlds, vary dramatically. Smith (1991) categorized factors that influence problem solving as external and internal. External factors are those related to the nature of the problem as encountered in the world. Internal factors are related to the personal characteristics of the problem solver, such as prior experience, prior knowledge, or strategies used. Problem solving varies both externally (the problem as it exists in the world) and internally (how the individual conceptualizes and resolves the problem). I will first describe external problem factors and later explicate some internal factors that are important to problem solving.

What External Factors Mediate Problem Solving?

Problems, as they are encountered in the world, differ in several important ways. Bassok (2003) described two important external attributes of problems: abstraction and continuity. Abstraction refers to the representation of the content and context of a problem that either facilitates or impedes analogical transfer of one problem to another. Most classroom problems are more abstract than most everyday problems, which are embedded in various contexts. Continuity of the problem is the degree to which attributes of problems remain the same or change over time (described later as dynamicity). High-continuity problems are more easily solved and transferred.

The primary reason for distinguishing among different kinds of problems is the assumption that solving different kinds of problems calls on distinctly different sets of skills (Greeno, 1980). Solving different kinds of problems entails different levels of certainty and risk (Wood, 1983). Given that different kinds of problems require different sets of skills, then learning to solve different kinds of problems will require different forms of instruction. In order to better understand how problems differ, I describe five external characteristics of problems:

1. structuredness;

2. context;

3. complexity;

4. dynamicity;

5. domain specificity.

What Is Structuredness of Problems?

Foremost among the differences among problems is the continuum of structuredness, between well-structured and ill-structured problems (Arlin, 1989; Jonassen, 1997, 2000c; Newell & Simon, 1972; Voss & Post, 1988). Most problems encountered in formal education are well-structured problems, while problems that occur in our everyday and professional lives tend to be more ill structured. It is important to note that structuredness represents a continuum, not a dichotomous variable. While well-structured problems tend to be associated with formal education and ill-structured problems tend to occur in the everyday world, that is not necessarily the case.

The most commonly encountered problems in formal educational contexts are well-structured problems. Typically found at the end of textbook chapters and on examinations, well-structured problems present all of the information needed to solve the problems in the problem representation; they require the application of a limited number of regular and circumscribed rules and principles that are organized in a predictive and prescriptive way; possess correct, convergent answers; and have a preferred, prescribed solution process (Wood, 1983). These problems have also been referred to as transformation problems (Greeno, 1980) ...

Table of contents

• CONTENTS
• ILLUSTRATIONS
• READ ME FIRST
• PREFACE
• ACKNOWLEDGMENTS
• ABBREVIATIONS
• 1 HOW DOES PROBLEM SOLVING VARY?
• PART I PROBLEM-SPECIFIC DESIGN MODELS
• PART II CASES: THE BUILDING BLOCKS OF PROBLEM-SOLVING LEARNING ENVIRONMENTS
• PART III COGNITIVE SKILLS IN PROBLEM SOLVING
• PART IV ASSESSING PROBLEM SOLVING
• REFERENCES
• INDEX