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PROBLEM SOLVING AND CREATIVITY: MULTIPLE SOLUTION METHODS IN A CROSSCULTURAL STUDY IN MIDDLE LEVEL MATHEMATICS BY KATHERYN KIRK FOUCHE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1993 ACKNOWLEDGMENTS The success of my graduate years was due to the support of many people. Thanks in great part to my committee, my experiences were positive and productive. It was an honor to have been taught by the masters. Dr. Mary Grace Kantowski, my major advisor, mentor, and friend, shared her expertise, resources, and art. She challenged me to become a better problem solver, teacher, and researcher. While department chair, Dr. Eugene Todd recruited me and provided encouragement. Dr. Elroy Bolduc always had time to lend advice and a sympathetic ear. Dr. Nelson contributed support from the Department of Mathematics "across the street," and Dr. David Miller always found time to design the next phase of the study. The support and love of God and my family continues to be my source of energy and inspiration. Thanks to Kevin and David who may deservedly claim part of this degree in exchange for their good nature, flexibility, and understanding. To the rest of my family, Mother, Bob, Barbara, Chuck, Lindy, Chad, Joe, Betsy, Megan, Katie, and, Lindy, thanks for believing in me and for your encouragement. Thanks to my friends in Japan, Professor Toshiakira Fujii and Professor Tadao Ishida, who arranged for the JapanU.S. comparisons and to Christina Carter for her hours spent coding the Japanese data. Thanks to the panel of experts who ranked the problem solving methods, Dr. Carolyn Ehr of Fort Hays State University, Dr. Peggy House of the University of Minnesota, Dr. Mary Grace Kantowski of the University of Florida, Dr. William Moulds of Towson State University and Dr. James Wilson of the University of Georgia. Thanks to Dr. Jerry Becker of Southern Illinois University for his help and resources. Thanks to Alisan Hardman who graded each of the Torrance Tests of Creativity and to Sally Scudder for helping me code the original marble problem. Thanks to my friends, Cindy King, Donna Otzel, Luke Reckamp, Joan Donnelly, and their students at P. K. Yonge, Vanguard High, and Osceola Middle for welcoming me into their classrooms, and thanks to the teachers and students at Gainesville High, Ft. Clarke Middle, Ft. King Middle, and Westwood Middle who shared their solution methods. A special thanks is reserved for Dr. John Zbikowski who coached, edited, and consoled. Thanks to Stephanie Robinson and Donna Otzel, my friends and fellow graduate students, to Susan Starks who stayed with me until the wee hours during that final copying session, and to the Norman Hall gang. This dissertation is dedicated to my dad, Joseph L. Kirk, Jr. He would have been so proud. TABLE OF CONTENTS ACKNOW LEDGM ENTS............................................................................................ iia ABSTRACT .............................................................................. .................................. vi CHAPTERS I INIRODUCTION......................................................... .......................... 1 Purpose of Study...................................................... ............................. 2 Rationale ........................................................ .......................................... 4 Research Questions ............................................................................. 6 Definition of Key Term s ....................................................................... 7 Summ ary..................................................................................................... 8 II REVIEW OF RELATED LITERATURE................................................ 10 Problem Solving....................................................................................... 10 Summary of Related Problem Solving Research................. 11 Constructivist Theory................................................................... 12 Teaching and Learning from a Constructivist Perspective ................................................... .............................. 20 Creativity.................................................................... .............................. 24 M them atics Education in Japan......................................................... 30 Sum mary......................................................... ......................................... 34 III PROCEDURES ....................................................... ................................ 36 Hypotheses........................................................... .................................. 36 The Main Study....................................................................................... 39 The Subjects .................................................................................... 39 Instrum entation for the M ain Study........................................ 40 The M arble Problem ..................................................................... 40 Rationale for Reexamination of the Marble Problem.......... 44 Adm inistration of Instrum ents................................................. 46 Group Assignm ent ..................................................................... 47 Treatm ent Procedures................................................................. 48 Analysis of Creativity Data for American Students............ 51 Weight Assignment for Solution Methods............................. 52 The U.S.Japan Comparisons.............................................................. 53 Summary................................................................................................... 55 IV DATA ANALYSIS AND RESULTS OF THE STUDY...................... 56 Review of the Design and Data Collection Procedures.............. 56 Results of Hypotheses Testing............................................................ 59 Number of Solution Methods .................................................... 59 Complexity of Solution Methods.............................................. 63 Pretest/Posttest Comparisons .................................................. 66 U.S.Japan Comparisons.............................................................. 69 Other Results............................................................................................ 71 Summary........................................ ........................................................ 73 V CONCLUSIONS AND IMPLICATIONS.............................................. 77 Overview of Study.................................................... ............................. 77 Discussion of Results.......................................................................... 80 Implications .............................................................................................. 85 Suggestions for Future Research ...................................................... 88 Limitations ................................................................................................. 90 Conclusions................................................................................................ 92 LIST OF REFERENCES..................................................................................... 94 APPENDICES A PROBLEMS FOR TREATMENT .......................................................... 102 B THE MARBLE PROBLEM ................................................................... 106 C PANEL TO DETERMINE COMPLEXITY WEIGHTS......................113 D DESCRIPTION OF SOLUTION METHODS....................................1... 19 E PARENTAL CONSENT FORMS ........................................................... 122 F THE PILOT MARBLE PROBLEM ........................................................ 135 BIOGRAPHICAL SKETCH .................................................................................... 140 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PROBLEM SOLVING AND CREATIVITY: MULTIPLE SOLUTION METHODS IN A CROSSCULTURAL STUDY IN MIDDLE LEVEL MATHEMATICS Katheryn Kirk Fouche August, 1993 Chairman: Eleanore L. Kantowski Major Department: Instruction and Curriculum The purpose of this study was to investigate the solution methods employed by middle level students engaged in solving nonroutine mathematical problems and the role of creativity during the reflection process. The solution methods of students who participated in problem solving sessions designed to elicit multiple solution methods for nonroutine mathematical problems were compared to methods employed by students who received no such deliberate practice. The solution methods were compared by number of different methods and attained level of complexity. Complexity was determined by the level of understanding indicated by the solution method and its generalizability. Additionally, this study included an international perspective. The solution methods of students in Japan were compared to those of American students. Comparisons of both the number of alternative methods employed and the attained level of complexity were made between nations and between grades. The major findings from the study suggest that both the number of solution methods students were able to generate for a single nonroutine mathematical problem and the level of complexity of those methods were significantly increased when students were given the opportunity to engage in reflective problem solving activities. Also, the initial solution method tended to be unsophisticated with the level of complexity increasing with each new method generated. The number of solution methods students were able to generate for a single nonroutine mathematical problem was positively related to their measure of creativity, but there was no significant relationship between attained level of complexity and measure of creativity. For the U.S.Japan comparisons, American eighth grade students were able to produce a greater number of alternative methods than either the Japanese sixth or eighth grade students or the American sixth grade sample. Although the American eighth grade students were able to generate more alternative solution methods than their Japanese counterparts, there were no differences between attained level of complexity. CHAPTER I INTRODUCTION In 1989 the National Council of Teachers of Mathematics (NCTM) published its Curriculum and Evaluation Standards for School Mathematics, which outlined recommendations for the next phase of moving mathematics into the Information Age of the 21st century. The goals for all students, as expressed in the Standards, include providing opportunities to "become mathematically literate. This term denotes an individual's ability to explore, to conjecture, and to reason logically, as well as to use a variety of mathematical methods effectively to solve problems" (p. 6). In 1991 NCTM's Professional Standards for Teaching Mathematics recommended several ways to promote problem solving. The ideal mathematics classroom community is described as one in which students rely on mathematical reasoning, not just memorization; solve problems, not just mechanically find answers; and connect mathematics instead of treating mathematics as a body of isolated concepts and procedures. The challenges presented in the curriculum Standards and the teaching Standards eloquently blend the goals of an international host of visionaries involved in mathematics education. Although a seemingly novel idea for teachers who persist in using the lecturepractice method, the idea of encouraging multiple solution methods for a single problem is not a recent development. In 1942 Brownell suggested that to be most fruitful, practice in problem solving should not consist of repeated experiences in solving the same problems with the same techniques, but should consist of the solution of different problems by the same techniques and the application of different techniques to the same problems. (p. 439) Teachers are still being encouraged to model and discuss with their students a variety of strategies that can be used to solve a single problem (NCTM, 1991). Providing an opportunity to reflect on a problem, after attempting a solution, in order to find additional methods is part of the looking back phase of problem solving described by Polya (1957). Although the practice of finding multiple solution methods for a single problem has been suggested as a technique for teaching problem solving, as a method of alternative assessment, and as a way to foster development of the creativity that is a component of problem solving (House et al., 1983; NCTM, 1989, 1991; Thompson, 1991), there is little evidence to suggest that the looking back phase of problem solving is a regular procedure in the American mathematics classroom (Stigler & Perry, 1988). When students are required to look for alternative solution methods for a single problem, what characterizes their methods? Is there a difference in the methods employed by students who have practiced reflecting during problem solving and those who have not? What characteristics of the problem solver are correlated with the complexity and number of solution methods a student is able to find? Purpose of Study The purpose of this study was to investigate solution methods employed by students in problem solving and the role of creativity during the looking back phase of problem solving. Multiple solution methods for a single nonroutine mathematical problem were compared by number of different solution methods and level of complexity. These comparisons involved students who participated in a treatment designed to practice searching for multiple solution methods for a single problem and those students who received no such deliberate practice. Additionally, an international comparison was conducted in which the solution methods of Japanese students were compared to those of U.S. students. Teaching students to reflect on the problem solving process is a routine part of the Japanese elementary mathematics classroom (Azuma & Hess, 1991; Becker et al., 1990; Nagasaki & Becker, 1993; Stigler & Stevenson, 1991). Reportedly, students in Japan often spend an entire classroom period on a single problem, while American teachers too often strive to complete as many problems as possible. Since this study will document underlying mechanisms of problem solving as well as differences in achievement such as level of complexity represented in the method, crosscultural comparative research was especially useful for this research agenda. Both Standards suggest that students be challenged by tasks that promote divergent yet sound mathematical thinking: tasks that allow creative ideas to flourish. The possibility of a correlation between creativity in problem solving and solution methods was explored for the United States sample. Creativity was defined in terms of fluency, originality, abstractness, elaboration, and resistance to closure, categories in the Torrance Test of Creative Thinking (Torrance et al., 1992). Rationale In her address at the New Orleans NCTM Annual Conference (April, 1991), Iris Carl, NCTM President, described the 1990s as the most exciting time in history for mathematics education. The challenges outlined by NCTM in Problem Solving in School Mathematics (1980) and The Agenda in Action (1983) have moved into the next phase of reform. According to Carl, the curriculum Standards and the teaching Standards outline a more extensive educational reform than ever suggested in history. The major thrust of the reform movement as defined by both Standards involves fostering the students' ability to think, reason, and communicate mathematically in order to facilitate a deeper understanding. With the implementation of the current reform movement still in its early stages, there exists a need for research that adds to the body of knowledge concerning how students solve problems. Silver (1985) outlined the strides in the area of mathematical problem solving performance to that date and summarized suggestions for the research agenda of this decade. One of the most prevalent themes in the document involves the need to view problem solving and problem solving activity within a broader context than traditionally found in schools. This broader context would provide experiences for students "to do mathematics rather than having it done to them" (p. 276). That is, students need to engage in generative mathematical inquiry and activity so that mathematics might be more stimulating. Additionally, there exists a need to research some specific areas of problem solving. The establishment of a positive correlation between some measure of creativity and the ability to generate alternative solution methods for a mathematical problem would provide impetus to the theory that teaching should tap and develop the creative potential of every learner. Examining methods of solution for a nonroutine mathematical problem and the outcomes when alternative methods are sought involves problem solving assessment. Silver (1985) insists that there is overwhelming agreement concerning the need to investigate the lesserknown area of assessing problem solving. Focusing on the process of problem solving, as in this study, shifts the evaluation emphasis from the product to the process used to obtain the solution. Silver and Kilpatrick (1985) note that although the assessment of problem solving should provide testers with information needed to make instructional decisions, assessment activity has not received attention proportionate to need. After two decades of research on how students solve problems, they argue, very little research has had direct influence on problem solving assessment. Finally, this study was enriched by the unique opportunity to add a crosscultural dimension. Research involving comparisons to other cultures has been suggested as a method of improving understanding of education in one's own culture. According to Sowder (1989), American researchers are habituated to American classrooms and remain unaware of some of their most obvious aspects. Examining classrooms and educational systems of other cultures can give us a fresh set of lenses to use in viewing American classrooms. Of the few crosscultural studies that have been undertaken, the results often document differences in achievement across cultures without tracking the underlying mechanisms that may produce the differences. Crosscultural research is difficult and costly to conduct, yet it provides unique opportunities to add to our basic understanding of the teaching and learning of mathematics. (p. 35) Having visited Japanese classrooms and having assisted with the U.S. Japan Crossnational Research on Students' Problem Solving Behaviors, this researcher was afforded a rare opportunity to conduct problem solving research via an established network of communication. Research Questions This study addressed the following questions: 1. What is the relationship between the number and complexity of methods employed in the solution of a nonroutine mathematical problem and training specifically designed to elicit multiple solution methods? 2. Is there a relationship between the number and complexity of methods employed in the solution of a nonroutine mathematical problem and a student's measure of creativity? 3. When comparing the responses of Japanese students to those of American students to a nonroutine mathematical problem, is there a difference in the number or complexity of methods found? 4. When comparing the responses of Japanese students to those of American students who have spent about the same amount of time studying mathematics in school, is there a difference in the number or complexity of methods found? Definition of Key Terms Creativity is possessing the flexibility to formulate different representations for the same problem and the originality to combine elements of a problem in novel ways. Creativity measure is the mean of standardized scores for fluency, originality, elaboration, abstractness of titles, and resistance to premature closure as determined by the Torrance Test of Creative Thinking: Figural A (Torrance, 1990). Generalizability is the degree to which a solution method for a specific case of a problem solution can be extended to solve the general case. Looking back is the stage of problem solving described by Polya (1957) as a period of reflection when one might check a solution, relate the solution method to other problems, extend the problem, or look for alternative solution methods. This research concentrated on the aspect of looking back that involves the search for alternative solution methods. A method of solution is the process employed by a student to resolve a problem. A nonroutine problem is a situation for which the individual who confronts it has no algorithm that will guarantee a solution. That person's relevant knowledge must be put together in a new way to solve the problem (Kantowski, 1974). Problem solving is "(a) the process, or set of behaviors or activities that direct the search for the solution, and (b) the product, or the actual solution" (Kantowski, 1977, p. 163). Sophistication or complexity of a solution method is an indication of the level of intellect engaged in the solution of a problem and the generalizability of the method. Summary The curriculum Standards (1989) describe the ideal mathematics classroom as one where mathematically literate students deepen their understanding of mathematics by exploring, conjecturing, reasoning logically, and using a variety of mathematical methods effectively to solve problems. This research explored the nature of these important activities. Although it is impossible to isolate these activities, this research focused on one observable process associated with deeper understanding and reasoning, the process of finding alternative solution methods for a given problem. The solution methods of students who practiced the activity of searching for alternative methods were compared with the methods of students who received no such deliberate practice. In addition to the number of different solution methods a student was able to find, the complexity or sophistication of the methods was appraised. The degree of sophistication or complexity was based on a solution method's generalizability and the level of understanding represented by the method. In order to explore a possible correlation between creativity and solution methods, a measure of a student's creativity was compared to the number and complexity of solution methods employed. An international perspective was added to this study by the inclusion of the solution methods of Japanese students in the comparisons. When asked to look for another way to solve a given problem, students and even teachers often respond, why? The attitude that one correct answer is sufficient may seem logical to many. Is a student who can find four different ways to solve a problem a better problem solver than the student who can only think of one method? When solving a problem should students be encouraged to use only the most efficient method or have their choice of any method that employs sound mathematical procedures? The motivation to incorporate creative problem solving into an overcrowded curriculum must come from sound research that provides evidence that it is a necessity. How do teachers stimulate the students' creative potential, and what effect does that stimulation have on the solution methods employed during problem solving? These questions were addressed in this research project. CHAPTER II REVIEW OF RELATED LITERATURE The review of literature presented in this chapter will cover the following topics related to the present study: (a) problem solving, (b) creativity, and (c) a comparison of mathematics education in Japan and the U.S. These lines of research are related to the present study in the following manner. Problem solving has been the focus of numerous important research activities, but there exist gaps in the research that connect problem solving instruction to learning. One aspect of problem solving that is of particular interest in the present study is that of creativity and its role in the search for alternative solution methods. Finally, the investigation of alternative solution methods was enhanced in the present study by U.S.Japan comparisons. Problem Solving The purpose of this section is threefold. First, a summary of related problem solving research to date will be presented. The summary indicates that the teaching and learning of mathematical problem solving have generally been studied as separate topics. There Is a need for additional research that combines both of these classroom elements as did the present study. Secondly, the constructivist theory, which provided the framework for the present study, will be discussed. Finally, support for the treatment format implemented in the present study will be examined through a review of teaching and learning research from a constructive perspective. Summary of Related Problem Solving Research By 1980 when NCTM recommended that problem solving should be the focus of mathematics education for that decade, several important research questions had been explored. A generalized summary of problem solving research through that date by Suydam (1982) suggests that 1. Problem solving strategies or heuristics can be specifically taught, and when they are, they are used more and students achieve correct solutions more frequently. 2. There is no optimal strategy for solving problems. 3. Students should be a) faced with problems in which the approach to solving the problem is not apparent and b) encouraged to generate and test many alternative approaches. 4. Teaching children strategies for problem solving provides them with a repertoire from which they can draw as they meet the wide variety of problems that exist. 5. Students choose to employ some strategies more frequently than others, with various strategies used at different stages of the problem solving process. 6. A child's problem solving achievement is related to development level. 7. Problem solving skills are improved by incorporating them throughout the curriculum. The bulk of this research focused on the teaching of problem solving heuristics. During the 1980s researchers intensified their investigation of how students solve problems. Romberg (1992) contends that the intense controversy and reflection that took place during this period provided the impetus for the radical reform period in which we are now engaged. The focus of research, however, needs to be more unified. Although substantial knowledge has been gained, Romberg and Carpenter (1986) point out that most research on problem solving can be classified as either research on teaching or research on learning. Research on teaching has been focused on instruction and on the group as a whole with less emphasis on what Is taught, whereas research on learning has primarily been concerned with individual students and their cognitions, not with how learning occurs. Carpenter and Fennema (1991) suggest a new research paradigm "that blends what is known about students' learning, thinking, and problem solving with what is known about teachers as active, thoughtful professionals" (p. 7). Such a paradigm would make it possible to integrate research on teaching and learning by accounting for teachers' cognition, students' cognition, and the learningteaching process within a seamless framework (Cobb et al., 1991). A description of the learning and instruction research on which the present study was based follows. Constructivist Theory Most mathematics curricula of the 1950s and 1960s were based on the stimulusresponse theory (Davis, 1990). The mental processes of the students were thought to be unscientific. Therefore, the dominant teaching strategy involved showing or telling students what to do, supervising their practice, and evaluating their ability to regurgitate facts and imitate rituals. Since World War II there have been various projects and reform movements based on the notion that more effective models of teaching and learning mathematics exist. Davis contends that encouraging students to think creatively about mathematics and helping them understand what they are doing have been common goals for all of these reform movements. Although the origins of the constructivist theory is uncertain, it remains, according to Davis, central to those involved in the major reform efforts in mathematics education since World War II. Noddings (1990) has noted that although there exist conceptual differences in the constructivist theories currently influencing mathematics education, the following themes are generally accepted: 1. All knowledge is constructed. Mathematical knowledge is constructed, at least in part, through a process of reflective abstraction. 2. There exist cognitive structures that are activated in the processes of construction. These structures account for the construction; that is, they explain the result of cognitive activity in roughly the way a computer program accounts for the output of a computer. 3. Cognitive structures are under continual development. Purposive activity induces transformation of existing structures. The environment presses the organism to adapt. 4. Acknowledgment of constructivism as a cognitive position leads to the adoption of methodological constructivism. a Methodological constructivism in research develops methods of study consonant with the assumption of cognitive constructivism. b. Pedagogical constructivism suggests methods of teaching consonant with cognitive constructivism. (p. 10) About the time that the evaluation Standards were published, the National Research Council encouraged a strong mathematics program for each student in its document Everybody Counts (1989). One of its more profound statements suggests that In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. .... students learn mathematics well only when they construct their own mathematical understanding. (p. 58) In a recent review of literature, Cobb et al., (1991) concluded, 'The notion that students learn mathematics by actively reorganizing their own experiences is almost universally accepted by the mathematics research community" (p. 84). Teaching and learning theories under the umbrella of this notion have been termed constructivism (Lochhead, 1985). The idea was born long before the label existed. Dewey (1956) suggested that people cannot convey ideas from one to another. He contends that when someone shares an idea, it is to the receiver a fact. The fact only becomes an idea when the receiver wrestles with the conditions of the problem by thinking it out individually. Piaget (1973) believed that the goal of education is not to promote the memorization of readymade facts, but instead to help students build their own set of ideas through activities designed for inquiry and discovery. Richards and von Glasersfeld (1980) felt that the act of searching for alternative solution methods to a problem reflects the essence of constructivism inherent in Piaget's model of adaptation and accommodation. It is likely that few mathematics educators would argue with the notion that actively constructing knowledge is preferable to passively receiving knowledge from external sources. Von Glasersfeld (1990) extended constructivism to the extreme when he described radical constructivism. His theory of knowledge is based on the belief that one can only construct knowledge within one's own world of experiences and, therefore, has no way to know the truth that exists in the reality of the outside world. Kilpatrick (1987) warned that the radical constructivist view operates on a negative feedback or blind view toward the real world. If the only reality that one can come to know excludes the reality outside of one's self, then one can only learn about the world's constraints, what does not work. With neither knowledge nor information flowing in or out, one cannot communicate meanings, therefore knowledge is subjective. Kilpatrick argued that the radical constructivist cannot make inferences about what the student is thinking and must rely on overt responses as clues about the restraints controlling the student's internal processes. While learning from errors can be effective, students cannot be expected to respond to constant negative feedback that is precipitated by failure. As suggested by Kilpatrick, the constructivist theory adapted for this research does not put "sanitizing quotation marks" around terms such as problem solving, but instead builds upon the basic principles of the constructivist theory that are inherent in the mathematics education reform movement presently under way, Vygotsky (1978) identified the range of cognitive activities within which learning occurs as the "zone of proximal development." Vygotsky defines this zone as "the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers" (p. 86). Abilities that are in the embryonic state today are the abilities that will mature and internalize tomorrow. Vygotsky's zone of proximal development represents a helpful model for problem solving instruction. Vygotsky's theory places the teacher in the important position of decision maker and model. Activities that only tap a student's actual developmental abilities do not require reconstruction and are exercises, not problems. In order to balance motivation, self confidence and frustration, teachers must choose problems that are within each student's zone and then carefully decide when and how to model the problem solving process. Choosing good problems to challenge yet not overwhelm each student could prove to be impossible, especially for a class with diverse abilities or experiences. One remedy lies in the practice of requiring students to provide as many alternative solution methods as possible for each problem. Thus, students could begin within their actual developmental level and then explore their zone through cooperative learning or modeling by the teacher or peers. Newell and Simon (1972) described problem solving as a search through a "problem solving space" until a solution is found. The problem space is where a student internally represents the initial situation and the desired goal situation. Trial and error occurs when the search is random as opposed to a search generated from some mathematical principle. Trial and error is often the search method employed by inexperienced problem solvers. Experienced problem solvers tend to use heuristics, or more sophisticated rules for selecting search paths. It is possible, according to Newell and Simon, for one to become engaged in a mind "set." When the mind becomes fixed in a particular path, the problem solver can find no other way to "think" about the problem. One's mind could become set in an incorrect path and thus the problem would remain unsolved, or the mind set could occur when attempting an alternative solution method. When additional solution methods lie within students' levels of development, on which type of solution methods do they tend to become set? This question was explored in this research. Newell and Simon (1972) described the mental processing of most problem solving activity as serial rather than parallel steps limited by the problem solver's capacity for processing. Thus, different steps in the problem solving process compete for the solver's limited mental resources. Bruner et al. (1956) see this competition as a cause for a conflict of goals. One goal is to complete the task efficiently and the other goal is to minimize cognitive strain. The ideal solution method provides an efficient strategy with minimal cognitive strain. However, a problem solver faced with a conflict might choose a less efficient solution method in order to keep cognitive strain within acceptable bounds. A problem solver always begins with an initial representation of the problem. This initial representation may be constructed by the individual or be the result of how the problem was presented. Simon and Hayes (1976) indicated that the initial representation of the problem has a strong effect on the solution methods chosen by the solver. They suggest that the problem solver will change representations that place too heavy a burden on shortterm memory. The case may be, therefore, that a mind set could be the result of a conscious or unconscious avoidance of cognitive strain. As a result of the belief that students learn mathematics well only when they construct their own mathematical understandings, both the curriculum and evaluation Standards (1989) and the teaching Standards (1991) are filled with verbs such as explore, justify, represent, solve, construct, discuss, use, investigate, describe, develop, and predict. The student is not considered a passive receiver of information, but as an active participant both mentally and physically. This active involvement of the student during problem solving is reflected in the constructivist theory of learning and teaching. With each new experience students reformulate their individual picture of the world by adjusting and adapting to some changes while initiating others (Confrey, 1990). The depth of students' understandings relies to a great extent on their ability to defend their position. Confrey emphasizes the importance of flexibility in this process. To assist students in constructing their own understanding teachers must show interest in and value each student's response. Teachers must approach even the foreign or unexpected response with interest while probing for possible implications. When a student adopts a solution method based on an inadequate or incorrect knowledge base, the teacher must first understand the student's way of thinking before attempting to assist the student In restructuring those views. Reflection is necessary for the construction process. According to Confrey (1990), the most fundamental quality of a student's construction is reflected in the student's own belief in the construction. Constructivists contend that students must believe in order to know. Other clues exist for teachers to judge the power of students' constructions. Among the 10 qualities of a powerful construction listed by Confrey are the following: 1. A convergence among multiple forms and contexts of representations; 2. An ability to be reflected on and described; 3. A potential to act as a tool for further constructions; 4. An ability to be justified and defended. (p. 111112) Teachers in a constructivist classroom must be committed to helping students create powerful constructions. As a facilitator, teachers use three levels of questioning during problem solving sessions to develop students' reflective process (Confrey, 1990). At the first level teachers use questioning to evaluate and increase students' level of understanding of the problem. Asking students to restate the problem gives clues about their understanding. Time spent at this level can be time curtailed in developing a plan. At the second level teachers use questions to help students deepen their own understanding of the cognitive strategies. Instead of "filling in the blanks" for students who are not able to describe what they are doing, through proper questioning the teacher can request more precision. After a student understands the problem and can describe the solution method, teachers' third level of questioning should require the student to defend the solution. The rigor of questioning is based on the teacher's knowledge of the students and the complexity of the method being employed. These three levels of questioning dealing with the student's (a) interpretation of the problem, (b) cognitive strategies, and (c) justification of the strategies become the teacher's tools for developing each student's reflective process. Teaching and Learning from a Constructivist Perspective Given that each student in a classroom has a unique combination of mathematical experiences and abilities, how does the teacher provide opportunities for each student to individually create knowledge and understanding? Carpenter and Fennema (1991) suggest that teaching is problem solving and that both the teacher and each student enter the classroom with a unique knowledge base. Therefore, instead of following an outline of "prescribed" procedures, it is up to the teacher to make informed decisions during the instructionthinkinglearning process. To cultivate good problem solvers, teachers must challenge students with carefully chosen activities that encourage students to create and test new rules, to reconstruct their own mental schemas. Summarizing research on teaching mathematics for understanding and on students' understanding of mathematics, Lampert (1991) has suggested that two of the teacher's most important roles involve (a) choosing and posing good problems and (b) developing tools for communication between teachers and learners. A good problem, according to Lampert, is one that creates a learning environment conducive to students expressing their mathematical thinking. Given that each student enters the classroom with a unique combination of mathematical knowledge and experiences, a good problem is one for which a range of alternative solution methods exist, at least one of which lies within each student's ability to solve but also admits of solution methods that extend and deepen the student's understanding. Good problems allow for communication between students and teachers and thus establish communication tools that can be used to connect a student's familiar ways of solving problems with other less familiar methods. Students who prepare a discourse of defense for their own solutions exhibit more mathematical understanding than those who accept mathematical principles by virtue of the teacher's authority. Finding multiple solution methods for good nonroutine problems has the potential of placing students in control of their mathematical thinking and focuses attention on the process rather than the product. Knowledge of the mathematical assumptions that underlie each strategy employed by students becomes an integral part of the teacher's evaluation and decision making. The present study focused on encouraging students to search for additional solution methods for nonroutine mathematical problems. If students are to take charge of their learning through active involvement, how should problem solving sessions be structured in order to maximize results? Kantowski (1980) suggested that a teacher's role in problem solving instruction should be based on the student's developmental level. For a novice problem solver, the teacher assumes the role of a model. As a student progresses in his or her understanding and problem solving skills, the teacher becomes more of a crutch, then a problem provider, and finally, a facilitator. At each level it is the responsibility of the teacher to choose good problems or evaluate those posed by the students. It is assumed that spending time on reflection about a problem solution method allows a student the opportunity to gain a deeper understanding of the problem. As a result of an increased level of understanding can the teacher expect to see evidence of a higher level of sophistication in the solution methods? Vygotsky (1978) suggested that the zone where learning takes place is bounded by what one can master alone and what one can accomplish with the assistance of an adult or more capable peer. While grouping students for cooperative learning has been promoted as one method for helping students become actively involved in their own learning, the debate about how students should be grouped is ongoing. Tudge (1990) conducted a study to determine the effects of various grouping arrangements on students' solution methods to a nonroutine problem. He was particularly interested in investigating what happened to the more capable or equally capable peer when students were grouped for cooperative learning. Does the zone of proximal learning allow for regression as well as advancement? Each student in the study was given an individual pretest to establish which of the six methods of solution each subject would employ. The methods were assigned a level of sophistication according to the complexity of the method. Acting on the advice that students should be paired with peers who are within their own zone of proximal development (Mugny & Doise, 1978), Tudge assigned each subject to a treatment group identified as follows: (a) those who continued to work individually, (b) those who were paired with a student who had chosen a solution method at an equal level of sophistication, (c) those who were paired with a student working at a level of sophistication one or two below, or (d) those who were paired with a student working at a level of sophistication one or two above. Students were asked to predict the results of several scenarios involving weights being placed on a balance scale. Students arrived at their predictions through discussion with their partner without the benefit of testing their conjectures. The results indicated regression for over onethird of the subjects whose partners used the same method and almost half of those whose partners had used a less sophisticated method. Additionally, the students who regressed to a lower method than the one used on the pretest continued to use that less sophisticated method in subsequent individual testing. Only subjects who worked with more capable peers benefited from the cooperative learning problem solving activity. Tudge (1990) conducted a followup study to determine what factors are most critical in determining whether students adopt a method of a higher or a lower level of sophistication. He discovered that a student's degree of confidence was very influential during interaction. Methods of less sophistication could often be used more confidently than more complex rules, therefore students often abandoned methods that were generated through a deeper understanding of the problem. In subsequent research Tudge (1990) repeated his former study with the addition of informative feedback. In this study students were encouraged to test their conjectures using the weights and balance scale. The results of this study highlighted the importance of feedback for each student in the group. Feedback not only benefited students in the selection of a method identified as more sophisticated, it overshadowed the effects of discussion with a partner. Dewey (1956) emphasizes the importance of time for students to interpret, infer, share observations, and reflect. He sees this sharing time as "the social clearinghouse, where experiences and ideas are exchanged and subjected to criticism, where misconceptions are corrected, and new lines of thought and inquiry are set up" (p. 55). Dewey cautioned that this sharing time should not be a time when a student is forced to say something, but instead a time offered for the student who has something to say. For the present study feedback was offered not only during cooperative learning time, but also each session concluded with a time for students to share solution methods with the class. Problem solving research should blend research on student cognition and instructional methods of problem solving. The present study focused on not only student cognitions as represented by their solution methods but also the instructional process of asking students to search for alternative methods after having solved the problem. Creativity The notion of creativity encompasses a vast array of definitions and theories. Creativity can be thought of in terms of individuals who possess certain talents, or in terms of the actual resulting product. Although differences exist in interpretations, all of the definitions have in common the notion of originality. Beyer (1987) separated critical thinking from creative thinking. He claims that critical thinking is a skill involving examining and breaking down reality in order to understand a situation; whereas creative thinking is a talent that requires the combining of elements of reality in novel ways to formulate new understandings. Parker (1963) defined creativity as the "art of seeking out, trying out and combining knowledge in new ways" (p. 170). Creativity has been identified as occasions when one combines previously unrelated structures in such a way that the whole is greater than its parts (Koestler, 1964; Parnes et al., 1977) In their review of creativity, Mumford and Gustafson (1988) concentrated on the product of creativity by focusing on studies that involved the production of a creative outcome. Hocevar and Bachelor (1989) agreed but were quick to point out that studying creative thinking and creative personalities is valid not only because of their interest, but also because there is evidence to suggest that they are potential causes of creative productions. Mayer (1989) suggested that creativity is the production of something that is new to the individual. This definition is problematic; it could be used to characterize all learning as creative. For this study, a good definition of creativity in problem solving could be used to identify those who possess the flexibility to formulate different representations for the same problem and the originality to combine elements of a problem in novel ways. A creative solution method reflects thinking that is determined to be unusual for the sample under consideration. In summary of a wide range of theories, Brown (1989) listed four general views of creativity. In these four views creativity is (a) an associative process; (b) an aspect of intelligence; (c) a largely unconscious process; and (d) an aspect of problem solving. These perspectives will be reviewed briefly below. The associative view was held by several theorists who based many of their ideas on the Spearman Principle (Brown, 1989). Spearman's model "involves an active process in which associations with an initial idea can be freed from their relation to it and thus lead to something wholly new" (p. 5). This theory was the basis for important subsequent studies including those of divergent thought. Haensly and Reynolds (1989) reviewed the debate concerning the connection between intelligence and creativity that has evolved over the past halfcentury and concluded that the most prevalent current view is that "creativity is a distinct category of mental functioning that has limited overlap with intelligence, both in the processes used and in the characteristics of individuals who exhibit them" (p. 111). Haensly and Reynolds believed that trying to determine how much intelligence is associated with how much creativity would result in a simplistic view of a vast array of mental capabilities. They suggest that an individual's most intelligent responses could still be considered ordinary. The creative response extends into the realm of the extraordinary. Guilford (1959) emphasized a curriculum that promotes creative thinking as a means to prepare our students for a changing world. The debate over creativity and intelligence would not be relevant in mathematics education with a curriculum based on the ideals outlined in the curriculum and evaluation Standards and teaching Standards. The framework for such a curriculum would reflect Guilford's (1965) attitude that "creativity is not a special gift of the select few. It is instead a property shared by all humanity, to a greater or smaller degree" (p. 7). All students need to be provided with opportunities and the motivation to think. Although helping students learn to be good problem solvers appears first in the list of goals for mathematics education (NCTM, 1989), the ultimate goal is engaging students in creative teaching and learning. The third view of creativity, that it is unconscious thought, might be extracted from a model described by Wallas in 1926. The stages of the model are (a) preparation, which includes the gathering of requisite knowledge and skills, (b) incubation, the phase in which retrospective thought is unconscious (c) illumination, the occurrence of insight, and (d) verification, the process of correcting or revising an idea. Armbruster (1989) supported the use of the model because it has been an implied part of so many respected studies over the years and because it is a useful means of organizing a discussion of creativity. Armbruster warned, however, that Wallas' writing implies that creative thought is linear, whereas research suggests that creative thought is more interactive and iterative, with communication among the stages. No one knows exactly what occurs during the stage of incubation, but Armbruster concluded that flexible knowledge that has been acquired during the preparation stage is restructured into new mental structures. Though incubation proceeds unconsciously, the creative individual may have a metacognitive skill that allows for an efficient and effective control over the reconstruction of their schemas. The stages of creative thinking described by Wallas (1926) closely parallel the phases of problem solving detailed by Polya (1957): understanding the problem, devising a plan, carrying out the plan, and looking back. During the first phase, understanding the problem, a problem solver prepares to solve the problem by identifying as much information about the problem, known and unknown, as possible. The more information a solver can establish about the problem, the deeper the understanding. Spiro et al. (1987) believed that the creative individual can think independently while gathering knowledge and representing it in a flexible and productive schema. Spiro and Myers (1984) further suggested that the ability to maintain flexibility in representation might be attained by consciously considering the same information using many different models. In summary, it is during this important phase of understanding and preparation that a problem solver determines the probability of a correct, and possibly creative, solution. Polya's second phase of problem solving, devising a plan, is related to but different from incubation, the second phase of creativity. Whereas incubation is unconscious, devising a plan suggests a conscious search of one's knowledge base in order to find or reconstruct a solution method that will eventually work. There is no way to estimate how much time an individual might require for incubation, but time is a necessary ingredient for successful incubation. Polya stressed the importance of allowing students adequate time to think about a problem. The joy that accompanies finding the "answer" during Polya's third stage of problem solving could be equated to the excitement that coexists with the third stage of creativity, illumination. Although illumination is often described as a point when an idea spontaneously springs from the unconscious to the conscious. Spiro et al. (1987) believe it is the result of unconscious flexible representation. Polya's (1957) fourth stage of problem solving, looking back, is similar to the verification phase of creativity. These reflective phases of problem solving and creativity require the student to relate the present situation to previous situations, think in terms of "what if," and look at the situation from a variety of viewpoints. These reflective phases of problem solving and creativity also require a student to shift from the newly acquired sense of relief to additional mental strain (Armbruster, 1989). This task is often overlooked or consciously avoided by students and teachers. Even students who have been trained to look back at a problem do not systematically do so (Kantowski, 1977). With creativity and problem solving so closely paralleled what, if anything, can be done to improve a student's creativity? Torrance and Torrance (1973) collected evidence to support their belief that although creativity is a natural process, teaching can make a difference. They identified skills involved in creative problem solving that require practice and can be enhanced by teaching. Although Torrance and Torrance issued no guarantee with even the best teaching situations, they promote a classroom that provides for the deliberate teaching of skills involved in creative problem solving in order to increase the probability that creative development would occur. According to Torrance, to develop these skills, which involve both cognitive and emotional functioning, one must be provided with adequate structure, motivation, involvement, practice, and interaction with teachers and other students. If, in fact, everyone possesses some degree of creativity that needs to be fostered, decision makers must insist upon a curriculum that provides each student with appropriate activities and experiences. In summary, the topic of creativity has been an interest for numerous researchers over the past century. Creativity, whether identified as a personality trait or a product, is considered a valuable asset in business, industry (Scott, 1992), leisure activities, and parenting. Thus, it becomes the duty of educational systems to produce educated, creative graduates. The educator's task of nurturing the creative potential possessed to some degree by each individual has been aided by Wallas's description of identifiable stages that interact during creativity. The strong similarities between Wallas's stages of creativity and Polya's phases of problem solving give justification to research focused on their relationship. Knowing the processes through which a creative problem solver passes, and recognizing the products of creativity help educators plan for, manage, and evaluate the problem solving curriculum outlined in the curriculum and evaluation Standards. The present study investigated the possibility that encouraging students to represent a problem in as many different schemas as possible taps their potential for creative problem solving. The present study compared the solution methods selected by students identified as creative with those who show less of an inclination for the type of creativity surveyed. Polya's suggestion that students need time to think about a problem was incorporated into the present study. The possibility that the pursuit of several alternative representations for a given problem leads to more sophisticated solution methods was also examined. Mathematics Education in Japan Achievements of the Japanese in business and education have captured worldwide attention. On the Second International Mathematics Study (SIMS) (McKnight et al., 1987), Japanese eighth grade students ranked first among 20 countries in all computational and noncomputational categories for that age group. Ironically, many of the educational reform ideas that have brought Japan to its present competitive status were ideas borrowed from other countries including the United States (Shimahara, 1992; U.S. Department of Education, 1987). Japan's accomplishments in mathematics education made the prospect of including their students in a study of problem solving very intriguing. The fact that Japanese students far outperform students in the United States on international tests of mathematics achievement is well documented (Husen, 1967; McKnight et al., 1987; Stevenson et al., 1990). While American educators Investigate numerous variables that might explain the discrepancy in scores, Japanese educators are carefully analyzing American schools (Burstein & Hawkins, 1992). Americans are understandably fascinated by the educational achievements of Japanese students, but one may be puzzled by Japanese educators' interest in American instructional practices. Matthew Prophet, superintendent of the Portland School District, where 1 of 5 Japanese immersion programs in the U.S. is offered, thinks that "the Japanese are trying to be more like us" (quoted in Graves, 1991, p. 14). One of the reasons for Japanese interest in U.S. schools is that American classrooms are viewed as more conducive to creative and assertive behavior. Ikuo Idaka, a Japanese consul for cultural affairs and education in New York City, stated that Japanese educators are coming to the United States "to find out how the individual student character is being respected" (quoted in Graves, 1991, p. 14). A Japanese commission established to evaluate Japan's educational system reported, Despite its merit, the main thrust of this country's education has been to have students memorize information and facts. The development of the ability to think and judge on one's own and the development of creativity have been hampered. Too many stereotyped persons without marked individuality have been produced. (Haberman, 1985, p. Fl) Japanese mathematics educators, members of a virtually singleculture nation, are curious about how the diverse American students solve problems (Becker, 1992; Nohda, 1989). Partly due to an interest in U.S. mathematics education, a team of Japanese researchers assisted in the design of the present study and two of the team's mathematics educators invested their own time to manage the data collection process in Tokyo and Hiroshima. Many factors could contribute to the disparity in the mathematics achievement of the U.S. and Japan. Contrasts in the home environment and other environmental differences (Chen, 1991; Stevenson et al., 1986), amount of instruction in mathematics (Stigler et al., 1987), textbooks (Stigler et al., 1982), and other factors have been shown to influence achievement. While the effects are difficult to isolate, they are each correlated to some degree with differences in achievement. Lynn (1982) even tried to show that the gap was due to a significant difference in the cognitive abilities between Asian and American children. Lynn's results were later refuted (Stevenson et al., 1985). Although no systematic relationship between time spent on mathematics and achievement could be established when considering all 20 countries in the SIMS study (McKnight et al., 1987), when comparing U.S. and Japan scores specifically, time spent on mathematics was identified as a major contributing factor (Stigler et al., 1987). American students spend less time each year in school, less time each day in classes, less time each day in mathematics class, and less time in the mathematics classroom receiving instruction than Japanese students. American students, according to this study, spend more time working alone at their seats on material that they do not understand well, engage more often in irrelevant activities, and spend more time in transition from one activity to another. The fact that Japanese seventh grade students have spent 40% more time studying mathematics (an equivalent of 2.4 more years) than seventh grade students in the United States was also established as a contributing factor to Japanese students' superior performance in relation to U.S. counterparts by Iben (1988). Since solution methods employ skills that exist in a student's knowledge base, a student who has had the equivalent of between two and three years of extra time spent on mathematics might be expected to produce more sophisticated solution methods. Results of a U.S.Japan Crossnational Research on Students' Problem Solving Behaviors (Becker, 1992) suggest that although students in the United States may use similar solution methods for nonroutine problems, Japanese students tend to employ more mathematically sophisticated methods at an earlier age. Although data analyzed qualitatively give some support to the hypothesis that the complexity and sophistication of solution methods of Japanese students could be equated with American students two grades their senior, to date there is still little statistical evidence to support such a claim. With mathematical computation scores that rank at the top globally, the opportunity to examine the solution methods of Japanese students for nonroutine problems added a timely and valuable dimension to problem solving research. The present study was partially based on one portion of a former project, the U.S.Japan Crossnational Research on Students' Problem Solving Behaviors, henceforth referred to as the U.S.Japan. Cross national Research. (Becker, 1992; Becker & Miwa. 1987; Nohda, 1989). The objectives of the research dealt with various aspects of problem solving that could be compared between the two cultures. Summary Mathematics education has been the target of many reform movements since World War II; none as radical as the changes now underway with the learning and teaching of mathematics being brought together as a single focus. One of the common themes of each of the major reform movements is that students should be given the opportunity to learn mathematics by actively constructing their own meaning. In the role of facilitator, the teacher inherits the task of choosing problems so that each student is challenged but not overwhelmed, problems for which a variety of solution methods lie within each student's zone of proximal development, and problems for which the reflective process can be guided through questioning techniques. Constructivist teachers encourage students to think creatively about mathematics and help them understand what they are doing. Students are being challenged to not only solve problems, but to be creative problem solvers. While much attention has been awarded international tests of computational skills, there is interest in 35 how students in the U.S. and Japan compare on methods used to solve nonroutine mathematical problems. This chapter contains a summary of problem solving research to date, a review of learning and teaching theories adopted by those who come under the umbrella of constructivists, and a comparison of mathematics education in the U.S. and Japan. The next chapter will describe the treatment format for students in the main study and the procedures used to test the hypotheses. CHAPTER III PROCEDURES This chapter includes a description of the data collection techniques, the treatment, and method of analysis for (a) the main study which examined the number and complexity of solution methods students employed for a nonroutine mathematical problem and (b) the U.S.Japan comparisons that served as a supplement to the main research.. Hypotheses The purpose of the present study was to investigate methods of solution used to solve a nonroutine mathematical problem and the role of creativity in the reflective phase of problem solving. Problem solving methods of students who participated in sessions designed to encourage the construction of alternative solution methods were compared to the methods employed by students who had received no such deliberate practice. The study was enhanced by the inclusion of Japanese students for comparisons. Methods of solution for a nonroutine, pattern finding problem were compared and contrasted in order to subject the following hypotheses to statistical tests: Hol: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and the treatment received. Ho2: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and receiving the pretest. H03: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and the measure of creativity. H04: The relationship between the number of different solution methods employed for a nonroutine mathematical problem and treatment received does not differ by pretest. Ho5: The relationship between the number of different solution methods employed for a nonroutine mathematical problem and the measure of creativity does not differ by pretest. Ho6: The relationship between the number of different solution methods employed for a nonroutine mathematical problem and measure of creativity does not differ by treatment. Ho7: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and the treatment received. Ho8: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and receiving the pretest. Ho9: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and the measure of creativity. Ho10: The relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and treatment received does not differ by pretest. Holl: The relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and the measure of creativity does not differ by pretest. Ho12: The relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and measure of creativity does not differ by treatment. Ho13: There is no significant difference between the number of different solution methods employed for a nonroutine mathematical problem on the premeasure and on the postmeasure. Ho14: There is no significant difference between the complexity of different solution methods employed for a nonroutine mathematical problem on the premeasure and on the postmeasure. H015: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and grade level. Ho16: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and nationality. Ho17: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and grade level. Ho18: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and nationality. Analysis of variance was used to test the hypotheses. The alpha level set for rejection of the null hypotheses was 0.05. The Main Study Following the description of the design of the main study will be a description of the U.S.Japan comparisons that served as a supplement to the main study. The Subjects A total of 595 students were involved in one or both parts of the present study. The survey sample of the main study consisted of 217 eighth grade American students from four algebra and four prealgebra intact classes. The eight classes were from four different middle schools located in Central Florida. Most of the students in three of the schools were from middle class backgrounds, whereas the fourth school maintains the same racial and SES balance as the state of Florida. Only algebra and prealgebra students, who typically are more skilled than students in general mathematics classes, were included in the present study since it was assumed that students who possess the greater number of mathematical skills have more potential for providing multiple solution methods than would those students whose pool of mathematical resources is not as rich. Also, algebra and prealgebra students were asked to participate because 171 students from the main portion of this study were also included in the U.S. Japan comparisons of the present study. Since all eighth grade students in Japan take algebra, any differences in solution methods might be masked by an imbalance of computational skills between Japanese students, who have all practiced these skills, and American students who do not take algebra and who therefore cannot be assumed to have practiced the same computational skills as Japanese students. Instrumentation for the Main Study The two instruments used in the present study include a revised version (figures 31 and 32) of the marble problem which was part of a former study, the U.S.Japan Crossnational Research (Appendix F) and the Figural Booklet A version of the Torrance Test of Creative Thinking: Thinking Creatively with Pictures (Torrance et al., 1992). The Marble Problem The purpose of the U.S.Japan Crossnational Research (Becker, 1992) was to compare and contrast how students in each country solve nonroutine mathematical problems. Analysis of student's performance on one of the five mathematical problems included in the research, the marble problem, was used as a basis for part of the present study. The U.S. and Japanese mathematics educators who conducted the research considered a wide range of problems before deciding on the five to be included in their complete survey. Problems selected were nonroutine: the solution would not be immediately obvious to the solver but would be within the solver's ability range. Also, each of the selected problems afforded a variety of alternative solution methods. The problems chosen were judged to be the best for the two samples in terms of simplicity, understandability, and the inclusion of the necessary skills for solving the problems in the curriculums of both groups. The marble problem, figures 31 and 32, is a nonroutine, patternfinding problem. The marble problem was chosen because the solution, although simple, can be found by students of varying mathematical abilities using numerous different methods. Beginning with the simple case shown in Part 1, the problem becomes progressively more generalized in Parts 2 and 3. The marble problem was administered to sixth, eighth, and eleventh grade students in the United States and sixth and eighth grade students in Japan. It was the only problem of the five that was included at three different grade levels. Having been assigned the task of assisting in the analysis of the student's solutions of the marble problem for the U.S.Japan Cross national Research, this researcher began by studying the problem and recording solution methods that might be expected from the students. When the researcher's list was exhausted, the student responses were skimmed first and then classified according to which method was used. Methods generally included enumeration, the identification of several different patterns, various grouping techniques, and a formula. The list of identified solution methods was subsequently extended to include additional creative methods supplied by the students. Each identified method of solution was evaluated independently. To provide an objective check on coding, the researcher refined a list of solution methods and the characteristics that could be used to Part 1: Marbles are arranged as follows: first second third fourth place place place place Do not erase anything you write down, just draw a line through anything you feel is in error. (1) If you were to continue building marble structures, how many marbles would there be in the fourth place? FIND THE NUMBER OF MARBLES USING AS MANY DIFFERENT METHODS AS YOU CAN. Show your method and the number of marbles. FIGURE 31 Part 1 of the Revised Marble Problem first second third fourth place place place place * * . Part 2: (2) How many marbles would there be in the sixteenth (16th) place? Show one method of solution and the number of marbles. Number of marbles Part 3: (3) How many marbles would there be in the one hundredth (100th) place? Show one method of solution and the number of marbles. Number of marbles FIGURE 32 Parts 2 and 3 of the Revised Marble Problem. distinguish each method. Results of the marble problem data analysis suggested that Japanese students found methods of solution that, although of the same type, were identified as higher level more often than did their American counterparts (Fouche & Kantowski, 1992). Rationale for Reexamination of the Marble Problem One of the stated objectives of the U.S.Japan Crossnational Research was to compare several aspects of how students in each culture solve nonroutine mathematical problems. The examination of the Japanese marble problem data, as reported by Ishida (1991), combined with the analysis of the American sample, provided insights that elicited interesting speculations and generated additional questions. As for using the marble problem data as a basis for comparisons between countries, only the few general observations could be documented. Additionally, the compatibility of the analysis techniques employed by each team member and the channel of communication available between team members was a factor in the comparison process. In the case of the marble problem, crossnational data comparisons were limited because coding, analysis, and reporting techniques were not uniform between countries. Determination of significant differences was unachievable since Ishida (1991) did not report any statistical analysis of the Japanese marble problem data. The frequency with which each identified strategy was employed could be obtained from the Japanese report but there was no indication of the students' mean number of solution methods. There was not a consistent opinion of what constitutes a generalizable solution method and comparisons of a student's initial solution method, to a subsequent method differed between reports. In addition, the present tense wording on the American survey for Part 1 which read, "How many marbles are there in the fourth place?", proved to be confusing for many of the American sixth grade students (See Appendix F). Responses such as "There are no marbles in the fourth place." were not uncommon. Others indicated that the five dashes in line with four marbles from Stage 3 were related to the five marbles that would go on line two of that stage. Imprecise wording of the question reduced the validity of even basic comparisons such as the percent of students obtaining the correct solution. However, the marble problem data did provide interesting results that warranted further investigation. While computational skills have been the focus of many crosscultural studies, there have been few studies designed to compare the problem solving methods for a nonroutine mathematical problem. With the nation involved in a radical reform movement in mathematics education, valuable insights for curriculum, teaching, and evaluation could be found in research focused on creative problem solving. An indication of the need to revise the original marble problem was suggested by results of the U.S.Japan Crossnational Research. The present tense wording of the Part 1 question, "How many marbles are in the fourth place?" was changed to, "If you were to continue building marble structures, how many marbles would there be in the fourth place?." Also, to avoid possible confusion, the dashes in the fourth place in the original problem were replaced with a question mark. To ensure that students did not spend all of their allotted time on Part 1, students were allowed 8 minutes for Part 1 and 8 minutes for Parts 2 and 3 combined. To eliminate the need for turning back to Part 1 to refer to the problem, the problem and diagram were repeated for Parts 2 and 3. The text of the revised version of the marble problem appears in figures 31 and 32 with the entire problem booklet included in Appendix B. Administration of Instruments The revised marble problem was used as both a pre and posttest measure. This researcher began each pretest by reading a script (Appendix E) to the class that further explained the purpose of the research and the importance of their participation. The survey booklets were then distributed, the problem and procedures explained, and the survey administered. The students were monitored throughout the survey to maintain an atmosphere conducive to concentration and to discourage the sharing of ideas. The entire process for the marble problem took about 2025 minutes of the first half of a class period. As a posttest, the revised marble problem was administered in the same manner. To obtain a measure of students' creativity, the Torrance Tests of Creativity Thinking: Figural A (Torrance et al., 1992) was selected for this study. Given that creativity has been defined to include a wide range of personality traits and resulting products and given that much of the creative process is unconscious, trying to measure creativity is problematic. However, the Torrance Tests of Creative Thinking (TTCT) have shown significant validity and reliability in the areas of fluency, flexibility, originality, and elaboration (Torrance, 1990). Although there remains a question as to how much originality the Torrance or any other test can actually measure (Cooper, 1991), the TTCT has played an important part in research and group assessment for over two decades. Norms for that test were developed using a representative sample of students in the U.S. in grades K12+ in all major regions of the country. Raterreliability coefficients for grade eight with the streamline scoring procedure for fluency, originality, abstractness of titles, elaboration, and resistance to premature closure consistently stay well above .90 (Torrance, 1990). The Figural A version of the TTCT was administered according to the author's guidelines to every subject included in the main study. To provide for uniformity in the recording instrument used and to motivate students, the researcher gave a new, sharpened cedar wood pencil to each student. Students were timed for ten minutes on each of three sections with instructions for each section delivered verbally as well as included in the test booklet. Group Assignment The major factor that effected group placement of the eight classes was teacher cooperation and interest in the study. The teachers who were the most enthusiastic about the study were those who were convinced that the 1011 class periods that would be devoted to the study would provide valuable learning experiences for their students and would promote the problem solving skills incorporated in their existing curriculum. The five teachers who volunteered their students for the treatment group were themselves interested in problem solving and in how their students would respond to the lessons. The eight classes were in schools that service a similar population of students and parents. Therefore, the teacher was the deciding factor, not the students. As the result of a project just prior to the time of this study, this researcher had worked closely with three of the five teachers who offered their students as subjects in the treatment group. The four treatment classes were therefore chosen from those taught by these three teachers. Treatment Procedures For the investigation of creativity the eight classes were divided into four groups with one algebra and one prealgebra class in each group. A quasiexperimental, Solomon fourgroup design was employed. Group identification is shown in Table 31. TABLE 31 Treatment yes no Group 1 Group 2 yes Pretest Group 3 Group 4 no The two treatment groups received eight full class periods of problem solving instruction over a period of ten weeks during the spring of 1992. The objective of the problem solving instruction and planned practice was to elicit multiple solution methods for each of the eight problems. The problems used in this study appear in Appendix A. Classes were planned and taught by the researcher with occasional assistance. All instruments were administered by the researcher. For each problem solving class the instructor served as a facilitator and the procedures were as follows. 5 minutes Introduction to the problem of the day 10 minutes Students work individually to produce as many solution methods as possible 20 minutes Students work cooperatively in small groups to produce additional solution methods 15 minutes Solution methods shared with the class by individual students via an overhead transparency on which they had recorded their solution method. Deviations from this routine occurred with the 'Ten People in a Boat", the "Beans, Toothpicks, and Cubes", and the "Kids With Beans and Kids With Figs" problems. Since students could not be expected, in a single class, to solve the 'Ten People in a Boat" problem and practice the correct moves with enough accuracy to recognize patterns in their moves, students were introduced to the problem the week before it was to be used as an inclass problem solving activity. Thus, any student who had not mastered the correct sequence of moves required by the problem was given directions and an opportunity to practice during the time regularly devoted to the introduction to the problem. The "Kids With Beans and Kids With Figs" problems were designed as cooperative learning activities (Erickson, 1989) so students did not have the ten minutes to work alone on the problem. Instead, each student was given a clue card and the four students in the group had to use the four clues to solve the problem. The rules for the activity included directions that no student could show their clue card to another group member. If someone wanted to have a particular clue repeated, the owner of the clue was to reread it to the group, not pass it around. Once the group found the solution they were asked to think of other ways to find the answer. In addition to alternative solution methods, the "Kids with Figs" problem had four possible solutions. Instead of sharing with the class, each group showed the researcher their method before looking for an additional method. When they had found all the possible solution methods with or without hints, they were given the second problem. A third problem was planned, but no group was able to exhaust more than two problems. The third deviation was not scheduled prior to the study as were the other deviations. After administering the "Beans, Toothpicks, and Cubes" problem to the two classes in the pretesttreatment group, it was determined that the problem was too difficult for the majority of students to solve in more than one way during a single class period. Therefore, for the no pretesttreatment group the problem was introduced and practiced in class and then the students were given four days to work on the problem outside of class before sharing solution methods. The eight problems chosen for this study shown in Appendix A were selected because each was considered to be (a) motivational, (b) problems for which several alternative solution methods existed within the ability range of the subjects, and (c) problems that were familiar to the researcher who had successfully used them with other groups. The order in which the problems were administered is as follows: Week 1: Horses and Ducks Introduction to Ten People in a Boat Week 2: Ten People in a Boat Week 3: Diagonals in a Polygon Week 4: Dots on the Side of a Triangle Week 5: Telephone Lines Week 6: Triangular Numbers Week 7: Kids With Beans and Kids With Figs Week 8: Beans, Toothpicks, and Cubes Students' work was collected each class period but was not graded since students were encouraged to work together and the groups were not necessarily the same each week. Additionally, without grades it was hoped that students would be more free to think about the process instead of the product. Analysis of Creativity Data for American Students Given students who perform at approximately the same mathematical skill level, those who possess the higher measure of creativity might be expected to exhibit greater problem solving potential than those who demonstrate a less promising capacity for creativity. In order to test this conjecture, students in the United States sample were administered the Figural A version of the TTCT: Thinking Creatively with Pictures. All tests were graded by one graduate student who had previously been trained to use the TTCT. Standardized scores for fluency, originality, elaboration, abstractness of titles, and resistance to premature closure were averaged to obtain the mean measure of creativity variable. A relationship between students' measure of creativity and the number and complexity of different methods of solution found on the marble problem was then investigated. Weight Assignment for Solution Methods Each student response was coded as zero if Irrelevant computations that did not relate to the problem were shown, the solution was incorrect, or there was no response. Correct solution methods were given a weighted decimal value from one to five depending on the level of sophistication. Responses were recorded sequentially for Part 1. No credit was given for a response that repeated a previously used method. To assist the researcher in assigning a sophistication value to each solution method, a panel of five experts in the field of problem solving in mathematics education independently ranked each method. A copy of the letter requesting this assistance and a list of those who responded are included in Appendix C. The weighted value of each response represents its ranking according to its predetermined level of sophistication. Sophistication rankings for solution methods are based on the degree of intellectual engagement indicated and on generalizability. Detailed descriptions of each method of solution appear in the results section and in Appendix D. Methods of solution fall into the general categories of enumeration, pattern recognition, grouping, or a formula. The U.S.Japan Comparisons For the U.S.Japan comparisons, results of 100 sixth and 100 eighth grade participating Japanese students were included in the study. Because the Japanese educational system does not promote ability grouping, no instructions were given to proctors concerning the selection of classes for the survey. The two schools chosen, one school in Hiroshima and one in Tokyo, are both public schools and therefore represent the Japanese population more closely than would a private school with very selective entrance requirements. Although no classes identified as "gifted" were part of the survey, it is assumed that students from both countries had experienced some problem solving activities. In addition to the 217 American eighth grade students who were included in the sample for the creativity portion of the study, 96 eleventh grade and 94 sixth grade students were included in the sample of U.S. students for the U.S.Japan comparisons. The sixth and eleventh grade students were from one additional local middle school and one high school; both with students of comparable backgrounds to the eighth grade American subjects The eleventh grade students were enrolled in a trigonometry class while the sixth grade subjects were in a regular sixth grade heterogeneous mathematics class. The data collection for the Japanese sample was administrated by two colleagues of the researcher, Professor Toshiakira Fujii of the University of Yamanashi and Professor Tadao Ishida of Hiroshima University, both members of the U.S.Japan Crossnational Research (Becker, 1992). These Japanese professors and the researcher worked together on the collaborative research project during their visit to the United States and this researcher's visit to Japan in 1990. As a result of the above mentioned collaborative research project when Japanese researchers visited U.S. classrooms and participated in joint discussions on problem solving, a communication network was formulated and friendships established. Professors Fujii and Ishida each collected one half of the Japanese data. The Japanese sample included 100 sixth and 100 eighth grade students equally divided between Tokyo and Hiroshima. Three intact classes from each location were actually tested and papers were chosen at random by the coder, Christina Carter, for inclusion in the study. Mrs. Carter, personal friend of the researcher and University of Florida graduate student in mathematics education and a resident of Hiroshima during the data collection process, has special expertise in Japanese education. For the past three years, Mrs. Carter has been very involved with the Japanese elementary school that two of her children attended and is fluent enough in Japanese to communicate both orally and in writing. Both the number of solution methods employed for the nonroutine mathematical problem and the complexity of those solution methods were compared between the U.S. and Japanese sixth and eighth grade data. Since middle level students in Japan have spent the equivalent of approximately two to three more years of classroom time studying mathematics than their American counterparts, different grade comparisons were investigated. In particular, the sixth grade Japanese results were compared to the eighth grade U.S. results and the eighth grade Japanese results to the eleventh grade data from the U.S. Summary In the spring of 1992 data were collected from 217 American eighth grade students to investigate solution methods for a nonroutine mathematical problem. A relationship between a measure of creativity as determined by the Torrance Test of Creative Thinking: Figural A, and the number and complexity of solution methods employed was the focus of the main study. A Solomon fourgroup design was used with one prealgebra and one algebra class in each group. The two treatment groups each received eight full sessions of problem solving designed to elicit multiple alternative solution methods for the treatment problems. The marble problem (Appendix B) was used as both a pre and posttest measure. The main study was enhanced by a crossnational comparison of solution methods employed for the marble problem. In addition to the marble problem results collected for the main study, 96 eleventh grade and 94 sixth grade American students and 100 sixth and 100 eighth grade Japanese students were administered the marble problem. Comparisons of number and complexity of solution methods were not only made between nations but also samegrade and differentgrade comparisons were investigated. CHAPTER IV DATA ANALYSIS AND RESULTS OF THE STUDY The purpose of the main study was to investigate a possible relationship between a student's measure of creativity and both the number and complexity of solution methods employed for a nonroutine mathematical problem. Comparisons of solution methods of students in the United States and those of Japanese students were also made. This chapter begins with a review of the data collection procedures for both the main study and the USJapan comparisons, followed by a report of the analyses. Review of the Design and Data Collection Procedures A quasiexperimental Solomon fourgroup design was used for the main study involving 217 subjects. Each of the four groups consisted of one algebra and one prealgebra class of eighth grade students from four middle schools in two Central Florida counties. The number of subjects assigned to pretest and/or treatment groups is shown in Table 41. TABLE 41 Number of Subjects in Each Group Pretest No Pretest Treatment 64 46 No Treatment 54 53 The protests and posttests were identical and consisted of one nonroutine mathematical problem. The marble problem, shown in Appendix B, was chosen because, even though the answer can be easily obtained, it can be solved using numerous different methods that vary in level of sophistication. This problem was also one of five problems chosen for inclusion in an earlier study; the U.S.Japan Crossnational Research (Becker, 1992). The weighted level of sophistication for each solution method was determined by a panel of five experts in the area of problem solving who independently ranked each solution method on a decimal scale from one to five according to the degree of understanding represented by the method and its generalizability. The mean of the five rankings became the method's level of sophistication or complexity. The solution methods generally fall into the categories of pattern finding, grouping, and a formula. The 13 different solution methods for Part 1 appear in Appendix D along with the corresponding level of sophistication and a description of each method. To obtain a measure of creativity, the Torrance Test of Creative Thinking: Figural A was administered to each student in the main study. The creativity measure was computed by calculating the mean of each student's standardized scores for fluency, originality, flexibility, abstractness of titles, and resistance to closure. The treatment consisted of eight class periods of problem solving in which one problem (Appendix A) was solved during each session using as many different solution methods as possible. Students spent time working individually and in small groups to devise and carry out their solution plans and then took turns presenting representative methods to the class via overhead transparencies. For the USJapan comparisons results of 496 subjects were analyzed. The number of subjects in each grade by country is shown in Table 42. For the eighth grade American sample, the pretest results of the 118 students in the pretest groups of the main study were used. The Japanese sample consisted of 100 sixth and 100 eighth grade students from public schools in Hiroshima and Tokyo. Data collected in Japan were used for comparisons only since it was not possible for this researcher to obtain a measure of creativity from the Japanese subjects and a matching variable was not available. TABLE 42 Number of Subjects in the U.S.Japan Comparisons Country Grade Subjects U.S. 6 91 U.S. 8 118 U.S. 11 87 Japan 6 100 Japan 8 100 In the analyses that follows, each hypothesis is stated in null form followed by the decision to reject or not reject and a description of the results. An alpha level of 0.05 was used for all tests. The results section divides the hypotheses into four parts, (a) hypotheses 1 6 were used to analyze the number of solution methods employed, (b) hypotheses 712 were used to analyze the complexity of the solution methods, (c) a repeated measure was used to test hypotheses 1314 which addressed the changes in mean number and complexity of solution methods over time, and (d) hypotheses 1518 concerned the U.S.Japan comparisons by nation and grade for both number and complexity of the pretest solution methods. Results of Hypotheses Testing Number of Solution Methods An analysis of variance was used in analyzing the three main effects (treatment, pretest, and creativity) on the total number of solution methods generated on the posttest. The results shown in Table 43 confirm that the overall model accounts for a significant amount of variation in the mean number of solution methods. At least one of the main effects was significantly related to the differences in mean number of solution methods between the four groups. TABLE 43 __ANOVA Model of Main Effects Source df SS MS F Between Groups 3 29.72 9.91 9.17 ** Within Groups 213 230.21 1.08 Iotal 216 259.93 p<.05 *p<.01 The first three hypotheses were used to examine each of the three main effects. The results are shown in Table 44. TABLE 44 Summary of ANOVA Posttest Number of Solution Methods Source DF Type III SS F Treatment 1 20.89 19.33 ** Pretest 1 2.79 2.58 Creativity 1 9.11 8.43 ** Error 213 *p<.05 **p<.01 Hol: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and the treatment received. This hypothesis was rejected. The results shown in Table 44 suggest that the mean number of solution methods found on the posttest was significantly related to the treatment effect. The mean number of solution methods employed on the posttest by the nontreatment groups was 2.00 with a standard deviation of 1.07, while the treatment groups generated a mean of 2.60 solution methods with a standard deviation of 1.04. Ho2: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and receiving the pretest. This hypothesis was not rejected. As shown in Table 44, receiving the pretest did not significantly effect the outcome of the posttest. The mean number of solution methods generated by each group on the posttest is shown by group in Table 45. TABLE 45 Mean Number of Solution Methods on Pretest and Posttest Group n Pretest SD Posttest SD PretestTreatment 64 2.05 0.84 2.61 0.99 PretestNo Treatment 54 2.06 1.16 2.12 1.08 No PretestTreatment 46 2.59 1.13 No PretestNo Treatment 53 1.87 1.06 H03: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and the measure of creativity. This hypothesis was rejected. Table 44 shows that the mean number of solution methods found on the posttest was significantly related to the student's measure of creativity. A test for curvilinearity was also conducted but no such relationship was found. Table 46 shows the mean creativity score for each total number of solution methods ranging from zero to five. TABLE 46 Mean of Creativity Measure for Posttest Number of Solution Methods number creativity standard of methods mean deviation 0 100.50 19.63 1 99.02 14.82 2 99.19 14.40 3 101.36 14.30 4 105.04 13.00 5 115.33 11.85 The results of the next three hypotheses ruled out the possibility of twoway interactions among the three main effects. None of the three hypotheses were rejected. H04: The relationship between the number of different solution methods employed for a nonroutine mathematical problem and treatment received does not differ by pretest. Ho5: The relationship between the number of different solution methods employed for a nonroutine mathematical problem and the measure of creativity does not differ by pretest. Ho6: The relationship between the number of different solution methods employed for a nonroutine mathematical problem and measure of creativity does not differ by treatment. TABLE 47 Summary of ANOVA Posttest Number of Solution Methods TwoWay Interactions Source DF Type III SS F Treatment*Pretest 1 0.31 0.29 Creativity*Pretest 1 0.67 0.62 Creativity*Treatment 1 2.60 2.41 Error 210 *p<.05 **p<.01 The mean number of solution methods generated on the posttest was not significantly effected by the interaction of pretest with treatment, the interaction of the measure of creativity with the experience of taking the pretest, or the interaction of creativity and the treatment. These results are shown in Table 47. Complexity of Solution Methods An analysis of variance was used in analyzing the three main effects (treatment, pretest, and creativity) on the level of complexity attained on the posttest solution methods. The results shown in Table 48 confirm that the overall model accounts for a significant amount of variation in the level of complexity attained on the posttest solution methods. At least one of the main effects was significantly related to the differences in the mean level of complexity attained by each group on the posttest solution methods. TABLE 48 ANOVA Model of Main Effects on the Complexity of Solution Methods Source df SS MS F Between Groups 3 44.54 14.85 11.77 ** Within Groups 213 268.69 1.26 total 216 313.23 p<.05 *p<.01 Hypotheses 79 were used to examine the relationship between each of the three main effects on the level of complexity attained on the posttest solution methods. The results are shown in Table 49. H07: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and the treatment received. TABLE 49 Summary of ANOVA Posttest Complexity of Solution Methods Source DF Type III SS F Treatment 1 33.54 26.59** Pretest 1 8.30 6.58* Creativity 1 4.08 3.23 Error 213_ *p<.05 ** p < .01 This hypothesis was rejected. Table 49 indicates that the level of complexity attained on the posttest solution methods was significantly related to the experience of receiving the treatment. The mean level of complexity attained by the treatment groups (on a scale of zero to five) was 3.36 (standard deviation of 1.11), while the mean for the nontreatment groups was 2.56 (standard deviation of 1.17). H08: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and receiving the pretest. This hypothesis was rejected. The experience of taking the pretest was significantly related to the level of complexity attained on the posttest solution methods. The mean level of complexity attained for those who took the pretest was 3.15 (standard deviation of 1.18), while the mean of those who did not take the pretest was 2.75 (standard deviation of 1.21). H09: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and the measure of creativity. This hypothesis was not rejected. The mean level of complexity attained on the posttest solution methods was not significantly related to the individual's measure of creativity. The results of hypotheses 1012 ruled out the possibility of two way interactions among the three main effects. None of these hypotheses were rejected. Ho10: The relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and treatment received does not differ by pretest. Holl: The relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and the measure of creativity does not differ by pretest. Ho12: The relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and measure of creativity does not differ by treatment. TABLE 410 Summary of ANOVA Posttest Complexity Level of Solution Methods TwoWay Interactions Source DF Type III SS F Treatment*Pretest 1 0.18 0.14 Creativity*Pretest 1 0.02 0.02 Creativity*Treatment 1 0.61 0.48 Error 210 p < .05 ** p < .01 The mean level of complexity attained on the posttest solution methods was not significantly related to the interaction of pretest with treatment, the interaction of the creativity measure with the experience of taking the pretest, or the interaction of creativity and the treatment. These results are shown in Table 410. Pretest/Posttest Comparisons Hypotheses 1314 were used to compare the number and level of complexity of solution methods on the pretest and the corresponding number and level of complexity on the posttest. Both were tested using an analysis of variance with repeated measures. Ho13: There is no significant difference between the number of different solution methods employed for a nonroutine mathematical problem on the premeasure and on the postmeasure. This hypothesis was rejected. While the mean number of solution methods on the pretest was approximately the same for subjects in both the treatment and nontreatment groups, when averaged across the pretest and the posttest (F = 2.56, df =1, 116), there was a significant interaction found between time and treatment. The graph of the of the two pre/post lines shown in Figure 41 illustrates the significantly different slopes for treatment and control groups. treatment no treatment pI~t~t poattest pretest posttest FIGURE 41 Interaction of Time and Treatment on Number of Solution Methods The change in the number of solution methods found on the pretest and the number generated on the posttest was different depending on whether the student experienced the treatment (F = 4.42, df = 1, 116, p < 0.05). The treatment and control groups were almost equal on mean number of solution methods found on the pretest, but after the experience of the treatment, the treatment group showed dramatic gains while the control group remained approximately the same. Table 411 shows the mean number of solution methods for the treatment groups and the nontreatment groups on both the pre and posttest measure. TABLE 411 Mean Number of Solution Methods Over Time By Treatment Means Treatment SD Control SD Pretest Number of Methods 2.05 0.84 2.06 1.16 Posttest Number of Methods 2.61 0.99 2.13 1.08 Ho14: There is no significant difference between the complexity of solution methods employed for a nonroutine mathematical problem on the premeasure and on the postmeasure. This hypothesis was rejected. The graph of the of the two pre/post lines shown in Figure 42 illustrates the absence of significant interaction for treatment and control groups over time. pretest posttest FIGURE 42 Time and Treatment on Complexity of Solution Methods However, both main effects, treatment and the pretest, significantly affected the mean level of complexity attained on posttest solution methods. The linear model showed that the effect of the treatment was not the same for treatment and nontreatment group (F = 20.20, df = 1, 116, p < 0.01). While there was a significant difference within subjects on the complexity attained on the pretest and on the posttest (F = 5.98, df = 1, 116. p < 0.05). the significance of the interaction found between time and treatment showed that within subjects, this difference did not significantly depend on whether the student received the treatment (F = 0.90, df = 1, 116). Table 412 shows the TABLE 412 Mean Complexity Level Over Time By Treatment Means Treatment SD No Treatment SD Pretest Complexity Level 3.07 0.92 2.52 1.19 Posttest Complexity Level 3.51 1.12 2.71 1.10 treatment no treatment mean complexity level of solution methods for the treatment groups and the nontreatment groups on both the pre and posttest measure. USJapan Comparisons The last four hypotheses were tested using the pretest results of the U.S. eighth grade students who were in one of the main study pretest groups, a sample of U.S. sixth grade students, and the results from 100 Japanese students in each of the grades six and eight. Since U.S. students, who by their middle to high school grades, have spent approximately the same amount of time studying mathematics as have Japanese students two to three grades their juniors, comparisons were made between both the mean number and complexity level of solution methods for grades six and eight. HO15: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and grade level. This hypothesis was not rejected (F = 1.08, df = 1, 406). There were no significant differences when comparing all eighth grade students to all sixth grade students. When compared by grade and by nation, the U.S. eighth grade students found significantly more solution methods than did the Japanese eighth grade students but not more than either sixth grade sample. The means are shown in table 413. H016: There is no significant relationship between the number of different solution methods employed for a nonroutine mathematical problem and nationality. TABLE 413 Mean Number of Solution Methods By Grade and Nation Country Grade n Mean SD U.S. 6 91 2.01 0.91 U.S. 8 118 2.05 0.99 Japan 6 100 1.90 1.02 Japan 8 100 1.66 0.84 This hypothesis was rejected (F = 7.62, df = 1, 406, p < 0.01). The U.S. sixth and eighth grade students found significantly more solution methods than did students in the Japanese sample. The mean number of solutions for the combined U.S. sixth and eighth grade students was 2.03 (standard deviation of 0.96), while the Japanese mean was 1.70 (standard deviation of 0.94). Results of comparisons by grade and nation showed that the eighth grade U.S. students found significantly more solutions than did the eighth grade Japanese students, but not more than the American or Japanese sixth grade students. The means are shown in Table 413. Ho17: There is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and grade level. This hypothesis was not rejected (F = 1.44, df=l, 406). There were no significant differences when comparing all eighth grade students to all sixth grade students. Followup comparisons of sixth and eighth grade students yielded no significant differences in complexity level for either grade by nation. The means are shown in table 414. TABLE 414 Mean Complexity Level of Solution Methods By Grade and Nation Country Grade n Mean SD U.S. 6 91 2.78 0.97 U.S. 8 118 2.81 1.09 Japan 6 100 2.86 1.13 Japan 8 100 2.56 1.15 Ho18: There Is no significant relationship between the complexity of different solution methods employed for a nonroutine mathematical problem and nationality. This hypothesis was not rejected (F = 0.81, df=l, 406). There was no significant difference in the level of complexity attained on the solution methods between the U.S. sample of students and the Japanese sample. The means are shown in Table 414. Other Results During the coding of the marble problem results and the treatment experience it was observed that students tended to initially solve the marble problem with a solution method that was less complex than subsequent methods. The conjecture was made that the complexity level of student's solution methods tends to increase with the generation of additional methods. To test this conjecture the mean complexity level of the first solution method for all students who generated at least one method on the pretest was computed. Then the mean level was computed on the second solution method for all students who generated at least two methods, etc. An ANOVA, designed so that independence of groups was not violated, was used to test for significant differences in the means for each individual student. The results indicated that as the number of solution methods increased, the mean level of complexity was significantly different (F = 9.58, df = 4, 127, p < 0.01). The mean level of complexity for each solution in the order generated on the pretest is shown in Table 415. As students generated additional solution methods, the complexity of their methods tended to increase. TABLE 415 Mean Level of Solution Method Complexity in the Order Generated on Pretest Ordinal Position n Mean SD 1st Solution 111 2.04 0.93 2nd Solution 89 2.69 0.94 3rd Solution 32 2.85 1.08 4th Solution 8 3.10 1.16 5th Solution 2 3.73 0.04 Since this trend was discovered on the pretest, the question was then directed to the treatment. Would this trend still hold for students who had experienced the treatment? To answer this question the same ANOVA was applied to the posttest results for those students who participated in the treatment. The results again indicated that as the number of solution methods increased, the mean level of complexity was significantly different (F = 15.41, df = 4, 175, p < 0.01). The mean level of complexity for each solution in the order generated on the posttest is shown in Table 416. As students generated additional solution methods, the complexity of their methods tended to increase. The increase was more pronounced for those who had participated in the treatment than for those in the control group. TABLE 416 Mean Level of Solution Method Complexity in the Order Generated on Posttest Ordinal Position n Mean SD 1st Solution 107 1.92 1.09 2nd Solution 97 2.80 1.11 3rd Solution 58 2.95 1.09 4th Solution 21 3.61 0.79 5th Solution 3 3.13 0.29 Summary The purpose of this research was to investigate solution methods used by students to solve a nonroutine mathematical problem. The main study investigated a possible relationship between creativity and both the number and complexity of solution methods employed by eighth grade students. Additionally, comparisons of solution methods of students in the U.S. and those of Japanese students were made. A quasiexperimental Solomon fourgroup design was used for the main study. The four groups consisted of a pretesttreatment group, a pretestno treatment group, a no pretesttreatment group, and a no pretestno treatment group. The treatment consisted of eight class periods of problem solving designed to elicit multiple solution methods for a single nonroutine mathematical problem. The protests and the posttests were identical and consisted of one nonroutine mathematical problem. Each test was evaluated for the total number of different solution methods employed and the level of complexity attained. Hypotheses 16 were used to investigate the number of solution methods employed in relation to three independent variables (treatment, pretest, and creativity) and any possible twoway interactions. Results showed that while there was no significant pretest effect or interactions, the number of solution methods generated on the posttest by students in the treatment groups (2.07) was significantly greater than those who did not experience the treatment (2.00). The posttest number of solution methods was also positively related to the student's creativity as measured by the Torrance Test of Creative Thinking: Figural A. The mean creativity measure for those students who found more than two solution methods was significantly greater than the mean of those who found two or fewer methods. Hypotheses 712 were used to investigate the level of complexity attained in the solution methods in relation to the three independent variables (treatment, pretest, and creativity) and any possible twoway interactions. Results showed that there was a significant pretest and treatment effect, but no significant relationship between a student's measure of creativity and the level of complexity attained. There were no twoway Interactions. The level of complexity attained on the posttest solution methods was significantly related to the experience of receiving the treatment. The mean level of complexity attained by the treatment groups (on a scale of zero to five) was 3.36, while the mean for the nontreatment group was 2.56. There was a significant pretest effect for the complexity level attained on the posttest. The experience of taking the pretest significantly increased the level of complexity attained on the posttest. A repeated measures procedure was employed for hypotheses 13 and 14 to test the treatment effects over time on both the number and complexity of solution methods. The change in the number of solution methods found on the pretest and the number generated on the posttest was different depending on whether the student experienced the treatment. The treatment and control groups were almost equal on mean number of solution methods found on the pretest, but after the experience of the treatment, the treatment group showed dramatic gains while the control group remained approximately the same. The results of the repeated measure for level of complexity was quiet different. While both the treatment and the pretest significantly affected the level of complexity attained on posttest solution methods, the effect of the treatment was not the same for all students who received the treatment. While there was a significant difference within subjects on the complexity attained on the pretest and on the posttest, the lack of significant interaction between time and treatment showed that within subjects, this difference did not significantly depend on whether the student received the treatment. Hypotheses 1518 were used to compare by grade and by nation the number of solution methods and level of complexity attained by students in the U.S. with students in Japan. The U.S. sixth and eighth grade students found significantly more solution methods than did the sixth and eighth grade Japanese students. Followup comparisons revealed that the U.S. eighth grade students found significantly more solution methods than did the Japanese eighth grade sample, but not more than either sixth grade group. There were no significant differences in attained level of complexity when the means were compared by grade or by nation. Additionally, an important finding was made concerning the complexity of students' solution methods. A student's initial solution method tended to be less sophisticated than the other methods generated. As the number of solution methods increased, the mean level of complexity tended to increase for subsequent methods. This trend existed on the pretest and was even more pronounced on the posttest for students in the treatment group. CHAPTER V CONCLUSIONS AND IMPLICATIONS Overview of Study There have been a number of reform movements in mathematics education in the last 40 years but none so radical and encompassing as the reforms outlined in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989), and Professional Standards for Teaching Mathematics (1991). In the ideal classroom every student is guaranteed the opportunity to construct mathematical concepts and develop his or her power to communicate mathematically, reason mathematically, make mathematical connections, and solve problems. This study incorporated each of these strands by focusing on the teaching and learning of reflective problem solving. The purpose of this study was to investigate solution methods employed by middle level students engaged in solving nonroutine mathematical problems and the role of creativity during the reflection process. The solution methods of students who participated in problem solving sessions designed to elicit multiple solution methods for nonroutine mathematical problems were compared to methods employed by students who received no such deliberate practice. The solution methods were compared by number of different methods and attained level of complexity. Additionally, this study included an international perspective. The solution methods of students in Japan in grades six and eight were compared to those of American students. Comparisons of both the number of alternative methods employed and the attained level of complexity were made between nations and between grades. The solution methods of Japanese students for nonroutine mathematical problems provided for a particularly interesting comparison because Japanese achievement scores have received worldwide attention and because of Japanese elementary educators' emphasis on the looking back phase of problem solving. The opportunity to conduct crossnational problem solving research with the Japanese provided a rare and timely opportunity to view the teaching and learning of problem solving through a different set of lenses. For the main study the comparison groups were composed of intact eighth grade algebra and prealgebra classes arranged in a quasi experimental, Solomon fourgroup design. The 217 students were all enrolled in middle level schools in Central Florida. Each of the four groups was composed of one algebra and one prealgebra class. Students in the treatment groups participated in eight problem solving sessions spanning a 10week period. The premeasure and postmeasure consisted of a single nonroutine mathematical problem for which each student was to generate as many solution methods as possible in a given time frame. The problem was one of five included in a previous international study, the U.S.Japan Crossnational Research (Becker, 1992). The format of the problem solving treatment sessions was based on the constructivist theory that supports the claim ". .. students learn mathematics well only when they construct their own mathematical understanding" (National Research Council, 1989, p. 58). In the constructive classroom the teacher does not attempt to dispense knowledge but instead acts as a model and facilitator. The key element in the problem solving sessions was the significance awarded time spent on reflection. Thus, attention was shifted from the product to the process. Since some students generate solution methods that can be classified as innovative when compared to methods employed by their peers, the role of creativity during problem solving was addressed. Each student in the main study was assigned a measure of creativity as determined by the Torrance Test of Creative Thinking: Figural A. Standardized scores for each of the five categories (fluency, flexibility, originality, elaboration, and resistance to premature closure) were used to compute a mean standardized creativity score. The 13 different solution methods generated by the survey sample were ranked according to sophistication or complexity. The rank, based on the level of intellect engaged in the solution method and the generalizability of the method, represents the mean of rankings offered by five mathematics educators considered experts in the field of problem solving. For the U.S.Japan comparisons premeasure results of 100 eighth and 100 sixth grade students were chosen at random from the intact classes who participated in the study. Each of the schools involved, one in Tokyo and one in Hiroshima, are public schools, therefore they represented the Japanese population more closely than would a private school with very selective entrance requirements. Discussion of Results The major findings that resulted from the data analyses were as follows: 1. Both the number of solution methods students were able to generate for a single nonroutine mathematical problem and the level of complexity of those methods were significantly increased when the eighth grade students in the main study were given the opportunity to engage in reflective problem solving activities. The constructivists agree that students need opportunities to construct their own knowledge. The theme of the curriculum and teaching Standards is that students should be involved in actively "doing" mathematics. As a result of incorporating these standards in the treatment, students were not only able to generate a greater number of alternative solution methods but were also able to move to a more generalizable, thus a more sophisticated, method. Even though the treatment problems did not require a formula, some students progressed to that high form of generalization during the reflection process. The number of students who could generate a formula continued to increase throughout the treatment period. This increase could be attributed to a deeper understanding of the problem which developed during reflection as well as to feedback. At the end of each problem solving session students shared individually or in groups the full range of solution methods generated during the session. This communication time was intentionally arranged so that methods would be shared by order of increasing complexity. Thus students were able to witness the extension of many solution methods to the most general case; a formula. While the percentage of students who generated a formula on the pretest and posttest remained at about two percent for the nontreatment group, the percentage of students in the treatment group went from three percent on the pretest to seventeen percent on the posttest. However, once a student obtained a formula, it was often difficult for that student to generate less sophisticated solution methods. During the treatment it was observed that the algebra students were more likely to produce a formula early in the problem solving process, then loose their flexibility to a mind set. The prealgebra students, on the other hand, remained flexible for a longer period and were actually able to generate a greater number of less complex solution methods. For example, this was especially evident during the "horses and ducks " problem (Appendix A) which states that, among the farmers collection of horses and ducks there are 9 heads and 26 feet. Students were to find the number of horses and ducks in the collection. While the algebra students who had just completed a unit on simultaneous equations had difficulty finding a method other than applying their newly acquired skill, the prealgebra students who had not been introduced to this algorithm, generated many creative methods. To the students a formula represented the ultimate solution method. Thus, after finding a formula, they seemed to lose their flexibility and motivation for continuing the search. The algebra students were also more reluctant to recognize unsophisticated methods such as counting or drawing a picture as bonafide. Those who used the less complex solution methods often wished to remain anonymous and only began to look for unsophisticated methods when they realized that such methods would "count." While the percentage of students in the control groups who used counting as a method of solution remained at approximately 48% on both the pre and posttest measures, the percentage of students in the treatment groups who used counting as a method of solution went from approximately 30% on the pretest to 68% on the posttest. Algebra students in search of a formula to represent a problem could gain valuable insights to the problem by constructing a less sophisticated solution method which might then provide the framework for the desired formula. Results from this research suggested that even when algebra students are unable to produce a formula, they do not systematically look for or value less sophisticated solution methods. Results also indicated that students can be trained to employ alternative solution methods as a means for producing a more generalizable solution. 2. Students' initial solution methods tended to be of low complexity with the level of complexity increasing with each new method generated. Although the task of searching for alternative solution methods for a single problem is not a new suggestion (Brownell, 1942; Polya, 1957), justification for investing the time required for such an activity is not apparent in many American classrooms where the objective is "answergetting." Results of this study indicated that with the exception of a few students of algebra, when students are asked to find the answer to a nonroutine mathematical problem, initially they tend to employ a solution method rooted in a basic level of understanding of the problem. When required to search for additional methods, these students are given the opportunity to increase their understanding of the problem's mathematical principles. A deeper understanding of the problem could promote the generation of more complex and thus more generalizable solution methods. Results of this study showed that for the marble problem there was an upward trend in level of complexity with each additional method generated. This phenomenon occurred on the pretest and was even more pronounced on the posttest. Additionally it was observed during the treatment sessions that when students were given problems for which there exists a variety of solution methods within their zone of proximal development, they tended to initially employ solution methods that only required a basic understanding of the problem's mathematical structure. While probing for additional methods students deepened their understanding of the problem and constructed more complex solution methods. 3. The number of solution methods students were able to generate for a single nonroutine mathematical problem was positively related to their measure of creativity but there was no significant relationship between attained level of complexity and measure of creativity. With a shift from the procedural/rote learning model of mathematics education to the constructive model, there is increased opportunity to exercise mathematical creativity. Krutetskit (1976) insists that students must not only master mathematical skills but must also demonstrate mathematical creativity. While Krutetskii tends to associate mathematical creativity with mathematical giftedness, the challenge put forth in today's classroom is that every student be given the opportunity to exercise his or her mathematical creativity. Research conducted by Torrance (1973) demonstrated that creativity can be improved through training. Since younger students have been observed inventing more sophisticated methods of counting on their own (Groen & Resnick, 1977), older creative students might be expected to also invent more sophisticated methods of solving problems. While creativity was positively related to the number of solution methods students were able to generate and the level of complexity tended to increase with each additional method, for this study there was no significant relationship between creativity and attained level of complexity. Since creativity allowed students the flexibility necessary to approach problems from different perspectives, why did creativity not account for a significant amount of the increase in complexity? The attained level of complexity in students' solution methods may be effected more by their opportunity to engage in reflective problem solving than their measure of creativity. This claim was supported by the fact that the level of complexity attained on the posttest was significantly effected by whether the student took the pretest. The increase in level of complexity that resulted from the pretest experience possibly masked differences that may have been related to creativity. It may also indicate that no matter how creative, the sophistication of students' solution methods Is limited by their level of mathematical expertise. Creativity alone cannot substitute for sound mathematical principles and the opportunity to engage in reflective problem solving. 4. When the number of solution methods generated by students in Japan and America were compared on the pretest, American eighth grade students were able to produce a greater number of alternative methods than either the Japanese sixth or eighth grade students or the American sixth grade sample. 5. Although the American eighth grade students were able to generate more alternative solution methods than their Japanese counterparts, there were no differences between the level of complexity attained in the methods. There was no significant difference in the level of complexity attained by any sixth or eighth grade group in the entire study sample. Implications This research adds support for the suggestions outlined in the curriculum and evaluation and the teaching Standards; suggestions that form the basis for the most aggressive reform movement in the history of mathematics education in the United States. Research on teaching and research on learning were brought together as a single focus with ramifications for practitioners at all levels and researchers. This research demonstrated how the role of the teacher must change in order for the students to "do mathematics instead of having it done to them." The format of each lesson allowed students the opportunity to reason mathematically, to make mathematical connections, to communicate mathematically, and to reflect on their problem solving methods. The teacher whose only method of instruction involves an attempt to dispense knowledge in a direct teaching mode is not allowing students the opportunity to construct their own meaningful mathematics. The advances of technology and the rapid growth of information awaiting students who will be entering the work force of the twentyfirst century make it imperative that our students be equipped with more than a vast array of computational skills. One of the most important tasks for teachers in this role of model, guide, and coach is to provide good problems. Good nonroutine mathematical problems are those that can be solved using many alternative methods, at least one of which is within each student's zone of proximal development. Why should a student who has successfully solved a problem be prompted to formulate alternative methods? Results of this research suggested that generally a student's initial solution method was of low complexity. The complexity level tended to increase as additional methods were generated. Therefore, each student can begin solving the problem at his or her individual level of understanding and then progress to more complex solution methods by constructing his or her own mathematical meaning. In addition to becoming a better problem solver, each student must be given the opportunity to develop his or her creative potential. Like any other talent, creativity needs to be nurtured. In a longrange study conducted by Torrance (1972), creative achievement differences between the more creative and less creative subjects tended to widen as time elapsed. In an era when employers are placing an increased emphasis on creative thinking and ideas are, in many cases, as important as technical skills, students must be offered a curriculum designed to exercise their creative potential. Problem solving provides a challenging outlet for creative construction of mathematical principles. Asking students to generate multiple solution methods for good problems allows them to use their creativity to probe for a deeper understanding of the problem's mathematical structure. Headlines have painted a discouraging picture of mathematics education in the United States. However, it is important that educators not make hasty decisions in response to these reports. In his testimony before the U.S. House of Representatives, Huelskamp (1993), for example, warned the Committee on Education and Labor about the importance of decisions based on sound data. His report concluded that "the low opinion educators hold of themselves and the poor public perception of teachers are based on misinterpretations of simplistic data, such as average SAT scores and international comparisons. This unfortunate cycle of low selfesteem, followed by unfounded criticism from the public, raises the specter of a downward spiral in future educational quality" (p. 720). Berliner (1993) has also been concerned that administrative decisions, based on the negative publicity of the failures of American schools, are potentially dangerous. His evaluation of our educational system sheds a much brighter light with claims that the American system has logged remarkable successes for many of our students and parents. While he is able to defend against many of the unfair comparisons that have spawned an attitude of panic, he also concludes that our students are capable of learning more mathematics at an earlier age. If opportunities for students to exercise creative problem solving and critical thinking talents are being postponed until the skills deemed necessary for such activities are internalized, then valuable time for construction of meaningful mathematics is possibly being exchanged for fragmented memorization of rules and computational skills; computation that could often be accomplished with a simple calculator. With differences that exist in individual creative talent widening as children mature (Torrance, 1972), it is imperative that each student be given opportunity to become a creative problem solver. This study provides evidence suggesting that when American students are given the opportunity to unleash their creative talents in reflective problem solving activities, they are able to construct impressive mathematical understanding. Suggestions For Future Research This study not only provided valuable insights about reflective problem solving, it also generated many questions for further research. Since the U.S. sample included only algebra and prealgebra students, the study could be repeated with heterogeneous classes and with classes at different grade levels. Qualitative studies that focus on studentstudent, studentteacher interactions and individual constructions would provide additional clues about how sessions should be structured to maximize creative problem solving behaviors. Although the pre and postmeasure for this research included sections which extended the marble problem to the sixteenth and one hundredth place, those results were not included in the analyses. Questions remain about how students generalize solutions. The treatment problems began with the simplest case and progressed to the more general. When students are directed initially to the general case do they loose the flexibility to generate alternative solution methods? What long range results might we expect if algebra students were systematically trained to solve a simpler case before attempting to find a formula. Would the outcomes be the same for the more capable algebra students and the less able learner? What are the attitudes of students who are required to exercise their creative talents? Students in this study were excited about their problem solving sessions. It was difficult to sort the origins of that attitude but it seemed to be a combination of many factors. Middle level students, characterized by their energy and quick minds, usually welcome a change in their routine no matter how rewarding that routine might be. The subjects in this study felt that our sessions were something other than a mathematics class. This was probably due in part to the fact that there were no grades assigned but they were also aware of the absence of their mathematics book and worksheets. When the creative potential of students is unleashed it becomes the teacher's responsibility to channel the talents in the proper direction. The students needed little encouragement to begin their search for solution methods but required coaching to sustain the effort. Their attitudes about problem solving showed marked Improvement over the course of the treatment with the exception of the sharing sessions. While they were able to realize the importance of sharing methods, they too often found that time boring. They competed for the opportunity to share their own methods but were impatient when it was their classmates' opportunity. This may have been due to the change of pace that occurred during sharing. Students often had difficulty expressing their findings. They did not have the rhetorical skills or the mathematical vocabulary necessary to translate their methods into coherent explanations. Researchers might further explore methods for training students to articulate their mathematical ideas. There continues to be a need for joint international research. The goal should not be to model ourselves after someone else or to impose our system on another culture. Instead, cooperative efforts establish "what is" so that each country can blend a workable model for "what can be." With each possibility there are informed choices to be made. Cooperative research must remain open and honest. The United States and Japan should continue their cooperative research in mathematics education. Each country has much to learn from each other's successes and failures. Duke (1986) believes that the economic growth in Japan has depended extensively on the role of the school. He suggests that Japan could offer the U.S. lessons about perseverance, high expectations for all students, and commitment to the group without forfeiture of individual creativity or freedoms. For lessons Japan might learn from the U.S., Duke points to our class size, diverse teaching methods, flexibility, and alternative evaluation methods. Before we can incorporate lessons from Japan or any other country, researchers must carefully analyze what changes are feasible, make informed predictions of the expected results, and construct workable plans of action. The Standards have built our foundation but there are many unanswered questions for the researcher. Limitations The use of intact classrooms was a limitation in this study. Even thought the students were from similar socioeconomiceconomic backgrounds and shared similar mathematics curriculums and skill levels, there were no common standardized tests that could be used for matching students on their mathematics proficiency. A better crosssection of students on SES and mathematical ability would have increased validity and reliability for this research. Time was also a limitation in this study. While each treatment session was awarded an entire period of instruction, time for students to think about the problem and incubate their ideas was limited. How might the results have been different if students were introduced to a nonroutine problem and then encouraged to think about it over a longer period? Illumination often comes after sleep or at unexpected moments. Creative thinking cannot always be scheduled. The time for treatment was also limited. Ideally, students should be creatively challenged on a daily basis and encouraged to regularly engage in reflective problem solving. While significant differences resulted after only eight problem solving sessions, the results were not intended to imply that problem solving should be scheduled for a specific time frame. The hypotheses tested in this research could be the basis for a longitudinal study. Although interesting, results of the international comparisons in this study were limited in statistical merit since there was no way to match subjects. Effort was made to choose students who could be compared fairly but there was no instrument to determine the degree of compatibility. No measure of the Japanese student's creative talent was available. The international component of this research was only possible because of the communication and cooperation of those who assisted from Japan. The lack of controls compounded by distance and language barriers limited the validity of the crossnational findings. Conclusions The purpose of this research was to investigate methods employed by middle level students in the solution of a nonroutine mathematical problem. The number of alternative methods generated and the level of complexity attained was compared between students who participated in a treatment and those who received no such deliberate practice. Since some students are able to generate solution methods that are unusual when compared to those of their peers, the role of creativity was investigated. An international perspective was added to this research with the inclusion of data from students in Japan. During the eight treatment sessions designed to encourage the generation of multiple solution methods, students not only significantly increased the number of alternative solutions they were able to generate, but also the level of complexity in their solution methods. While a student's measure of creativity was positively related to the number of methods generated, a relationship between attained level of complexity and creativity could not be established. Complexity level was significantly related to the experience of taking the pretest. With the exception of a few algebra students who often began their solution search with an attempt to set up a formula, a student's initial solution method tended to be less complex than their subsequent methods. Thus, when students do not reflect on a problem with the intention of finding alternative solution methods, their single solution method likely represents only a basic understanding of the problem's mathematical principles. The level of understanding represented and the generalizability of the method tended to increase with each new method generated. This phenomenon was established on the pretest and was even more evident on the posttest results. Even though middle level students in Japan have spent about the same amount of time studying mathematics as American students two to three grades their senior, the results of this research did not support the hypothesis that Japanese sixth grade students and American eighth grade students demonstrate approximately the same level of problem solving ability. American eleventh grade students found significantly more solution methods and attained a higher level of complexity in their methods than did any other group in the study. At a time when problems of American mathematics education are a frequent theme of the headlines, we need to evaluate what is positive about our system and capitalize on our discoveries. This research reinforced the belief that students are capable of constructing their own mathematical meanings. Their creativity allowed them the flexibility to consider problems from a variety of perspectives and generate more generalizable solution methods through a deeper understanding of the problem. While creativity cannot replace sound mathematical principles, it can be harnessed to personalize mathematics comprehension. 