Mathematical creativity : a developmental perspective
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Bibliographic Information
Mathematical creativity : a developmental perspective
（Research in mathematics education / series editors, Jinfa Cai, James Middleton）
Springer, c2022
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
This book is important and makes a unique contribution in the field of mathematics education and creativity. The book comprises the most recent research by renowned international experts and scholars, as well as a comprehensive up to date literature review. The developmental lens applied to the research presented makes it unique in the field. Also, this book provides a discussion of future directions for research to complement what is already known in the field of mathematical creativity. Finally, a critical discussion of the importance of the literature in relation to development of learners and accordingly pragmatic applications for educators is provided.
Many books provide the former (2) foci, but omit the final discussion of the research in relation to developmental needs of learners in the domain of mathematics. Currently, educators are expected to implement best practices and illustrate how their adopted approaches are supported by research. The authors and editors of this book have invested significant effort in merging theory with practice to further this field and develop it for future generations of mathematics learners, teachers and researchers.
Table of Contents
- Foreword/Preface: Todd Kettler (Baylor University)Section I: History and background of mathematical creativityChapter 1: Innovation and invention: Hadamard, Poincare, and the pioneers of mathematical creativity(authors: Peter Liljedahl and colleagues). A focus of this chapter pertains to sharing early research in thefield of mathematical creativity. As illustrated, early research dates to at least the 1900 era whenPoincare discussed the concept of invention and innovation. Subsequently, Hadamard (circa 1945)followed Poincare's seminal address, writings, and overall message. Hadamard and Poincare, as anexample, discuss how such thinking comes to pass among mathematicians and mathematical thinkers.Their thoughts and writings are quite antiquated, but likely as revelatory today as they were 75-100years ago. The principal reason for infusing a chapter on the history of creativity is that many of thefundamental principles that undergird creativity are those that have shaped the direction of research acentury later. Hence, in highlighting the literature and discussing where and how the research baseoriginated, readers will be better able to make sense of why creativity research is going in the directionthan it is today. The gap from the early 1900s to current is discussed in this chapter and much of thegroundwork is therefore laid for making sense of mathematical creativity in the book. The psychologicalconstruct of development as a lens through which to view mathematical creativity, is also discussed in aconcise manner.Chapter 2: Conceptions of mathematical creativity and their implicit and explicit value in society arediscussed (authors: Scott Chamberlin and colleagues). In this chapter, the construct of mathematicalcreativity is discussed in relation to its value to greater society, as well as mathematics educators andpolicy makers. In specific, the chasm between how society views mathematical creativity (i.e., as quiteimportant), the manner in which it is emphasized in mathematics standards documents (again, as quiteimportant), and lack of resources invested in it from educational administrators, enjoys a rather largedisparity. In short, policy makers and academicians assert the relative importance of mathematicalcreativity, but scant resources are invested when it comes to engendering it in the mathematicsclassroom. In this chapter, this chasm is explored. In addition, the organizational framework for thebook is outlined and readers will gain a deepened conception of mathematical creativity through thelens of development. Creativity in mathematics is not as homogeneous as perhaps thought when oneanalyzes the processes, as well as accompanying products, involved in ages 5-12, 13-18, and 19-23. Thisis because mathematics becomes increasingly sophisticated as learners age and the domains ofmathematics are subtly altered from predominately number sense and operations to more complexdomains such as calculus, algebra, and geometry, in addition to requiring myriad types of reasoning andprocesses, such as mathematical modeling.Chapter 3: In this jointly written chapter (authors: Scott Chamberlin, Peter Liljedahl, and Milos Savic),the authors carefully elucidate their position regarding the relationship of development andmathematical creativity. In specific, their thesis is that development has not been considered as afundamental psychological construct relative to mathematical creativity and it may hold promise formaking sense of its emergence at various age levels. As an example, mathematical creativity is multifaceted,and variables involved with precipitating it (e.g., process and product) in mathematical learningepisodes should not be left to happenstance. When mathematics educators become intentional infostering mathematical creativity with learners, it may enhance the probability of it emerging.Chapter 4: In this chapter (author Alane Starko), commentary is provided on the three chapters thatconstitute section I. All commentary chapters for the book contain a summary of the section, along withpositive and negative attributes of the passages for reader interpretation. Commentator perspectivesare offered to provide readers with a point of view that may be in agreement or disagreement with theauthors.NOTE: The first two chapters of this section of the book will use literature to put forth a thesis that iselucidated in chapter 3. Consequently, in the first section, literature is utilized as a means to form andsupport the developmental thesis along the lines of a literature review as articulated in Liljedahl (2019).Section II: Synthesis of literature findings for researchersChapter 5: The focus of this chapter pertains to considerations of how mathematical creativity literatureinfluences and directs researcher efforts for mathematics education, ages 5-12 (Authors: ScottChamberlin and colleagues). In addition, the authors highlight research and how and what it informs thefield of mathematical creativity to do in order to provide an answer to the field's most immediatequestions. Contrary to many academic books, this chapter, and the following two chapters, will arise asa direct result of what is found in the research and construction of the first three chapters. Chapters 4,5, and 6 cannot be written until an exhaustive investigation of extant literature is provided and allchapters in this section are designed to provide insight to researchers regarding next steps inmathematical creativity research endeavors. This chapter, as is true for chapters five and six, is guidedby a synthesis of literature, as explicated in Newman and Gough (2020).Chapter 6: The focus of this chapter pertains to considerations of how mathematical creativity literatureinfluences and directs researcher efforts for mathematics education, ages 13-18 (Authors: Peter Liljedahland colleagues). Chapter 5 will follow the same format as that provided in the description for chapter 4,but be specific to ages 13-18. In this sense, research on mathematical creativity between the three agegroups is not necessarily in harmony. Needs and nuances of students in secondary settings differs fromtheir peers in early grades, just as it does from students in later grades (college for instance). In soviewing the ages as interrelated, yet somewhat discrete, the focus of the book, Mathematical creativity:A developmental perspective, is truly realized. That is to say, the domain of mathematical creativity hasbeen done a disservice by considering all students to be identical in their approach to mathematicalcreativity. As stated in the chapter four description, this chapter will maintain the structure of asynthesis of literature, as provided in commentary from Newman and Gough (2020).Chapter 7: In this chapter, a consideration of how mathematical creativity literature influences anddirects researcher efforts for mathematics education, ages 19-23 is provided (Authors: Milos Savic andcolleagues). Similar to the prior two chapters, mathematical creativity enjoys a varied conception intertiary grades than it does in earlier (i.e., ages 5-18) ages. This is because creativity is not precisely thesame at this age as it was in the previous two age ranges in part due to learner changes in development,school environments that can be more independent of instructors, and significantly more complexmathematics than in primary and secondary grades. Consequently, research needs are varied in ages 19-23, just as they are in ages 5-12, and 13-18. Commentary by Dr. Savic and colleagues precisely isolatesthe needs of researchers for the next decade(s) in this chapter. This chapter, as well, will be a synthesisof literature.Chapter 8: In this chapter (proposed author Mehdi Nadjafikhah), commentary is provided on the threechapters that constitute section II. All commentary chapters for the book contain a summary of thesection, along with positive and negative attributes of the passages for reader interpretation.Commentator perspectives are offered to provide readers with a point of view that may be inagreement or disagreement with the authors.NOTE: The first three chapters in this section attempt to uncover what is already known, but yetsynthesized, about creativity as developmental. As such, the literature reviews will be a comprehensivesurvey of research previously conducted in the three different age/grade bands: primary, secondary,tertiary. To achieve this synthesis, a slightly adapted approach of the methodology laid out by Newmanand Gough (2020) will be employed. Given the limited scope of empirical research in K-16 mathematicseducation, selection criteria may be mitigated.Section III: Recently completed empirical studies in mathematics educationChapter 9: In this chapter (written by Sandra Crespo and Higinio Dominguez), the authors illustrate howdominant perspectives for theorizing and researching children's mathematical thinking have renderedinvisible important aspects of children's creative mathematical thinking. The authors unpack keyassumptions that traditional research on children's thinking make and how these assumptions getchallenged when researchers work closely with children and learn with and from them about how theyplay with and invent mathematical ideas. Using examples from their research with children inelementary schools in the U.S. and in Latin America, they illustrate how children's creative mathematicsbecome invisible and visible when researchers adopt different theoretical lenses. The authors invitereaders to consider what they can see children doing and saying in a selected video episode and thenshow how different theoretical lenses can help describe and interpret different aspects of mathematicalcreativity. The chapter concludes by offering implications for mathematics education research andpractice.Chapter 10: In this chapter (written by Isabelle DeVink and Eveline Schoevers), a concise overview ofliterature relevant to the study is presented, as well as the method employed, the results, and theimplications. The objective of this chapter is to share recent findings with readers, so that they will haveaccess to new findings in the realm of mathematical creativity in early grades. In this chapter, authorsqualitatively examine how 24 5th grade students (either with excellent mathematical performance ormathematical difficulties) solve a problem posing, multiple-solution and routine mathematics task.Differences between children and between the different tasks are described, as well as children's use ofcreative thinking, specifically divergent and convergent thinking. Insight into the different steps that aretaken to apply creative thinking in a mathematics task is pivotal to fully understand the relation betweenthe two and advise teachers on how to implement creativity in the mathematics classroom.Chapter 11: In this chapter (written by Peter Liljedahl) instances of collaborative creativity in asecondary classroom are investigated while students work in a pedagogical space called a ThinkingClassroom (Liljedahl, 2020). Continuing with the focus of this book on the creative process this chapterprovides an in depth look at the process of creativity as observed while students work collaboratively ona series of tasks designed to elicit flow (Csikszentmihalyi, 1998, 1996, 1990). The data are analyzedthrough the lens of 'burstiness' (Riedl & Williams Wooley, 2018) - a construct from psychology thatdescribes the way collaborative groups talk over top of each other when they are being creative. Theresults will show that burstiness is an effective proxy for creativity that focuses on the creative process -as opposed to most proxies which focus on creative products.Chapter 12: In this chapter (written by Gulden Karakok, Emily Cilli-Turner, Gail Tang, Milos Savic,Houssein El Turkey, and Rani Satyam), the Creativity Research Group shares results from a recent studyin which undergraduate students' perspective of mathematical creativity in an introduction-to-proofscourse were explored. The course was intentionally designed to explicitly value students' creativitythroughout the in-class activities and out-of-class assignments. The Creativity-in-Progress Reflection(CPR) on proving formative assessment tool was introduced to students and used in several classsessions and assignments. In this qualitative study, the main question considered was: In what ways dostudents' perspectives of mathematical creativity evolve over the period of the course and how doestheir perception of instructional design play a role in such evolvement? To answer this question,students' perspectives of mathematical creativity in pre- and post-course surveys were documented andresearchers captured students' perception of instructional design that emphasized mathematicalcreativity in an end-of-semester interview. Following grounded theory methodology, survey andinterview data were coded inductively and researchers triangulated analysis with other data sourcessuch as in-class video recording and students' written work on assignments. Results of the data analysisindicate that even though some students continued to view creativity as an ability that a person hasfrom birth, by the end of the semester other students identified the possibility of development ofcreativity.Chapter 13: In this chapter, the authors (Ceire Monahan and Mika Munakata) report on the role ofcreativity in the adaptations instructors made in designing instructional tasks for an onlinemathematics course for non-mathematics majors. Because the focus of the course was to encouragestudents to think differently about mathematics and engage in mathematically creativetasks, instructors were challenged to rework existing modules designed for in-person teachingto encourage creativity in an online setting while thinking about what it means to teach creatively.Chapter 14: In this chapter (author Esther S. Levenson), commentary is provided on the three chaptersthat constitute section III. All commentary chapters for the book contain a summary of the section,along with positive and negative attributes of the passages for reader interpretation. Commentatorperspectives are offered to provide readers with a point of view that may be in agreement ordisagreement with the authors.Section IV: Research application and editors' summative considerationsChapter 15: In this chapter, the emphasis is on applying research to ages 5-12, 13-18, and 19-23(Authors: Peter Liljedahl, Milos Savic, and Scott Chamberlin). To accomplish this objective, a synthesis ofliterature presented in the previous eight chapters is taken into consideration relative to whatcould/should be done to generate ideal learning episodes for mathematics students at all three ageranges. The intent is to have practitioners realize the importance of topics such as creative process,creative person, educational value in development, as well as affect [feelings, emotions, and dispositions(McLeod & Adams, 1989)], and cognition as a manner in which to shape mathematical environments(Hiebert, et al., 1997
- Wood, Merkel, Uerkwitz, 1996). Practical implications will be offered forresearchers as well as mathematics educators working with students on a daily basis. This chapter is acall to action for educators and researchers alike.Chapter 16: In this concluding chapter, the three editors summarize findings presented in the book andoffer a new paradigm for viewing mathematical creativity (Authors: Milos Savic, Scott Chamberlin, andPeter Liljedahl). In specific, the authors campaign that mathematical creativity is not as unidimensionalas advertised. From this perspective, they communicate that mathematical creativity is not identical inprocess, product, and individual's affect for a six year old grade one student as it is for a 19 year oldstudent in second year at university. Hence, the practicality of the research is interpreted in a mannerthat enables stakeholders the opportunity to utilize the scholarly works and literature of peers in theirresearch and/or teaching efforts to forge new paths and respond to areas of interest in creativity thatare yet to be explored.Chapter 17: In this chapter (author Tracy Zager), commentary is provided on the three chapters thatconstitute section IV. All commentary chapters for the book contain a summary of the section, alongwith positive and negative attributes of the passages for reader interpretation. Commentatorperspectives are offered to provide readers with a point of view that may be in agreement ordisagreement with the authors.*Each chapter can be as long as 25 pages, including title page, text, tables, figures, graphs, diagrams,references, and appendices).
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