Selective Problem Solving (Sps): a Model for Teaching Creative Problem- Solving

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Selective Problem Solving (sps): A Model for Teaching Creative Problem- Solving

Article in Gifted Education International · January 2011 DOI: 10.1177/026142941102700310

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Ahmed Mohamed Assiut University, Egypt and C. June Maker The University of Arizona, USA creative storytelling: evaluating problem solving in children's invented stories

Abstract

One of the most important functions of storytelling is to help individuals solve problems that can be of significance in their daily lives. The purpose of this study was to discover new criteria to use when assessing the quality of children's invented stories as examples of their linguistic problem solving ability. Three children's stories, told during a Discovering Intellectual Strengths and Capabilities while Observing Varied Ethnic Responses (DISCOVER) Assessment and analyzed by trained observers, were re-analyzed using the high point micro analysis (Labov, 1972; Peterson & McCabe, 1983) and the Narrative Assessment Profile (Stubbs, 1983). Teachers and other observers, who assigned ratings to children's work focused on certain linguistic conventions, and because of this focus, missed other important qualities of stories. A new list of characteristics, including such traits as using "openers and closers", using adverbs and quantifiers, and varied linguistic forms was proposed. This list of characteristics can be helpful to teachers who assess the quality of children's invented stories as well as observers who assess students' linguistic problem solving during the DISCOVER Assessment debriefing sessions.

Keywords: DISCOVER, invented stories, problem solving, storytelling.

Narrative or Storytelling Functions of Stories

Narrative is a pragmatic language skill. It Storytelling has various functions. It involves understanding and reproducing de enhances children's conceptual and writing contextualized information or content. Any development (Dwyer & Bain, 1999), is a narrative is characterized by being driving force in transmitting heritage, structured, having a beginning, middle, and civilization, and historical events. Story a conclusion or an end, and it contains a telling enhances children's imagination, problem (Fein, 1995). Narrative also usually enables self-representation, integrates them is about real or imagined memories, and is into their own culture and unique ways of told in the past tense {McCabe & Bliss, 2003). using language, and conveys emotions and

Ugur Sak Anadolu University, Turkey selective problem solving (sps): a model for teaching creative problem-solving

Abstract

Problem solving is an integral part of human life from mathematics and science to business, marketing, arts, technology and more. It is sometimes a form of creativity with discoveries and aesthetics. In fact, wherever a genuine problem exists novel behavior is needed on the part of the problem solver. The aim of this article was to describe and discuss the Selective Problem Solving (SPS) model that can be used for creatively solving problems that are confronted in many disciplines. The SPS was developed based on an integration of research on creativity and teachings of Polya on problem solving. The SPS model is a revision and extension of the Polya model with an inclusion of problem finding, problem identification, problem construction, and selective thinking. It consists of six steps: Problem definition, problem identification, problem solution, problem construction, problem solution, and reflection.

Keywords: problem solving, creativity in math, creative problem solving, creativity.

Selection, the way all life becomes perfect, problem solving (Polya, 1954). A filter can be has been the unchanging law of nature. As a good analogy to a selectively working Darwin put forward, all living things, more mathematician. S/he filters out what is or less, have experienced natural selection. related to the problem under investigation Those who were not selective vanished as a and what is unrelated to it. Then, s/he works result of maladaptation. Likewise, selection with the information that is promising to the should be the habit of human thinking for solution. perfection. In fact, the eminent mathe matician Poincare (Gould, 2001) asserted Second, selection occurs in constructions. that discovery is selection. A creative A creative mathematician, for instance, mathematician, for example, makes selective works selectively while constructing comparisons among mathematical elements mathematical combinations. According to and selects those that have promise for Poincare (Gould, 2001), a creative useful combinations. mathematician selects the most useful combination from among numerous others. First, selection occurs when attempting In mathematics, the number of samples is so to understand the problem (Davidson & numerous that the entire life of a Sternberg, 1984). A good mathematical mind, mathematician would not be enough to for example, becomes selective during examine all samples to make combinations

Volume 27 No 3, 2011, 349 that are promising. What a mathematician out the plan, and looking back. In the theory needs to do is to choose among numerous of insightful thinking, Davidson and samples or combinations with a view to Sternberg (1984) suggested that three eliminating those that are useless. According knowledge-acquisition components form the to Poincare, this selective combination can be bases for three different kinds of potentially felt with insight. One road that leads to creative ideas: Selective encoding, selective insight is thinking in the form of analogies. combination, and selective comparison. An insight of selective encoding involves sorting Third, selection also has an important relevant from irrelevant information. An role in finding analogies. One of the insight of selective combination involves fundamental characteristics distinguishing combining facts and ideas into a relevant creative people from others is their unique and integrated whole. An insight of selective capacity to discover analogies. Indeed, comparison involves relating newly analogy and selection are believed to be acquired information to information fundamental tools for discovery. A acquired in the past. Furthermore, research mathematician, for example, selectively on creativity has shown that problem looks for underlying structures between a identification, problem definition, and target problem and a source problem. problem construction that are believed to be According to Poincare (Gould, 2001), a more important than problem solving are creative mathematician discerns harmonious found to be the true roots of creative relations hidden among mathematical problem solving (Runco, 2006). In his model, elements, and according to Polya (1957; Polya emphasizes the use of analogy in 1968), almost no problems exist that are problem solving. The SPS model is a revision unrelated to formerly solved problems. and extension of the Polya model with an Polya (1954) further asserted that analogy inclusion of problem finding, problem has the lion's share in many discoveries. identification, and problem construction Furthermore, many creative insights seem to from research on creativity and selective have resulted from analogical thinking encoding, and selective comparison from the (Michalko, 2001; Runco, 2006). For example, theory of insightful thinking. the benzene ring was discovered by thinking of an analogy to a snake biting its own tail; the steam engine was an analogy to tea SPS Components kettles; and telegraph stations was an analogy to stations where post horses were The purpose of the SPS is to develop creative changed. thinking and problem solving ability through the use of analogical, insightful, and In this article, the author presents a selective thinking, and to enrich an description of the Selective Problem Solving individual's knowledge repertoire so that it (SPS) model inspired by the problem solving is transferable to different problem model proposed by the mathematician Polya situations. The model includes six problem (1957), the theory of insightful thinking solving stages: Problem definition, problem proposed by Davidson and Sternberg (1984), identification, problem solution, problem and research on creativity. Polya' s model construction, problem solution, and includes four stages in mathematical reflection (Table 1). problem solving: Understanding the problem, devising a solution plan, carrying

350, Gifted Education Intemational Stage 1: Problem Definition write necessary notations and draw figures if Defining problems usually is the first step of needed. Note that problems presented to creative problem solving when a problem is students at this stage should be more recognized. This is a very important stage advanced than their current knowledge. For because altering the problem definition example, in Figure 1, a problem related to the could lead to unexpected solutions. In fact, area of a cube is presented. Students are in creative accomplishments defining assumed to know algorithms to find the area problems is believed to be more influential of a square or rectangle but not that of a than solving problems (Runco, 1994; Runco cube. & Dow, 1999). A problem definition provides an understanding of the problem and its Stage 2: Problem Identification components. To understand a problem one needs to get acquainted with it and encode Problem identification involves recognizing its parts. When understanding a problem, the existence of a problem when a task is two types of encodings are employed. The simply recognized but not made operational first is the description and interpretation of (Runco & Dow, 1999) and/ or finding what each part of a problem means. The contextual problems by making selective second is called selective encoding whereby comparisons between problems. Selective relevant information is separated from comparison involves relating newly irrelevant information (Davidson & acquired information to information Sternberg, 1984). Significant problems acquired in the past (Davidson & Sternberg, usually have available large amounts of 1984), information learned in the past to information, but only some of this recently encountered information, and information may be relevant to the problem recently learned information to information solution. to be learned in the future. Selective comparison processes are responsible for The purpose of this stage of the SPS is to determining which information from long help students acquire a full understanding of term memory is relevant for the solution of a the problem and define it to make it problem and will be retrieved and stored in workable. At this stage of the SPS, students working memory (Sternberg, 1986). Problem should consider principal parts of the solving by analogy is an instance of selective problem, such as knowns and unknowns, comparison. By analogy, one realizes that from various perspectives, and define the new information is similar to old problem from their own point of view with information in some ways. One important their own words. To initiate the problem part of using analogy is retrieving a useful solving activity, the teacher should ask the source analog from memory (Holyoak & following questions after presenting a target Nisbett, 1988}. However, some aspects of the problem to solve (see Table 1): What is the target problem must have retrieval hints to problem? What is known? What is remind the problem solver of an analog unknown? The teacher should ask students (Schank, 1982). the following questions to identify relevant and irrelevant information for the solution if The students' task at this stage is to irrelevant information exists: What identify or select simpler analogous information is required to solve this problems that have similarities with the problem? What information is not required target problem and that could be useful in to solve this problem? The teacher should the solution of the target problem. At this

Volume 27 No 3, 2011, 351 Table 1: SPS discussion form: SPS problem solving steps, student behaviors involved in each step and student and teacher roles.

Steps Behaviors Focus Questions Student Role Teacher Role 1. Define the problem. What is the problem? Separate Present a target Problem Identify known and What is known? various parts problem. Definition unknown information. What is unknown? of a problem List data. Identify information What information is required to Introduce required for the solve this problem? notations. solution. What information is not required Draw a figure Identify information to solve this problem? if needed. irrelevant for the How is the condition sufficient to solution. solve the problem?

2. Identify a 1) Have you seen this problem Find an Elicit students' Problem similar Identify before? analogous prior Identifi- problem. an Have you seen the same problem problem. knowledge. cation Compare analogous in a different form? Solve the problems. problem Do you know a similar problem? analogous Infer relations How can you solve this problem problem. between you just have found? problems and their 2) Which one of the two problems is Select the Present two components. Select an similar to the target problem? correct problems if Recognize analogou How are they similar? analogous needed. relationships sproblem Which one of the two problems problem. Monitor between can be used in the solution of the Solve the problem relations. target problem? analogous solving Select the How can you solve this problem problem. process. correct you have selected? analogous problem. 3. Problem Apply knowledge in How could you use the solution Solve the target Monitor Solution different problem method of the analogous problem problem. problem situations. in the solution of the target Check solution solving Analyze solution steps. problem? steps. process. How can you prove that each step is correct?

4. Problem Develop a similar What are other more advanced Construct an Present a Construe- problem. problems than the target problem advance problem if tion Compare problems. that you can solve using the analogous needed. Infer relations between strategies and methods you used problem. problems. to solve the target problem? Recognize relationships What is the problem? between relations. How are the solutions to these two problems similar? How is this new problem more advanced than the target problem?

352, Gifted Education International Steps Behaviors Focus Questions Student Role Teacher Role 5. Problem Apply knowledge in How could you use the solution Solve the new Monitor Solution different problem method of the target problem in problem. problem situations. the solution of the new problem? Check solution solving Analyze solution steps. How can you prove that each step steps. process. is correct?

6. Explain analogous What have you learned while Share problem Encourage Reflection problem solving. solving problems? solving students' Explain selective How does an analogy work in experience. expressions. problem solving. solving problems? How do you use analogies to develop novel problems? How can you be selective while solving problems? stage, first, the teacher should ask the Stage 3: Problem Solution following question: Do you know a related problem? Keep in mind that numerous Once students identify a correct analogous problems might be related to the problem problem, make correct comparisons between under discussion. In such cases, we need to the target problem and the analogous think of a problem having the same or a problem and solve it correctly, they can more similar unknown. At this time, the teacher easily transfer their knowledge to solve the may ask the second question: Do you know target problem provided the analogous a problem having the same or similar problem has structural similarities with the unknown? If students still cannot relate a target problem. However, note that the formerly solved problem to the new instruction required to access a good source problem, the teacher should present two or analog could fail to identify a good source more problems, of which only one is related analog when it is superficially dissimilar to to and useful for the solution of the target the target (Holyoak & Nisbett, 1988). In this problem. The task of students is to compare case, similarity between the source problem parts of the teacher-presented problems with (analogous problem) and the target problem those of the target problem. Then the teacher should be increased. Increased similarity can should ask the following questions: Which lead to an increase in spontaneous analogical problem is similar to our target problem? transfer in problem solving (Holyoak & Koh, After students select a problem the teacher 1987). should ask the following questions: What similarities do you see between this problem At this stage, the teacher should and the target problem? What similarities do encourage students to use the methods and you see between their unknowns? The procedures they use in the solution of the teacher should continue questioning until analogous problem in the solution of the students select the correct problem and make target problem, and should ask the following correct comparisons. For example, in Figure question to initiate the stage: How could you 1, a square is presented or identified by use the solution method of the analogous students as a correct analogy for the cube problem in the solution of the target problem. problem? Once students start to solve the problem, they need to pay attention to the

Volume 27 No 3, 2011, 353 target problem but more advanced than it by analogical experience gained during the applying the methods or procedures that solutions of the analogous problem and the they acquire in the solution of the target target problem is transferred to the solution problem to new problems. This stage of the advanced analogous problem. The requires the use of analogy and selective teacher should ask the following question to comparison. Some problems generated by start this stage: How could you use the students may be very similar to the target solution method of the target problem in the problem, with little or no novelty because solution of the new problem? Students may students could develop a mental block make errors in the construction of an toward the target problem and not be able to analogical problem. If the problem students overcome it at the initial period of the construct is not a correct analogous problem, problem construction stage. After they they would fail to solve it correctly. If they systematically are exposed to this stage, they construct a correct analogous but very are likely to deviate from the usual cannons difficult problem at the initial period of the of thinking and develop more advanced and problem construction stage, they still could original problems. To initiate this stage, the fail to solve it because the analogy between teacher should ask the following question: the two problems could be very distant and What are other more advanced problems they could lack the knowledge they need to than the target problem that you can solve apply to solve the new problem. If students using the strategies and methods you used fail to solve the advanced analogous to solve the target problem? For example, as problem, they can be presented with or seen in Figure 1, several problems could be prompted to find a simple analogous constructed as analogies to the target problem that could be useful in the solution problem. After students construct one or of the advanced analogous problem. For several problems, the teacher should ask example, as seen in Figure 1, after students students to define the new problem and successfully complete Stage 4 by compare it to the target problem. The constructing several problems, they can be thought provoking questions are the asked to find an analogy for the rectangular following (Table 1): What is the problem? cuboid. In this case, the analogy would be a How are the solutions to these two problems rectangle. When students are solving the similar? How is this new problem more advanced analogous problem the teacher advanced than the target problem? The needs to keep track of where they are, how teacher should encourage students to the solution process is going and whether imagine cases in which they could use the use of analogy is successful. The teacher procedures they employ in the solution of should ask the following question to the target problem. By doing this several provoke students to check their solutions: times, students consolidate their knowledge How can you prove that each step is correct? and develop their ability to transfer knowledge to deal with novel problems. Stage 6: Reflection

Stage 5: Problem Solution The purpose of this stage is to learn from experience for further development. At this This stage is somewhat similar to Stage 3 in stage, students evaluate problem solving which experience gained in the solution of procedures they carry out from the first stage the analogous problem is transferred to the to the fifth stage and the experience they solution of the target problem. At this stage, acquire during these stages, and they reflect

Volume 27 No 3, 2011, 355 upon their thinking. The teacher should ask References the following question to start students to think about their experience and what they Davidson, J. E., & Sternberg, R. J. (1984). The role have learned during the stages of the SPS: of insight in intellectual giftedness. Gifted Child What have you learned while solving Quarterly, 28, 58-64. problems? The teacher should encourage students to reflect on the whole process of Einstein, A., & Infeld, L. (1938). The evolution of problem solving, and then should ask the physics. New York: Simon and Schuster. following questions: How does an analogy Getzels, J. W. (1975). Problem finding and the work in solving problems? How do you use inventiveness of solutions. Journal of Creative analogies to develop novel problems? How Behavior, 9, 89-118. can you be selective while solving problems? As metaphorically seen in Figure 1, students Gould, S. J. (Ed.). (2001). The value of science: should be encouraged to think upon Essential writings of Henri Poincare. New York: The analogical reasoning, analogical transfer and Modem Library. selective thinking to find analogies. That Holyoak, K. J., & Nisbett, R. E. (1988). Induction. students learn the value of the SPS in creative In R. J. Sternberg & E. E. Smith (Eds.), The problem solving and the ways it works is an psychology of human thought (pp. 50-91). important learning outcome for students. Cambridge: Cambrdige University Press.

Holyoak, K. J., & Koh, K. (1987). Surface and Conclusions structural similarity in analogical transfer. Memory & Cognition, 15, 332-340. Polya said "a great discovery solves a great Michalko, M. (2001). Cracking creativity: The secrets problem but there is a grain of discovery in of creative genius. Berkeley, CA: Ten Speed Press. the solution of any problem" (1957, p. V). That is, many ways exist to solve problems Polya, D. (1954). Induction and analogy in but each method of problem solving can mathematics. Princeton, NJ: Princeton University uncover new discoveries. Analogy and Press. selection are methods used as major instruments for discoveries. In scientific Polya, D. (1957). How to solve it. 2nd ed. NJ: domains, for example, analogy provides an Princeton University Press. additional top-down mechanism for concept Polya, D. {1968). Mathematics and plausible formation. Scientists working in a reasoning. (2nd ed.). Princeton, NJ: Princeton problematic new domain usually look to University Press. areas they already understand as a source of transportable concepts and problem solving Runco, M. A. (2006). Creativity theories and themes: techniques (Holyoak & Nisbett, 1988). Two Research, development, and practice. San Diego, CA: fundamental characteristics of the Selective Academic Press. Problem Solving model are selection and Runco, M. A. (1994). Conclusions concerning analogy. Higher-order thinking skills are problem finding, problem solving, and creativity. tapped as the teacher takes students from In M. A. Runco (Ed.), Problem finding, problem simple analogical thinking to advanced solving, and creativity, 272-290. Norwood, NJ: analogical thinking and from trial-error to Ablex Publishing. selectiveness in thinking. Runco, M.A., & Dow, G. {1999). Problem finding.

356, Gifted Education International In M.A. Runco & S.R. Pritzker (Eds.), Encyclopedia Sternberg, R. J. (1985). Beyond IQ: A triarchic theory of Creativity (Vol. 2, pp. 443-445). San Diego, CA: of human intelligence. New York: Cambridge Academic Press. University Press.

Schank, R. C. (1982). Dynamic memory. of Cambridge: Cambridge University Press. Sternberg, R. J. (1986). Toward a unified theory human reasoning. Intelligence, 10, 281-314.

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