The Relationship of Affect and Creativity in Mathematics: How the Five Legs of Creativity Influence Math Talent

By Scott Chamberlin and Eric L. Mann

The Relationship of Affect and Creativity in Mathematics explores the five legs of creativity—Iconoclasm, Impartiality, Investment, Intuition, and Inquisitiveness—as they relate to mathematical giftedness. This book:

• Discusses these affective components relevant to mathematical learning experiences.
• Shares how affective components impact students' creative processes and products.
• Shows the influence of learning facilitators, including teachers, afterschool mentors, and parents.
• Describes facilitating environments that may enhance the likelihood that creative process and ultimately product emerge.
• Utilizes the expertise of two young scholars to discuss the practical effects of affect and creativity in learning experiences.

This practical, research-based book is a must-read for stakeholders in gifted education, as many advanced students are underidentified in the area of creativity in mathematics.

• Language: English
• Publisher: Sourcebooks
• Release date: Jan 15, 2021
• ISBN: 9781646320769

Scott Chamberlin

Scott A. Chamberlin, Ph.D., is a professor of Elementary and Early Childhood Education at the University of Wyoming. His research interests concern how learners solve problems, their affective ratings during the activities, and creative products generated during such episodes. His main discipline interest in mathematics is statistics and probability.

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Introduction

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When affective ratings for Iconoclasm, Impartiality, Investment, Intuition, and Inquisitiveness are low, the likelihood of creative output emerging is extremely poor and nearly impossible according to this theory.

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The purpose of this book is threefold. First, a discussion of extant literature regarding what precisely affect and creativity are in relation to mathematics will be provided. Second, an evolving theory relevant to creative output and mental states in mathematics and, in particular, problem-solving episodes will be elucidated. Third, a rather laconic section will consider pragmatic implications in the classroom. The third section will remain brief for two reasons. Primarily, the theory is evolving and in the process of being empirically tested; hence, overgeneralizations should be avoided. Second, it is patently egocentric for four university researchers, albeit all with K–12 teaching experience, to make suggestions with respect to how teachers in urban Salt Lake City, UT, suburban Bonita Springs, FL, and still smaller Bremen, IN, should operate their class, having never met the students, teachers, parents, or school administrators.

Much consideration in creativity and mathematics has been invested in indicators of creativity, such as originality, fluency, flexibility, and elaboration (Nadjafikhah et al., 2012). In this book, the authors discuss affective preconditions that are believed to enhance the likelihood of creative process and, therefore, output. The relationship of mathematical affect and mathematical creativity is not an entry-level one in the domain of mathematics and gifted education. In fact, few experts in the field of mathematics education, gifted education, or the smaller domain of mathematics gifted education have invested thought regarding any prospective relationship between the two psychological constructs. Recently, the two lead authors, Scott A. Chamberlin and Eric L. Mann, invested considerable time in discussions about the two constructs.

Although it will be explicated in much greater detail later in the book, the Five Legs of Creativity Theory, hereafter referred to as the Five Legs Theory, is one in which affective mental states are said to dramatically influence creative process in mathematics and subsequently creative output in mathematics. The five legs of creativity are Iconoclasm, Impartiality, Investment, Intuition, and Inquisitiveness. In short, when considering the five legs, when all are at an optimal (high) level, the creative process and product output will be maximized. When the five legs are somewhat mixed, the prospect for creativity emerging is good but not perhaps great. When all five legs are at low levels, the prospect for creativity emerging is low. What does this mean in practicality? To researchers studying and teachers trying to elicit creative process and product, the emergence of creativity is not as much happenstance as perhaps initially believed.

This conception enjoys a cursory relationship to what Mann (2020) discussed when he referred to the four Ps: person, product, process, and press (Pitta-Pantazi et al., 2018), along with the fifth P (problem). Further, the Five Legs Theory is a merger of empirical literature on affect (student mental states) and creativity that has helped shape understandings in mathematical psychology. The penultimate purpose of the theory is to provide a framework for researchers to guide the field of affect and creativity research, and the ultimate purpose is to assist teachers, learning facilitators, and curriculum coordinators so that they can manipulate environmental conditions of mathematics classrooms to enhance the likelihood of creative process and product.

The authors have an enduring interest in these topics. Dr. Mann completed his dissertation on creativity in mathematics at the University of Connecticut in gifted education. Dr. Chamberlin completed his dissertation on affect in mathematics at the Gifted Education Resource Institute at Purdue University. Shortly after graduating from Purdue, Chamberlin was introduced to an incoming faculty member (Mann). A working relationship was formed because, ironically, Chamberlin wanted to learn about creativity in mathematics and Mann wanted to learn about affect in mathematics. The relationship, perhaps initially thought of as one of convenience, enabled each researcher to intellectually challenge the other, and before long, they were emailing, holding Skype and Zoom conferences, and meeting up at conferences to discuss what has now come to be known as the Five Legs of Creativity Theory.

In this chapter, a cursory overview of the book is provided. The Five Legs Theory is elucidated in Chapters 1–5. In Chapter 6, mathematics tasks that promote creativity are discussed, and in Chapter 7, classroom climates that promote creativity are discussed. The discussion in Chapter 7 focuses on components of mathematics classrooms that promote creativity, and not classrooms in general. In Chapter 8, a pragmatic application of the Five Legs Theory is applied to mathematically gifted and general population students. In the Conclusion, some final thoughts about the theory and future direction of the research are provided. The Appendix includes discussion questions for each of the chapters to encourage inquiry and dialogue among readers.

Counter to much of the previous literature on mathematical creativity in which indicators of creativity are discussed, the focus of this model is on preconditions for creative output (product). Certainly, creative process (i.e., cognition) is subsumed in this model discussion because creative thought (process) is an antecedent to creative output (product). The content emphasis is mathematics, although it is hypothesized that this model may apply to the study of creativity in other content areas (e.g., engineering, literacy, art, science, social sciences, etc.). In this introduction, the five legs (or preconditions) of creativity are explicated. As a caveat, not all five legs of creativity are necessary for creative output to appear, but the greater their prevalence, the greater the propensity for emergence of creativity. Much of this theory is based on literature and was not developed from empirical data per se.

General Overview of Creativity in Mathematics

In mathematical psychology (i.e., the study of cognition, affect, and conation in mathematics), creativity is typically considered multifaceted. In fact, most educational psychologists consider the construct of creativity to be comprised of four subconstructs: originality, fluency, flexibility, and elaboration. In the context of mathematics, perhaps the best discussion of originality, fluency, and flexibility was provided more than 40 years ago in Krutetskii’s (1976) The Psychology of Mathematical Abilities in Schoolchildren. Others have clarified all four subconstructs more recently (e.g., Haylock, 1997; Imai, 2000; H. Kim et al., 2003; Mann, 2005). A highly insightful discussion of these four components is presented in Sriraman’s (2004, 2008) commentary of what precisely mathematical creativity is, which entails a strong discussion of its history. Also of note is Nadjafikhah et al.’s (2012) discussion of up-to-date definitions and characteristics associated with mathematical creativity. Other researchers of note in this field are Leikin, Singer, Karp, and Pitta-Pantazi (Leikin & Pitta-Pantazi, 2013; Leikin et al., 2015; Singer, 2018).

Originality

Originality (Chassell, 1916) typically refers to the generation of a highly novel or unique response to a mathematical problem. An example of this would be a new proof of the Pythagorean theorem, despite the multiple current proofs (Bogomolny, n.d.). In fact, it may be common for neophytes to creativity to think of originality as the only indicator or component of creativity.

Fluency

Fluency typically refers to the volume or number of responses to a mathematical solution. For this reason, the Torrance Tests of Creative Thinking focus on respondents generating as many solutions to a problem as possible (Krumm et al., 2014). Examples of fluency in mathematical problems would be the creation of several solutions to an engaging mathematics problem, using multiple content areas in mathematics, or creating multiple solution paths to arrive at a solution.

Flexibility

Closely related to fluency is flexibility. Perhaps the best conception of flexibility may be one’s ability to yield impulses or reject previously agreed-upon methods to identify new or alternative directions in mathematical solutions (Romo, 1997). An antonym of flexibility in mathematical thinking is fixation on a set path (Duncker, 1945; Luchins, 1942; Wertheimer, 1961). An example of this would be attempting to solve a problem with an approach that may be influenced by a previous discussion with a mentor or instructor. However, upon learning that insufficient knowledge has been accumulated about the topic, that approach ceases, and another approach is adopted, which then proves useful. For instance, one may solve a statistics problem using algebra given a greater familiarity with algebra than statistics, perhaps as a result of previous coursework or aptitude in algebra.

Elaboration

Elaboration in the world of mathematics pertains to one’s ability to provide in-depth explanations of mathematical solutions, make connections, generalize, and extend as an expert might (Imai, 2000). An example of elaboration would be a discussion of a mathematics concept (e.g., measures of central tendency, such as mode, median, and mean) with an in-depth discussion of optimal situations in which to utilize it.

Much of the field may agree that the aforementioned subconstructs that comprise creativity in mathematics have been identified for years. What remains to be clearly articulated are the preconditions that enhance the likelihood of creative products surfacing in mathematics.

Relationship of Creativity and Affect in Mathematics

It is imperative to mention several caveats about the proposed Five Legs Theory. First, as previously mentioned, this theory is just that: a theory. It is not a proven set of principles on which one should base monumental learning decisions. Considering that, much of the theory presented is based on literature in which experts in educational and mathematical psychology have promoted various hypotheses.

Second, in mathematics, the psychological constructs of affect and creativity have often been discussed in isolation, not in harmony. With this theory, it is hypothesized that the emergence of creative products need not be as accidental as initially considered. Currently, some individuals, such as mathematical psychologists, mathematics instructors, and/or mathematics educators, may simply wait for creative products to emerge as students are engaged in mathematical episodes such as problem solving. Utilizing critical thinking approaches and curricula, such as open-ended problems and problem posing, may enhance the likelihood of creativity emerging but is not as systematic as investing attention in student affect. This is because when the proposed subconstructs of affect (i.e., anxiety, aspiration, attitude, interest, locus of control, self-efficacy, self-esteem, and value [Anderson & Bourke, 2000]) are considered, the probability of creativity emerging is not as much happenstance as it may have been previously considered (see Table 0.1).

Third, the relationship of creativity and affect in mathematics may, in fact, be symbiotic. That is to say, individuals with an interest in and talent for mathematical thinking and a modicum of persistence may be most inclined to realize creative output (Renzulli, 1978, 1998), which may then engender high affective ratings. This is a bidirectional relationship in that high affective ratings in the proposed areas may engender creative output while one is engaged in mathematical episodes. Likewise, creative output may serve to motivate problem solvers to persist with engagement and facilitate positive affect in problem-solving scenarios. In this sense, the relationship of creativity and affect in mathematics can be considered symbiotic or bidirectional (See Figure 0.1).

TABLE 0.1

Subconstructs of Affect (Anderson & Bourke, 2000)

Fourth, although cognition and affect are closely related (Pessoa, 2008), the focus of this theory is exclusively on the relationship of affect and creativity. However, creativity cannot emerge without a requisite amount of intellectual capacity. Decades ago, Torrance (Haensly & Reynolds, 1989; Torrance, 1962; Yamamoto, 1964), as well as Getzels and Jackson (1962), forwarded the Threshold Theory of Creativity; this theory asserted that a requisite level of intellect must exist for creative products to emerge. Much has been written about this theory, and like many theories that enjoy widespread attention, eventually it came under heavy criticism. Experts in the domain of creativity appear to disagree about this theory (Jauk et al., 2013; K. H. Kim, 2005; Preckel et al., 2006) and, as such, have not reached consensus regarding this topic. One reason for disagreement with respect to the Threshold Theory of Creativity is that although intellect may be quantified by intelligence quotient assessments, such as the Weschler Intelligence Scale for Children (WISC-V) or the Stanford-Binet test, the precise level of intellect that must exist in an individual for creativity to emerge cannot be precisely quantified.

Figure 0.1

Bidirectional Relationship Between Affect and Creativity

Boden (2004) argued that a modicum of experience and knowledge in a field must exist for creative products to emerge. In relation to creative output in schools, it may be argued that pupils’ contributions to creativity are worthy of consideration for creative output because children are not intimately acquainted with what a profession has delivered. In questioning this theory, it is apparent that when suitable levels of intellect are considered, the theory starts to be realized. However, after a certain level is reached, creativity is not guaranteed, or the likelihood of prevalence is not enhanced relative to peers that exceed the very basic threshold of around 85 IQ (Jauk et al., 2013). Moreover, according to Jauk and colleagues, personality factors serve as a better predictor of creativity emerging than intelligence, once the IQ threshold has been reached.

As an example, problem solvers with high intellectual levels may be able to contribute creative products in mathematics, although extremely high levels of intellect do not guarantee creative output in mathematical problem-solving episodes. To the contrary, students with extremely low levels of intellect may be precluded from contributing creative products. Relative to creative output, additional barriers that may negatively influence creativity may include economically