Mathematical Creativity

Defining Mathematical Creativity

Attempts to define mathematical creativity have yielded a multitude of definitions. Some apply the concepts of fluency, flexibility, and creativity to the way students approach problem solving. Others consider the manner in which students formulate problems, or find new relationships, and test their theories. Mathematical creativity can also be viewed in terms of an individual's ability to elaborate on what is known by extending or improving problem-solving methods. An aspect of sensitivity is involved as well; the ability to see beauty or efficiency in the mathematics employed, a level of conceptual understanding necessary to assess cause and effect within a mathematical context, and the ability to offer constructive criticism of standard mathematical problem-solving methods. Regardless of the definition accepted, creativity in mathematics is essential for the advancement of the discipline and in solving problems encountered in the real world.

Academic Ability and Creativity

Although debate exists concerning whether the elements of mathematical creativity are general in nature, domain specific, or some combination, scholars agree that some mathematical knowledge is required for mathematical creativity to emerge. Yet, simply possessing mathematical knowledge does not imply creativity. Students may be able to apply a variety of problem-solving strategies to converge on the same solution, yet never evaluate the appropriateness of these strategies or explore alternate methods. An unwillingness to take risks or the attitude that there is one right way to solve a problem often causes students to fixate on rules and procedures rather than the nature of the problem. When mathematics is approached in this way, creativity is limited; students develop fixed dispositions in their responses to and interpretations of problems.

Henri Poincaré described the work of mathematicians not as the simple application of rules, but rather as the selective choice of ideas to create useful new ways to solve problems. He viewed the process as a period of hard work followed by a period of rest in which the idea incubates within the subconscious. The incubation period is followed by [Page 544]illumination during which the mathematician solves the problem and confirmatory work in which he or she seeks to extend the methods developed to a wider set of problems. The use of standardized test scores as the only means to identify mathematical giftedness runs counter to Poincaré's process, reducing the concept of mathematical ability to simply accuracy of computation and speed of response. Such tests neglect the value of sustained effort and time needed for reflection that provides the fertile environment necessary for creativity to flourish.

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