Toward a Unified Theory of Problem Solving

Views From the Content Domains

Mike U. Smith

• Publisher: Routledge
• Year: 2012
• eBook ISBN:
  9781136467165
• Topic: Education
• Subtopic: Education General

One of the most active fields of educational research in recent years has been the investigation of problem-solving performance. Two opposing views of current research -- one suggesting that there are more differences than similarities within different domains, and the other stating that there is great similarity -- lead to a variety of questions:

* Is problem solving a single construct?

* Are there aspects of problem-solving performance that are similar across a variety of content domains?

* What problem-solving skills learned within one context can be expected to transfer to other domains?

The purpose of this book is to serve as the basis for the productive exchange of information that will help to answer these questions -- by drawing together preliminary theoretical understandings, sparking debate and disagreement, raising new questions and directions, and perhaps developing new world views.

1	A VIEW FROM BIOLOGY

Mike U.Smith

Mercer University

INTRODUCTION

This chapter is an attempt to critically analyze problem-solving research within the domain of biology, especially genetics, and to combine the conclusions of that analysis with a critical analysis of research in other disciplines so as to produce a unified theory of problem solving that would apply across content domains. This is truly a formidable task, and the result should be considered as only a first approximation of a statement of such a theory. The purpose of this work, therefore, is not to produce a definitive statement of a unified theory of problem solving, but to provide a target for discussion, criticism, and debate among problem-solving researchers and theorists. Such criticism is a routine and necessary part of the evolution of any theoretical construct

Before any attempt is made to develop a statement of theory, one must first ask, what is a theory and how is the merit of a theory to be judged? According to Popper (1959), “Theories are [sets of] universal statements” (p. 27) or propositions. They serve two essential functions: explanation and prediction. Popper maintains that these statements delineate “universal concepts or names,” (p. 74) that each statement must be a necessary part of the theory, that the statements taken collectively must be a sufficient summary of the theory, and that the theory must be free from contradiction. Turchin (1977) further suggests that a theory should be judged by 1) how well the statements describe reality, 2) the generality/predictive power of the statements, and 3) their “dynamic” nature, i.e., their ability to open new regions of study. The adequacy of the theory statement to be presented below should therefore be judged by whether or not it

1)proposes appropriate consensus definitions of critical terms,

2)adequately explains prior observations,

3)synthesizes observational data into a series of universal statements,

4)is parsimonious, and

5)leads to empirically testable predictions.

DEFINITION OF TERMS

One of the more troubling issues about which problem-solving researchers disagree is the exact meaning of some of the most basic terms that we use. Among these are such terms as “expert,” “novice,” “algorithm,” “heuristic,” “problem,” and even “problem solving” itself, to name but a few. As any science student knows, the heart of any scientific discipline is a set of terms that have specific and widely accepted meanings, i.e., force, pressure, gene, acid, etc. Science courses that focus on acquiring these terms have, in fact, often been criticized as being equivalent to a course in a foreign language.

Why are terms, and specifically consensus definitions of terms, so important in the sciences? The basic answer to this question is simple: Shared meanings for terms allow accurate communication. When one physicist speaks to another about an “atom,” there is no confusion about the meaning of the term. And if both these physicists report studies of nuclear magnetic resonance, you can be assured that they are referring to the same phenomenon.

It is not so in problem-solving research. The literature refers to social problems, alcohol and drug problems, students solving textbook problems, physicists solving “real” problems, etc. Any two problem-solving researchers may therefore be using the same word but with meanings that differ in substantive ways. Consider for example the following question: Do expert problem solvers usually use a forward-working or a backward-working (means-ends analysis) approach to problems? The answer to that question depends entirely upon your definition of the term “problem.” Much of the literature reports that experts do indeed use a forward-working approach on typical textbook tasks that might be called “exercises” for these individuals. For non-routine tasks, however, experts often revert to an approach that more closely resembles the means-ends analysis approach of novices in the field. Therefore, if the definition of “problem” includes exercises, the proper answer is that experts may use both approaches depending on the nature of the problem presented. If only non-routine, non-exercises are defined as problems, however, experts most often use a general means-ends analysis strategy. Therefore, if we are to be able to communicate with each other, we must make explicit our definitions of vague terms. And if we are to be able to compare the results of our research, we must achieve considerable agreement about what these definitions are. If we are to seek the commonalities in our findings, we must have common delimiters of the boundaries of the topic or at least an understanding of how our definitions vary. Without such definitions accurate communication is impossible.

The most important of the terms that are being used in our literature with varying interpretations is the term “problem” itself. There are almost as many definitions in print as there are researchers in the area, but no definition of the term has received wide acceptance. The following are some of the better known examples:

A problem is a “stimulus situation for which an organism does not have a response,” ... a problem arises “when the individual cannot immediately and effectively respond to the situation.” (Woods, Crow, Hoffman, & Wright, 1985, p. 1)

“A person is confronted with a problem when he wants something and does not know immediately what series of actions he can perform to get it.” (Newell & Simon, 1972, p. 72)

“Whenever there is a gap between where you are now and where you want to be, and you don't know how to find a way to cross that gap, you have a problem.” (Hayes, 1980, p. i)

In their study of the arithmetic activity involved in everyday grocery shopping, Lave, Murtaugh, and de la Rocha (1984) propose that the shopper expects the task to proceed “unproblematically and effortlessly... It is in relation to this expectation that ‘problems’ take on meaning; they are viewed assnags or interruptions” (p. 79, emphasis added).

The definitions noted above appear to focus not on the nature of the task itself but upon the distance between the problem and the solver—the solver's lack of knowledge of an appropriate solution method. These “gap” definitions all sound quite similar. The principal difficulty with such definitions, however, is that in practice problem-solving research has not historically been so narrowly limited. Much of what is generally considered to be problem-solving research details the performance of subjects on classroom “problems” that are well structured and would typically be considered to be exercises. It is the solution of such exercises that is now well understood to include the application of domain-specific algorithms, the application of chunked procedural and conceptual knowledge, “pop-up” knowledge access, etc. If this narrow definition is accepted, then much of what we know about the performance of novices on such tasks and most of what we know about expertise is not accurately included in problem-solving theory. Most of the work to date (including my own) has compared the performance of novices and experts on the solution of problems that can only be considered to be “exercises” for the expert. Similarly, the extensive literature (see Davis, 1985) detailing our understanding of “routine” vs. “non-routine” problem solving would have to be eliminated since these definitions do not allow for “routine” problems.

The question is whether routine and non-routine problems are best considered to be apples and oranges or merely two varieties of the same fruit. Granted, there are fundamental differences in the performance of individuals on routine and non-routine tasks, but there are also fundamental similarities. Individuals completing both kinds of tasks use many of the same cognitive tools for both. They plan as necessary, break the problem into parts and attack parts separately, use domain-specific algorithms, attempt to bring their relevant domain knowledge to bear on the problem, use qualitative representations, check the accuracy of their work, etc. (see Smith, 1983). The question, therefore, is whether to focus on the similarities or the differences.

The issue here is a fundamental one: Is the status of a task as a problem an “innate characteristic of a task... [or a] subtle interaction between the task and the individual” (Bodner, this volume)? Put more simply, should “exercises” (tasks with which the solver is generally familiar and adept) be considered problems? Historically, the answer is clear: We have excluded exercises from our formal definitions but included them in our research. I would argue that we should use the more inclusive definition implied by our practice. Gap definitions are difficult to apply because they require a “floating” assessment Tasks that are “problems” for one individual may not be for a different individual, and tasks that are problems for a person today may not be problems tomorrow. This may be an acceptable state of affairs, but it certainly makes the operational definition more difficult to apply.

A second set of issues revolves around the confusion that arises when terms (such as “problem“) that are in common usage are also used by researchers. The multiple meanings of such terms in popular culture will necessarily cloud our understanding of the narrower scientific meanings of the terms. In genetics, for example, researchers have long recognized the student confusion that occurs from our technical usage of the term “dominant,” a usage that is at variance with the common understanding of the term (e.g., dominant form of a gene vs. “dominant” person). Two common meanings of the term “problem” contribute to our confusion: 1) “difficult to deal with” and 2) “a source of perplexity or vexation” (Webster's Seventh New Collegiate Dictionary, 1971).

In the first sense, any task that is not difficult is not a problem. For example, “Making an A in his class is no problem.” Or the one seen on student sweatshirts: “I don't have a drinking problem. I drink. I get drunk. I fall down. No problem.” Presumably routine tasks (exercises) would not be difficult and thus would not be considered problems to individuals familiar with these tasks. Also in this sense, only tasks that are sufficiently complex would be considered to be “real” problems.

In the second sense of everyday usage, if my wife and I are having marital difficulties, that is a “problem”. The student with the sweatshirt may not think he has a drinking problem, but his/her parents may view the situation as “problematic.” The central characteristic of this kind of “problem” is the perplexity encountered by the solver—the person doesn't immediately know what to do. For purposes of our research, however, I maintain that perplexity is not a necessary component of problem solving. The principal reason for this position is the fact that problem solvers often exhibit many common behaviors whether they find the task perplexing or not. If the challenge is so perplexing as to be overwhelming, in fact, this aspect of the problem may paralyze the system so that effective problem-solving techniques cannot be appropriately summoned and applied.

Problems that are perplexing, for which we have no “immediate and effective” response, do indeed call forth additional and sometimes altogether different problem-solving behaviors. On the other hand, I also find various genetics exercises to be challenging “problems” even though I have a ready store of strategies, heuristics, and algorithms with which I can “immediately and effectively” respond. That such exercises do not perplex me seems an artificial and extraneous constraint on the definition.

A third issue related to common usage of the term “problem” that has crept into the literature is the fact that the word is often equated with the term “question.” Not all questions, however, should be considered to be problems. For example, is “What does 2 + 2 equal?” a problem? Or, “What phenotypic ratios are expected among the offspring of a monohybrid cross between two heterozygotes?” This latter type of item has occasionally been included in studies of genetics “problem solving” and is often included on typical genetics exams in a section of “problems,” but I would argue that such items should not be defined as problems. How do such questions differ from any other memorization item? Would you consider “What is the capital of the United States?” to be a problem? Probably not.

Perhaps again the common usage of the term causes the confusion. If a student is faced with the question on a test but doesn't know that the expected ratio is three to one or that the U .S. capital is Washington, DC, then he does indeed have a “problem”/difficulty. The vast majority of the mental tools that can be applied to problems that cannot be solved by memorization, however, cannot be applied to ameliorate this difficulty. This is not a “problem” in what must be called the “technical” meaning of the term (vs. “common usage“). The solution of a problem must require more than simple recognition or recall from memory.

Similarly, problems cannot be solved algorithmically, i.e., with little or no understanding of what has been done or why it was correct. Landa (1972) defines an algorithm as a “completely determined... ready-made prescription on how to act” (p. 21,23,24). Lochhead and Collura (1981) add that algorithms can be “black boxes used to produce answers” (p. 47) with little or no understanding. By analogy, is reproducing a diagram that may have appeared in the text or on the blackboard during class solving a problem? Is identically repeating a series of steps solving a problem? I maintain that it is not Developing the ability to reproduce a pattern and do it appropriately may indeed be learning, but performing the task is not problem solving.

Algorithms are a basic part of the procedural repertoire of any skilled problem solver. This issue of concern here is whether or not the algorithm is applied mindlessly (“algorithmically”) or with understanding. My earlier position (cited by Bodner in this volume) was that tasks that could be solved by a single algorithm should not be considered to be problems. The selection of appropriate algorithms and their modification to accommodate the unique aspects of a problem, however, are often important aspects of problem solving. The mindless application of algorithms, often to situations in which they are inappropriate, is in fact a recognized distinction ...

Table of contents

• Front Cover
• Title Page
• Copyright
• Dedication
• Contents
• Preface
• 1 A VIEW FROM BIOLOGY
• 2 A VIEW FROM CHEMISTRY
• 3 A VIEW FROM MEDICINE
• 4 A VIEW FROM PROGRAMMING
• 5 A VIEW OF MATHEMATICAL PROBLEM SOLVING IN SCHOOL
• 6 A VIEW FROM PHYSICS
• 7 A VIEW FROM TROUBLESHOOTING
• Author Index
• Subject Index