Mathematical Creativity

Foreword/Preface: Todd Kettler (Baylor University)<div>Section I: History and background of mathematical creativity</div><div>Chapter 1: Innovation and invention: Hadamard, Poincaré, and the pioneers of mathematical creativity</div><div>(authors: Peter Liljedahl and colleagues). A focus of this chapter pertains to sharing early research in the</div><div>field of mathematical creativity. As illustrated, early research dates to at least the 1900 era when</div><div>Poincaré discussed the concept of invention and innovation. Subsequently, Hadamard (circa 1945)</div><div>followed Poincaré’s seminal address, writings, and overall message. Hadamard and Poincaré, as an</div><div>example, discuss how such thinking comes to pass among mathematicians and mathematical thinkers.</div><div>Their thoughts and writings are quite antiquated, but likely as revelatory today as they were 75-100</div><div>years ago. The principal reason for infusing a chapter on the history of creativity is that many of the</div><div>fundamental principles that undergird creativity are those that have shaped the direction of research a</div><div>century later. Hence, in highlighting the literature and discussing where and how the research base</div><div>originated, readers will be better able to make sense of why creativity research is going in the direction</div><div>than it is today. The gap from the early 1900s to current is discussed in this chapter and much of the</div><div>groundwork is therefore laid for making sense of mathematical creativity in the book. The psychological</div><div>construct of development as a lens through which to view mathematical creativity, is also discussed in a</div><div>concise manner.</div><div>Chapter 2: Conceptions of mathematical creativity and their implicit and explicit value in society are</div><div>discussed (authors: Scott Chamberlin and colleagues). In this chapter, the construct of mathematical</div><div>creativity is discussed in relation to its value to greater society, as well as mathematics educators and</div><div>policy makers. In specific, the chasm between how society views mathematical creativity (i.e., as quite</div><div>important), the manner in which it is emphasized in mathematics standards documents (again, as quite</div><div>important), and lack of resources invested in it from educational administrators, enjoys a rather large</div><div>disparity. In short, policy makers and academicians assert the relative importance of mathematical</div><div>creativity, but scant resources are invested when it comes to engendering it in the mathematics</div><div>classroom. In this chapter, this chasm is explored. In addition, the organizational framework for the</div><div>book is outlined and readers will gain a deepened conception of mathematical creativity through the</div><div>lens of development. Creativity in mathematics is not as homogeneous as perhaps thought when one</div><div>analyzes the processes, as well as accompanying products, involved in ages 5-12, 13-18, and 19-23. This</div><div>is because mathematics becomes increasingly sophisticated as learners age and the domains of</div><div>mathematics are subtly altered from predominately number sense and operations to more complex</div><div>domains such as calculus, algebra, and geometry, in addition to requiring myriad types of reasoning and</div><div>processes, such as mathematical modeling.</div><div>Chapter 3: In this jointly written chapter (authors: Scott Chamberlin, Peter Liljedahl, and Miloš Savić),</div><div>the authors carefully elucidate their position regarding the relationship of development and</div><div>mathematical creativity. In specific, their thesis is that development has not been considered as a</div><div>fundamental psychological construct relative to mathematical creativity and it may hold promise for</div><div>making sense of its emergence at various age levels. As an example, mathematical creativity is multifaceted,</div><div>and variables involved with precipitating it (e.g., process and product) in mathematical learning</div><div>episodes should not be left to happenstance. When mathematics educators become intentional in</div><div>fostering mathematical creativity with learners, it may enhance the probability of it emerging.</div><div>Chapter 4: In this chapter (author Alane Starko), commentary is provided on the three chapters that</div><div>constitute section I. All commentary chapters for the book contain a summary of the section, along with</div><div>positive and negative attributes of the passages for reader interpretation. Commentator perspectives</div>are offered to provide readers with a point of view that may be in agreement or disagreement with the<div>authors.</div><div>NOTE: The first two chapters of this section of the book will use literature to put forth a thesis that is</div><div>elucidated in chapter 3. Consequently, in the first section, literature is utilized as a means to form and</div><div>support the developmental thesis along the lines of a literature review as articulated in Liljedahl (2019).</div>Section II: Synthesis of literature findings for researchers<div>Chapter 5: The focus of this chapter pertains to considerations of how mathematical creativity literature</div><div>influences and directs researcher efforts for mathematics education, ages 5-12 (Authors: Scott</div><div>Chamberlin and colleagues). In addition, the authors highlight research and how and what it informs the</div><div>field of mathematical creativity to do in order to provide an answer to the field’s most immediate</div>questions. Contrary to many academic books, this chapter, and the following two chapters, will arise as<div>a direct result of what is found in the research and construction of the first three chapters. Chapters 4,</div><div>5, and 6 cannot be written until an exhaustive investigation of extant literature is provided and all</div><div>chapters in this section are designed to provide insight to researchers regarding next steps in</div><div>mathematical creativity research endeavors. This chapter, as is true for chapters five and six, is guided</div><div>by a synthesis of literature, as explicated in Newman and Gough (2020).</div><div>Chapter 6: The focus of this chapter pertains to considerations of how mathematical creativity literature</div><div>influences and directs researcher efforts for mathematics education, ages 13-18 (Authors: Peter Liljedahl</div><div>and colleagues). Chapter 5 will follow the same format as that provided in the description for chapter 4,</div><div>but be specific to ages 13-18. In this sense, research on mathematical creativity between the three age</div><div>groups is not necessarily in harmony. Needs and nuances of students in secondary settings differs from</div><div>their peers in early grades, just as it does from students in later grades (college for instance). In so</div><div>viewing the ages as interrelated, yet somewhat discrete, the focus of the book, Mathematical creativity:</div><div>A developmental perspective, is truly realized. That is to say, the domain of mathematical creativity has</div><div>been done a disservice by considering all students to be identical in their approach to mathematical</div><div>creativity. As stated in the chapter four description, this chapter will maintain the structure of a</div><div>synthesis of literature, as provided in commentary from Newman and Gough (2020).</div><div>Chapter 7: In this chapter, a consideration of how mathematical creativity literature influences and</div><div>directs researcher efforts for mathematics education, ages 19-23 is provided (Authors: Miloš Savić and</div><div>colleagues). Similar to the prior two chapters, mathematical creativity enjoys a varied conception in</div><div>tertiary grades than it does in earlier (i.e., ages 5-18) ages. This is because creativity is not precisely the</div><div>same at this age as it was in the previous two age ranges in part due to learner changes in development,</div><div>school environments that can be more independent of instructors, and significantly more complex</div><div>mathematics than in primary and secondary grades. Consequently, research needs are varied in ages 19-</div><div>23, just as they are in ages 5-12, and 13-18. Commentary by Dr. Savić and colleagues precisely isolates</div><div>the needs of researchers for the next decade(s) in this chapter. This chapter, as well, will be a synthesis</div><div>of literature.</div><div>Chapter 8: In this chapter (proposed author Mehdi Nadjafikhah), commentary is provided on the three</div><div>chapters that constitute section II. All commentary chapters for the book contain a summary of the</div>section, along with positive and negative attributes of the passages for reader interpretation.<div>Commentator perspectives are offered to provide readers with a point of view that may be in</div><div>agreement or disagreement with the authors.</div><div>NOTE: The first three chapters in this section attempt to uncover what is already known, but yet</div><div>synthesized, about creativity as developmental. As such, the literature reviews will be a comprehensive</div>survey of research previously conducted in the three different age/grade bands: primary, secondary,<div>tertiary. To achieve this synthesis, a slightly adapted approach of the methodology laid out by Newman</div><div>and Gough (2020) will be employed. Given the limited scope of empirical research in K-16 mathematics</div><div>education, selection criteria may be mitigated.</div><div>Section III: Recently completed empirical studies in mathematics education</div><div>Chapter 9: In this chapter (written by Sandra Crespo and Higinio Dominguez), the authors illustrate how</div><div>dominant perspectives for theorizing and researching children’s mathematical thinking have rendered</div><div>invisible important aspects of children’s creative mathematical thinking. The authors unpack key</div><div>assumptions that traditional research on children’s thinking make and how these assumptions get</div><div>challenged when researchers work closely with children and learn with and from them about how they</div>play with and invent mathematical ideas. Using examples from their research with children in<div>elementary schools in the U.S. and in Latin America, they illustrate how children’s creative mathematics</div><div>become invisible and visible when researchers adopt different theoretical lenses. The authors invite</div><div>readers to consider what they can see children doing and saying in a selected video episode and then</div><div>show how different theoretical lenses can help describe and interpret different aspects of mathematical</div><div>creativity. The chapter concludes by offering implications for mathematics education research and</div><div>practice.</div><div>Chapter 10: In this chapter (written by Isabelle DeVink and Eveline Schoevers), a concise overview of</div><div>literature relevant to the study is presented, as well as the method employed, the results, and the</div><div>implications. The objective of this chapter is to share recent findings with readers, so that they will have</div>access to new findings in the realm of mathematical creativity in early grades. In this chapter, authors<div>qualitatively examine how 24 5th grade students (either with excellent mathematical performance or</div><div>mathematical difficulties) solve a problem posing, multiple-solution and routine mathematics task.</div><div>Differences between children and between the different tasks are described, as well as children’s use of</div><div>creative thinking, specifically divergent and convergent thinking. Insight into the different steps that are</div><div>taken to apply creative thinking in a mathematics task is pivotal to fully understand the relation between</div><div>the two and advise teachers on how to implement creativity in the mathematics classroom.</div><div>Chapter 11: In this chapter (written by Peter Liljedahl) instances of collaborative creativity in a</div><div>secondary classroom are investigated while students work in a pedagogical space called a Thinking</div><div>Classroom (Liljedahl, 2020). Continuing with the focus of this book on the creative process this chapter</div><div>provides an in depth look at the process of creativity as observed while students work collaboratively on</div><div>a series of tasks designed to elicit flow (Csíkszentmihályi, 1998, 1996, 1990). The data are analyzed</div><div>through the lens of 'burstiness' (Riedl & Williams Wooley, 2018) - a construct from psychology that</div><div>describes the way collaborative groups talk over top of each other when they are being creative. The</div><div>results will show that burstiness is an effective proxy for creativity that focuses on the creative process -</div><div>as opposed to most proxies which focus on creative products.</div><div>Chapter 12: In this chapter (written by Gulden Karakok, Emily Cilli-Turner, Gail Tang, Miloš Savić,</div><div>Houssein El Turkey, and Rani Satyam), the Creativity Research Group shares results from a recent study</div><div>in which undergraduate students’ perspective of mathematical creativity in an introduction-to-proofs</div><div>course were explored. The course was intentionally designed to explicitly value students’ creativity</div><div>throughout the in-class activities and out-of-class assignments. The Creativity-in-Progress Reflection</div><div>(CPR) on proving formative assessment tool was introduced to students and used in several class</div><div>sessions and assignments. In this qualitative study, the main question considered was: In what ways do</div><div>students’ perspectives of mathematical creativity evolve over the period of the course and how does</div><div>their perception of instructional design play a role in such evolvement? To answer this question,</div><div>students’ perspectives of mathematical creativity in pre- and post-course surveys were documented and</div><div>researchers captured students’ perception of instructional design that emphasized mathematical</div><div>creativity in an end-of-semester interview. Following grounded theory methodology, survey and</div><div>interview data were coded inductively and researchers triangulated analysis with other data sources</div>such as in-class video recording and students’ written work on assignments. Results of the data analysis<div>indicate that even though some students continued to view creativity as an ability that a person has</div><div>from birth, by the end of the semester other students identified the possibility of development of</div><div>creativity.</div><div>Chapter 13: In this chapter, the authors (Ceire Monahan and Mika Munakata) report on the role of</div><div>creativity in the adaptations instructors made in designing instructional tasks for an online</div><div>mathematics course for non-mathematics majors. Because the focus of the course was to encourage</div><div>students to think differently about mathematics and engage in mathematically creative</div><div>tasks, instructors were challenged to rework existing modules designed for in-person teaching</div><div>to encourage creativity in an online setting while thinking about what it means to teach creatively.</div>Chapter 14: In this chapter (author Esther S. Levenson), commentary is provided on the three chapters<div>that constitute section III. All commentary chapters for the book contain a summary of the section,</div><div>along with positive and negative attributes of the passages for reader interpretation. Commentator</div><div>perspectives are offered to provide readers with a point of view that may be in agreement or</div><div>disagreement with the authors.</div><div>Section IV: Research application and editors’ summative considerations</div><div>Chapter 15: In this chapter, the emphasis is on applying research to ages 5-12, 13-18, and 19-23</div><div>(Authors: Peter Liljedahl, Miloš Savić, and Scott Chamberlin). To accomplish this objective, a synthesis of</div><div>literature presented in the previous eight chapters is taken into consideration relative to what</div><div>could/should be done to generate ideal learning episodes for mathematics students at all three age</div>ranges. The intent is to have practitioners realize the importance of topics such as creative process,<div>creative person, educational value in development, as well as affect [feelings, emotions, and dispositions</div><div>(McLeod & Adams, 1989)], and cognition as a manner in which to shape mathematical environments</div><div>(Hiebert, et al., 1997; Wood, Merkel, Uerkwitz, 1996). Practical implications will be offered for</div><div>researchers as well as mathematics educators working with students on a daily basis. This chapter is a</div><div>call to action for educators and researchers alike.</div><div>Chapter 16: In this concluding chapter, the three editors summarize findings presented in the book and</div><div>offer a new paradigm for viewing mathematical creativity (Authors: Miloš Savić, Scott Chamberlin, and</div><div>Peter Liljedahl). In specific, the authors campaign that mathematical creativity is not as unidimensional</div><div>as advertised. From this perspective, they communicate that mathematical creativity is not identical in</div><div>process, product, and individual’s affect for a six year old grade one student as it is for a 19 year old</div><div>student in second year at university. Hence, the practicality of the research is interpreted in a manner</div><div>that enables stakeholders the opportunity to utilize the scholarly works and literature of peers in their</div><div>research and/or teaching efforts to forge new paths and respond to areas of interest in creativity that</div><div>are yet to be explored.</div><div>Chapter 17: In this chapter (author Tracy Zager), commentary is provided on the three chapters that</div><div>constitute section IV. All commentary chapters for the book contain a summary of the section, along</div><div>with positive and negative attributes of the passages for reader interpretation. Commentator</div><div>perspectives are offered to provide readers with a point of view that may be in agreement or</div><div>disagreement with the authors.</div>*Each chapter can be as long as 25 pages, including title page, text, tables, figures, graphs, diagrams,<div>references, and appendices).</div>