## The Characteristics of Mathematical Creativity

Citations: | 1 - 0 self |

### BibTeX

@MISC{Sriraman_thecharacteristics,

author = {Bharath Sriraman},

title = {The Characteristics of Mathematical Creativity},

year = {}

}

### OpenURL

### Abstract

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 mathematics education. 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 field of mathematics as a whole. The constant increase in the number of journals devoted to mathematics research bears evidence to the growth of mathematics. Yet what lies at the essence of this growth, the creativity of the mathematician, has not been the subject of much research. It is usually the case that most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation (Ervynck, 1991). The earliest known attempt to study mathematical creativity was an extensive questionnaire published in the French periodical L'Enseigement Mathematique (1902). This questionnaire and a lecture on creativity given by the renowned 20th century mathematician Henri Poincaré to the Societé de Psychologie inspired his colleague Jacques Hadamard, another prominent 20th century mathematician, to investigate the psychology of mathematical creativity

### Citations

636 |
The Society of Mind
- Minsky
- 1985
(Show Context)
Citation Context ...esult 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. The social-personality approach The social-personality approach to studying creativity focuses on personality ... |

124 |
The art of thought
- Wallas
- 1926
(Show Context)
Citation Context ...he mental images used in doing mathematics. Hadamard (1945), influenced by the Gestalt psychology of his time, theorized that mathematicians’ creative processes followed the four-stage Gestalt model (=-=Wallas, 1926-=-) of preparation-incubation-illumination-verification. As we will see, the four-stage Gestalt model is a characterization of the mathematician's creative process, but it does not define creativity per... |

119 | Mathematical problem solving - Schoenfeld - 1985 |

116 |
Qualitative research & evaluation methods. Thousand Oaks
- Patton
(Show Context)
Citation Context ...im. Data Analysis Since creativity is an extremely complex construct involving a wide range of interacting behaviors, I believe it should be studied holistically. The principle of analytic induction (=-=Patton, 2002-=-) was applied to the interview transcripts to discover dominant themes that described the behavior under study. According to Patton (2002), "analytic induction, in contrast to grounded theory, begins ... |

98 |
Handbook of creativity
- Sternberg
- 1999
(Show Context)
Citation Context ...gy, and mathematics education. It is only in the last twenty-five years that there has been a renewed interest in the phenomenon of creativity in the psychology community. The Handbook of Creativity (=-=Sternberg, 2000-=-), which contains a comprehensive review of all research then available in the field of creativity, suggests that most of the approaches used in the study of creativity can be subsumed under six categ... |

85 |
Frames of Mind
- Gardner
- 1993
(Show Context)
Citation Context ... Creativity as a feature of mathematical thinking is not a patent of the mathematician! (Krutetskii, 1976); and although most studies on creativity have focused on eminent individuals (Arnheim, 1962; =-=Gardner, 1993-=-, 1997; Gruber, 1981), I suggest that contemporary models from creativity research can be adapted for studying samples of creativity such as are produced by high school students. Such studies would re... |

76 | The Psychology of Invention in the Mathematical Field - Hadamard - 1945 |

58 |
Productive thinking
- Wertheimer
- 1959
(Show Context)
Citation Context ...ch 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 ... |

56 |
The mathematical experience
- Davis, Hersh
- 1981
(Show Context)
Citation Context ... 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 mathemati... |

55 |
The Social Psychology of Creativity: A Componential Conceptualization
- Amabile
- 1983
(Show Context)
Citation Context ...h for a solution (Polya, 1954; Skemp, 1986). 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 o... |

50 |
Contemporary Abstract Algebra
- Gallian
- 1994
(Show Context)
Citation Context ...ascal 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... |

48 | Adventures of a mathematician - Ulam - 1976 |

45 |
The psychology of learning mathematics
- Skemp
- 1986
(Show Context)
Citation Context ...t among mathematicians is the reliance on particular cases, isomorphic reformulations, or analogous problems that simulate the original problem situations in their search for a solution (Polya, 1954; =-=Skemp, 1986-=-). 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;... |

44 |
The concept of creativity: Prospects and paradigm
- Sternberg, Lubart
- 1999
(Show Context)
Citation Context ...mething unique? In this model creative work is defined as that which is novel and has value. This definition is consistent with that used by current researchers in creativity (Csikszentmihalyi, 2000; =-=Sternberg & Lubart, 2000-=-). Gruber and Wallace (2000) also claim that creative work is always the result of purposeful behavior and that creative work is usually a long undertaking “reckoned in months, years and decades” (p. ... |

43 |
Society, culture, and person: A systems view of creativity. In: The nature of creativity
- Csikszentmihalyi
- 1988
(Show Context)
Citation Context ...s cultural diversity, war, availability of role models, availability of financial support, and competitors in a domain” (p. 9). 22 Mathematical CreativityMost 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... |

42 |
The psychology of mathematical abilities in school children
- Krutetskii
- 1976
(Show Context)
Citation Context ... problem in mathematics and a work of invention (creation)…there is only a difference of degree" (Polya, 1954). Creativity as a feature of mathematical thinking is not a patent of the mathematician! (=-=Krutetskii, 1976-=-); and although most studies on creativity have focused on eminent individuals (Arnheim, 1962; Gardner, 1993, 1997; Gruber, 1981), I suggest that contemporary models from creativity research can be ad... |

37 |
Introduction to qualitative research methods: The search for meanings
- Taylor, Bogdan
- 1984
(Show Context)
Citation Context ...tion, in contrast to grounded theory, begins with an analyst's deduced propositions or theory-derived hypotheses and is a procedure for verifying theories and propositions based on qualitative data” (=-=Taylor and Bogdan, 1984-=-, p. 127). Following the principles of analytic induction, the data was carefully analyzed in order to extract common strands. These strands were then compared to theoretical constructs in the existin... |

35 |
Implications of a systems perspective for the study of creativity
- Csikszentmihalyi
- 1999
(Show Context)
Citation Context ...sources to accomplish something unique? In this model creative work is defined as that which is novel and has value. This definition is consistent with that used by current researchers in creativity (=-=Csikszentmihalyi, 2000-=-; Sternberg & Lubart, 2000). Gruber and Wallace (2000) also claim that creative work is always the result of purposeful behavior and that creative work is usually a long undertaking “reckoned in month... |

35 |
The Philosophy of Mathematics Education
- Ernest
- 1991
(Show Context)
Citation Context ...old that “all concepts of mathematics can ultimately be reduced to logical concepts” which implies that “all mathematical truths can be proved from the axioms and rules of inference and logic alone” (=-=Ernest, 1991-=-). Formalists do not believe that mathematics is discovered; they believe mathematics is simply a game, created by mathematicians, based on strings of symbols that have no meaning (Davis & Hersh, 1981... |

32 |
Science and Method
- Poincaré
- 1908
(Show Context)
Citation Context ...ult also constitute creativity? These are among the areas of exploration in this paper. The Problem Of Defining Creativity Mathematical creativity has been simply described as discernment, or choice (=-=Poincaré, 1948-=-). According to Poincaré (1948), to create consists precisely in not making useless combinations and in making those which are useful and which are only a small minority. Poincaré is referring to the ... |

29 | Creativity: Beyond the myth of genius - Weisberg - 1993 |

21 | A Course in Mathematical Logic - Manin - 1977 |

13 | Torrance Tests of Creative Thinking: Norms-technical manual. Bensenville, IL: Scholastic Testing Service - Torrance - 1974 |

12 | Basics of qualitative research. Thousand - Strauss, Corbin - 1998 |

10 |
Mathematics and plausible reasoning I: Induction and analogy in mathematics
- Polya
- 1954
(Show Context)
Citation Context ...ny 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 ga... |

9 |
Complex problem solving: principles and mechanisms
- Sternberg, Frensch
- 1991
(Show Context)
Citation Context ...vity.” Recently psychologists have attempted to link creativity to measures of intelligence (Sternberg, 1985) and to the ability to abstract, generalize (Sternberg, 1985), and solve complex problems (=-=Frensch & Sternberg, 1992-=-). Sternberg and Lubart (2000) define creativity as the ability to produce unexpected original work that is useful and adaptive. Mathematicians would raise several arguments with this definition, simp... |

8 |
From case studies to robust generalizations: an approach to the study of creativity
- POLICASTRO, GARDNER
- 1999
(Show Context)
Citation Context ...n (Polya, 1954; Skemp, 1986). 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 an... |

6 | Extraordinary Minds - Gardner - 1998 |

6 |
The case study method and evolving systems approach for understanding unique creative people at work
- Gruber, Wallace
- 1999
(Show Context)
Citation Context ...ilability of role models, availability of financial support, and competitors in a domain” (p. 9). 22 Mathematical CreativityMost 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 cre... |

6 |
Methodological considerations in research on mathematical problem solving
- Lester
- 1985
(Show Context)
Citation Context ...e 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... |

5 |
Mathematics and Psychology
- Birkhoff
- 1969
(Show Context)
Citation Context ...oducts 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. The social-personality approach The social-personality approach to studying creativity focuses o... |

5 |
Reasoning and proof
- Yackel, Hanna
- 2003
(Show Context)
Citation Context ...erving as gatekeepers of the field. A different example is that of the role of proof. Proof is the social process through which the mathematical community validates the mathematician's creative work (=-=Hanna, 1991-=-). The Russian logician Manin (1977) said "A proof becomes a proof after the social act of accepting it as a proof. This is true of mathematics as it is of physics, linguistics, and biology." Bharath ... |

3 | The development of the idea of mathematical proof: A 5-year case study - Maher, Martino - 1996 |

3 |
Ed.) Human Abilities: An Information- Processing Approach
- Sternberg
- 1985
(Show Context)
Citation Context ...omenon of creativity has resulted in literature that attempts to define and operationalize the word “creativity.” Recently psychologists have attempted to link creativity to measures of intelligence (=-=Sternberg, 1985-=-) and to the ability to abstract, generalize (Sternberg, 1985), and solve complex problems (Frensch & Sternberg, 1992). Sternberg and Lubart (2000) define creativity as the ability to produce unexpect... |

3 |
Resolving the continuing dilemmas in school geometry
- Usiskin
- 1987
(Show Context)
Citation Context ...st of the mathematicians in this study mentioned that the last thing they looked at was a formal proof. This is consistent with the literature on the role of formal proof in mathematics (Polya, 1954; =-=Usiskin, 1987-=-). Most of the mathematicians mentioned the need for coherence with other results in the area. The mathematician’s responses to the posed question follow. Excerpt 6 B. I think I would go for repeated ... |

2 | Children's strategies for solving two- and three-dimensional combinatorial problems - English - 1993 |

2 | Conversation as a metaphor for mathematics and learning
- Ernest
- 1994
(Show Context)
Citation Context ...lying metaphor for the wholly isolated individual mind. Instead, the underlying metaphor is that of persons in conversation, persons who participate in meaningful linguistic interaction and dialogue (=-=Ernest, 1994-=-). Language is the shaper, as well as being the “summative” product, of individual minds (Wittgenstein, 1978). The recent literature in psychology acknowledges these social dimensions of human activit... |

2 |
Creativity
- Ervynck
- 1991
(Show Context)
Citation Context ...the mathematician, has not been the subject of much research. It is usually the case that most mathematicians are uninterested in analyzing the thought processes that result in mathematical creation (=-=Ervynck, 1991-=-). The earliest known attempt to study mathematical creativity was an extensive questionnaire published in the French periodical L'Enseigement Mathematique (1902). This questionnaire and a lecture on ... |

2 |
Mathematical giftedness, problem solving, and the ability to formulate generalizations
- Sriraman
- 2003
(Show Context)
Citation Context ...olved problems to draw comparisons between various isomorphic problems (English, 1991, 1993; Hung, 2000; Maher & Kiczek, 32 Mathematical Creativity2000; Maher & Martino, 1997; Maher & Speiser, 1996; =-=Sriraman, 2003-=-; Sriraman, 2004b). In addition, encouraging students to look for similarities in a class of problems fosters "mathematical" behavior (Polya, 1954), leading some students to discover sophisticated mat... |

1 |
Picasso’s guernica
- Arnheim
- 1962
(Show Context)
Citation Context ... (Polya, 1954). Creativity as a feature of mathematical thinking is not a patent of the mathematician! (Krutetskii, 1976); and although most studies on creativity have focused on eminent individuals (=-=Arnheim, 1962-=-; Gardner, 1993, 1997; Gruber, 1981), I suggest that contemporary models from creativity research can be adapted for studying samples of creativity such as are produced by high school students. Such s... |

1 |
Young children's combinatoric strategies
- English
- 1991
(Show Context)
Citation Context ... considerable reflection. This implies that educators should recognize the value of allowing students to reflect on previously solved problems to draw comparisons between various isomorphic problems (=-=English, 1991-=-, 1993; Hung, 2000; Maher & Kiczek, 32 Mathematical Creativity2000; Maher & Martino, 1997; Maher & Speiser, 1996; Sriraman, 2003; Sriraman, 2004b). In addition, encouraging students to look for simil... |

1 |
Some insights into the generalizations of mathematical meanings
- Hung
- 2000
(Show Context)
Citation Context ...ion. This implies that educators should recognize the value of allowing students to reflect on previously solved problems to draw comparisons between various isomorphic problems (English, 1991, 1993; =-=Hung, 2000-=-; Maher & Kiczek, 32 Mathematical Creativity2000; Maher & Martino, 1997; Maher & Speiser, 1996; Sriraman, 2003; Sriraman, 2004b). In addition, encouraging students to look for similarities in a class... |

1 | Long term building of mathematical ideas related to proof making. Contributions to Paolo - Maher, Kiczek - 2000 |

1 | How far can you go with block towers? Stephanie's intellectual development - Maher, Speiser - 1997 |

1 |
The psychology of mathematical creativity
- Muir
- 1988
(Show Context)
Citation Context ...2000), primarily in the form of publication in a prominent journal. However, the acceptance of a mathematical result, the end product of creation, does not ensure its survival in the Darwinian sense (=-=Muir, 1988-=-). The mathematical result may or may not be picked up by other mathematicians. If the mathematical community picks it up as a viable result, then it is likely to undergo mutations and lead to new mat... |

1 | Biologie de l'invention - Nicolle - 1932 |

1 | The influence of Platonism on mathematics research and theological beliefs - Sriraman |

1 | Discovering a mathematical principle: The case of Matt - Sriraman |

1 |
Remarks on the foundations of mathematics (Rev. Ed.).Cambridge: Massachusetts Institute of
- Wittgenstein
- 1978
(Show Context)
Citation Context ...rsons in conversation, persons who participate in meaningful linguistic interaction and dialogue (Ernest, 1994). Language is the shaper, as well as being the “summative” product, of individual minds (=-=Wittgenstein, 1978-=-). The recent literature in psychology acknowledges these social dimensions of human activity as being instrumental in the creative process. The Notion of Creativity in Psychology As stated earlier, r... |