Mathematical Problem-Solving via Wallas’ Four Stages of Creativity: Implications for the Undergraduate Classroom

IntroductionProblem solving is a process that is declared important for mathematics and mathematics education. Schoenfeld (1992), after citing national studies by the National Council of Teachers of Mathematics and the National Research Council, stated that "there is general acceptance of the idea that the primary goal of mathematics instruction should be to have students become competent problem solvers" (p. 3). While mathematics education researchers continue to investigate problem solving to understand mechanisms that a solver goes through, many components of this problem-solving process built upon seminal work of Polya (1957) that provided four stages: (i) understanding the problem, (ii) developing a plan, (iii) carrying out the plan, and (iv) looking back. Polya's four-stage framework influenced other problem-solving frameworks, including Carlson and Bloom's (2005) Multidimensional Problem-solving Framework. However, the term "problem solving" has been defined in many ways, to the point where Chamberlin (2008) stated: "There is rarely an agreed upon definition of mathematical problem solving and reaching consensus on a conceptual definition would provide direction to subsequent research and curricular decisions" (p. 1). Similarly, the term "mathematical creativity" has been defined in many ways, to the point where Mann (2006) stated: "An examination of the research that has attempted to define mathematical creativity found that the lack of an accepted definition for mathematical creativity has hindered research efforts" (p. 238). However, Mann (2006) also claimed that not investigating mathematical creativity to enhance students' efforts could "drive the creatively talented underground or, worse yet, cause them to give up the study of mathematics altogether" (p. 239). Mathematical creativity is either a process or a product (depending on the definition) that is declared important for mathematics and mathematics education. The Committee on Undergraduate Programs in Mathematics (Schumacher & Siegel, 2015) stated that, "A successful major offers a program of courses to gradually and intentionally leads students from basic to advanced levels of critical and analytical thinking, while encouraging creativity and excitement about mathematics" (p. 9). According to Liljedahl (2009), "it is through mathematical creativity that we see the essence of what it means to 'do' and learn mathematics. " (p. 239). While mathematical creativity has been researched (e.g., Sriraman, 2004), many components of the creative process come from the psychologist Wallas (1926). Wallas stated that there are four stages of creativity: i) preparation (thoroughly understanding a problem), ii) incubation (when the mind goes about solving a problem subconsciously and/or automatically), iii) illumination (internally generating an idea after the incubation process, sometimes known as the AHA! experience), and iv) verification (determining whether that idea is correct). Mathematicians (Hadamard, 1945; Poincare, 1946) have stated that they have experienced similar stages in their mathematical process, and primary and secondary school mathematics education researchers have used Wallas' stages to explore problem solving (e.g., Prusak, 2015). As discussed in the previous two paragraphs, there seems to be a connection between both problem solving and mathematical creativity, which evokes the following research question: are stages of "problem solving" and "mathematical creativity" equal sets, or is one a subset of another? In this article, the stance is that the stages of problem solving are a subset of mathematical creativity, applying Wallas' four-stage process as a basis for discussion of problem solving. Utilizing the psychodynamic lens, the consideration of mathematical creativity is more in the process of problem solving (e.g., Guilford, 1967; Pelczer & Rodriguez, 2011) and less in the product created by said process (e.g., Runco & Jaeger, 2012). …