The Characteristics of Mathematical Creativity Quotes

“In terms of the mathematician’s beliefs about the nature of mathematics and its influence on their research, the study revealed that four of the mathematicians leaned towards Platonism, in contrast to the popular notion that Platonism is an exception today. A detailed discussion of this aspect of the research is beyond the scope of this paper; however, I have found that beliefs regarding the nature of mathematics not only influenced how these mathematicians conducted research but also were deeply connected to their theological beliefs (Sriraman, 2004a).”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“Creating original mathematics requires a very high level of motivation, persistence, and reflection, all of which are considered indicators of creativity (Amabile, 1983; Policastro & Gardner, 2000; Gardner, 1993). The literature suggests that most creative individuals tend to be attracted to complexity, of which most school mathematics curricula has very little to offer. Classroom practices and math curricula rarely use problems with the sort of underlying mathematical structure that would necessitate students’ having a prolonged period of engagement and the independence to formulate solutions. It is my conjecture that in order for mathematical creativity to manifest itself in the classroom, students should be given the opportunity to tackle non-routine problems with complexity and structure - problems which require not only motivation and persistence but also considerable reflection.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The transition from incubation to illumination often occurred when least expected. Many reported the breakthrough occurring as they were going to bed, or walking, or sometimes as a result of speaking to someone else about the problem. One mathematician illustrated this transition with the following: "You talk to somebody and they say just something that might have been very ordinary a month before but if they say it when you are ready for it, and Oh yeah, I can do it that way, can’t I! But you have to be ready for it. Opportunity knocks but you have to be able to answer the door.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“All of the mathematicians in this study worked on more than one problem at a given moment. This is consistent with the investment theory view of creativity (Sternberg & Lubart, 1996). The mathematicians invested an optimal amount of time on a given problem, but switched to a different problem if no breakthrough was forthcoming. All the mathematicians in this study considered this as the most important and difficult stage of creativity. The prolonged hard work was followed by a period of incubation where the problem was put aside, often while the preparatory stage is repeated for a different problem; and thus, there is a transition in the mind from conscious to unconscious work on the problem. One mathematician cited this as the stage at which the "problem begins to talk to you." Another offered that the intuitive side of the brain begins communicating with the logical side at this stage and conjectured that this communication was not possible at a conscious level.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“Besides revealing the difficulty of describing mental imagery, all the mathematicians reported that they did not use computers in their work. This characteristic of the pure mathematician's work is echoed in Poincaré's (1948) use of the “choice” metaphor and Ervynck's (1991) use of the term “nonalgorithmic decision making.” The doubts expressed by the mathematicians about the incapability of machines to do their work brings to mind the reported words of Garrett Birkhoff, one of the great applied mathematicians of our time. In his retirement presidential address to the Society for Industrial and Applied Mathematics, Birkhoff (1969) addressed the role of machines in human creative endeavors. In particular, part of this address was devoted to discussing the psychology of the mathematicians (and hence of mathematics). Birkhoff (1969) said: The remarkable recent achievements of computers have partially fulfilled an old dream. These achievements have led some people to speculate that tomorrow's computers will be even more "intelligent" than humans, especially in their powers of mathematical reasoning...the ability of good mathematicians to sense the significant and to avoid undue repetition seems, however, hard to computerize; without it, the computer has to pursue millions of fruitless paths avoided by experienced human mathematicians. (pp. 430-438)”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The preceding excerpt indicates that mathematicians tend to work on more than one problem at a given time. Do mathematicians switch back and forth between problems in a completely random manner, or do they employ and exhaust a systematic train of thought about a problem before switching to a different problem? Many of the mathematicians reported using heuristic reasoning, trying to prove something one day and disprove it the next day, looking for both examples and counterexamples, the use of "manipulations" (Polya, 1954) to gain an insight into the problem. This indicates that mathematicians do employ some of the heuristics made explicit by Polya. It was unclear whether the mathematicians made use of computers to gain an experimental or computational insight into the problem.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“These responses indicate that the mathematician spends a considerable amount of time researching the context of the problem. This is primarily done by reading the existing literature and by talking to other mathematicians in the new area. This finding is consistent with the systems model, which suggests that creativity is a dynamic process involving the interaction between the individual, domain, and field (Csikzentmihalyi, 2000).”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“This concludes the review of three commonly cited prototypical confluence theories of creativity, namely the systems approach (Csikszentmihalyi, 2000), which suggests that creativity is a sociocultural process involving the interaction between the individual, domain, and field; Gruber & Wallace’s (2000) model that treats each individual case study as a unique evolving system of creativity; and investment theory (Sternberg & Lubart, 1996), which suggests that creativity is the result of the convergence of six elements (intelligence, knowledge, thinking styles, personality, motivation, and environment).”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The investment theory model
Bharath Sriraman 25
suggests that creativity is more than a simple sum of the attained level of functioning in each of the six elements. Regardless of the functioning levels in other elements, a certain level or threshold of knowledge is required without which creativity is impossible. High levels of intelligence and motivation can positively enhance creativity, and compensations can occur to counteract weaknesses in other elements. For example, one could be in an environment that is non-supportive of creative efforts, but a high level of motivation could possibly overcome this and encourage the pursuit of creative endeavors.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“Investment theory claims that the convergence of six elements constitutes creativity. The six elements are intelligence, knowledge, thinking styles, personality, motivation, and environment. It is important that the reader not mistake the word intelligence for an IQ score. On the contrary, Sternberg (1985) suggests a triarchic theory of intelligence that consists of synthetic (ability to generate novel, task appropriate ideas), analytic, and practical abilities. Knowledge is defined as knowing enough about a particular field to move it forward. Thinking styles are defined as a preference for thinking in original ways of one’s choosing, the ability to think globally as well as locally, and the ability to distinguish questions of importance from those that are not important. Personality attributes that foster creative functioning are the willingness to take risks, overcome obstacles, and tolerate ambiguity. Finally, motivation and an environment that is supportive and rewarding are essential elements of creativity (Sternberg, 1985).”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The Riemann hypothesis states that the roots of the zeta function (complex numbers z, at which the zeta function equals zero) lie on the line parallel to the imaginary axis and half a unit to the right of it. This is perhaps the most outstanding unproved conjecture in mathematics with numerous implications. The analyst Levinson undertook a determined calculation on his deathbed that increased the credibility of the Riemann-hypothesis. This is another example of creative work that falls within Gruber and Wallace's (2000) model.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The case study as an evolving system has the following components. First, it views creative work as multi-faceted. So, in constructing a case study of a creative work, one must distill the facets that are relevant and construct the case study based on the chosen facets. Some facets that can be used to construct an evolving system case study are: (1) uniqueness of the work; (2) a narrative of what the creator achieved; (3) systems of belief; (4) multiple time-scales (construct the time-scales involved in the production of the creative work); (5) problem solving; and (6) contextual frame such as family, schooling, and teacher’s influences (Gruber & Wallace, 2000). In summary, constructing a case study of a creative work as an evolving system entails incorporating the many facets suggested by Gruber & Wallace (2000). One could also evaluate a case study involving creative work by looking for the above mentioned facets.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“Most of the recent literature on creativity (Csikszentmihalyi, 1988, 2000; Gruber & Wallace, 2000; Sternberg & Lubart, 1996) suggests that creativity is the result of a confluence of one or more of the factors from these six aforementioned categories. The “confluence” approach to the study of creativity has gained credibility, and the research literature has numerous confluence theories for better understanding the process of creativity. A review of the most commonly cited confluence theories of creativity and a description of the methodology employed for data collection and data analysis in this study follow.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The social-personality approach to studying creativity focuses on personality and motivational variables as well as the socio-cultural environment as sources of creativity. Sternberg (2000) states that numerous studies conducted at the societal level indicate that “eminent levels of creativity over large spans of time are statistically linked to variables such as cultural diversity, war, availability of role models, availability of financial support, and competitors in a domain” (p. 9).”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The cognitive approach to the study of creativity focuses on understanding the “mental representations and processes underlying human thought” (Sternberg, 2000, p. 7). Weisberg (1993) suggests that creativity entails the use of ordinary cognitive processes and results in original and extraordinary products. These products are the result of cognitive processes acting on the knowledge already stored in the memory of the individual. There is a significant amount of literature in the area of information processing (Birkhoff, 1969; Minsky, 1985) that attempts to isolate and explain cognitive processes in terms of machine metaphors.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The psychometric approach to studying creativity entails quantifying the notion of creativity with the aid of paper and pencil tasks. An example of this would be the Torrance Tests of Creative Thinking, developed by Torrance (1974), that are used by many gifted programs in middle and high schools to identify students that are gifted/creative. These tests consist of several verbal and figural tasks that call for problemsolving skills and divergent thinking. The test is scored for fluency, flexibility, originality (the statistical rarity of a response), and elaboration (Sternberg, 2000). Sternberg (2000) states that there are positive and negative sides to the psychometric approach. On the positive side, these tests allow for research with noneminent people, are easy to administer, and objectively scored. The negative side is that numerical scores fail to capture the concept of creativity because they are based on brief paper and pencil tests. Researchers call for using more significant productions such as writing samples, drawings, etc., subjectively evaluated by a panel of experts, instead of simply relying on a numerical measure.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The psychodynamic approach to studying creativity is based on the idea that creativity arises from the tension between conscious reality and unconscious drives (Hadamard, 1945; Poincaré, 1948, Sternberg, 2000, Wallas, 1926; Wertheimer, 1945). The four-step Gestalt model (preparation-incubationillumination-verification) is an example of the use of a psychodynamic approach to studying creativity. It should be noted that the gestalt model has served as kindling for many contemporary problem-solving models (Polya, 1945; Schoenfeld, 1985; Lester, 1985). Early psychodynamic approaches to creativity were used to construct case studies of eminent creators such
as Albert Einstein, but the behaviorists criticized this approach because of the difficulty in measuring proposed theoretical constructs.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The pragmatic approach entails “being concerned primarily with developing creativity” (Sternberg, 2000, p. 5), as opposed to understanding it. Polya’s (1954) emphasis on the use of a variety of heuristics for solving mathematical problems of varying complexity is an example of a pragmatic approach. Thus, heuristics can be viewed as a decision-making mechanism which leads the mathematician down a certain path, the outcome of which may or may not be fruitful. The popular technique of brainstorming, often used in corporate or other business settings, is another example of inducing creativity by seeking as many ideas or solutions as possible in a non-critical setting.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity

“The mystical approach to studying creativity suggests that creativity is the result of divine inspiration or is a spiritual process. In the history of mathematics, Blaise Pascal claimed that many of his mathematical insights came directly from God. The renowned 19th century algebraist Leopold Kronecker said that “God made the integers, all the rest is the work of man” (Gallian, 1994). Kronecker believed that all other numbers, being the work of man, were to be avoided; and although his radical beliefs did not attract many supporters, the intuitionists advocated his beliefs about constructive proofs many years after his death. There have been attempts to explore possible relationships between mathematicians’ beliefs about the nature of mathematics and their creativity (Davis and Hersh, 1981; Hadamard, 1945; Poincaré, 1948; Sriraman, 2004a). These studies indicate that such a relationship does exist. It is commonly believed that the neo-Platonist view is helpful to the research mathematician because of the innate belief that the sought after result/relationship already exists.”
― Bharath Sriraman, The Characteristics of Mathematical Creativity