The Mathematics Educator 2004, Vol. 14, No. 1, 19–34 The Characteristics of Mathematical Bharath Sriraman

Mathematical creativity ensures the growth of mathematics as a whole. However, the source of this growth, the creativity of the mathematician, is a relatively unexplored area in mathematics and . In order to investigate how mathematicians create mathematics, a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts and to verify the theory driven hypotheses. The results indicate that, in general, the mathematicians’ creative processes followed the four-stage Gestalt model of preparation-incubation-illumination-verification. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. Additionally, contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity .

Mathematical creativity ensures the growth of the does one define creativity? In particular what exactly is field of mathematics as a whole. The constant increase mathematical creativity? Is it the discovery of a new in the of journals devoted to mathematics theorem by a research mathematician? Does student research bears evidence to the growth of mathematics. discovery of a hitherto known result also constitute Yet what lies at the essence of this growth, the creativity? These are among the areas of exploration in creativity of the mathematician, has not been the this paper. subject of much research. It is usually the case that The Problem Of Defining Creativity most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation Mathematical creativity has been simply described (Ervynck, 1991). The earliest known attempt to study as discernment, or choice (Poincaré, 1948). According mathematical creativity was an extensive questionnaire to Poincaré (1948), to create consists precisely in not published in the French periodical L'Enseigement making useless combinations and in making those Mathematique (1902). This questionnaire and a lecture which are useful and which are only a small minority. on creativity given by the renowned 20th century Poincaré is referring to the fact that the “proper” mathematician Henri Poincaré to the Societé de combination of only a small minority of ideas results in Psychologie inspired his colleague Jacques Hadamard, a creative insight whereas a majority of such another prominent 20th century mathematician, to combinations does not result in a creative outcome. investigate the psychology of mathematical creativity This may seem like a vague characterization of (Hadamard, 1945). Hadamard (1945) undertook an mathematical creativity. One can interpret Poincaré's informal inquiry among prominent mathematicians and "choice" metaphor to mean the ability of the scientists in America, including George Birkhoff, mathematician to choose carefully between questions George Polya, and Albert Einstein, about the mental (or problems) that bear fruition, as opposed to those images used in doing mathematics. Hadamard (1945), that lead to nothing new. But this interpretation does influenced by the Gestalt psychology of his time, not resolve the fact that Poincaré’s definition of theorized that mathematicians’ creative processes creativity overlooks the problem of novelty. In other followed the four-stage Gestalt model (Wallas, 1926) words, characterizing mathematical creativity as the of preparation-incubation-illumination-verification. ability to choose between useful and useless As we will see, the four-stage Gestalt model is a combinations is akin to characterizing the art of characterization of the mathematician's creative sculpting as a process of cutting away the unnecessary! process, but it does not define creativity per se. How Poincaré's (1948) definition of creativity was a result of the circumstances under which he stumbled Bharath Sriraman is an assistant professor of mathematics and upon deep results in Fuchsian functions. The first stage mathematics education at the University of Montana. His in creativity consists of working hard to get an insight publications and research interests are in the areas of cognition, foundational issues, mathematical creativity, problem-solving, into the problem at hand. Poincaré (1948) called this proof, and gifted education. the preliminary period of conscious work. This period is also referred to as the preparatory stage (Hadamard,

Bharath Sriraman 19 1945). In the second, or incubatory, stage (Hadamard, generalize mathematical content. There is also an 1945), the problem is put aside for a period of time and outstanding example of a mathematician (George the mind is occupied with other problems. In the third Polya) attempting to give heuristics to tackle problems stage the solution suddenly appears while the in a manner akin to the methods used by trained mathematician is perhaps engaged in other unrelated mathematicians. Polya (1954) observed that in "trying activities. "This appearance of sudden illumination is a to solve a problem, we consider different aspects of it manifest sign of long, unconscious prior work" in turn, we roll it over and over in our minds; variation (Poincaré, 1948). Hadamard (1945) referred to this as of the problem is essential to our work." Polya (1954) the illuminatory stage. However, the creative process emphasized the use of a variety of heuristics for does not end here. There is a fourth and final stage, solving mathematical problems of varying complexity. which consists of expressing the results in language or In examining the plausibility of a mathematical writing. At this stage, one verifies the result, makes it conjecture, mathematicians use a variety of strategies. precise, and looks for possible extensions through In looking for conspicuous patterns, mathematicians utilization of the result. The “Gestalt model” has some use such heuristics as (1) verifying consequences, (2) shortcomings. First, the model mainly applies to successively verifying several consequences, (3) problems that have been posed a priori by verifying an improbable consequence, (4) inferring mathematicians, thereby ignoring the fascinating from analogy, and (5) deepening the analogy. process by which the actual questions evolve. As is evident in the preceding paragraphs, the Additionally, the model attributes a large portion of problem of defining creativity is by no means an easy what “happens” in the incubatory and illuminatory one. However, psychologists’ renewed interest in the phases to subconscious drives. The first of these phenomenon of creativity has resulted in literature that shortcomings, the problem of how questions are attempts to define and operationalize the word developed, is partially addressed by Ervynck (1991) in “creativity.” Recently psychologists have attempted to his three-stage model. link creativity to measures of intelligence (Sternberg, Ervynck (1991) described mathematical creativity 1985) and to the ability to abstract, generalize in terms of three stages. The first stage (Stage 0) is (Sternberg, 1985), and solve complex problems referred to as the preliminary technical stage, which (Frensch & Sternberg, 1992). Sternberg and Lubart consists of "some kind of technical or practical (2000) define creativity as the ability to produce application of mathematical rules and procedures, unexpected original work that is useful and adaptive. without the user having any awareness of the Mathematicians would raise several arguments with theoretical foundation" (p. 42). The second stage this definition, simply because the results of creative (Stage 1) is that of algorithmic activity, which consists work may not always have implications that are primarily of performing mathematical techniques, such “useful” in terms of applicability in the contemporary as explicitly applying an algorithm repeatedly. The world. A recent example that comes to mind is Andrew third stage (Stage 2) is referred to as creative Wiles’ proof of Fermat’s Last Theorem. The (conceptual, constructive) activity. This is the stage in mathematical community views his work as creative. It which true mathematical creativity occurs and consists was unexpected and original but had no applicability in of non-algorithmic decision making. "The decisions the sense Sternberg and Lubart (2000) suggest. Hence, that have to be taken may be of a widely divergent I think it is sufficient to define creativity as the ability nature and always involve a choice" (p. 43). Although to produce novel or original work, which is compatible Ervynck (1991) tries to describe the process by which a with my personal definition of mathematical creativity mathematician arrives at the questions through his as the process that results in unusual and insightful characterizations of Stage 0 and Stage 1, his solutions to a given problem, irrespective of the level description of mathematical creativity is very similar to of complexity. In the context of this study involving those of Poincaré and Hadamard. In particular his use professional mathematicians, mathematical creativity is of the term “non-algorithmic decision making” is defined as the publication of original results in analogous to Poincaré’s use of the “choice” metaphor. prominent mathematics research journals. The mathematics education literature indicates that The Motivation For Studying Creativity very few attempts have been made to explicitly define mathematical creativity. There are references made to The lack of recent mathematics education literature creativity by the Soviet researcher Krutetskii (1976) in on the subject of creativity was one of the motivations the context of students’ abilities to abstract and for conducting this study. Fifteen years ago Muir

20 Mathematical Creativity (1988) invited mathematicians to complete a modified (intuitionist) viewpoint is that “human mathematical and updated version of the survey that appeared in activity is fundamental in the creation of new L'Enseigement Mathematique (1902) but the results of knowledge and that both mathematical truths and the this endeavor are as yet unknown. The purpose of this existence of mathematical objects must be established study was to gain insight into the nature of by constructive methods" (Ernest, 1991, p. 29). mathematical creativity. I was interested in distilling Contradictions like Russell’s Paradox were a major common attributes of the creative process to see if blow to the absolutist view of mathematical there were any underlying themes that characterized knowledge, for if mathematics is certain and all its mathematical creativity. The specific questions of theorems are certain, how can there be contradictions exploration in this study were: among its theorems? The early constructivists in mathematics were the intuitionists Brouwer and Is the Gestalt model of mathematical creativity still applicable today? Heyting. Constructivists claim that both mathematical truths and the existence of mathematical objects must What are the characteristics of the creative process be established by constructivist methods. in mathematics? The question then is how does a mathematician go Does the study of mathematical creativity have any about conducting mathematics research? Do the implications for the classroom? questions appear out of the blue, or is there a mode of Literature Review thinking or inquiry that leads to meaningful questions and to the methodology for tackling these questions? I Any study on the nature of mathematical creativity contend that the types of questions asked are begs the question as to whether the mathematician determined to a large extent by the culture in which the discovers or invents mathematics. Therefore, this mathematician lives and works. Simply put, it is review begins with a brief description of the four most impossible for an individual to acquire knowledge of popular viewpoints on the nature of mathematics. This the external world without social interaction. is followed by a comprehensive review of According to Ernest (1994) there is no underlying contemporary models of creativity from psychology. metaphor for the wholly isolated individual mind. The Nature of Mathematics Instead, the underlying metaphor is that of persons in conversation, persons who participate in meaningful Mathematicians actively involved in research have linguistic interaction and dialogue (Ernest, 1994). certain beliefs about the ontological nature of Language is the shaper, as well as being the mathematics that influence their approach to research “summative” product, of individual minds (Davis & Hersh, 1981; Sriraman, 2004a). The Platonist (Wittgenstein, 1978). The recent literature in viewpoint is that mathematical objects exist prior to psychology acknowledges these social dimensions of their discovery and that “any meaningful question human activity as being instrumental in the creative about a mathematical object has a definite answer, process. whether we are able to determine it or not” (Davis & Hersh, 1981). According to this view, mathematicians The Notion of Creativity in Psychology do not invent or create mathematics - they discover As stated earlier, research on creativity has been on mathematics. Logicists hold that “all concepts of the fringes of psychology, educational psychology, and mathematics can ultimately be reduced to logical mathematics education. It is only in the last twenty-five concepts” which implies that “all mathematical truths years that there has been a renewed interest in the can be proved from the axioms and rules of inference phenomenon of creativity in the psychology and logic alone” (Ernest, 1991). Formalists do not community. The Handbook of Creativity (Sternberg, believe that mathematics is discovered; they believe 2000), which contains a comprehensive review of all mathematics is simply a game, created by research then available in the field of creativity, mathematicians, based on strings of symbols that have suggests that most of the approaches used in the study no meaning (Davis & Hersh, 1981). of creativity can be subsumed under six categories: Constructivism (incorporating Intuitionism) is one mystical, pragmatic, psychodynamic, psychometric, of the major schools of thought (besides Platonism, cognitive, and social-personality. Each of these Logicism and Formalism) that arose due to the approaches is briefly reviewed. contradictions that emerged in the development of the theory of sets and the theory of functions during the early part of the 20th century. The constructivist

Bharath Sriraman 21 The mystical approach as Albert Einstein, but the behaviorists criticized this The mystical approach to studying creativity approach because of the difficulty in measuring suggests that creativity is the result of divine proposed theoretical constructs. inspiration or is a spiritual process. In the history of The psychometric approach mathematics, Blaise Pascal claimed that many of his The psychometric approach to studying creativity mathematical insights came directly from God. The entails quantifying the notion of creativity with the aid renowned 19th century algebraist Leopold Kronecker of paper and pencil tasks. An example of this would be said that “God made the integers, all the rest is the the Torrance Tests of Creative Thinking, developed by work of man” (Gallian, 1994). Kronecker believed that Torrance (1974), that are used by many gifted all other , being the work of man, were to be programs in middle and high schools to identify avoided; and although his radical beliefs did not attract students that are gifted/creative. These tests consist of many supporters, the intuitionists advocated his beliefs several verbal and figural tasks that call for problem- about constructive proofs many years after his death. solving skills and divergent thinking. The test is scored There have been attempts to explore possible for fluency, flexibility, originality (the statistical rarity relationships between mathematicians’ beliefs about of a response), and elaboration (Sternberg, 2000). the nature of mathematics and their creativity (Davis Sternberg (2000) states that there are positive and and Hersh, 1981; Hadamard, 1945; Poincaré, 1948; negative sides to the psychometric approach. On the Sriraman, 2004a). These studies indicate that such a positive side, these tests allow for research with non- relationship does exist. It is commonly believed that eminent people, are easy to administer, and objectively the neo-Platonist view is helpful to the research scored. The negative side is that numerical scores fail mathematician because of the innate belief that the to capture the concept of creativity because they are sought after result/relationship already exists. based on brief paper and pencil tests. Researchers call The pragmatic approach for using more significant productions such as writing The pragmatic approach entails “being concerned samples, drawings, etc., subjectively evaluated by a primarily with developing creativity” (Sternberg, 2000, panel of experts, instead of simply relying on a p. 5), as opposed to understanding it. Polya’s (1954) numerical measure. emphasis on the use of a variety of heuristics for The cognitive approach solving mathematical problems of varying complexity The cognitive approach to the study of creativity is an example of a pragmatic approach. Thus, focuses on understanding the “mental representations heuristics can be viewed as a decision-making and processes underlying human thought” (Sternberg, mechanism which leads the mathematician down a 2000, p. 7). Weisberg (1993) suggests that creativity certain path, the outcome of which may or may not be entails the use of ordinary cognitive processes and fruitful. The popular technique of brainstorming, often results in original and extraordinary products. These used in corporate or other business settings, is another products are the result of cognitive processes acting on example of inducing creativity by seeking as many the knowledge already stored in the memory of the ideas or solutions as possible in a non-critical setting. individual. There is a significant amount of literature in The psychodynamic approach the area of information processing (Birkhoff, 1969; The psychodynamic approach to studying Minsky, 1985) that attempts to isolate and explain creativity is based on the idea that creativity arises cognitive processes in terms of machine metaphors. from the tension between conscious reality and The social-personality approach unconscious drives (Hadamard, 1945; Poincaré, 1948, The social-personality approach to studying Sternberg, 2000, Wallas, 1926; Wertheimer, 1945). creativity focuses on personality and motivational The four-step Gestalt model (preparation-incubation- variables as well as the socio-cultural environment as illumination-verification) is an example of the use of a sources of creativity. Sternberg (2000) states that psychodynamic approach to studying creativity. It numerous studies conducted at the societal level should be noted that the gestalt model has served as indicate that “eminent levels of creativity over large kindling for many contemporary problem-solving spans of time are statistically linked to variables such models (Polya, 1945; Schoenfeld, 1985; Lester, 1985). as cultural diversity, war, availability of role models, Early psychodynamic approaches to creativity were availability of financial support, and competitors in a used to construct case studies of eminent creators such domain” (p. 9).

22 Mathematical Creativity Most of the recent literature on creativity (Csikzentmihalyi, 2000). These three components - (Csikszentmihalyi, 1988, 2000; Gruber & Wallace, individual, domain, and field - are necessary because 2000; Sternberg & Lubart, 1996) suggests that the individual operates from a cultural or symbolic creativity is the result of a confluence of one or more (domain) aspect as well as a social (field) aspect. of the factors from these six aforementioned “The domain is a necessary component of categories. The “confluence” approach to the study of creativity because it is impossible to introduce a creativity has gained credibility, and the research variation without reference to an existing pattern. New literature has numerous confluence theories for better is meaningful only in reference to the old” understanding the process of creativity. A review of the (Csikzentmihalyi, 2000). Thus, creativity occurs when most commonly cited confluence theories of creativity an individual proposes a change in a given domain, and a description of the methodology employed for which is then transmitted by the field through time. data collection and data analysis in this study follow. The personal background of an individual and his position in a domain naturally influence the likelihood Confluence Theories of Creativity of his making a contribution. For example, a The three most commonly cited “confluence” mathematician working at a research university is more approaches to the study of creativity are the “systems likely to produce research papers because of the time approach” (Csikszentmihalyi, 1988, 2000); “the case available for “thinking” as well as the creative study as evolving systems approach” (Gruber & influence of being immersed in a culture where ideas Wallace, 2000), and the “investment theory approach” flourish. It is no coincidence that in the history of (Sternberg & Lubart, 1996). The case study as an science, there are significant contributions from evolving system has the following components. First, it clergymen such as Pascal and Mendel because they views creative work as multi-faceted. So, in had the means and the leisure to “think.” constructing a case study of a creative work, one must Csikszentmihalyi (2000) argues that novel ideas, which distill the facets that are relevant and construct the case could result in significant changes, are unlikely to be study based on the chosen facets. Some facets that can adopted unless they are sanctioned by the experts. be used to construct an evolving system case study are: These “gatekeepers” (experts) constitute the field. For (1) uniqueness of the work; (2) a narrative of what the example, in mathematics, the opinion of a very small creator achieved; (3) systems of belief; (4) multiple number of leading researchers was enough to certify time-scales (construct the time-scales involved in the the validity of Andrew Wiles’ proof of Fermat’s Last production of the creative work); (5) problem solving; Theorem. and (6) contextual frame such as family, schooling, and There are numerous examples in the history of teacher’s influences (Gruber & Wallace, 2000). In mathematics that fall within the systems model. For summary, constructing a case study of a creative work instance, the Bourbaki, a group of mostly French as an evolving system entails incorporating the many mathematicians who began meeting in the 1930s, facets suggested by Gruber & Wallace (2000). One aimed to write a thorough unified account of all could also evaluate a case study involving creative mathematics. The Bourbaki were essentially a group of work by looking for the above mentioned facets. expert mathematicians that tried to unify all of The systems approach mathematics and become the gatekeepers of the field, so to speak, by setting the standard for rigor. Although The systems approach takes into account the social the Bourbakists failed in their attempt, students of the and cultural dimensions of creativity instead of simply Bourbakists, who are editors of certain prominent viewing creativity as an individualistic psychological journals, to this day demand a very high degree of rigor process. The systems approach studies the interaction in submitted articles, thereby serving as gatekeepers of between the individual, domain, and field. The field the field. consists of people who have influence over a domain. A different example is that of the role of proof. For example, editors of mathematics research journals Proof is the social process through which the have influence over the domain of mathematics. The mathematical community validates the mathematician's domain is in a sense a cultural organism that preserves creative work (Hanna, 1991). The Russian logician and transmits creative products to individuals in the Manin (1977) said "A proof becomes a proof after the field. The systems model suggests that creativity is a social act of accepting it as a proof. This is true of process that is observable at the “intersection where mathematics as it is of physics, linguistics, and individuals, domains and fields interact” biology."

Bharath Sriraman 23 In summary, the systems model of creativity years of numerical calculations. Andrew Wiles’ proof suggests that for creativity to occur, a set of rules and of Fermat’s Last Theorem was a seven-year practices must be transmitted from the domain to the undertaking. The states that the individual. The individual then must produce a novel roots of the zeta function (complex numbers z, at variation in the content of the domain, and this which the zeta function equals zero) lie on the line variation must be selected by the field for inclusion in parallel to the imaginary axis and half a unit to the the domain. right of it. This is perhaps the most outstanding unproved conjecture in mathematics with numerous Gruber and Wallace’s case study as evolving implications. The analyst Levinson undertook a systems approach determined calculation on his deathbed that increased In contrast to Csikszentmihalyi’s (2000) argument the credibility of the Riemann-hypothesis. This is calling for a focus on communities in which creativity another example of creative work that falls within manifests itself, Gruber and Wallace (2000) propose a Gruber and Wallace's (2000) model. model that treats each individual as a unique evolving system of creativity and ideas; and, therefore, each The investment theory approach individual’s creative work must be studied on its own. According to the investment theory model, creative This viewpoint of Gruber and Wallace (2000) is a people are like good investors; that is, they buy low belated victory of sorts for the Gestaltists, who and sell high (Sternberg & Lubart, 1996). The context essentially proclaimed the same thing almost a century here is naturally in the realm of ideas. Creative people ago. Gruber and Wallace’s (2000) use of terminology conjure up ideas that are either unpopular or that jibes with current trends in psychology seems to disrespected and invest considerable time convincing make their ideas more acceptable. They propose a other people about the intrinsic worth of these ideas model that calls for “detailed analytic and sometimes (Sternberg & Lubart, 1996). They sell high in the sense narrative descriptions of each case and efforts to that they let other people pursue their ideas while they understand each case as a unique functioning system move on to the next idea. Investment theory claims that (Gruber & Wallace, 2000, p. 93). It is important to note the convergence of six elements constitutes creativity. that the emphasis of this model is not to explain the The six elements are intelligence, knowledge, thinking origins of creativity, nor is it the personality of the styles, personality, motivation, and environment. It is creative individual, but on “how creative work works” important that the reader not mistake the word (p. 94). The questions of concern to Gruber and intelligence for an IQ score. On the contrary, Sternberg Wallace are: (1) What do creative people do when they (1985) suggests a triarchic theory of intelligence that are being creative? and (2) How do creative people consists of synthetic (ability to generate novel, task deploy available resources to accomplish something appropriate ideas), analytic, and practical abilities. unique? In this model creative work is defined as that Knowledge is defined as knowing enough about a which is novel and has value. This definition is particular field to move it forward. Thinking styles are consistent with that used by current researchers in defined as a preference for thinking in original ways of creativity (Csikszentmihalyi, 2000; Sternberg & one’s choosing, the ability to think globally as well as Lubart, 2000). Gruber and Wallace (2000) also claim locally, and the ability to distinguish questions of that creative work is always the result of purposeful importance from those that are not important. behavior and that creative work is usually a long Personality attributes that foster creative functioning undertaking “reckoned in months, years and decades” are the willingness to take risks, overcome obstacles, (p. 94). and tolerate ambiguity. Finally, motivation and an I do not agree with the claim that creative work is environment that is supportive and rewarding are always the result of purposeful behavior. One essential elements of creativity (Sternberg, 1985). counterexample that comes to mind is the discovery of In investment theory, creativity involves the penicillin. The discovery of penicillin could be interaction between a person, task, and environment. attributed purely to chance. On the other hand, there This is, in a sense, a particular case of the systems are numerous examples that support the claim that model (Csikszentmihalyi, 2000). The implication of creative work sometimes entails work that spans years, viewing creativity as the interaction between person, and in mathematical folklore there are numerous task, and environment is that what is considered novel examples of such creative work. For example, Kepler’s or original may vary from one person, task, and laws of planetary motion were the result of twenty environment to another. The investment theory model

24 Mathematical Creativity suggests that creativity is more than a simple sum of Background of the Subjects the attained level of functioning in each of the six Five mathematicians from the mathematical elements. Regardless of the functioning levels in other sciences faculty at a large Ph.D. granting mid-western elements, a certain level or threshold of knowledge is university were selected. These mathematicians were required without which creativity is impossible. High chosen based on their accomplishments and the levels of intelligence and motivation can positively diversity of the mathematical areas in which they enhance creativity, and compensations can occur to worked, measured by counting the number of counteract weaknesses in other elements. For example, published papers in prominent journals, as well as one could be in an environment that is non-supportive noting the variety of mathematical domains in which of creative efforts, but a high level of motivation could they conducted research. Four of the mathematicians possibly overcome this and encourage the pursuit of were tenured full professors, each of whom had been creative endeavors. professional mathematicians for more than 30 years. This concludes the review of three commonly cited One of the mathematicians was considerably younger prototypical confluence theories of creativity, namely but was a tenured associate professor. All interviews the systems approach (Csikszentmihalyi, 2000), which were conducted formally, in a closed door setting, in suggests that creativity is a sociocultural process each mathematician’s office. The interviews were involving the interaction between the individual, audiotaped and transcribed verbatim. domain, and field; Gruber & Wallace’s (2000) model that treats each individual case study as a unique Data Analysis evolving system of creativity; and investment theory Since creativity is an extremely complex construct (Sternberg & Lubart, 1996), which suggests that involving a wide range of interacting behaviors, I creativity is the result of the convergence of six believe it should be studied holistically. The principle elements (intelligence, knowledge, thinking styles, of analytic induction (Patton, 2002) was applied to the personality, motivation, and environment). interview transcripts to discover dominant themes that Having reviewed the research literature on described the behavior under study. According to creativity, the focus is shifted to the methodology Patton (2002), "analytic induction, in contrast to employed for studying mathematical creativity. grounded theory, begins with an analyst's deduced Methodology propositions or theory-derived hypotheses and is a procedure for verifying theories and propositions based The Interview Instrument on qualitative data” (Taylor and Bogdan, 1984, p. 127). The purpose of this study was to gain an insight Following the principles of analytic induction, the data into the nature of mathematical creativity. In an effort was carefully analyzed in order to extract common to determine some of the characteristics of the creative strands. These strands were then compared to process, I was interested in distilling common theoretical constructs in the existing literature with the attributes in the ways mathematicians create explicit purpose of verifying whether the Gestalt model mathematics. Additionally, I was interested in testing was applicable to this qualitative data as well as to the applicability of the Gestalt model. Because the extract themes that characterized the mathematician’s main focus of the study was to ascertain qualitative creative process. If an emerging theme could not be aspects of creativity, a formal interview methodology classified or named because I was unable to grasp its was selected as the primary method of data collection. properties or significance, then theoretical comparisons The interview instrument (Appendix A) was developed were made. Corbin and Strauss (1998) state that “using by modifying questions from questionnaires in comparisons brings out properties, which in turn can be L’Enseigement Mathematique (1902) and Muir (1988). used to examine the incident or object in the data. The The rationale behind using this modified questionnaire specific incidents, objects, or actions that we use when was to allow the mathematicians to express themselves making theoretical comparisons can be derived from freely while responding to questions of a general the literature and experience. It is not that we use nature and to enable me to test the applicability of the experience or literature as data “but rather that we use four-stage Gestalt model of creativity. Therefore, the the properties and dimensions derived from the existing instruments were modified to operationalize comparative incidents to examine the data in front of the Gestalt theory and to encourage the natural flow of us” (p. 80). Themes that emerged were social ideas, thereby forming the basis of a thesis that would interaction, preparation, use of heuristics, imagery, emerge from this exploration. incubation, illumination, verification, intuition, and

Bharath Sriraman 25 proof. Excerpts from interviews that highlight these valued the interaction they had with their graduate characteristics are reconstructed in the next section students. Excerpts of individual responses follow.1 along with commentaries that incorporate the wider Excerpt 1 conversation, and a continuous discussion of connections to the existing literature. A. I've had only one graduate student per semester and she is just finishing up her PhD right now, Results, Commentaries & Discussion and I'd say it has been a very good interaction The mathematicians in this study worked in to see somebody else get interested in the academic environments and regularly fulfilled teaching subject and come up with new ideas, and and committee duties. The mathematicians were free to exploring those ideas with her. choose their areas of research and the problems on B. I have had a couple of students who have sort which they focused. Four of the five mathematicians of started but who haven't continued on to a had worked and published as individuals and as PhD, so I really can't speak to that. But the members of occasional joint ventures with interaction was positive. mathematicians from other universities. Only one of the mathematicians had done extensive collaborative C. Of course, I have a lot of collaborators, these work. All but one of the mathematicians were unable are my former students you know…I am to formally structure their time for research, primarily always all the time working with students, this due to family commitments and teaching is normal situation. responsibilities during the regular school year. All the D. That is difficult to answer (silence)…it is mathematicians found it easier to concentrate on positive because it is good to interact with research in the summers because of lighter or non- other people. It is negative because it can take existent teaching responsibilities during that time. Two a lot of time. As you get older your brain of the mathematicians showed a pre-disposition doesn't work as well as it used to towards mathematics at the early secondary school and…younger people by and large their minds level. The others became interested in mathematics are more open, there is less garbage in there later, during their university education. The already. So, it is exciting to work with younger mathematicians who participated in this study did not people who are in their most creative time. report any immediate family influence that was of When you are older, you have more primary importance in their mathematical experience, when you are younger your mind development. Four of the mathematicians recalled works faster …not as fettered. being influenced by particular teachers, and one reported being influenced by a textbook. The three E. Oh…it is a positive factor I think, because it mathematicians who worked primarily in analysis continues to stimulate ideas …talking about made a conscious effort to obtain a broad overview of things and it also reviews things for you in the mathematics not necessarily of immediate relevance to process, puts things in perspective, and keep their main interests. The two algebraists expressed the big picture. It is helpful really in your own interest in other areas of mathematics but were research to supervise students. primarily active in their chosen field. Commentary on Excerpt 1 Supervision Of Research & Social Interaction The responses of the mathematicians in the As noted earlier, all the mathematicians in this preceding excerpt are focused on research supervision; study were tenured professors in a research university. however, all of the mathematicians acknowledged the In addition to teaching, conducting research, and role of social interaction in general as an important fulfilling committee obligations, many mathematicians aspect that stimulated creative work. Many of the play a big role in mentoring graduate students mathematicians mentioned the advantages of being interested in their areas of research. Research able to e-mail colleagues and going to research supervision is an aspect of creativity because any conferences and other professional meetings. This is interaction between human beings is an ideal setting further explored in the following section, which for the exchange of ideas. During this interaction the focuses on preparation. mathematician is exposed to different perspectives on the subject, and all of the mathematicians in this study

26 Mathematical Creativity Preparation and the Use of Heuristics own. That doesn't mean that I don't When mathematicians are about to investigate a simultaneously try to work on something. new topic, there is usually a body of existing research Commentary on Excerpt 2 in the area of the new topic. One of goals of this study These responses indicate that the mathematician was to find out how creative mathematicians spends a considerable amount of time researching the approached a new topic or a problem. Did they try their context of the problem. This is primarily done by own approach, or did they first attempt to assimilate reading the existing literature and by talking to other what was already known about that topic? Did the mathematicians in the new area. This finding is mathematicians make use of computers to gain insight consistent with the systems model, which suggests that into the problem? What were the various modes of creativity is a dynamic process involving the approaching a new topic or problem? The responses interaction between the individual, domain, and field indicate that a variety of approaches were used. (Csikzentmihalyi, 2000). At this stage, it is reasonable Excerpt 2 to ask whether a mathematician works on a single problem until a breakthrough occurs or does a A. Talk to people who have been doing this topic. mathematician work on several problems concurrently? Learn the types of questions that come up. It was found that each of the mathematicians worked Then I do basic research on the main ideas. I on several problems concurrently, using a back and find that talking to people helps a lot more than forth approach. reading because you get more of a feel for what the motivation is beneath everything. Excerpt 3 B. What might happen for me, is that I may start A. I work on several different problems for a reading something, and, if feel I can do a better protracted period of time… there have been job, then I would strike off on my own. But for times when I have felt, yes, I should be able to the most part I would like to not have to prove this result, then I would concentrate on reinvent a lot that is already there. So, a lot of that thing for a while but they tend to be what has motivated my research has been the several different things that I was thinking desire to understand an area. So, if somebody about a particular stage. has already laid the groundwork then it's B. I probably tend to work on several problems at helpful. Still I think a large part of doing the same time. There are several different research is to read the work that other people questions that I am working have done. on…mm…probably the real question is how C. It is connected with one thing that simply…my often do you change the focus? Do I work on style was that I worked very much and I even two different problems on the same day? And work when I could not work. Simply the that is probably up to whatever comes to mind problems that I solve attract me so much, that in that particular time frame. I might start the question was who will die working on one rather than the other. But I first…mathematics or me? It was never clear would tend to focus on one particular problem who would die. for a period of weeks, then you switch to something else. Probably what happens is that D. Try and find out what is known. I won't say I work on something and I reach a dead end assimilate…try and find out what's known and then I may shift gears and work on a different get an overview, and try and let the problem problem for a while, reach a dead end there speak…mostly by reading because you don't and come back to the original problem, so it’s have that much immediate contact with other back and forth. people in the field. But I find that I get more from listening to talks that other people are C. I must simply think on one thing and not giving than reading. switch so much. E. Well! I have been taught to be a good scholar. D. I find that I probably work on one. There A good scholar attempts to find out what is might be a couple of things floating around but first known about something or other before I am working on one and if I am not getting they spend their time simply going it on their

Bharath Sriraman 27 anywhere, then I might work on the other and work. I have a very geometrically based intuition and then go back. uhh…so very definitely I do a lot of manipulations. E. I usually have couple of things going. When I A. That is a problem because of the particular get stale on one, then I will pick up the other, area I am in. I can't draw any diagrams, things and bounce back and forth. Usually I have one are infinite, so I would love to be able to get that is primarily my focus at a given time, and some kind of a computer diagram to show the I will spend time on it over another; but it is complexity for a particular ring… to have not uncommon for me to have a couple of something like the Julia sets problems going at a given time. Sometimes or…mmm…fractal images, things which are when I am looking for an example that is not infinite but you can focus in closer and closer coming, instead of spending my time beating to see possible relationships. I have thought my head against the wall, looking for that about that with possibilities on the computer. example is not a very good use of time. To think about the most basic ring, you would Working on another helps to generate ideas have to think of the ring of integers and all of that I can bring back to the other problem. the relationships for divisibility, so how do you somehow describe this tree of divisibility for Commentary on Excerpt 3 integers…it is infinite. The preceding excerpt indicates that mathematicians tend to work on more than one B. Science is language, you think through problem at a given time. Do mathematicians switch language. But it is language simply; you put back and forth between problems in a completely together theorems by logic. You first see the random manner, or do they employ and exhaust a theorem in nature…you must see that systematic train of thought about a problem before somewhat is reasonable and then you go and switching to a different problem? Many of the begin and then of course there is big, big, big mathematicians reported using heuristic reasoning, work to just come to some theorem in non- trying to prove something one day and disprove it the linear elliptic equations… next day, looking for both examples and C. A lot of mathematics, whether we are teaching counterexamples, the use of "manipulations" (Polya, or doing, is attaching meaning to what we are 1954) to gain an insight into the problem. This doing and this is going back to the earlier indicates that mathematicians do employ some of the question when you talked about how do you do heuristics made explicit by Polya. It was unclear it, what kind of heuristics do you use? What whether the mathematicians made use of computers to kind of images do you have that you are using? gain an experimental or computational insight into the A lot of doing mathematics is creating these problem. I was also interested in knowing the types of abstract images that connect things and then imagery used by mathematicians in their work. The making sense of them but that doesn't appear mathematicians in this study were queried about this, in proofs either. and the following excerpt gives us an insight into that D. Pictorial, linguistic, kinesthetic...any of them is aspect of mathematical creativity. the point right! Sometimes you think of one, Imagery sometimes another. It really depends on the The mathematicians in this study were asked about problem you are looking at, they are very the kinds of imagery they used to think about much…often I think of functions as very mathematical objects. Their responses are reported kinesthetic, moving things from here to there. here to give the reader a glimpse of the ways Other approaches you are talking about is mathematicians think of mathematical objects. Their going to vary from problem to problem, or responses also highlight the difficulty of explicitly even day to day. Sometimes when I am describing imagery. working on research, I try to view things in as many different ways as possible, to see what is Excerpt 4 really happening. So there are a variety of Yes I do, yes I do, I tend to draw a lot of pictures approaches. when I am doing research, I tend to manipulate things in the air, you know to try to figure out how things

28 Mathematical Creativity Commentary on Excerpt 4 preparation, so that the sub-conscious or Besides revealing the difficulty of describing intuitive side may work on it and the answer mental imagery, all the mathematicians reported that comes back but you can't really tell when. You they did not use computers in their work. This have to be open to this, lay the groundwork, characteristic of the pure mathematician's work is think about it and then these flashes of echoed in Poincaré's (1948) use of the “choice” intuition come and they represent the other metaphor and Ervynck's (1991) use of the term “non- side of the brain communicating with you at algorithmic decision making.” The doubts expressed whatever odd time. by the mathematicians about the incapability of D. I am not sure you can really separate them machines to do their work brings to mind the reported because they are somewhat connected. You words of Garrett Birkhoff, one of the great applied spend a lot of time working on something and mathematicians of our time. In his retirement you are not getting anywhere with it…with the presidential address to the Society for Industrial and deliberate effort, then I think your mind Applied Mathematics, Birkhoff (1969) addressed the continues to work and organize. And maybe role of machines in human creative endeavors. In when the pressure is off the idea comes…but particular, part of this address was devoted to the idea comes because of the hard work. discussing the psychology of the mathematicians (and E. Usually they come after I have worked very hence of mathematics). Birkhoff (1969) said: hard on something or another, but they may The remarkable recent achievements of computers come at an odd moment. They may come into have partially fulfilled an old dream. These my head before I go to bed …What do I do at achievements have led some people to speculate that point? Yes I write it down (laughing). that tomorrow's computers will be even more Sometimes when I am walking somewhere, the "intelligent" than humans, especially in their mind flows back to it (the problem) and says powers of mathematical reasoning...the ability of good mathematicians to sense the significant and to what about that, why don't you try that. That avoid undue repetition seems, however, hard to sort of thing happens. One of the best ideas I computerize; without it, the computer has to pursue had was when I was working on my thesis millions of fruitless paths avoided by experienced …Saturday night, having worked on it quite a human mathematicians. (pp. 430-438) bit, sitting back and saying why don't I think Incubation and Illumination about it again…and ping! There it was…I knew what it was, I could do that. Often ideas Having reported on the role of research supervision are handed to you from the outside, but they and social interaction, the use of heuristics and don't come until you have worked on it long imagery, all of which can be viewed as aspects of the enough. preparatory stage of mathematical creativity, it is natural to ask what occurs next. As the literature Commentary on Excerpt 5 suggests, after the mathematician works hard to gain As is evident in the preceding excerpt, three out of insight into a problem, there is usually a transition the five mathematicians reported experiences period (conscious work on the problem ceases and consistent with the Gestalt model. Mathematician C unconscious work begins), during which the problem is attributed his breakthroughs on problems to his put aside before the breakthrough occurs. The unflinching will to never give up and to divine mathematicians in this study reported experiences that inspiration, echoing the voice of Pascal in a sense. are consistent with the existing literature (Hadamard, However, Mathematician A attributed breakthroughs to 1945; Poincaré, 1948). chance. In other words, making the appropriate Excerpt 5 (psychological) connections by pure chance which eventually result in the sought after result. B. One of the problems is first one does some I think it is necessary to comment about the preparatory work, that has to be the left side unusual view of mathematician A. Chance plays an [of the brain], and then you let it sit. I don't important role in mathematical creativity. Great ideas think you get ideas out of nowhere, you have and insights may be the result of chance such as the to do the groundwork first, okay. This is why discovery of penicillin. Ulam (1976) estimated that people will say, now we have worked on this there is a yearly output of 200,000 theorems in problem, so let us sleep on it. So you do the

Bharath Sriraman 29 mathematics. Chance plays a role in what is considered field, is one instance of a unification of apparently important in mathematical research since only a random fragments because the proof involves algebra, handful of results and techniques survive out of the , and . volumes of published research. I wish to draw a Polya (1954) addresses the role of chance in a distinction between chance in the "Darwinian" sense probabilistic sense. It often occurs in mathematics that (as to what survives), and chance in the psychological a series of mathematical trials (involving computation) sense (which results in discovery/invention). The role generate numbers that are close to a Platonic ideal. The of chance is addressed by Muir (1988) as follows. classic example is Euler's investigation of the infinite series 1 + 1/4 + 1/9 + 1/16 +…+ 1/n2 +…. Euler The act of creation of new entities has two aspects: the generation of new possibilities, for which we obtained an approximate numerical value for the sum might attempt a stochastic description, and the of the series using various transformations of the selection of what is valuable from among them. series. The numerical approximation was 1.644934. However the importation of biological metaphors Euler confidently guessed the sum of the series to be to explain cultural evolution is dubious…both π2/6. Although the numerical value obtained by Euler creation and selection are acts of design within a and the value of π2/6 coincided up to seven decimal social context. (p. 33) places, such a coincidence could be attributed to Thus, Muir (1988) rejects the Darwinian chance. However, a simple calculation shows that the explanation. On the other hand, Nicolle (1932) in probability of seven digits coinciding is one in ten Biologie de L'Invention does not acknowledge the role million! Hence, Euler did not attribute this coincidence of unconsciously present prior work in the creative to chance but boldly conjectured that the sum of this process. He attributes breakthroughs to pure chance. series was indeed π2/6 and later proved his conjecture to be true (Polya, 1954, pp. 95-96). By a streak of lightning, the hitherto obscure problem, which no ordinary feeble lamp would Intuition, Verification and Proof have revealed, is at once flooded in light. It is like a creation. Contrary to progressive acquirements, Once illumination has occurred, whether through such an act owes nothing to logic or to reason. The sheer chance, incubation, or divine intervention, act of discovery is an accident. (Hadamard, 1945) mathematicians usually try to verify that their intuitions were correct with the construction of a proof. Nicolle's Darwinian explanation was rejected by The following section discusses how these Hadamard on the grounds that to claim creation occurs mathematicians went about the business of verifying by pure chance is equivalent to asserting that there are their intuitions and the role of formal proof in the effects without causes. Hadamard further argued that creative process. They were asked whether they relied although Poincaré attributed his particular on repeatedly checking a formal proof, used multiple breakthrough in Fuchsian functions to chance, Poincaré converging partial proofs, looked first for coherence did acknowledge that there was a considerable amount with other results in the area, or looked at applications. of previous conscious effort, followed by a period of Most of the mathematicians in this study mentioned unconscious work. Hadamard (1945) further argued that the last thing they looked at was a formal proof. that even if Poincaré's breakthrough was the result of This is consistent with the literature on the role of chance alone, chance alone was insufficient to explain formal proof in mathematics (Polya, 1954; Usiskin, the considerable body of creative work credited to 1987). Most of the mathematicians mentioned the need Poincaré in almost every area of mathematics. The for coherence with other results in the area. The question then is how does (psychological) chance mathematician’s responses to the posed question work? follow. It is my conjecture that the mind throws out fragments (ideas) that are products of past experience. Excerpt 6 Some of these fragments can be juxtaposed and B. I think I would go for repeated checking of the combined in a meaningful way. For example, if one formal proof…but I don't think that that is reads a complicated proof consisting of a thousand really enough. All of the others have to also be steps, a thousand random fragments may not be enough taken into account. I mean, you can believe to construct a meaningful proof. However the mind that something is true although you may not chooses relevant fragments from these random fully understand it. This is the point that was fragments and links them into something meaningful. made in the lecture by … of … University on Wedderburn's Theorem, that a finite division ring is a

30 Mathematical Creativity Dirichlet series. He was saying that we have process. “Mathematics in the making resembles any had a formal proof for some time, but that is other human knowledge in the making. The result of not to say that it is really understood, and what the mathematician’s creative work is demonstrative did he mean by that? Not that the proof wasn't reasoning, a proof; but the proof is discovered by understood, but it was the implications of the plausible reasoning, by guessing” (Polya, 1954). How result that are not understood, their mathematicians approached proof in this study was connections with other results, applications very different from the logical approach found in proof and why things really work. But probably the in most textbooks. The logical approach is an artificial first thing that I would really want to do is reconstruction of discoveries that are being forced into check the formal proof to my satisfaction, so a deductive system, and in this process the intuition that I believe that it is correct although at that that guided the discovery process gets lost. point I really do not understand its Conclusions implications… it is safe to say that it is my surest guide. The goal of this study was to gain an insight into mathematical creativity. As suggested by the literature

C. First you must see it in the nature, something, review, the existing literature on mathematical first you must see that this theorem creativity is relatively sparse. In trying to better corresponds to something in nature, then if you understand the process of creativity, I find that the have this impression, it is something relatively Gestalt model proposed by Hadamard (1945) is still reasonable, then you go to proofs…and of applicable today. This study has attempted to add some course I have also several theorems and proofs detail to the preparation-incubation-illumination- that are wrong, but the major amount of proofs verification model of Gestalt by taking into account the and theorems are right. role of imagery, the role of intuition, the role of social D. The last thing that comes is the formal proof. I interaction, the use of heuristics, and the necessity of look for analogies with other things… How proof in the creative process. your results that you think might be true would The mathematicians worked in a setting that was illuminate other things and would fit in the conducive to prolonged research. There was a general structure. convergence of intelligence, knowledge, thinking styles, personality, motivation and environment that E. Since I work in an area of basic research, it is enabled them to work creatively (Sternberg, 2000; usually coherence with other things, that is Sternberg & Lubart, 1996, 2000). The preparatory probably more than anything else. Yes, one stage of mathematical creativity consists of various could go back and check the proof and that sort approaches used by the mathematician to lay the of thing but usually the applications are yet to groundwork. These include reading the existing come, they aren't there already. Usually what literature, talking to other mathematicians in the guides the choice of the problem is the particular mathematical domain (Csikzentmihalyi, potential for application, part of what 1988; 2000), trying a variety of heuristics (Polya, represents good problems is their potential for 1954), and using a back-and-forth approach of use. So, you certainly look to see if it makes plausible guessing. One of the mathematicians said that sense in the big picture…that is a coherence he first looked to see if the sought after relationships phenomenon. Among those you've given me, corresponded to natural phenomenon. that’s probably the most that fits. All of the mathematicians in this study worked on Commentary on Excerpt 6 more than one problem at a given moment. This is This excerpt indicates that for mathematicians, consistent with the investment theory view of creativity valid proofs have varied degrees of rigor. “Among (Sternberg & Lubart, 1996). The mathematicians mathematicians, rigor varies depending on time and invested an optimal amount of time on a given circumstance, and few proofs in mathematics journals problem, but switched to a different problem if no meet the criteria used by secondary school geometry breakthrough was forthcoming. All the mathematicians in this study considered this as the most important and teachers (each statement of proof is backed by reasons). Generally one increases rigor only when the difficult stage of creativity. The prolonged hard work result does not seem to be correct” (Usiskin, 1987). was followed by a period of incubation where the Proofs are in most cases the final step in this testing problem was put aside, often while the preparatory

Bharath Sriraman 31 stage is repeated for a different problem; and thus, Implications there is a transition in the mind from conscious to It is in the best interest of the field of mathematics unconscious work on the problem. One mathematician education that we identify and nurture creative talent in cited this as the stage at which the "problem begins to the mathematics classroom. "Between the work of a talk to you." Another offered that the intuitive side of student who tries to solve a difficult problem in the brain begins communicating with the logical side at mathematics and a work of invention (creation)…there this stage and conjectured that this communication was is only a difference of degree" (Polya, 1954). not possible at a conscious level. Creativity as a feature of mathematical thinking is not a The transition from incubation to illumination patent of the mathematician! (Krutetskii, 1976); and often occurred when least expected. Many reported the although most studies on creativity have focused on breakthrough occurring as they were going to bed, or eminent individuals (Arnheim, 1962; Gardner, 1993, walking, or sometimes as a result of speaking to 1997; Gruber, 1981), I suggest that contemporary someone else about the problem. One mathematician models from creativity research can be adapted for illustrated this transition with the following: "You talk studying samples of creativity such as are produced by to somebody and they say just something that might high school students. Such studies would reveal more have been very ordinary a month before but if they say about creativity in the classroom to the mathematics it when you are ready for it, and Oh yeah, I can do it education research community. Educators could that way, can’t I! But you have to be ready for it. consider how often mathematical creativity is Opportunity knocks but you have to be able to answer manifested in the school classroom and how teachers the door." might identify creative work. One plausible way to Illumination is followed by the mathematician’s approach these concerns is to reconstruct and evaluate verifying the result. In this study, most of the student work as a unique evolving system of creativity mathematicians looked for coherence of the result with (Gruber & Wallace, 2000) or to incorporate some of other existing results in the area of research. If the the facets suggested by Gruber & Wallace (2000). This result cohered with other results and fit the general necessitates the need to find suitable problems at the structure of the area, only then did the mathematician appropriate levels to stimulate student creativity. try to construct a formal proof. In terms of the A common trait among mathematicians is the mathematician’s beliefs about the nature of reliance on particular cases, isomorphic reformulations, mathematics and its influence on their research, the or analogous problems that simulate the original study revealed that four of the mathematicians leaned problem situations in their search for a solution (Polya, towards Platonism, in contrast to the popular notion 1954; Skemp, 1986). Creating original mathematics that Platonism is an exception today. A detailed requires a very high level of motivation, persistence, discussion of this aspect of the research is beyond the and reflection, all of which are considered indicators of scope of this paper; however, I have found that beliefs creativity (Amabile, 1983; Policastro & Gardner, 2000; regarding the nature of mathematics not only Gardner, 1993). The literature suggests that most influenced how these mathematicians conducted creative individuals tend to be attracted to complexity, research but also were deeply connected to their of which most school mathematics curricula has very theological beliefs (Sriraman, 2004a). little to offer. Classroom practices and math curricula The mathematicians hoped that the results of their rarely use problems with the sort of underlying creative work would be sanctioned by a group of mathematical structure that would necessitate students’ experts in order to get the work included in the domain having a prolonged period of engagement and the (Csikzentmihalyi, 1988, 2000), primarily in the form of independence to formulate solutions. It is my publication in a prominent journal. However, the conjecture that in order for mathematical creativity to acceptance of a mathematical result, the end product of manifest itself in the classroom, students should be creation, does not ensure its survival in the Darwinian given the opportunity to tackle non-routine problems sense (Muir, 1988). The mathematical result may or with complexity and structure - problems which may not be picked up by other mathematicians. If the require not only motivation and persistence but also mathematical community picks it up as a viable result, considerable reflection. This implies that educators then it is likely to undergo mutations and lead to new should recognize the value of allowing students to mathematics. This, however, is determined by chance! reflect on previously solved problems to draw comparisons between various isomorphic problems (English, 1991, 1993; Hung, 2000; Maher & Kiczek,

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New York: Basic Books. Skemp, R. (1986). The psychology of learning mathematics. Gruber, H. E. (1981). Darwin on man. Chicago: University of Middlesex, UK: Penguin Books. Chicago Press. Sriraman, B. (2003). Mathematical giftedness, problem solving, Gruber, H. E., & Wallace, D. B. (2000). The case study method and and the ability to formulate generalizations. The Journal of evolving systems approach for understanding unique creative Secondary Gifted Education. XIV(3), 151−165. people at work. In R. J. Sternberg (Ed.), Handbook of Sriraman, B. (2004a). The influence of Platonism on mathematics creativity (pp. 93-115). Cambridge UK: Cambridge University research and theological beliefs. Theology and Science, 2(1), Press. 131−147.

Bharath Sriraman 33 Sriraman, B. (2004b). Discovering a mathematical principle: The APPENDIX A: Interview Protocol case of Matt. Mathematics in School (UK), 3(2), 25−31. The interview instrument was developed by modifying questions Sternberg, R. J. (1985). Human abilities: An information from questionnaires in L’Enseigement Mathematique (1902) and processing approach. New York: W. H. Freeman. Muir (1988). Sternberg, R. J. (2000). Handbook of creativity. Cambridge, UK: 1. Describe your place of work and your role within it. Cambridge University Press. 2. Are you free to choose the mathematical problems you tackle Sternberg, R. J., & Lubart, T. I. (1996). Investing in creativity. or are they determined by your work place? American Psychologist, 51, 677−688. 3. Do you work and publish mainly as an individual or as part of a group? Sternberg, R. J., & Lubart, T. I. (2000). The concept of creativity: 4. Is supervision of research a positive or negative factor in your Prospects and paradigms. In R. J. Sternberg (Ed.), Handbook work? of creativity (pp. 93 115). Cambridge, UK: Cambridge − 5. Do you structure your time for mathematics? University Press. 6. What are your favorite leisure activities apart from Taylor, S. J., & Bogdan, R. (1984). Introduction to qualitative mathematics? research methods: The search for meanings. New York: John 7. Do you recall any immediate family influences, teachers, Wiley & Sons. colleagues or texts, of primary importance in your Torrance, E. P. (1974). Torrance tests of creative thinking: Norms- mathematical development? technical manual. Lexington, MA: Ginn. 8. In which areas were you initially self-educated? In which areas do you work now? If different, what have been the Ulam, S. (1976). Adventures of a mathematician. New York: Scribners. reasons for changing? 9. Do you strive to obtain a broad overview of mathematics not Usiskin, Z. P. (1987). Resolving the continuing dilemmas in school of immediate relevance to your area of research? geometry. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning 10. Do you make a distinction between thought processes in and teaching geometry, K-12: 1987 yearbook (pp. 17−31). learning and research? Reston, VA: National Council of Teachers of Mathematics. 11. When you are about to begin a new topic, do you prefer to Wallas, G. (1926). The art of thought. New York: Harcourt, Brace assimilate what is known first or do you try your own & Jovanovich. approach? Weisberg, R. W. (1993). Creativity: Beyond the myth of genius. 12. Do you concentrate on one problem for a protracted period of New York: Freeman. time or on several problems at the same time? 13. Have your best ideas been the result of prolonged deliberate Wertheimer, M. (1945). Productive thinking. New York: Harper. effort or have they occurred when you were engaged in other Wittgenstein, L. (1978). Remarks on the foundations of unrelated tasks? mathematics (Rev. Ed.).Cambridge: Massachusetts Institute of 14. How do you form an intuition about the truth of a proposition? Technology Press. 15. Do computers play a role in your creative work (mathematical thinking)? 16. What types of mental imagery do you use when thinking about mathematical objects?

Note: Questions regarding foundational and theological issues have been omitted in this protocol. The discussion resulting from these questions are reported in Sriraman (2004a).

34 Mathematical Creativity