problem.solving concepts theories

7/28/2019 Problem.solving Concepts Theories

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E d u c a t i o n a l S t r a t e g i e s

P r o b l e m S o l v i n g C o n c e p t s a n d T h e o r i e s

L a u r a E . H a r d i n

A B S T R A C T

M a n y e d u c a t o r s , e s p e c i a l l y t h o s e i n v o l v e d i n p r o f e s s i o n a l c u r r i c u l a , a r e i n t e r e s t e d i n p r o b l e m s o l v i n g a n d i n h o w t o s u p p o r t s t u -

d e n t s d e v e l o p m e n t i n t o s u c c e s s f u l p r o b l e m s o l v e r s . T h e f o l l o w i n g a r t i c l e s e r v e s a s a n o v e r v i e w o f e d u c a t i o n a l r e s e a r c h o n

p r o b l e m s o l v i n g . S e v e r a l c o n c e p t s a r e d e f i n e d a n d t h e t r a n s i t i o n f r o m o n e t h e o r y t o a n o t h e r i s d i s c u s s e d . E d u c a t i o n a l t h e o r i e s

d e s c r i b i n g p r o b l e m s o l v i n g i n t h e c o n t e x t o f b e h a v i o r a l , c o g n i t i v e , a n d i n f o r m a t i o n - p r o c e s s i n g p e d a g o g y a r e d i s c u s s e d . T h e

f i n a l s e c t i o n o f t h e a r t i c l e d e s c r i b e s p r i o r f i n d i n g s r e g a r d i n g e x p e r t n o v i c e d i f f e r e n c e s i n p r o b l e m s o l v i n g o f v a r i o u s k i n d s .

Any problem has at least three components: givens, goal,and operations. Givens are the facts or pieces of informationpresented to describe the problem. Goal is the desired endstate of the problem. Operations are the actions to be per-

formed in reaching the desired goal.1

Problems are categorized as ill defined or well defined,based on how problem and goal are represented. Problemswith complex representations and/or more than one solu-

tion are termedill defined

. Problems with discrete represen-tations and finite goals are termed well defined. Thedistinction between ill defined and well defined is a contin-uum, based on the complexity of the problem and what isrequired cognitively to solve it.

Problem-solving knowledge is, conceptually, of two kinds.Declarative knowledge is knowing that something is the case.It is knowledge of facts, theories, events, and objects. Proce-dural knowledge is knowing how to do something. It includesmotor skills, cognitive skills, and cognitive strategies. Bothdeclarative and procedural knowledge are activated inworking memory as problem solving occurs. Psychologistswho distinguish between declarative and proceduralknowledge believe that the two forms of knowledge are

both distinct and interdependent. Declarative and proce-dural knowledge interact in a variety of ways during prob-lem solving.

A basic unit of declarative knowledge in the human infor-mation-processing system is the proposition. This expressesor proposes the relationships among concepts. For instance,the phrase, the man fixed the tire, depicts a complete idea.A proposition always contains two elements: a relation and

one or more arguments.2

There are three attributes that are commonly used to differ-entiate expert from novice problem-solving characteristics.These attributes are conceptual understanding; basic, auto-mated skills; and domain-specific strategies. The following

paragraphs explain these attributes and how they are usedto distinguish expert from novice problem-solving charac-teristics.

Conceptual understanding refers to both the actual informa-tion in memory and the organization of that information inmemory. Conceptual understanding is closely related toschema theory, in which information is considered to bestored in memory as frameworks or structures that, onceinstantiated, provide a lens through which to view newinformation. Having a conceptual understanding of adomain means that an individual can make meaning of

domain-specific situations or problems, based on priorknowledge of that domain.

Basic, automated skills in any domain are those that allow anindividual to perform necessary and routine operationswithout much thought. These skills are overlearned to thepoint that they become habitual and even unconscious,enabling individuals to operate quickly and accuratelywithout taxing their short-term memories. Automaticityallows individuals to focus their attention on the more com-plex tasks associated with a specific domain and is a generalattribute associated with experts in a domain. Automaticitysupports the experts speed and skill of execution.

Unlike basic, automated skills, which occur unconsciouslyand thus do not tax short-term memory, domain-specificstrategies remain under conscious control. They are the pro-cesses and procedures in a domain that an individual, evenan expert, must consciously think about in order to solve aproblem. They are, in other words, the procedural knowl-edge associated with a domain.

Expertnovice differences have been studied and describedwithin the context of these three attributes: Experts (1)exhibit better conceptual understanding of their domain; (2)use more automated skills and domain-specific strategies;and (3) have a conceptual understanding that is declarative,

while basic skills and strategies are procedural.3

H I S T O R I C A L P R O G R E S S I O N O F P R O B L E M S O L V I N G

T H E O R I E S

As theories of how learning occurred evolved, the under-standing of the problem-solving process also evolved. Thefollowing section will present the model of problem solvingassociated with each concurrent learning theory. The domi-nant learning theories discussed are within the conceptualdomains of behaviorism, cognitive psychology, and infor-mation processing. Behaviorists view problem solving as a

process that develops through positive and negative rein-forcement mechanisms. Cognitive psychologists view prob-lem solving as a process that includes introspection,observation, and the development of heuristics. The infor-mation-processing view of problem solving is based on gen-eral problem solving skills and artificial intelligence.

B e h a v i o r i s t

Behaviorists understanding of learning was based on causeand effect. In this conceptualization, a behavior was fol-lowed by reinforcement. If the behavior was followed bypositive reinforcement, then the behavior was more likely to


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be repeated; if there was negative reinforcement, the behav-ior was less likely to be repeated.

Two problem-solving methodologies explain the problem-solving process within the framework of behaviorist learn-ing theory. One such method is trial and error. This involvesattacking the problem by various methods until a solution isfound. Young children solving a jigsaw puzzle exhibit thistype of problem-solving behavior. The children try fittingdifferent pieces into the same spot until eventually they find

the piece that fits.4

Another method consistent with behaviorist learning is

Hulls response hierarchy.5 This method involves learnedresponses that are applied to a situation in a hierarchicalmanner. The hierarchy is based on the response for whichhabit strength is strongest. Stimuli in a problem situationmay evoke several different responses, and responses willbe produced, one at a time, in order of strength, until eitherthe problem is solved or the organism exhausts its reper-toire of responses.

In their emphasis on trial-and-error learning and habitstrength, behaviorists focused on the role that stimulus

response interactions might play on problem solving.2

These early conceptions of learning and problem solving

described the observable characteristics of the process anddid not seek to elaborate on the cognitive mechanisms of thesubject.

C o g n i t i v e

As cognitive psychology progressed as a discipline, moreinterest and effort was directed toward the mental processesof learning and problem solving. An early cognitiveapproach to problem solving was to identify the mentalstages through which problem solving proceeded. Twonoted cognitive psychologists, Wallas and Polya, developeda four-stage model of problem solving. The four stages ofproblem solving identified by Wallas were (1) prepara-tiondefining the problem and gathering information rele-vant to it; (2) incubationthinking about the problem at asubconscious level; (3) inspirationhaving a suddeninsight into the solution of the problem; and (4) verifica-

tionchecking to be certain that the solution was correct.5

Similarly, Polya6 described the following four steps in theproblem-solving process: (1) understand the problem, (2)devise a plan, (3) carry out the plan, and (4) look backward.

C o g n i t i v e H e u r i s t i c s

Polya6 promoted the idea that the application of generalproblem-solving strategies was key to problem-solvingexpertise and intellectual performance. General problem-solving strategies have also been called heuristics. The wordheuristics comes from the Greek, heuriskin, meaning servingto discover. A commonly used synonym for heuristics isrule of thumb. In problem-solving literature, the term impliesthe general methods used in problem solving.

The heuristics Polya6 identifies in mathematical problemsolving are discussed within the framework of a four-stageproblem-solving model as discussed earlier. Some of theheuristics applied within this plan include understandingthe unknown, understanding the nature of the goal state,drawing a graph or diagram, thinking of structurally analo-gous problems, simplifying the problem, and generalizing

the problem. These heuristic methods can be applied to aproblem in any content domain; thus they are considered tobe general problem-solving skills.

In addition to the problem-solving processes already dis-cussed, other heuristics have been identified. People oftenhave to make decisions in the face of uncertainty, withsketchy information about the situation, on the basis of sug-gestive but inconclusive evidence. The reasoning processesused to resolve the uncertainty are often called judgmentheuristics. One form of judgment heuristic is similarity

judgment, where an instance is evaluated based on priorknowledge of a similar instance. A similar type of judgmentis representativeness, where an assumption is made basedon the belief that the characteristics of the individual arerepresentative of the group.

Another heuristic is the availability heuristic. In this case,judgments are made based on which elements can most eas-ily be retrieved from memory. Analogical reasoning isanother heuristic method, where judgment is made bydrawing similarities to events that have occurred previ-ously. Still another judgment heuristic is the developmentof a mental model (simulation) to predict the outcome of anevent.7 These heuristics are examples of general-purpose

thinking skills, which seem applicable to many domains.The heuristics approach emphasizes finding a good repre-sentation of the problem. While content-specific knowledgeis required to solve the problem, the belief that generalproblem-solving skills were also valuable was supported by

studies in the domains of math and computer science.8

I n f o r m a t i o n P r o c e s s i n g

As elaboration on problem solving and expertnovice dif-ferences continued, the information-processing theory oflearning emerged. This theory emphasizes the role of fac-tors such as working memory capacity, organization oflong-term memory, and cognitive retrieval of relevant infor-mation. The bulk of current research in problem solving

reflects inquiry into the nature of these cognitive processes.5

Newells early work in artificial intelligence1 (AI) sustainedthegeneral problem solving theory. AI is the study and devel-opment of computer programs to solve problems. An AIprogram uses a finite set of functions to work from the prob-lem state to the solution state. Many simple puzzles andproblems in logic were successfully completed with AI pro-grams, supporting the idea that problem-solving successwas directly related to general problem-solving skill.

Newell and colleagues proposed a theory of human prob-lem solving that emphasized the similarities between AIand human problem solving. There are four underlyingprinciples of this theory: (1) A few gross characteristics of

the problem-solving process are invariant over the task andthe problem solver, (2) the characteristics of the problem aresufficient to determine the problem space, (3) the structureof the task environment determines the possible structure ofthe problem space, and (4) the structure of the problemspace determines the possible programs (methods) that canbe used for problem solving.


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H I S T O R I C A L P R O G R E S S I O N O F E X P E R T N O V I C E

P R O B L E M - S O L V I N G T H E O R I E S

General problem solving knowledge was determined to bean incomplete explanation of how problem solvingoccurred. A close look at the expertnovice research revealsthat the expert does need domain knowledge, basic auto-mated skills, and domain-specific expertise to exhibit thecharacteristics of expert problem solving. Several studiesreveal that expertise relies on both domain-specific knowl-edge and problem-solving skill. The following examples,

from various domains, illustrate the body of research thathas been conducted to reach this conclusion.

C h u n k i n g

A key area of expertnovice research has been the study ofchess players. The game of chess is thought to require sub-stantial skill and cognitive ability for successful play. Cogni-tive researchers began studying expert and novice chessplayers to develop understanding of the underlying cogni-tive processes. Research by De Groot, as cited by Chase andSimon,9 indicated that master chess players are better atreconstructing chess positions due to chunking informationwithin a relational structure.

Chase and Simon9 conducted further research to discoverand evaluate the structures, or chunks, that are perceived bythe chess player. Three chess players (a master, a Class Aplayer, and a beginner) conducted memory tasks on posi-tions derived from 20 published chess games. Participantswere asked to view a game board layout, then reproduce,from memory, the layout on another board. Results of thisstudy indicated that the information extracted from a posi-tion, briefly exposed to the viewer, varies with playingstrength (the more expert player derives more information).The data also suggested that experienced players are able toencode the positions into larger chunks (than novices). Thenumber of chunks in short-term memory appeared consis-tent across levels of playing expertise, thereby strengthen-

ing the belief that content-specific relationships are chunkedin a cognitively accessible framework.

Contrary to original speculation, studies describing the cog-nitive processes of experts and novices have indicated thatexperts (chess players) do not consider plays farther aheadthan the novice; rather, experts choose among vastly supe-rior (complex) moves. The expert is able to chunk relevantinformation, while novices envision single pieces of infor-

mation.10

Researchers began to look beyond chess into other knowl-edge-rich domains. Studies have described the problem-solving process in physics, mathematics, computer pro-gramming, and medical diagnosis. Language skills, such as

reading and writing, and the question how students usedthese skills to acquire more knowledge were also studiedintensively.10, 11

S c h e m a T h e o r y

Differences between expert and novice teachers have alsobeen studied. Research has shown that teachers, like expertsin other fields, possess well-elaborated schemas that pro-vided a framework for meaningful interpretation of infor-mation. Expert teachers also had an understanding of whatto expect in the classroom and set up procedures and rules

for student behavior. A study comparing expert and noviceteachers also revealed that expert teachers thought aboutlearning from the perspective of the student and performeda cognitive analysis of each learning task, which theyadapted to students needs during teaching. Novice teach-ers used specific objectives to form lesson plans and did notadapt to student needs during teaching.12

Barba and Rubbas13 research with 30 expert (in-service) and30 novice (preservice) earth- and space-science teachers wasconducted to study the cognitive differences between

experts and novices.

Data from audio-taped interviews revealed that expertteachers were more accurate, verbalized more declarativeknowledge (facts and concepts), used fewer steps, and gen-erated more subroutines than their novice counterparts.Experts generated more alternative solution paths andmoved less between procedural (rules and strategies) anddeclarative knowledge than did the novices.

As the authors noted, these findings support Norman andGagnes theories of cognitive learningspecifically, thatknowledge is hierarchically arranged in schemata. Accord-ing to this theory, facts unite to form concepts, concepts jointo form rules, and rules join to form problem-solving struc-

tures. These findings also support Normans (1982) theorythat expert performance in procedural knowledge is charac-terized by smoothness, automaticity, and decreased mentaleffort, as compared to that of the novice.14

E x p e r i e n c e a n d T r a i n i n g

A study by Schoenfeld and Herrmann15 showed thatdomain-specific content is relevant to successful problemsolving. Nineteen college students participated in thisstudy, which evaluated the differences between experts andnovices in mathematical problem perception. Experts deter-mined the deep- and surface-structure properties of eachproblem prior to initiation of the study. All students wereasked to sort the 32 math problems, based on which, if any,were mathematically similar (each of the similar problemswould be solved in the same way). After completion of thefirst sort, subjects either participated in a course in mathe-matical problem-solving strategies that stressed a system-atic, organized approach to solving problems or took astructured computer programming course that taught astructural, hierarchical, and orderly way to solve non-math-ematical problems using the computer.

Upon completion of the course, subjects sorted an equiva-lent set of mathematical problems. Those completing themathematical problem-solving course were more accuratein identifying the appropriate deep and surface structuresof the problems than they had been prior to the course,

whereas those completing the programming course werenot. This study showed that training within the problemdomain affects problem solving more than training outsidethat domain.

M e t a c o g n i t i o n

Bruer summarizes the research on domain-specific problemsolving as follows: Expertise, these studies suggest, relieson highly organized, domain-specific knowledge that canarise only after extensive experience and practice in the

domain10


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Studies in domain-specific problem-solving expertise alsointroduce the underlying principle of metacognition. Meta-cognition is the ability to think about thinking, the self-awareness of problem solving, and the ability to monitorand control ones mental processing. According to Bruer,Bransfords self-study of physics revealed several aspects ofmetacognition, which ultimately constitute an awareness

and control of learning that can cross content domains.10

The ability to solve problems successfully depends on anumber of factors related to the human information-pro-

cessing (IP) system. This higher order learning theory elabo-rates the cognitive processes of problem solving. There aresix attributes that define expertnovice differences in prob-lem-solving skill within the IP framework, regardless ofcontent domain. Through years of study of the cognitiveprocesses applied during problem solving, six characteris-tics of expert performance have become widely accepted:

Experts perceive large, meaningful patterns in their domain.The ability to see meaningful patterns reflects organizationof the knowledge base. An expert has the knowledgerequired to solve a problem, and this knowledge is assess-able in a way that does not tax working memory.

Experts are faster and more accurate than novices at solving

problems within their domain. There are two likely reasonsfor this phenomenon: Experts have developed the basic,automated skills applicable to the problem, and they havean organized database from which to retrieve the solution.

Experts have superior short- and long-term memory,again based on superior memory organization, ratherthan volume.

Experts see and represent data at a more conceptual(principled) level than novices.

Experts spend more time analyzing and evaluating aproblem quantitatively before beginning to solve theproblem.

Experts have strong self-monitoring skills. They aremore aware when they make errors, why they fail tocomprehend, and when they need to check their

solutions.4

The preceding review of research in general problem-solv-ing methods and domain-specific problem-solving charac-teristics has concluded with a summary of thecharacteristics of expert problem-solving. The discussion ofboth general and content-specific problem-solvingattributes leads to the conclusion that both content knowl-edge and general problem-solving skill are necessary forexpert problem solving to occur. Content-specific knowl-edge allows the expert to perceive information in a way thatmaximizes memory and information is conceptualizedmore on the level of principles. Superior problem analysis

and evaluation and strong self-monitoring skills can be rec-ognized as general problem-solving expertise.

R E F E R E N C E S

1. Newell A, Simon H.Human Problem Solving. EnglewoodCliffs, NJ: Prentice Hall, 1972.

2. Gagn ED, Yekovich CW, Yekovich FR. The Cognitive Psy-chology of School Learning. New York: Harper Collins, 1993.

3. Miller K. Differences between expert and novice writers.Unpublished, 1996.

4. Chi MH, Glaser R, Farr MJ. The Nature of Expertise. Hills-dale, NJ: Lawrence Erlbaum, 1988.

5. Ormrod JE.Human Learning. Upper Saddle River, NJ:Prentice Hall, 1987.

6. Polya G.Mathematics and Plausible Reasoning. Princeton,NJ: Princeton University Press, 1954.

7. Glass AL, Holyoak KJ. Reasoning and decision making. InCognition. New York, NY: Newberry Awards Records, Inc,1996:333363.

8. Nickerson RS, Perkins DN, Smith EE. The Teaching ofThinking. Hillsdale, NJ: Lawrence Erlbaum, 1985.

9. Chase WG, Simon HA. Perception in chess.Cognit Psychol4:5581, 1973.

10. Bruer JT. The minds journey from novice to expert.AmEducator Summer:646, 1993 p15.

11. Perkins DN, Salomon G. Are cognitive skills context.Educ Res JanFeb:1625, 1989.

12. Westerman DA. Expert and novice teacher decisionmaking.J Teach Educ 42:292305, 1991.

13. Barba RH, Rubba PA. A comparison of preservice andin-service earth and space science teachers general mentalabilities, content knowledge, and problem-solving skills.J

Res Sci Teach, 29:10211035, 1992.

14. Norman GR, Learning and Memory. San Francisco, CA:W.H. Freemen, 1992.

15. Schoenfeld AH, Herrmann DJ. Problem perception andknowledge structure in expert and novice mathematicalproblem solvers. J Exp Psychol Learn Mem Cogn 8:484494,1982.

A U T H O R I N F O R M A T I O N

L a u r a E . H a r d i n , D V M , M S , P h D , i s a n A s s i s t a n t P r o f e s s o r o f

V e t e r i n a r y B a s i c S c i e n c e s , M i s s i s s i p p i S t a t e U n i v e r s i t y , C o l l e g e

o f V e t e r i n a r y M e d i c i n e . E - m a i l : S M T P : l h a r d i n @ c v m . m s s t a t e . e d u

problem.solving concepts theories

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