Toward a Meta-Theory of Problem Solving

I am embarking on a long journey. Most constructivist learning environments are designed to support some form of problem solving. It is unfortunate that none of the models for designing them makes an y explicit assumptions about the nature of the problems or the kinds of thinking required to solve them. I shall be devotong much of my energy in the future to explicating the nature of problem solving in all its forms and constructing models of instructional design for each. This paper is a crude beginning but should give you some idea what I am thinking. Stay tuned. More will follow.

Variations in Problem Solving

Gagne believed that "the central point of education is to teach people to think, to use their rational powers, to become better problem solvers" (1980, p. 85). Most educators, like Gagne, regard problem solving as the most important learning outcome from life. Why? Because most people, especially professionals and tradespeople, are rewarded in their careers for their abilities to solve problems. No one is paid for memorizing information and completing examinations. Unfortunately, very little education and training requires learners to solve problems, and virtually none engages the kinds of problem solving encountered in the real world. At best, education and training efforts engage learners in well-structured (textbook) problems, while real world problems are nearly always ill-structured.

The ability to solve problems, we all believe, is intellectually demanding and engages learners in higher-order thinking skills. Over the past three decades, a number of information processing models of problem solving, such as the classic General Problem Solver (Newell & Simon, 1972), have been promulgated to explain problem solving. 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, 1978) 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 conceive of all problems as equivalent, articulating a generalizable problem-solving procedure. These information-processing conceptions of problem solving assume that the same processes applied in different contexts yield the similar results. The culmination of this activity was an attempt to articulate a uniform theory of problem solving (Smith, 1991).

Problem solving is not a uniform activity. Problems are not equivalent, either in content, form, or process. 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. If the learner possesses a complete schema for any problem type, then constructing the problem representation is simply a matter mapping an existing problem schema onto a problem. Existing problem schemas result from previous experience in solving particular types of problems, enabling the learner to proceed directly to the implementation stage of problem solving (Gick, 1986) and trying out the activated solution. Experts are better problem solvers because they recognize different problem states which 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 problem schemas, are not able to recognize problem types, so they must rely on general problem solving strategies, such as the information processing approaches, which provide weak strategies for problem solutions.

As depicted in Figure 1, the ability to solve problems is a function of the nature of the problem, the way that the problem is represented to the solver, and a host of individual differences that mediate the process. Each of these factors will be addressed in turn.

 

 

Problem Variations

It is clear that problems vary in their nature, in their presentation, in their components, and certainly in the cognitive and affective requirements for solving them. Jonassen (1997) distinguished well-structured from ill-structured problems and articulated different kinds of cognitive processing engaged by each. Smith (1991) distinguished external factors, including domain and complexity, from internal characteristics of the problem solver. And Mayer and Wittrock (1996) described problems as ill-defined/well-defined and routine/nonroutine. There is increasing agreement that problems vary in substance, structure, and process. In the next section, I will attempt to describe all of the ways in which problems vary.

 

Problem Types

Structuredness

Jonassen (1997) distinguished well-structured from ill-structured problems and recommended different design models for each, because they call on distinctly different kinds of skills. The most commonly encountered problems, especially in schools and universities, are well-structured problems. Typically found at the end of textbook chapters, these well-structured "application problems" require the application of a finite number of concepts, rules, and principles being studied to a constrained problem situation. These problems have also been referred to as transformation problems (Greeno, 1978) which consist of a well-defined initial state, a known goal state, and constrained set of logical operators. Well-structured problems:

o Present all elements of the problem,

o Are presented to learners as well-defined problems with a probable solution (the parameters of problem specified in problem statement),

o Engage the application of a limited number of rules and principles that are organized in a predictive and prescriptive arrangement with well-defined, constrained parameters,

o Involve concepts and rules that appear regular and well-structured in a domain of knowledge that also appears well-structured and predictable,

o Possess correct, convergent answers,

o Possess knowable, comprehensible solutions where the relationship between decision choices and all problem states is known or probabilistic (Wood, 1983), and

o Have a preferred, prescribed solution process.

Ill-structured problems are the kinds of problems that are encountered in everyday practice, so they are typically emergent dilemmas. Because they are not constrained by the content domains being studied in classrooms, their solutions are not predictable or convergent. They may also require the integration of several content domains. Solutions to problems such as pollution may require components from math, science, political science, and psychology. There may be many alternative solutions to problems. However, because they are situated in everyday practice, they are much more interesting and meaningful to learners, who are required to define the problem and determine which information and skills are needed to help solve it. Ill-structured problems:

o Appear ill-defined because one or more of the problem elements are unknown or not known with any degree of confidence (Wood, 1983),

o Have vaguely defined or unclear goals and unstated constraints (Vons, 1988),

o Possess multiple solutions, solution paths, or no solutions at all (Kitchner, 1983), that is, no consensual agreement on the appropriate solution,

o Possess multiple criteria for evaluating solutions,

o Possess less manipulable parameters,

o Have no prototypic cases because case elements are differentially important in different contexts and because they interact (Spiro at al, 1987, 1988),

o Present uncertainty about which concepts, rules, and principles are necessary for the solution or how they are organized,

o Possess relationships between concepts, rules, and principles that are inconsistent between cases,

o Offer no general rules or principles for describing or predicting most of the cases

o Have no explicit means for determining appropriate action,

o Require learners to express personal opinions or beliefs about the problem, so ill-structured problems are uniquely human interpersonal activities (Meacham & Emont, 1989), and

o Require learners to make judgments about the problem and defend them.

Researchers have long assumed that learning to solve well-structured problems transfers positively to learning to solve ill-structured problems. Although information processing theories believed that "in general, the processes used to solve ill-structured problems are the same as those used to solve well structured problems" (Simon, 1978, p. 287), more recent research in situated and everyday problem solving makes clear distinctions between thinking required to solve convergent problems and everyday problems. Dunkle, Schraw, and Bendixen (1995) concluded that performance in solving well-defined problems is independent of performance on ill-defined tasks, with ill-defined problems engaging a different set of epistemic beliefs. Clearly more research is needed to substantiate this finding, yet it is obvious that well-structured and ill-structured problem solving engage different intellectual skills.

Complexity

Just as ill-structured problems are more difficult to solve than well-structured problems, complex problems are more difficult than simple ones. There are many potential definitions of problem complexity. For purpose of this study, complexity is assessed by the:

a) number of issues, functions, or variables involved in the problem

b) number of interactions among those issues, functions, or variables

c) predictability of the behavior of those issues, functions, or variables.

 

Although complexity and structuredness invariably overlap, complexity is more concerned with how many components are in the problem, how those components interact, and how consistently they behave. Complexity has more direct implications for working memory than for comprehension. The more complex a problem, the more difficult it will be for the problem solver to actively process the components of the problem. While ill-structured problems tend to be more complex, well-structured problems can be extremely complex and ill-structured problems fairly simple. Complexity is clearly related to structuredness, though it is a sufficiently independent factor to warrant consideration.

 

Meta-problems. Complexity occurs at several levels within problem solving. Discrete vs. meta-problems. An example of a metaproblem is computer programming, running a retail business

Domain Specificity (Abstract-Situated)

Contemporary research and theory on problem solving argue that problem solving skills are domain and context specific, that, problem solving activities are situated, embedded , and therefore dependent on the nature of the context or domain. This is because solving problems within a domain relies on cognitive strategies (strong methods) that are specific to that domain (Mayer, 1992; Smith, 1991; Sternberg & Frensch, 1991).

Ill-structured problems tend to be more situated, while well-structured problems tend to be more abstract. However, well-structured problems, in the form of story problems, can be quite situated while ill-structured problems, in the form of dilemmas, can be fairly abstract. Figure 2 illustrates the similarity of relationship among these three factors. They are definitely not orthogonal. Nor are they equivalent. There is a sufficient independence among the factors to warrant separate consideration.

This paper proposes that there are classes of problem solving types which assumes that there are similarities among problems-solving types. It also assumes that cognitive strategies can be generalized across domains and within problem types. This assumption contradicts the domain specificity that dominates current thinking.

In order to determine the context- or domain-specificity of problem solving outcomes, it is necessary to conduct activity analysis on the problems.

Figure 2. Relationship among problem factors.

 

Problem Representations

Problems also vary in how they are presented to the problem solver. Problems in the real world, of course, are embedded in their natural contexts, which requires the problem solver to distinguish important from irrelevant components and construct a problem space for generating solutions. Learning problems are almost always contrived or simulated, so instructional designers must decide which problem components to include and how to represent them. Designers provide or withhold contextual cues, prompts, or other clues about information that needs to mapped onto the problem space. How overt those cues are will determine problem difficulty and complexity. Additionally, designers make decisions about the modality for representing different problem components.

Perhaps the most important issue is the fidelity of the problem representation. Is the problem represented in its natural complexity and modality, or is it filtered when simulated. Should social pressures and time constraints be represented faithfully. That is, does the problem have to be solved in real time, or can it be solved in leisure time. What levels of cooperation or competition are represented in the problem. These are but a few of the decisions that designers must make when representing problems for learning.

 

Individual Differences

Smith (1991) distinguished between internal and external factors in problem solving. External factors are those that describe the problem. Internal factors are those that describe the problem solver.

 

Familiarity

Perhaps the strongest predictor of problem-solving ability is the solver's familiarity with the problem type. Experienced problem solvers have better developed problem schemas which can be employed more automatically (Sweller, 1988). Mayer and Wittrock (1996) refer to routine and nonroutine aspects of the problem. We believe that routineness is rather an aspect of the problem solver and is not endemic in the nature of the problem itself. Although familiarity with a type of problem will facilitate solving similar problems, that skill seldom transfers to other kinds of problems or even the same kind of problem represented in another way (Gick & Holyoak, 1980, 1983).

Nonroutine problems (those not familiar to the problem solver) and transfer of problems solving require high road transfer, which is effortful and conscious (Salomon & Perkins, 1989), whereas routine problems and near transfer of those skills rely more on low road transfer, which involves less conscious attention. Another danger of routineness, especially among experts, is that they lose their ability to reflect on and articulate the reasoning they use in solving problems.

 

Domain and Structural Knowledge

Another strong predictor of problem solving skills is the solvers level of domain knowledge. How much someone knows about a domain is important to understanding the problem and generating solutions. However, that domain knowledge must be well integrated in order to support problem solving. The integratedness of domain knowledge is best described as structural knowledge (Jonassen, Beissner, & Yacci, 1993). Structural knowledge is the knowledge of how concepts within a domain are interrelated. It is also known as cognitive structure, the organization of relationships among concepts in memory (Shavelson, 1972).

 

Domain-specific thinking skills. Domain knowledge and skills are very important in problem solving. Structural knowledge may be a stronger predictor of problem solving than familiarity. Robertson (1990) found that the extent to which think-aloud protocols contained relevant structural knowledge was a stronger predictor of how well learners would solve transfer problems in physics than either attitude or performance on a set of similar problems. Structural knowledge that connects formulas and important concepts in the knowledge base are important to understanding physics principles. Gordon and Gill (1989) found that the similarity of the learners' graphs (reflective of underlying cognitive structure) with the expert's was highly predictive of total problem solving scores (accounting for over 80% of the variance) as well as specific problem solving activities. Well integrated domain knowledge is essential to problem solving.

Likewise, previous experience in solving problems also supports problem solving.

 

Cognitive Controls

Individuals also vary in their cognitive controls, which represent patterns of thinking that control the ways that individuals process and reason about information (Jonassen & Grabowski, 1993). Field independents are better problem solvers (Davis & Haueisen, 1976; Maloney, 1981;Heller, 1982; Ronning, McCurdy, & Ballinger, 1984). There is little research, however, it is reasonable to predict that learners with higher cognitive flexibility and cognitive complexity will be better problem solvers because they consider more alternatives (Stewin & Anderson, 1974) and they are more analytical. The relationship between cognitive styles and controls needs to be better established.

 

Epistemological Beliefs

Problem solving, especially ill-structured problem solving, often requires the solver to consider the veracity of ideas and multiple perspectives in evaluating problems or solutions. The ability to do so depends partially on underlying beliefs about knowledge and how they come to develop it. That is, learners' epistemological beliefs about the nature of problem solving also affect the ways that they naturally tend to approach problems. A number of epistemological theories have been related to a broad range of learning outcomes (Hofer & Pintrich, 1997). The best known epistemological theory was developed by William Perry (1970). He distinguishes nine separate stages of intellectual development clustered into three periods. Dualist learners believe that knowledge is right or wrong; professors have the right knowledge, students do not. Their absolutist beliefs stress facts and truth. Multiplicity represents the acceptance of different perspectives and skepticism about expertise in general. Multiplists rely on methods and processes to establish truth. In the contextual relativism period, evaluative thinkers accept the role of judgment and wisdom in accommodating uncertainty, and that experts may provide better answers. However, ideas must be evaluated for their merits and the cultural and intellectual perspectives from which they derive.

More complex and ill-structured problems require higher levels of intellectual development, which most students have not yet developed. This is largely due to the ways they are taught. As a result of the preponderance of algorithmic teaching approaches in mathematics, for instance, there is "a belief by students that mathematical problems are solved by applying procedures that a person may or may not know" (Greeno, 1991, p. 83). There is a right and wrong way to do things. Solving more complex and ill-structured problems depends on multiplicitous and contextual relativistic thinking. Although little, if any, research has connected epistemological beliefs and problem solving, the relationship is obvious and needs to be examined especially in real world problem solving.

 

Affective and Conative

Mayer (1992) claims that the essential characteristics of problem solving are directed cognitive processing. Clearly, problem solving requires cognitive and metacognitive processes. Cognitive is a necessary but insufficient requirement for problems solving, which also requires significant affective and conative elements as well perseverance (Jonassen & Tessmer, 1996). Knowing how to solve problems and believing that you know how to solve problems are often dissonant. Problem solving also requires a number of affective, especially self-confidence and beliefs and biases about the knowledge domain. For example, Perkins, Hancock (1986) found that some students, when faced with a computer programming problem, would disengage immediately believing that it was too difficult while other would keep trying to find a solution. If problem solvers do not believe in their ability to solve problems, they will most likely not succeed. Their self-confidence of ability will predict the level of mindful effort and perseverance that will be applied to solving the problem, which provide evidence of motivation. Also, if problem solvers are predisposed to certain problem solutions because of personal beliefs, then they will be less effective because they over-rely on that solution.

Conative criteria relate to motivation to perform, which relates mostly to mindful effort and perseverance. Greeno (1991) claims that most students blieve that if math problems have not been solved in a few minutes, the problem is probably unsolvable and there is no point in continuing to try, despite the fact that mathematician often work for hours on a problem.

 

General Problem-Solving Skills

There is a general belief that some people are better problem solvers because they use more effective problem-solving strategies. That depends on the kind of strategies they use. Solvers who attempt to use weak strategies, such as general heuristics like means-ends analysis that can be applied across domains, generally fair no better than those who do not. However, solvers who use domain-specific, strong strategies are better problem solvers. Experts effectively use strong strategies, and some research has shown that less experienced solvers can also learn to use them (Singley & Anderson, 1989).

 

Individual vs. Group Problem Solving

The final individual difference relates to whether the problem is being solved by an individual or a group of people. One of the strongest predictors of problem solving success is the application of an appropriate problem schema. That is, has the problem solver constructed an adequate mental model of the problem and the system in which the problem occurs? A good conceptual model of the problem system along with the strategic knowledge to generate appropriate solutions and the procedural knowledge to carry them out will result in more successful problem solutions. When complex problems are solved by groups of people, sharing a similar mental model of the problem and system will facilitate solutions. When mental models are dissonant, more problems occur. So, team mental models must be constructed so that the members of the group are working with similar conceptions of the problem, its states, and solutions.

 

Typology of Problem Solving

 Table 1

Problem solving varies along three dimensions: problem type, problem representation, and individual differences. Problems vary by structuredness, complexity, and abstractness. Problem representations vary by context and modality. A host of individual differences mediate individuals' abilities to solve those problems. Although dichotomous descriptions of general types of problems are useful for clarifying attributes of problems, they are insufficient for suggesting specific cognitive processes and instructional strategies. Additional accuracy and clarity is needed to resolve specific problem types. A cognitive task analysis of the range of problem types has, to date, distinguished ten different kinds of problems (see Table 1).

Table 1 lists horizontally ten different types of problem solving outcome, including logical problems, algorithmic problems, story problems, rule-using problems, troubleshooting, diagnosis-solution problems, case method problems, design problems, and issue-based problems. This range of problem types describes a continuum of problem-solving outcomes from well-structured to ill-structured, abstract to concrete, and simple to complex. However, complexity is a more independent attribute, as well-or ill-structured problems may be simple or complex. The specific learning outcome for each of these problem types is described in the next row. Problem types are listed next. They briefly describe the cognitive processing requirements of each problem type. The next row lists solution types. The well-structured problems focus on correct, efficient solutions, while the ill-structured problems focus more on decision articulation and argumentation. Problems vary from very algorithms with exact solutions to conundrums or dilemmas with no verifiably correct solution. The role of problem context is listed next. The role of context becomes vitally important in defining ill-structured problems, while well-structured problems de-emphasize the role of context. Brief examples of each problem type are listed in Table 1. The bottom three rows describe the level of structuredness, abstractness, and complexity in each problem type. This table presents a brief overview of the different kinds of problems that practitioners and learners need to learn to solve. In this section of the paper, I will briefly describe each kind of problem solving. In Table 2, I provide a cognitive model for the processes involved in solving each kind of problem. These processes are based on a cognitive task analysis but need to be validated and further explicated by observation, interviewing, and artifact analysis of problem solutions. Those activities are describe later in the proposal.

 

Logical Problems

Logical problems tend to be abstract tests of logic that puzzle the learner. They are used to assess mental acuity, clarity, and logical reasoning. Classic mind game such as Missionaries and Cannibals or Tower of Hanoi challenge learners to find the most efficient (least number of movers) sequence of action. Rubic's Cube was a popular game in the 1970s requiring the user to rotate the rows and columns of a three-dimensional cube to form patterns. In each of these, there is a specific method of reasoning that will yield the most efficient solution. It is up to the learner to discover that method. More complex forms of logic problems include constructing geometry proofs.

Logical Problems can be decidedly more complex than these. Popular card games such as Bridge or Hearts and board game such as checkers and chess are more complex forms of logical problems. These games employ more complex rules and constraints. Current computer games, such a Pokemon, are also forms of complex logical problems. These more complex forms of problems also require other forms of problem solving, including rule-using, diagnosis/solution, and perhaps design. However, few if any logical problems are embedded in any common situation, making them necessarily more abstract.

 

Algorithmic Problems

One of the most common problem types encountered in schools is the algorithm. Most common in mathematics courses, students are taught to solve problems using a finite and rigid set of procedures with limited, predictive decisions. Solving equations requires learners to select and apply the correct sequence of operations to the formula. Such algorithmic approaches are also commonly used in science courses or home economics. Most recipes are algorithms for cooking. Many researchers, such as Smith (1991), argue that algorithms (repeating a series of steps) are, by nature, not problems. When learners are required to select and perhaps modify an algorithm for use in an exercise, it may become problem solving. Since algorithms are so generally considered to represent problems, for better or worse, they should be considered.

The primary limitation of algorithmic approaches is the over-reliance on procedural knowledge structures and the lack or absence of conceptual understanding of the procedure. Content that is learned only as a procedure can rarely be transferred because of a lack of conceptual understanding of the underlying processes. This is a common complaint about learning statistics, where professors focus on the algorithms and miss the purpose of studying the statistical analysis. Learners who are adept at abstract reasoning can learn increasingly complex algorithms, such as those encountered in calculus, trigonometry, and other mathematics domains. Average learners are limited in their ability to create such abstract representations of procedures, so they encounter problems.

 

Story Problems

In an attempt to situate algorithms in some kind of context, many textbook authors and teachers employ story problems. This usually takes the form of embedding the values needed to solve an algorithm into a brief narrative or situation. Learners are required to select the most appropriate formula for solving the problem, extract the values for the narrative and insert them into the formula, solving for the unknown quantity. This is a more complex cognitive process than simply following a procedure. Unfortunately, the story covers for the problems are too often uninteresting and irrelevant to students. So when they attempt to transfer story problem skills to other problems, they focus too closely on surface features or recall familiar solutions from previously solved problems (Woods, Hrymak, Marshall, Wood, Crowe, Hoffman, Wright, Taylor, Woodhouse, & Bouchard, 1997). They fail to understand the principles and the conceptual applications underlying the performance, so they are unable to transfer the ability to solve one kind of problem to problems with the same structure but dissimilar features.

 

Rule-Using Problems

Many problems have correct solutions but multiple solution paths or multiple rules governing the process. They tend to a have clear purpose or goal that is constrained but not restricted to a specific procedure or method. Rule-using problems can be as simple as expanding a recipe to accommodate more guests and as complex as completing tax return schedules. Using an online search system to locate a libraries holdings or using a search engine to find relevant information on the World Wide Web are examples of rule-using problems. The purpose is clear: find the most relevant information in the least amount of time. The user employs sets of rules and principles about search terms and combining them using Boolean connectors along with specific system rules. There are multiple solutions depending on the strategies and rules that were used. Many of this kind of problem also engage decision making, which is described next.

 

Decision-Making Problems

Decision-making problems are usually constrained to decisions with a limited number of solutions. For instance, do we move in order to accept a promotion? Which health plan do we accept? Which school is best for my children? Though these problems have limited number of solutions, the number of factors to be considered in deciding among those solutions can be very complex. Decision problems usually require comparing and contrasting the advantages and disadvantages of alternate solutions. Decisions are justified in terms of the weight those factors..

 

Troubleshooting Problems

Troubleshooting is one of the most common forms of everyday problem solving. More people get paid for troubleshooting than most other types. This is especially true in the military. Maintaining complex communications and avionics equipment requires troubleshooting skills. Debugging computer program and repairing computer equipment requires troubleshooting. The primary purpose of troubleshooting is fault state diagnosis. That is, some part of a system is not functioning properly, resulting in a set of symptoms, which have to be diagnosed and match with the user's knowledge of various fault states. Troubleshooters use symptoms to generate and test hypotheses about different fault states.

Troubleshooting skill requires a combination of domain and system knowledge (conceptual models of the system); troubleshooting strategies such as search-and-replace, serial elimination, and space splitting; and experience (represented in case-based reasoning). These skills are integrated and organized by the troubleshooter's experiences. The conceptual model consists of conceptual, functional, and declarative knowledge, including knowledge of system components and interactions, flow control, fault states (fault characteristics, symptoms, contextual information, and probabilities of occurrence), and fault testing procedures. The troubleshooting strategies consist of the strategic knowledge required to generate hypotheses and work plans.

 

Diagnosis-Solution Problems

Diagnosis-solution problems are similar to troubleshooting. Most diagnosis-solution problems require identifying a fault state, just like troubleshooting. However, in troubleshooting, the goal is to repair the fault and get the system back online as soon as possible, so the solution strategies are more restrictive. Diagnosis-solution problems usually begin with a fault state (e.g. symptoms of a sick person). The physician examines the patient and considers patient history before making an initial diagnosis. In a spiral of data collection, hypothesis generation, and testing, the physician focuses in a specific etiology and differential diagnosis of the patient's problem. At that point, the physician must suggest a solution. Frequently, there are multiple solutions and solution paths, so the physician must justify a particular solution. It is this ambiguity in solution paths that distinguishes diagnosis-solution problems from trouble shooting.

 

Strategic-Tactical Performance

 

To be completed

 

Situated Case Problems

Case problems are situations. What makes these problems difficult to solve is that it is not always clear what the problem is. Because defining the problem space is more ambiguous, these problems are more ill-structured. These are the most common types of problem solved in the real world. Case problems require the solver to articulate the nature of the problem and the different perspectives that impact the problem before suggesting solutions. They are more contextually bound that any kind of problem considered so far. That is, their solutions rely on an analysis of contextual factors. Solving business problems, including planning production, are common case problems. Deciding production levels, for instance, requires balancing human resources, technologies, inventory, and sales. Justifying decisions is among the most important processes in solving case problems.

 

Design Problems

One of the most ill-structured kinds of problems is designing something. Whether it be an electronic circuit, a house, a new dish for a restaurant, or any other product or system, designing requires applying a great deal of domain knowledge with a lot of strategic knowledge resulting in an original design. Design represents the farthest kind of transfer. Designing goes well beyond normal concepts of transfer of a specific problem-solving skills to generalization of a set of skills. Normally the designer is required to conduct a needs assessment and use domain knowledge to generate a design that will work within system constraints. Usually, most design problems have multiple if not infinite solutions. The criteria for the best solution are not always obvious, so skills in argumentation and justification help the designers to rationalize his/her design. Although designers always hope for the best solution, the best solution is seldom ever known. In addition to ill-structuredness, most design problems are complex, requiring the designer to balance many needs and constraints in the design. The importance of design problems cannot be diminished. Most professionals in the real world get paid for designing things (products, systems, etc.), not for taking examinations. More experience with design problems earlier is an important curricular revision.

 

Dilemmas

Dilemmas or issue-based problems are the most ill-structured and unpredictable, often because there is no solution that will ever be acceptable to a significant portion of the people affected by the problem. The current crisis in Kosovo is a prime example of a dilemma problem. Although there are many valuable perspectives on the situation (military, political, social, ethical, etc.), none is able to offer a reasonable solution to the crisis. It is so complex and unpredictable, that no best solution can ever be known. That does not mean that there are not many solutions, which can be attempted with variable degrees of success, none will ever meet the needs of the majority of people or escape the prospects of catastrophe. Dilemmas are often complex, social situations with conflicting perspectives, and they are usually the most vexing of problems.

 

Summary

Although these problem types have been described above as discrete classes, in reality, most problems constitute variations or hybrids of these classes. That is, these classes are not represented as discrete classes with no variants. Many complex problems may be described as meta-problems, which will involve a mix of these problem types. Many other problems may possess aspects of different classes of problems. We do not presume to have articulated discrete, predictable forms of problem-solving activity. Human cognition is seldom that predictable.

If we believe that the cognitive requirements of solving different kinds of problems varies, then so too must the nature of instruction that we use to support the development of problem-solving skills. Why? Among the most fundamental beliefs of instructional design is that different learning outcomes require different instructional conditions (Gagne, 1960). In the next section, we briefly describe an initial analysis of the cognitive and affective requirements for solving these kinds of problems and then later describe initial models for instructing these kind of problem solving.

 

 

Instructional Models to Support Transferable Problem-Solving

 

After verifying this problem typology, an important goal of this proposal is to articulate instructional design models for supporting the learning of each problem type. If we are able to articulate each problem type, its representation, and the kinds of cognitive and affective processes and individual differences that are required to solve it, articulating a model of instruction to support learning should be relatively simple. in the future, I will articulate those models. While many instructional methods between models overlap, each model recommends a different selection and sequence of instructional transactions. Generally, the more ill-structured the problem, the more open-ended and constructivistic the instructional model will be. The final goal of this proposal is to design and implement these different instructional models and assess their effectiveness in learning to solve problems ad to transfer those skills to new problems.

The meta-theory of problem solving that I have presented needs to be validated, refined, explicated, and evaluated. Although examples of each type of problem solving exist, it is not certain whether these problem types are exhaustive, whether they can accommodate problems in different domains, and what effects domain-specific reasoning may have on these problem types. That is, are the cognitive processes required for solving each type of problem the same in different domains? What are the instructional activities required to support learning each kind of problem solving. In order to answer these questions and to validate the theory and its methods, I propose the following research process.

 

Research Methods

 

The following research agenda is a complex, multi-year agenda.

 

1) Validate and refine typology. In years 1 and 2, we intend to:

 

a) Solicit and collect hundreds of examples of problem-solving situations from a variety of domains and job titles as well as from educational institutions. Interview the problem providers to assess their understanding of the problems. Classify them according to our problem typology. It is very likely that problems will emerge that cannot be classified using the typology, or that analysis of problem types suggest combing classes or expanding our typology. Our goal will be to adjust the typology in order to accommodate the range of problem examples.

b) Analyze the problem representations for each. Within the context they occur, we will perform cognitive task analyses on as many of these problem situations as possible. These analyses will include descriptions of the physical, social, and cultural aspects of the environment surrounding the problem. This part of each analysis will use qualitative methodologies such as observation, participant interviews, and artifact analysis.

c) These analyses will also articulate cognitive, affective, and conative requirements for each type of problem. Cognitive analysis will focus on procedural knowledge, strategic knowledge, and systemic or conceptual knowledge. That is, what do problem performers do? Why do they do it? What did they have to know in order to allow them to do it? Affective analysis will focus on the beliefs and attitudes about the problems and the processes possessed by problem performers as well as self-confidence. Analyze and articulate individual differences that may impact on performance. In order to analyze individual differences, we will use information gathered in the previous step and correlate it, wherever possible, with individual test results on cognitive control and personality measures.

d) Compare problem types within and across domains. We need to examine the same types of problems in different domains to assess the transferability of the instructional design models. It is likely that some problem types will be more domain-specific, which has implications for the nature of the instructional design process.

 

2) In years 2 and 3, we intend to articulate instructional models for each kind of problem. Table 3 provides an outline and an inexact beginning to the process of articulating instructional design models and methods for the different problem-solving outcomes. Until we are able to validate the processes for each type of problem solving, we will be unable to articulate these models.

 

3) In year 3 and beyond, we intend to test and validate instructional models in site-based applications. Each model must be tested in as many domains as possible. That entails designing and developing instruction for problems in different domains, engaging learners with them, and assessing the learning and transfer of problem solving. This is a long term process.

 

Given the long term nature of this research agenda, we seek to establish a Center for the Study of Problem Solving at the Pennsylvania State University. This Center will coordinate the activities of different researchers from education, psychology, management and information systems (to be identified) and provide an intellectual focus and identity for the participants.

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