# The Selective Problem Solving model (SPS) and its social validity in solving mathematical problems.

Problem solving is an all-day and all-life struggle of humankind. We confront easy problems, difficult problems and never-solved problems. That is, we face a continuum of problems in our life. As the type of problem varies, the way we approach problems also differs. Whatever the ways we use to solve problems, some of them result in success; but others result in a greater success surprising people and inspiring new ideas. In this article, first, we focus on creative problem solving, and then we review the Selective Problem Solving Model (SPS) followed by a research study we carried out on the social validity of the SPS. The purpose of the SPS is to develop creative thinking and problem solving ability through the use of analogical and selective thinking and to enrich knowledge repertoire so that it is transferable to different problem situations (Sak, 2011).Analogical Thinking in Creative Problem Solving

Analogy refers to the mapping of the relationship between two or more phenomena (Holyoak & Thagard 1995; Mumford & Porter, 1999). One of the fundamental characteristics distinguishing creative people from others is their unique capacity to discover analogies. Indeed, analogy and selection together are believed to be fundamental tools for making discoveries. A mathematician, for example, selectively looks for underlying structures between problems. According to Poincare (Gould, 2001), a creative mathematician discerns harmonious relations among mathematical elements, and according to Polya (1957), no problems exist that are unrelated to formerly solved problems. Many creative insights also seem to have benefited from analogical thinking (Boden, 2004; Holyoak & Thagard 1995; John-Steiner, 1997; Runco, 2006; Sawyer, 2006). One such classic example is Kekule's discovery of molecular structure of benzene.

Theoretical explanations underline two parts of an analogy: the target and the source. The phenomenon being explained is the target, and the analogical comparison is the source; or the target may be a problem that, for example, a mathematician attempts to solve. The source may be another problem that he or she uses to understand and solve the target problem. The process of making an analogy includes retrieval of a source from memory, finding similarities between the source and the target, and mapping features of the source onto those of the target (Dunbar, 1999). That is, first, an appropriate source has to be selected, and then the source has to be matched up with the target. The mapping of structural representations between the source analog and the target is crucial for successful analogical transfer. However, this process may not occur readily (Holyoak & Koh, 1987). According to Holyoak and Koh (1987), four processes may be required for analogical transfer to occur: constructing mental representations of the source and the target; selecting the correct source as a relevant analog; and mapping the structural similarities of the source and the analog; and extend the mapping to generate a solution for the target.

Research shows that the use of analogy is affected by ability. For example, novices in experiments tend to focus on superficial features between the source and the target. Unlike novices, experts use structural knowledge to discover structural similarities between seemingly unrelated problems (Forbus, Gentner & Law, 1995; Holyoak & Thagard 1995). Mapping the features of the source onto the target provides new insights into understanding the target problem; as a result, the problem solver discovers new features of the target while coming up with new ideas or creative solutions.

Selection in Creative Problem Solving

Selective thinking may be defined as a thinking process by which one focuses on useful or related facts or ideas while ignoring unrelated facts or ideas. Selection occurs in understanding the problem (Davidson & Sternberg, 1984). A good mathematical mind, for example, becomes selective during problem solving (Polya, 1954). A filter can be a good analogy to a selectively working mathematician. He or she filters out what is related to the problem under investigation and what is unrelated to it. Then, he/she works with the information that is promising to the solution. Selection also occurs in constructions. According to Poincare (Gould, 2001), a creative mathematician selects the useful combination among numerous others. In mathematics, the number of samples is so numerous that the entire life of a mathematician would not be enough to examine all samples to make combinations that are promising. He or she needs choose among numerous samples or combinations with a view to eliminating those that are useless. According to Poincare, the combinations that are potentially most useful are those that constitute remote analogies (Weisberg, 2006). Selection also plays an important role in finding analogies (Davidson & Sternberg, 1984; Gust, Krumnack, Kuhnberger & Schwering, 2008; Pereira de Barros, Primi, Koich Miguel & Almeida, 2010).

The Selective Problem Solving model (SPS)

The SPS model (Sak, 2011) was developed based on a synthesis of the problem solving model proposed by the mathematician Polya (1957), the theory of insightful thinking proposed by Davidson and Sternberg (1984), and the research on creativity. Polya's model includes four stages in mathematical problem solving: understanding the problem, devising a solution plan, carrying out the plan, and looking back. In the theory of insightful thinking, Davidson and Sternberg suggested that three knowledge-acquisition components form the bases for three different kinds of potentially creative ideas: selective encoding, selective combination, and selective comparison. Selective encoding involves sorting out relevant from irrelevant information. Selective combination involves combining facts and ideas into a relevant and integrated whole. Selective comparison involves relating newly acquired information to information acquired in the past. Furthermore, research on creativity has shown that problem identification, problem definition and problem construction are found to be the true roots of creative problem solving (Getzels, 1979; Getzels & Csikszentmihalyi, 1976; Runco, 2006). The SPS is composed of six problem solving stages (see table 1): definition of the target problem, identification of the source problem, solution of the target problem, construction of an original problem, solution of the original problem, and reflection. A description and importance of each stage are presented in the next sections of the review.

Definition of the Target Problem

Defining problems usually is the first step of creative problem solving when there exists a problem. This is a very important stage, because altering the problem definition could lead to unexpected solutions (Runco, 1994; Runco & Dow, 1999). A problem definition provides an understanding of the problem and its components. In understanding a problem, two types of encodings are employed. The first is the description and interpretation of what each part of a problem means. The second one is called selective encoding whereby relevant information is separated out from irrelevant information. Significant problems usually pose large amounts of information, but only some of this information may be relevant to problem solution (Davidson & Sternberg, 1984).

This stage of the SPS starts with a presentation of a target problem. The purpose of this stage is to help students acquire a full understanding of the problem and define it to make it workable. At this stage, students should consider principal parts of the target problem, such as knowns and unknowns, from various perspectives and define it from their own point of view. To initiate the problem solving activity, the teacher should ask the following questions after presenting a target problem (table 1): What is the problem? What is known? What is unknown? The teacher should ask students the following questions to identify relevant and irrelevant information for the solution if irrelevant information exists: What information is required to solve the problem? What information is not required to solve the problem? What is important at this stage is the difficulty of and level of complexity of the problem. Problems presented to students at this stage should be more advanced than their current knowledge. Otherwise, students could solve the target problem without analogical transfer.

Identification of the Source Problem

Problem identification involves both recognizing the existence of a problem where a task is simply recognized but not operationalized (Runco & Dow, 1999) and/or finding contextual problems by making selective comparisons between problems. Selective comparison involves relating newly acquired information to information acquired in the past (Davidson & Sternberg, 1984), information learned in the past to recently encountered information, and recently learned information to information to be learned in the future. Selective comparison processes are responsible for determining which information from long-term memory is relevant for the solution of a problem and will be retrieved and stored in the working memory (Sternberg, 1986). Problem solving by analogy is an instance of selective comparison. By analogy, one realizes that new information is similar to old information in some ways. One important part of using analogy is retrieving a useful source analog from memory (Holyoak & Nisbett, 1988). However, some aspects of the target problem must provide retrieval hints to remind the problem solver of an analog (Schank, 1982).

Students' task at this stage of the SPS is to identify or select simple analogous problems that have structural similarities with the target problem and that could be useful in the solution of the target problem. The ability to successfully discriminate between multiple potentially relevant source analogs when solving new problems was found to be crucial for proficiency in problem solving (Richland & McDonough, 2010). At this stage, the teacher should ask the following question: do you know a problem similar to our target problem in some ways? Keep in mind that there might be numerous problems related to the target problem under discussion. In such cases, the teacher may prefer to ask the second question: do you know a problem having the same or similar unknown? If students still cannot relate a formerly solved problem to the target problem, the teacher should present two or more problems, of which only one can be used as a source analog for the solution of the target problem. The task of students is to compare parts of the teacher-presented analogous problems with those of the target problem. Then the teacher should ask the following question: which problem is similar to the target problem? After students select a problem, the teacher should ask the following questions: what similarities do you see between the source problem and the target problem? Do you see any similarities between their unknowns? The teacher should keep asking similar questions to provide cues to support comparative reasoning until students select the correct source problem and discover correct and useful similarities. Providing cues for analogical transfer was found to be critical. For example, in experiments, cues presented to support comparative reasoning led to an increase in ability to discriminate between relevant analogs and to a better analogical transfer (Ngu & Yeung, 2012; Richland & McDonough, 2010).

Solution of the Target Problem

Once students identify a correct analogous problem, make correct comparisons between the target problem and the analogous problem and solve the analogous problem correctly, it is easier for them to transfer their knowledge to solve the target problem provided that the analogous problem has structural similarities with the target problem. However, the instruction required to identify a good source analog could result in a failure if the source analog is superficially dissimilar to the target (Holyoak & Nisbett, 1988). In this case, similarity between the source analog and the target problem should be increased or cues to support comparative reasoning should be increased. Increased similarity can lead to an increase in spontaneous analogical transfer in problem solving (Holyoak & Koh, 1987).

At this stage, the teacher should encourage students to use the methods and the procedures they use in solving the analogous problem for solving the target problem and should ask the following question to initiate the stage: how could you use the solution method of the source problem in the solution of the target problem? Once students start to solve the problem, they need to pay attention to the problem solving processes during this stage. The teacher should ask the following question to provoke students to check their solutions: can you prove that each step is correct? This stage is where students reexamine their solutions step by step, as well as holistically. By reexamining the solution, they internalize their knowledge and develop their ability to solve more advanced problems.

Construction of an Original Problem

Creativity researchers maintain that problem construction is as important as problem definition and identification in creativity. Getzels (1975), for example, put forward that the quality of a problem determines the quality of a solution. Creative people themselves also have pointed out the importance of problem construction in the creative process. Einstein, for example, asserted that "the formulation of a problem is often more essential than its solution...." (Einstein & Infeld, 1938, p. 83).

The fundamental characteristics of this stage are analogical transfer and novelty in problems students are expected to develop. First, students should construct problems that are analogical to the target problem presented to them in the beginning of the problem solving activity. Second, these problems should be novel for them or more advanced than the target problem. This stage requires the use of analogy and selective comparison. Some problems generated by students can be very similar to the target problem, with little or no novelty; because the target problem could become a mental block and prevent students' original thinking at the initial period of the problem construction stage. After they systematically are exposed to this stage, it is highly likely that they could deviate from the usual cannons of thinking and develop more advanced and original problems. To initiate this stage, the teacher should ask the following question: what are other more advanced problems than the target problem you can solve using the strategies and the methods you have used to solve the target problem? After students construct one or several problems, the teacher should ask students to define the new problem and compare it to the target problem. The thought provoking questions are the following (table 1): What is the problem? How are these problems similar in terms of their solutions? How is the new problem more advanced than the target problem? The teacher should encourage students to imagine cases in which they could utilize procedures they have used in the solution of the target problem. By doing this for several times, students consolidate their knowledge and develop their ability to transfer knowledge and to deal with novel problems.

Solution of the Original Problem

At this stage, analogical experience gained during the solution of the source problem and the target problem is transferred to the solution of the advanced analogous problem. The teacher should ask the following question to start this stage: How could you use the solution method of the target problem in the solution of the advanced problem? It is possible that students make errors in the construction of an analogical problem. If the problem students construct is not a correct analogous problem, they would fail to solve it correctly; it is because they are supposed construct a problem that is more advanced than their current knowledge. If they construct a correct analogous but very difficult problem at the initial period of the problem construction stage, they still could fail to solve it; this is because the analogy between the two problems could be very distant and they could lack sufficient knowledge to apply to solve the advanced problem. If students fail to solve the advanced analogous problem, they can be presented or prompted to find a simple analogous problem that could be useful in the solution of the advanced analogous problem. When students are solving the advanced analogous problem it is important to keep track of where they are, how the solution process is going and of whether the use of analogy is successful.

Reflection

The purpose of this stage is to learn from experience for further development. At this stage, students evaluate problem solving procedures they carry out from the first stage to the fifth stage and the experience they acquire during these stages and they reflect upon thinking. The teacher should ask the following question to start students to think about their experience and what they learn during the stages of the SPS: what have you learned while solving problems? The teacher should encourage students to reflect on the whole process of problem solving, and then should ask the following questions: How does an analogy work in solving problems? How do you use analogies to develop novel problems? How can you be selective while solving problems? That students understand the value of analogical and selective thinking in creative problem solving and the ways SPS helps them to solve problems creatively is an important learning outcome for them (Sak, 2011).

Rationale and Purpose of the Study

Because the SPS is a new model, research studies on its effectiveness and validity have currently been undertaken. An experimental study was carried out to investigate the effectiveness of the SPS on sixth, seventh and eighth-grade students' achievement in mathematics (Sak & Duman, 2012). One experimental classroom and one control classroom from each grade participated in the study. SPS lessons were used in the experimental classrooms for eight weeks whereas the control classrooms received teacher-directed instruction. After eight weeks of instruction, experimental and control groups' gain scores were compared. Preliminary findings revealed that gains scores of the seventh and eighth-grade experimental classrooms were much higher than the control classrooms. The difference between the gain scores of the eight graders was found to be significant, but not the seventh graders'. The experimental sixth-grade classroom had slightly a higher gain score than the control classroom. Preliminary findings show a partial support for the effectiveness of the SPS and suggest that a longer period of instruction than eight weeks might be necessary to obtain larger effects.

The purpose of the current study was to investigate student acceptability as an indicator of the social validity of the SPS model. The social validity assessment refers to the evaluation of the acceptability or viability of a programmed intervention and its use has become widespread in the behavioral literature (Schwartz & Baer, 1991). Particularly important for a new teaching practice developed to change behavior is its acceptability by students; this is because nonacceptance could lead student rejection of the practice (Wolf, 1978). The degree to which students hold positive perceptions about a teaching model and believe that it is effective in enhancing their skills can have a powerful influence on their involvement in learning activities; thus, the assessment of student satisfaction with the interventional model becomes essential. The assessment of student satisfaction with a teaching model is a type of social validation that can be accomplished by asking students to offer their perceptions of the model. In the social validity research, three types of questions usually are asked to consumers to evaluate interventions: goals, procedures and effects (Wolf). In fact, most social validity studies include questions related to effects of interventions (Hurley, 2012). In the current study, the measurement included questions about effects of the SPS teaching.

METHOD

Participants

Participants of the study included 210 students who were attending 6th and 7th grades in a major city in the mid part of Turkey. Of the total population, 125 students were 6th graders (43.2% female; 56.8% male) and 110 students were 7th graders (50% female; 50% male). Participants came from two different schools. Of the total sample, 103 students were from School A (6th grade = 44; 7th grade = 59) and 107 were from School B (6th grade = 56; 7th grade = 51). Schools and classrooms were selected based on availability and volunteer participation. Two sixth-grade and two seventh-grade classrooms from each school participated in the study. In total, research was carried out in eight classrooms.

Procedure

Instruction was carried out in students' regular mathematics classes. A four hour mathematics lesson plan consisting of two independent lessons was designed for each grade level using the SPS model. That is, two SPS lesson plans were used in each grade. Each lesson lasted 90 minutes with a break after 45 minutes. Only one lesson was taught in a day for all the classrooms. The two-lesson mathematics instruction was completed in two days for each classroom. The first lesson for the 6th grade was focused on sets and the second lesson on the greatest and the least common factors. The first lesson for the 7th grade was equations with one unknown and the second lesson was angles and arcs in circles (see Appendix for an abbreviated example of the SPS problem solving process). The objectives of the lessons were aligned with the national mathematics curriculum for 6th and 7th grades. All of the lessons were taught and guided by the first author of this study using the standard problem solving steps of the SPS presented in the SPS discussion form. The mathematics teachers of the classrooms were not present in the classrooms during the instructions.

Instrument

A pool of questionnaire items was developed by the researchers according to the purpose of the SPS and the current study to collect students' evaluative feedback about the use of the SPS. The questionnaire items were examined by two external reviewers for their content validity and appropriateness for the purpose of the study. One of the reviewers had experience in the SPS and the other reviewer was an expert in social validity research. After reviews, the number of the questionnaire items was dropped to 20 (see table 2). The items were related to students' evaluative perceptions about the effects of the SPS on their creative thinking and problem solving skills and self-confidence, learning and liking and interest in mathematics during the SPS instructions. Items were rated by the students using a 4-point Likert-type scale. Each item was assigned a score ranging from 0 to 3 (0 = do not agree; 1 = somewhat agree; 2 = mostly agree; 3 = strongly agree). A reliability analysis was run using data obtained from 210 participants of the study. Cronbach's alpha coefficient for internal consistency was found to be .91. Correlations between the items and the total score ranged from .55 to .69 and the correlations among the items ranged from .14 to .50. All of the correlations were significant (p < .01).

Data Collection

The questionnaire was administered to the students in the last 15 minutes of their last SPS lessons. All of the students had the same instruction to complete the questionnaire.

Data Analysis

Data analysis was conducted item by item using one-sample t-test. The test value (criterion) was set at 2.00 (mostly agreement in the questionnaire). Because we conducted multiple tests of significance (20 tests), we set a more strict alpha level (.0025) using the Bonferonni adjustment procedure rather than the traditional level of .05. A two-way between-groups analysis of variance was conducted to examine main and interaction effects of gender and grade on the total score obtained from the questionnaire.

RESULTS

Table 3 shows descriptive findings, as well as effects sizes and one-sample t-test results. All of the means were higher than 2.00 (t-test value). That is, students' overall agreements with the questionnaire items were above "mostly agreement." Item means ranged from 2.32 to 2.56 with an overall mean of 2.44, indicating a high level of student satisfaction about the effectiveness of the SPS in teaching and promoting their problem solving and creativity skills and self-confidence and liking in mathematics.

As the means were found to be higher than the test value, one-sample t-test analysis was conducted to test the significance of differences between the means and the test value. The analysis showed that all of the differences were statistically significant (p < 0.001). Follow up effect size analysis using the Cohen's d yielded moderate to large effects ranging from 0.33 to 0.80. The items 2, 3, 6, 7 and 16 had the highest effect sizes, whereas the items 4 and 17 had the lowest effects sizes.

A further analysis was carried out using two-way ANOVA to examine main and interaction effects, if any, of gender and grade on the total score. Levene's Test of Equality for Error Variance showed that the variance of the dependent variable across groups was not equal. Therefore, we set a more stringent significance level (.01) rather than the traditional one (.05). The results showed that there was a statistically significant main effect for grade [F(1, 206) = 12.28, p < .001) favoring the 6th grade, but not for gender [F(1, 206) = 4.39, p = .037). The interaction effect did not reach statistical significance. Effect sizes (Cohen's d) were small to moderate (.02 for gender and .05 for grade). Initially, we conjectured that perceptions of students about the effectiveness of the SPS did not differ by gender and grade. However, our conjecture was partially supported by the results as grade was found to have a significant main effect on the dependent variable.

DISCUSSION

In the current study, student acceptability as the social validity of the Selective Problem Solving (SPS) model was investigated in solving mathematical problems. The SPS is a new model to teach creative problem solving and to enhance creative capacity. Research findings show that the SPS model has high acceptability by students, an important finding for the social validity of the SPS. For example, students reported that they wanted the SPS to be used more frequently in mathematics instructions; they got better motivated when the SPS was used in problem solving and it helped them to make connections between their prior knowledge and new information.

The study yielded unexpected findings as well as expected findings. All of the means for the 20 items in the questionnaire were rated above the criterion (2: mostly agreement). We set such a criterion because we considered any mean rating that was under this criterion (mostly agree) would not be sufficient to support the social validity of the SPS. Besides high means, effect sizes also were found to range from low-medium to high. The items 2, 7, 8 and 20 related to liking the SPS (e.g., I like the SPS to be used more frequently in mathematics classes) and the items such as 3, 16 and 20 (e.g., I learned something new whenever the SPS was used in problem solving) received the highest effect sizes. Moreover, the item 4 had the lowest effect size compared to the other items; albeit, the rating still was above our criterion. This item asks students whether they can learn more than one topic at the same time when the SPS is used in solving problems. This finding seems to be unexpected initially, but, a detailed analysis of analogical problems used in the lessons shows that analogies between source problems and target problems were from similar areas of mathematics. If remote analogies, such as from remote areas of mathematics or from different disciplines, were used, students would be exposed to very distant topics and construct analogies between distant problems during the lessons. As a result, students could learn more than one topic during a problem solving process of the SPS.

Based on the findings briefly pointed out above, we can conclude that students had high satisfaction in terms of the use of the SPS in solving mathematical problems. However, one should note that in some studies, participants might report favorable responses if their responses bring about positive or negative results for them or if the questionnaire or the self-report includes sensitive items, such as pleasing the researcher or the instructor; thus, social desirability bias might contribute to research findings to some degree (Fisher, 1993). In the current study, participants' responses were not evaluated to make decisions about them; but, they might have wanted to please the instructor, and therefore, they might have overrated the questionnaire items. This limitation of the study should be taken into account while interpreting the results.

We did not expect any significant difference between the ratings by grade; however, sixth graders' overall ratings were higher than seventh graders' ratings. Although not very similar to this study, in previous research, eight and seventh graders did benefit more from the SPS instruction than did sixth graders (Sak & Duman, 2012). Among other reasons, that because we used different SPS lessons for sixth grade and seventh grade can be an important reason for the difference in results by grade. Particularly important was the difficulty of the lessons. Seventh graders' lessons (equations and arcs in circles) were much harder than sixth graders' lessons (sets and common factors). Furthermore, we can speculate that sixth graders might have benefited more from the instructions than seventh graders did, and therefore, sixth graders might have rated the questionnaire items higher than did seventh graders. This speculation can be validated by future research.

A few points as recommendations for future research should be noted. As the SPS is a new creative problem solving model, more research is needed to draw strong conclusions about its effectiveness on creative thinking and problem solving. First, a priority should be given to experimental studies involving experimental and control groups. In fact, a social validity research also can be included in this kind of experimental studies in which students' satisfaction can be compared based on their ability and achievement. Second, problem solving and creative ability of students who participate in experimental studies should be measured objectively. Measurement also should include skills such as the use of analogy, problem construction and selective thinking that are components of the SPS. Third, future studies also should focus on investigations on its effectiveness to improve creative thinking in sciences, as well as in mathematics; because, as noted earlier, selection and analogy play important roles in the sciences as well as in mathematics (Dunbar, 1999; John-Steiner, 1997; Sawyer, 2006; Weisberg, 2006). Lastly, the capacity of the SPS to improve creative ability also should be investigated with different populations; because the SPS requires very complex thinking and advanced abstract reasoning, such as the use of analogy and selective comparison in problem solving, students of different grade levels might benefit differently from SPS instructions.

In conclusion, selective and analogical thinking, two fundamental characteristics of the Selective Problem Solving model, have been indicated as the primary sources of inventions and discoveries in many disciplines (Byers, 2007; Dunbar, 1995; Feist, 2006; Holyoak & Thagard, 1995; Sawyer, 2006; Weisberg, 2006). Also analogical transfer is important for students to make their own discoveries because it helps students to build an understanding of new information, make connections between distant problems and build advanced analogous problems, and to progress from routine problem solving to advanced problem solving. Particularly in mathematics, analogical thinking enables the mapping of a learned problem to a new problem resulting in transfer of skills for solving novel problems because many problems in mathematics share similar structural features (Ngu & Yeung, 2012). Therefore, the use of the SPS in problem solving is expected to tap on higher-order thinking skills by leading students from simple comparative thinking to advance analogical thinking and from trial-error to selectiveness in problem solving (Sak, 2011).

Bilge Bal-Sezerel and Ugur Sak

Anadolu University, Turkey

Correspondence concerning this article should be addressed to Ugur Sak, Prof., & Chair, Gifted Education Division, Faculty of Education, Anadolu University, 26470 Eskisehir, Turkey. E-mail: usak@anadolu.edu.tr

This study was partly supported by a research grant from the Anadolu University Scientific Research Projects (Grant #1102E037).

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APPENDIX

An Abbreviated Example of the SPS Problem Solving Process in Sets in Mathematics

Stage I. Define the Target Problem

Target problem

* If A [intersection] B={a,c,e,f} and A [intersection] C={a,g,e}.

then what are the elements of the

set A [intersection] (B [union] C)?

Stage 2. Construct/Identify the Source Problem

* Construct a source problem.

* If a source problem cannot be constructed, choose the correct source problem among the given two problems below:

** Problem 1. If axb=270 and axc-290, then what is the solution of ax(b+c)?

** Problem 2: If a+b=T4 and c+d=17, then what is the solution of (a + c)+(b + d)?

* Solve the correct source problem (Problem 1). Use distributive property of multiplication as analogy: ax(b + c)=(axb) + (arc) =270+290 =560

Stage 3. Solve the Target Problem

* Map the solution of the source problem onto the solution of the target problem.

* Solve the target problem. Use distributive property of arithmetic as analogy:

A [intersection] (B [union] C) = (A [intersection] B) [union] (A [intersection] C)

= {a,c,e,f} [union] {a,g,c}

= {a,c,e,f,g}

Stage 4. Construct an Original Problem

* Construct an analogous problem to the target problem but more advanced than it. If an original problem cannot be constructed, solve the original problem given below:

If A={x: x [greater than or equal to] 120, x=3n, n [member of ] [Z.sup.+]}

B={x: x [greater than or equal to] 100, x=5n, n [member of ] [Z.sup.+]}

Stage 5. Solve the Original Problem

* Map the solution of target problem onto the solution of the original problem.

* Use distributive property in sets to solve the original problem:

A [intersection] (B [union] C) = (A [intersection] B) [union] (A [intersection] C)

={15,30,45,60,75,90} [union] (42,63,84,105}

={15,30,42,60,63,75,84,90,105}

* Check solution steps.

Stage 6: Reflection

Evaluate how analogy and selection work in solving new problems.

Table 1 SPS Discussion form: SPS Problem Solving Steps, Student Behaviors/ involved in Each Step And Student and Teacher Roles. Step Student Behaviors 1. Definition Define the problem. of the Target Problem Identify known and unknown information. Identify information required for the solution. Identify information irrelevant for the solution. 2. Identification Identify a similar 1. Identify a of the Source problem. source problem Problem Compare problems. Infer relations 2. Select a between problems source problem and their components. Recognize relationships between relations. Select the correct analogous problem. 3. Solution Apply knowledge in different of the Target problem situations. Problem Analyze solution steps. Develop similar problems. Compare problems. Infer relations between problems. Recognize relationships between 4. Construction Apply knowledge in different of an Original problem situations. Problem Analyze solution steps. 5. Solution of Explain analogous problem solving. the Original Problem 6. Reflection Explain selective problem solving. Step Focus Questions Student Role 1. Definition What is the problem? Separate various of the Target parts of a problem Problem What is known? What is unknown? What information is required to solve this problem? What information is not required to solve this problem? How is the condition sufficient to solve the problem? 2. Identification Have you seen this Find an analogous of the Source problem before? problem. Problem Have you seen the Solve the analogous same problem in a problem. different form? Do you know a similar problem? Can you solve this problem you just have found? Which one of the two Select the correct problems similar to analogous problem. the target problem? How are they Solve the analogous similar? Which one of the two problem. problems can be used in the solution of the target problem? Can you solve this problem you have selected? 3. Solution How could you use Solve the target of the Target the solution method problem. Problem of the analogous problem in the Check solution solution of the steps. target problem? Can you prove that each step is correct? 4. Construction What are other more Construct an of an Original advanced problems advanced analogous Problem than the target problem. problem you can solve using the strategies and methods How used to solve the target problem? What is the problem? How are these two problems similar in terms of their solutions? How is this new prohlem more advanced than the target problem? 5. Solution of How could you use Solve the advanced the Original the solution method problem. Problem of the target problem in the solution of the new problem? Can you prove that Check solution each step is steps. correct? 6. Reflection What have you Share problem learned while solving experience. solving problems? How does an analogy work in solving problems? How do you use analogies to develop novel problems? How can you be selective while solving problems? Step Teacher Role 1. Definition Present a target problem of the Target List data. Problem Introduce notations. Draw a figure if needed. 2. Identification Elicit students' prior of the Source knowledge. Problem Present two problems if needed. Monitor problem solving process. 3. Solution Monitor problem solving of the Target process. Problem 4. Construction Present a problem if of an Original needed. Problem 5. Solution of Monitor problem solving the Original process. Problem 6. Reflection Encourage students' expressions. Table 2 Questionnaire Items Statements 1. I believe that I can solve many problems using the SPS. 2 I think that I get better motivated in classes when the SPS is used in solving problems. 3. I understood after I used the SPS that problem construction is at least as important as problem solving. 4 I can learn more than one topic at the same time when the SPS is used in solving problems. 5. I got more interested in mathematical problems after I solved problems using the SPS. 6. After I used the SPS in problem solving, I understood the importance of separating out relevant information from irrelevant information for the solution of problems. 7. I like the SPS to be used more frequently in mathematics classes. 8. I like solving problems using the SPS. 9. I believe that I can solve problems using the SPS that I could not solve previously. 10. Solving problems using the SPS increases my self-confidence in mathematics. 11. I think that the SPS teaches an unusual method to solve problems. 12. I think that I learn topics better when problems are solved using the SPS. 13. Solving mathematical problems using the SPS is interesting to me. 14. I think that constructing analogies between problems while solving them step by step makes problem solving easier. 15. Seeing that difficult problems can be solved using the SPS as easy as solving simple problems reduces my fear against difficult problems. 16. I think that the SPS helps me make connections between my prior knowledge and new information. 17. I think that I explore new things when I use the SPS in problem construction. 18. I think that what I learn becomes sustainable when the SPS is used in solving problems. 19. I think that I can solve problems more creatively using the SPS. 20. I learned something new whenever the SPS was used in problem solving. Table 3 Means, Standard Deviations and One-Sample t-test Results Std. Error Mean Item Mean SD of Mean t Df Difference d * 1 2.47 .71 .04 9.48 ** 209 .47 .66 2 2.52 .67 .04 11.40 ** 209 .52 .77 3 2.55 .68 .04 11.80 ** 209 .55 .80 4 2.32 .83 .05 5.69 ** 209 .32 .38 5 2.44 .80 .05 8.10 ** 209 .44 .55 6 2.51 .71 .04 10.54 ** 209 .51 .71 7 2.56 .70 .04 11.66 ** 209 .56 .80 8 2.50 .80 .05 9.20 ** 209 .50 .62 9 2.44 .76 .05 8.49 ** 209 .44 .57 10 2.43 .73 .05 8.68 ** 209 .43 .58 11 2.34 .77 .05 6.41 ** 209 .34 .44 12 2.47 .73 .05 9.31 ** 209 .47 .64 13 2.37 .79 .05 6.82 ** 209 .37 .46 14 2.49 .79 .05 9.08 ** 209 .49 .62 15 2.40 .76 .05 7.65 ** 209 .40 .52 16 2.48 .68 .04 10.25 ** 209 .48 .70 17 2.32 .83 .05 5.65 ** 209 .32 .38 18 2.40 .79 .05 7.36 ** 209 .40 .50 19 2.41 .77 .05 7.70 ** 209 .41 .53 20 2.50 .79 .05 9.17 ** 209 .50 .63 Note. d * Cohen's d effect size; ** p < .001 (2-tailed)