Towards Recursive Mathematics Curricula: A Complexified Hermeneutic Journey by Lixin Luo A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Secondary Education University of Alberta © Lixin Luo, 2019 Available at https://era.library.ualberta.ca/items/2c6b842d-dbc6-405c-96d0-e692441f53d5 TOWARDS RECURSIVE MATHEMATICS CURRICULA ii Abstract Derived from Doll’s (1993) seminal conceptualization of a post-modern curriculum with the criteria of 4Rs (i.e., richness, relations, recursion and rigor), the present research continues the effort to complexify and theorize recursion and recursive curriculum. This study reconceptualizes mathematics curriculum as recursive through the lens of complexity thinking (Davis & Sumara, 2006), which studies complex systems that are adaptive, such as cognition and knowledge. A mathematics curriculum often seems be designed or delivered as linear: a sequence of predetermined, sometimes unrelated, topics with few chances for learners to revisit them from different perspectives. This suggests learning as accumulation with predictable outcomes. Learning, observed through a complexified world view, is neither linear nor predictable. Learning is a self-organizing process through which a learner and her environment co-evolve, and a recursive elaboration through which a learner transforms her previous understanding (Davis & Sumara, 2002; Davis, Sumara, & Luce-Kapler, 2008). Both learners and school subjects are complex systems with a biological structure (Davis & Sumara, 2002) that emerges. This view demands a recursive curriculum that centers on reviewing previously encountered ideas with an orientation towards newness and changes along its formation. What might such mathematics curriculum be like, particularly at high school level, in theory and practice is my research focus. The research methodology follows the tradition of hermeneutics (Gadamer, 1989/2013) that attends to language and emphasizes emerging understanding through iterative loops of interpretations. The interpretations in this research are informed by three kinds of entry texts, my personal reflections about recursive curriculum, teaching documents (i.e., programs of studies and textbooks), and conversations with teachers, serving to provoke my thinking and generate TOWARDS RECURSIVE MATHEMATICS CURRICULA iii further reflection subjected to new rounds of interpretations. Several high school mathematics teaching documents from Canada and China were examined to see in what ways a planned curriculum might afford recursion. Conversations with experienced high school mathematics teachers were conducted in professional development workshops and/or individual meetings. Teachers were invited to reflect on their learning and teaching experiences and comment on several teaching and learning practices (e.g., reviewing), and work with me to revise or generate curriculum materials to promote such practices orientated towards helping students to learn something new from what they have encountered before. This study aims to make a contribution in the field of mathematics education by addressing the gap between the perceived importance of recursive mathematics curricula and the insufficiency of research about them. I expect that this study speaks to a reinterpretation of reviewing, and potentially provokes learners (both teachers and students) to interpret mathematics and curriculum differently and inspires learners to (re)embark a complexified hermeneutic inquiry on recursive mathematics curricula for the purpose of both learning and teaching. This study has led to a metaphorical and iconic image (see the image on p. v or Figure 9.4.8) of recursive curricula that represents abundant curriculum possibilities rather than a fixed one. Such visualization can provide theoretical and practical references for mathematics educators and education researchers to draw inspirations from when designing towards recursive mathematics curricula. TOWARDS RECURSIVE MATHEMATICS CURRICULA iv Preface This research is an original study by Lixin Luo. The research project, of which this thesis is a part, received research ethics approval from the University of Alberta Research Ethics Board, (project name “Towards a Recursive Mathematics Curriculum”, No. 00054834) on March 10, 2015. The same project, under the title “Towards a Recursive Mathematics Curriculum: A Hermeneutic Inquiry”, also received research ethics approval from four school districts’ ethics boards in Edmonton AB, Canada: Elk Island Public Schools on April 17, 2015, Edmonton Catholic Schools on June 3, 2015, St Albert Public Schools on June 12, 2015, and Edmonton Public Schools on July 21, 2015. Parts of chapters 1, 2 and 3 of this thesis have been published in L. Luo, 2014, “Recursion in the mathematics curriculum,” Philosophy of Mathematics Education Journal, 28. Parts of chapter 8 have been published in L. Luo, 2015, “Repetition as a means of encouraging Tall’s met-befores,” Delta-k (Journal of the Mathematics Council of the Alberta Teachers’ Association), vol. 52, issue 2, 15-17. Parts of chapters 1, 2, 3, and 4 have been published in L. Luo, 2019, “A recursive path to infinity,” In M. Quinn (Ed.), Complexifying Curriculum Studies: Reflections on the Generative and Generous Gifts of William E. Doll, Jr. (pp. 94-101), New York, NY: Routledge. Parts of the abstract and the image on p. v or Figure 9.4.8 have been published in L. Luo, 2019, “Abundant recursive mathematics curricula possibilities” [Image participated in the Image of Research Competition 2019 at the University of Alberta]. Available from https://era.library.ualberta.ca/items/7c7edca1-b704-4c11-9467-980bda1af444 TOWARDS RECURSIVE MATHEMATICS CURRICULA This work is dedicated to Bill Doll Who taught me complexity thinking through his enactment and praxis Who cherished my recursive wrestling with recursion & Lixin Who struggled and survived v TOWARDS RECURSIVE MATHEMATICS CURRICULA vi Table of Contents 1 A Calling to Recursion........................................................................................................ 1 2 Theoretical Underpinning and Methodology ...................................................................... 8 2.1 Complexity Thinking ....................................................................................................... 8 The historical development of complexity thinking ........................................................... 8 Key ideas in complexity thinking ....................................................................................... 9 Complexity thinking and education .................................................................................. 11 2.2 Hermeneutic Inquiry ...................................................................................................... 15 Hermeneutics’ textual interest and central themes ........................................................... 15 The hermeneutic task ........................................................................................................ 18 Evaluating a hermeneutic inquiry ..................................................................................... 25 2.3 3 Connecting Complexity Thinking and Hermeneutic Inquiry......................................... 27 Recursion and Recursive Curriculum ............................................................................... 28 3.1 Recursion and Cognitive Development.......................................................................... 28 Recursion in mathematics and computer science ............................................................. 28 Recursion, as articulated by Doll ...................................................................................... 29 Recursion, as articulated by Bateson ................................................................................ 32 Recursion in complexity thinking ..................................................................................... 36 A working definition of recursion..................................................................................... 37 3.2 Recursion and the Growth of Mathematical Understanding .......................................... 39 The problems of a linear mathematics curriculum ........................................................... 39 The recursive nature of the growth in mathematical understanding ................................. 41 3.3 Recursive Curricula ........................................................................................................ 46 Doll’s post-modern curriculum ......................................................................................... 46 Bruner’s spiral curriculum ................................................................................................ 46 Davis and colleagues’ fractal-informed curriculum ......................................................... 48 Thom’s recursive curriculum (4Rs of recursion) .............................................................. 51 3.4 4 Research Necessity......................................................................................................... 54 Path Unfolding While Walking ........................................................................................ 55 4.1 Entry Texts ..................................................................................................................... 55 Autobiographical reflection .............................................................................................. 56 Document interpretation ................................................................................................... 58 Conversation with teachers ............................................................................................... 62 4.2 Phase Shifts .................................................................................................................... 65 TOWARDS RECURSIVE MATHEMATICS CURRICULA vii The resolution of my struggles in conversations .............................................................. 65 The emergence of an alternative interpretative framework and the unseen third dimension ........................................................................................................................................... 68 The arising of stories as attractors .................................................................................... 72 4.3 5 A Fractal-like Hermeneutic Inquiry ............................................................................... 74 Re(view): Re-view Reviewing .......................................................................................... 78 5.1 Make the Familiar Strange – Troubling Reviewing ....................................................... 78 5.2 Return to the Original Difficulty – What is Reviewing? ............................................... 82 5.3 From Reviewing to Re-viewing ..................................................................................... 89 6 Forms of Re-viewing ........................................................................................................ 91 6.0 Assumptions – Signs in Mathematics ............................................................................ 92 6.1 Re-languaging ................................................................................................................ 95 As shifting focus ............................................................................................................... 95 As repeatedly making language more meaningful and less arbitrary ............................... 98 As using equivalent languages to move inquiry forward and backward ........................ 105 As attending to and enlisting equivalent languages’ different affordances for thinking 108 Back to re-languaging as a whole ................................................................................... 110 6.2 Re-imaging ................................................................................................................... 111 As seeing/sensing the same image differently ................................................................ 114 As changing between equivalent images ........................................................................ 117 As learning from equivalent images’ affordances and limitations ................................. 119 Back to re-imaging as a whole ........................................................................................ 123 6.3 Re-inbodying ................................................................................................................ 124 As making sense of one’s bodily sense........................................................................... 126 As changing between equivalent physical engagements – a case of (re)enacting .......... 130 As invoking different ways of thinking and enlisting different modes of knowing through using the body differently or forming a different body .................................................. 135 Back to re-inbodying as a whole ..................................................................................... 139 6.4 7 Back to Re-viewing as a Whole ................................................................................... 140 Re(Re(view)): Re-view Re-viewing ............................................................................... 144 7.1 Mathematical Story ...................................................................................................... 146 7.2 Re-viewing Interpreted as Re-storying ........................................................................ 151 7.3 Space Opening Up ........................................................................................................ 155 8 Re-encountering Recursion and Recursive Curriculum ................................................. 160 8.1 Recursion ...................................................................................................................... 160 TOWARDS RECURSIVE MATHEMATICS CURRICULA viii Recursion as playing with equivalency........................................................................... 160 Recursion as thinking with the mediums of thought and living through the process of reencountering .................................................................................................................... 164 Recursion as unclogging and playing with time ............................................................. 167 8.2 Recursive Curriculum .................................................................................................. 170 Recursive curriculum as fractal-like and as a continuous interplay of part and whole .. 170 Recursive curriculum as process-oriented and as a biological structure in the state of becoming ......................................................................................................................... 178 9 Design towards Recursive Curricula .............................................................................. 184 9.1 Re-encountering Davis et al.’s Recursive Curriculum Design Process ....................... 185 9.2 Re-imaging Recursive Curriculum: Growing two Fractal Trees ................................. 188 9.3 Re-imaging Recursive Curricula: Growing a Fractal Tree-Spiral ............................... 189 9.4 Curricula Design as Recursive Learning (as thought experiments to enact, engender and experience recursive curricula) ............................................................................................... 194 Recursive curriculum visualization................................................................................. 195 Recursive curricula formation......................................................................................... 198 10 The End is Also the Beginning ....................................................................................... 209 References ................................................................................................................................... 221 Appendix A: Conversation Dates, Formats, and Topics............................................................. 237 Appendix B: Original Workshop and Individual Conversation Plans ........................................ 238 Appendix C: Examples of Revised Workshop and Individual Conversation Plans ................... 242 Appendix D: Textbook Excerpts ................................................................................................ 251 TOWARDS RECURSIVE MATHEMATICS CURRICULA List of Figures ix Figure 3.1.1. A recursive function example. ................................................................................ 28 Figure 3.1.2. A Koch snowflake formation. ................................................................................. 36 Figure 3.1.3. Four working connotations of recursion. ................................................................ 38 Figure 3.2.1. A recursive mathematical understanding model. .................................................... 44 Figure 3.3.1. A fractal tree............................................................................................................ 49 Figure 4.1.1. Two visualizations of recursion. ............................................................................. 61 Figure 4.2.1. A cognitive map & two recursive curriculum development models. ..................... 70 Figure 4.2.2. A holistic view of all workshops/conversations. .................................................... 71 Figure 4.3.1. A visualization of the recursive structure of a hermeneutic inquiry. ...................... 74 Figure 6.1.1. Re-languaging circle’s definition. ........................................................................... 97 Figure 6.1.2. A visualization of the statuses before and after re-languaging ............................. 101 Figure 6.1.3. Sign as two different types of pointers.................................................................. 102 Figure 6.1.4. Bill’s diagram. ....................................................................................................... 106 Figure 6.2.1. Maxine’s seeing a division bracket. ...................................................................... 114 Figure 6.2.2. Representing the sum of n consecutive numbers. ................................................. 117 Figure 6.2.3. A visualization of a re-imaging process. ............................................................... 118 Figure 6.2.4. Two kinds of change between equivalent images. ................................................ 119 Figure 6.2.5. Equivalent images. ................................................................................................ 120 Figure 6.4.1. A tentative categorization of the forms of re-viewing .......................................... 142 Figure 7.3.1. A renewed categorization of the forms of re-viewing. ......................................... 158 Figure 8.2.1. Two comparison design examples. ....................................................................... 172 Figure 8.2.2. Four equivalent comparisons. ............................................................................... 174 Figure 8.2.3. A fractal-like curriculum design. .......................................................................... 176 Figure 8.2.4. A visualization of three chapters about function. ................................................. 177 Figure 8.2.5. Curriculum development maps (set 1). ................................................................. 178 Figure 8.2.6. Curriculum development maps (set 2). ................................................................. 179 Figure 9.1.1. A fractal tree.......................................................................................................... 185 Figure 9.1.2. A visualization of growing a fractal tree. .............................................................. 186 Figure 9.2.1. Idea development & curriculum development as two fractal trees growing in reflection of each other. .............................................................................................................. 189 Figure 9.3.1. Blending three images (i.e., the spiral, the fractal-tree and the binary tree). ........ 190 Figure 9.3.2. A fractal tree with all nodes highlighted. .............................................................. 191 Figure 9.3.3. An infinite fractal tree-spiral. ................................................................................ 192 Figure 9.3.4. A blended representation of recursive curricula: A fractal tree-spiral. ................. 194 Figure 9.4.1. A recursive mathematics curriculum path (Seed: What is 3 x -4?). ..................... 195 Figure 9.4.2. A recursive poem writing curriculum path (Seed: some buttons). ....................... 196 Figure 9.4.3. A recursive curriculum formation ......................................................................... 199 Figure 9.4.4. A recursive curriculum path on the polynomial function binary tree. .................. 199 Figure 9.4.5. Two recursive curriculum formations. .................................................................. 202 TOWARDS RECURSIVE MATHEMATICS CURRICULA x Figure 9.4.6. Forming a self-similar idea and task. .................................................................... 204 Figure 9.4.7. A fractal view of a self-similar idea and a fractal view of a self-similar task. ..... 206 Figure 9.4.8. A fractal-like representation of recursive curricula. ............................................. 208 Figure 10.1. A visualization of perspective change. .................................................................. 215 Running Head: TOWARDS RECURSIVE MATHEAMTICS CURRICULA 1 1 A Calling to Recursion “There is always a story that happened once upon a time”, and such story is exactly where a hermeneutic work starts (Jardine, 2015, p. 238). I was called to the idea of recursion. The first time I encountered “recursion”, I was in Bill Doll’s graduate class. Along with richness, relations, and rigor, recursion was introduced to me as one of the four criteria of Doll’s (1993) post-modern curriculum. It reminded me of Confucius’s teaching in Analects (论语 lun yu): 温故知新 wen gu zhi xin, translated as “gain new knowledge by reviewing old; understand the present by reviewing the past” (Chinese Academy of Social Sciences, 2002, p. 2003). The concept of recursion appeared to me as iterations of the common emphasis of reflection (反思 fan si), review (复习 fu xi), and repetition (重复 chong fu) throughout my schooling and living in China. Still, revisiting these ideas and connecting them to recursion in a new context (i.e., postmodernism, complexity thinking), I thought of them differently (e.g., I learned to reflect by thinking critically and aim reflection towards relations and different understandings rather than the truth) and came up with some practical applications of recursion in teaching (see Luo, 2004). Yet, I found the concept of recursion suspiciously straight forward. The idea of recursion got more puzzling when I started to implement it in my mathematics classes. Interpreting recursion as recursive reflection, I tried out various methods to encourage students to reflect, including reflection journals, retests, learning method reviews and mathematics content reviews. It was my hope that through learning from what I modeled and promoted in class, students could become more reflective. Meanwhile, my professional growth benefited tremendously from constantly rereading the textbooks, reflecting on my practices and modifying them, and mostly by teaching the same course or courses related to each other in TOWARDS RECURSIVE MATHEMATICS CURRICULA 2 different classes over years. While my learning process seemed to be recursive, my students’ learning processes appeared to remain primarily linear: Many students showed little willingness and efficiency to learn from their previously learned topics and mistakes. The activities I used to encourage reflection remained at the margin of the lived curriculum. We spent most of class time rushing through topics: There seemed so little time for so many curriculum contents for which my students showed low readiness. I sensed, somehow, there was more to do to make a curriculum recursive than simply allocating space and time for reflection and offering time between reflections. Confused, I revisited Doll’s concept of recursion during the first year of my doctoral program. This reencounter with recursion brought forth new insights about the concept (see Luo, 2014). Subtly yet saliently, my focus, on my interpretation of recursion as “recursive reflection”, shifted from “reflection” to “recursive”. This emphasis, following complexity thinking, suggests recursion more as a continuous generative looping back movement. The question in recursion, then, is not just how to reflect, but also how to loop back. With this new interpretation, I reconsidered the idea of recursive curriculum, stressing practices with a structure of looping back. Reviewing, in which students go over what they have learned before, logically came to focus. I started to wonder, what kind of reviewing can afford recursion? More generally, how can one design looping back processes that help learners build connections and see something new from what they have encountered before? Or, what might a recursive curriculum be? These questions are worth asking. As shown in the literature review in Chapter 3, there is a gap in how little we (i.e., mathematics educators and mathematics education researchers) know about and enact a recursive curriculum and how important it is for mathematical learning. The word curriculum here refers to braid that includes (at least) planned, lived and hidden dimensions. TOWARDS RECURSIVE MATHEMATICS CURRICULA 3 Although mathematics curricula may be designed with a recursive quality, my teaching experience in Canadian high schools led me to a concern that the lived mathematics curricula tend to be linear: Mathematics topics tend to be covered with little chance of being revisited from different perspectives later, resulting in a fragmental view of mathematics knowledge. As such, mathematics learning can easily become a process of linearly accumulating disconnected topics. To enhance cognitive growth, a nonlinear curriculum centered on recursion is a promising direction through the lens of complexity thinking. A recursive mathematics curriculum fits with how mathematical understanding is thought to develop: According to Pirie and Kieren’s (1994) model, the growth in mathematical understanding follows a recursive path. Motivated by the rich potential of recursion, I embarked on a hermeneutic inquiry about recursive mathematics curricula. The leading question in this research is, “What might a high school recursive mathematics curriculum informed by complexity thinking be?” To answer this question, I sought inspirations through reinterpreting multiple texts, including autobiographical reflections, teaching documents (i.e., programs of studies and textbooks), and conversations with experienced teachers. Three sub-questions are used to guide my text generation and interpretation: 1) How do my teaching and learning experiences inform my understanding of recursive mathematics curriculum? 2) How do teaching documents inform my understanding of recursive mathematics curriculum? 3) How do conversations with experienced high school mathematics teachers inform my understanding of recursive mathematics curriculum? TOWARDS RECURSIVE MATHEMATICS CURRICULA 4 The purpose of the research is to enrich the interpretations of the concept of recursive mathematics curriculum. This research aims to contribute to mathematics education by narrowing the gap between the perceived importance of the recursive mathematics curriculum and the insufficiency of research about it. Given that I entered this study with an initial focus on the affordance of a recursive curriculum for students’ mathematics learning but was charmed later (as one can see in Chapter 4 on the research process) by a recursive curriculum’s affordance for the mathematics learning of teachers, I believe an inquiry about recursive curriculum is beneficial for the inquirer to develop mathematical and pedagogical understanding thus it is worthwhile for both teachers and students to take on. I use “learners” in this dissertation to refer to both students and teachers and use a learner’s pedagogical understanding to refer to an understanding of learning process that can be used to guide one’s and others’ learning. I expect that this research speaks to a reinterpretation of reviewing, and potentially provokes learners to interpret mathematics and curriculum differently and inspires learners to (re)embark on a hermeneutic inquiry on recursive mathematics curricula for the purpose of both learning and teaching. In short, the goal of this study is to inspire, to rekindle, and to continue growing the seed(ling) that has been planted. Before moving on to the details of the study, it is necessary to specify a few writing strategies used in this thesis. First, italics are frequently used as a way to interrupt, nudge and perturb, and to call for rereading, re-sensing and reconsideration, particularly towards any subtle differences that one’s reading might bring forth. Second, all the figures I created during this study were heuristic visualizations, meaning that their generation was my process of thinking through images instead of outputting completed thoughts. Therefore visual spatial elements, such as layout, line, curve, shape, color, font, empty space etc., were not used for decorative purpose TOWARDS RECURSIVE MATHEMATICS CURRICULA 5 but for eliciting unconscious knowing and incurring emerging possibilities. Particularly, I often used colors and fonts in certain ways without rational reasons until their appearance spoke to me some unnoticed relations, which prompted me to visualize again. Some figures turned out having multiple elements to signify the same message. For instance, I used both font and capitalization to differentiate class from case in Figure 8.2.3. In addition, many images are like doodling instead of clear-cut diagrams, signifying their incompleteness and organic becoming. The redundancy and incompleteness of the images are kept in this thesis to preserve the role of images as both medium and unfinished product of my thinking process, and to invite readers to review the images and visualize again. Third, to emphasize part-whole relationship, sections and figures are labeled in a way indicating their respective relations to a bigger whole where they locate (e.g., section 8.2 is the second section in Chapter 8, and Figure 8.2.3 is the third figure in section 8.2). One must note that although I am interested in nonlinearity in this study and the research (including writing) process is nothing but linear, this does not negate a “final” text of this study presented as a linear document. Given that spiral movements are well presented through multiple rounds of reinterpretations of the same idea (e.g., a form of re-viewing, re-viewing as a whole, recursion, recursive curriculum) in one chapter or across chapters, this linear form is recognized as an adequate one before a more effective form of writing is established. Given that I got to experience recursion in different ways in this study and consequently transformed my understanding of hermeneutic research process, recursion, recursive curriculum, mathematics, education, and self, it would not be surprising for a reader to hear similar stories resonating when reading different parts of the dissertation. Meanwhile, as a result of this study being influenced by hermeneutics and complexity thinking, each attempt of looking at the study as a whole or TOWARDS RECURSIVE MATHEMATICS CURRICULA 6 ending the writing led to a new round of reinterpretation. This is reflected particularly in the last three chapters, with each chapter trying to offer a holistic view of the study outcome yet only resulting into renewed interpretations of recursion and recursive curriculum. Another thing worth noting is that several metaphors (e.g., loops of spiral, hermeneutic circle, story, fractal tree-spiral) are used in this thesis as mediums for thinking, expressing, and playing with ideas. My affinity with metaphors came more from my experience of growing up in Chinese language and culture, in which analogies (including metaphors) are omnipresent, and less from literature related to metaphors. While I agree with Lakoff and Johnson (1980) that metaphors influence thoughts, how the particular metaphors used in my research influence my study and how metaphors in general influence thoughts is not the focus of the research at current stage. Rather, the returning to metaphors as mediums of thoughts and contemplating on their influences on thinking belong to the next research cycle. Here I give an overview of the rest of the thesis. In Chapter 2, I set up my study by articulating its theoretical framework (i.e., complexity thinking) and methodology (i.e., hermeneutic inquiry). In Chapter 3, I provide a working definition of recursion based on a preliminary concept study. I then establish the rationale of the study, by arguing for the importance of recursion in education through drawing support from complexity thinking and mathematics education research and by contrasting the insufficient amount of research about recursive curricula with their importance. In Chapter 4, I describe the research design and its dynamic process, reflect on how the process transformed my understanding of hermeneutic inquiry and recursion, and propose an interpretation of the hermeneutic study process as fractallike. Chapter 5 is a review of the process of reviewing through which I complexify it as reviewing, a form of recursion. I propose and conceptualize three forms of re-viewing (i.e., re- TOWARDS RECURSIVE MATHEMATICS CURRICULA 7 languaging, re-imaging, and re-inbodying) in Chapter 6 by drawing concepts and terminology from semiosis and by interpreting and reinterpreting some lived experiences of re-viewing. I then look at these three forms of re-viewing as a whole and reinterpret it as re-storying in Chapter 7. In Chapter 8, I re-theorize re-viewing as re-encountering and reinterpret recursion and recursive curriculum, before moving on to consider the implications of such interpretations in curriculum design in Chapter 9. This attempt brings forth new visualizations of recursive curricula. Chapter 10 looks into the practicality of recursive curriculum in classroom contexts where there often are overarching linear curriculum frameworks at play. This connects back to my personal struggling as a classroom teacher and my relationship with mathematics, resulting into transformations in both self-understanding and interpretations of recursive curriculum. TOWARDS RECURSIVE MATHEMATICS CURRICULA 2 8 Theoretical Underpinning and Methodology Complexity thinking is the theoretical framework for this study, and hermeneutic inquiry is chosen as a suitable methodology for it. In this chapter, I provide an overview of complexity thinking and an interpretation of education from this theoretical perspective. Then I examine the hermeneutic inquiry as a research methodology in general. Lastly, I connect complexity thinking and hermeneutics and explain why hermeneutic inquiry is an appropriate methodology for my research. 2.1 Complexity Thinking The historical development of complexity thinking Complexity thinking, Davis and Sumara (2006) explain, “arose in the confluence of several areas of Western research, including cybernetics, system theory, artificial intelligence, chaos theory, fractal geometry, and nonlinear dynamics” (p. 7). Although many of these lines of inquiry started to develop in the 1950s and 1960s, the origin of complexity thinking can be traced back to many earlier works, such as Giambattista Vico’s book New Science published in 1744, Henri Poincaré’s work related to the three-body problem in 1903, Ludwig Von Bertalanffy’s general system theory developed in the 1940s and 1950s, and Gregory Bateson’s work on cybernetics in the 1940s1. In the 1960s, the establishment of chaos theory supported Poincaré’s idea of unpredictability, a key notion in complexity thinking. Also, Prigogine’s study on thermodynamic systems furthered Bertalanffy’s ideas about open systems, and brought forth some key characteristics of complex systems (i.e., “dissipative structure”, “far from equilibrium”, and “self-organization”). 1 My understanding of the historical development of complexity thinking is largely informed by Davis and Sumara’s (2006), Doll’s (1986, 1989, 2002, 2008), Capra’s (1996), and Fleener’s (2005) works. TOWARDS RECURSIVE MATHEMATICS CURRICULA 9 Starting from the 1970s, complexity research gained great momentum through the use of computer technology. Hypotheses and conjectures related to complexity systems were tested. Through computer simulations of various systems, complexity theory started to emerge as a field of inquiry in the late 1970s and early 1980s. Particularly, Benoît Mandelbrot’s (1967, 1977, 1983) establishment of fractal geometry and its wide application in various disciplines further supported a new world view: “Nature embraces not simplicity but complexity” (Doll, 2002, p. 45). The development of complexity research sped up after 1986. Complexity thinking was popularized through the increasing use of computer technology in the 1980s and 1990s. By the 1990s, complexity research was a discernible domain. Its focus shifted to the stimulation and development of complex systems. At the end of the 1990s, complexity theory had developed such rigor that it was renamed as complexity science. Complexity thinking, commonly called complexity science, was not a distinct term until the 1990s. Based on Richardson and Cilliers (2001), complexity thinking refers to a school of thinking that “focuses on the epistemological consequences of assuming the ubiquity of complexity” (p. 7), which means an attitude that “is concerned with the philosophical and pragmatic implications of assuming a complex universe, and might thus be described as representing a way of thinking and acting” (Davis & Sumara, 2006, p. 18). Key ideas in complexity thinking Complexity thinking focuses on the study of complex systems, which are pervasive in the world. Anthills, climates, ecosystems, economies, cultures, brains and living units are all examples of complex system. A complex system is a self-organizing and adaptive system that exhibit attributes that are not possessed by any of its components (Capra, 1996; Davis & Sumara, TOWARDS RECURSIVE MATHEMATICS CURRICULA 10 2006). For a complex system, the whole is greater than the sum of the parts. Thus one needs to understand a complex system both holistically and analytically. Complex systems’ self-organization is sustained through the nonlinear feedback in them (Capra, 1996). The feedback loops enable the influence of one change in a system to loop back to the system itself. Complex systems are also open systems as they consistently exchange energy and matter with their environment (Davis & Sumara, 2006). Because of their selforganization and openness towards the outside world, complex systems can maintain their current structures while being open to emerging possibilities at the same time. They are unpredictable and nondeterministic: Changing one part of a system in one way does not guarantee certain behaviors from the other parts. A complex system is fractal-like, meaning that the system demonstrates self-similarity across various scales and its development is recursive (Davis & Sumara, 2006). Fractal was coined by Mandelbrot from the Latin adjective fractus, which means “irregular or fragmented” (Mandelbrot, 1977, p. 4) to refer to and depict a self-similar phenomenon. Self-similarity describes a signature property of fractals, that each portion of a fractal can be viewed as a reduced-scale image of the whole (Mandelbrot, 1967), or simply put, that you can see the whole through a part of it. While the similarities between parts and whole in a strictly self-similar fractal (e.g., the Koch snowflake in Figure 3.1.2) are exact, the ones in a fractal-like system are approximate—if you zoom in on different parts of it and you see slightly different copies of the whole. Fractals are formed by recursion through infinitely many stages (Mandelbrot, 1977, 1983). The fascinating and powerful aspect of fractals is that the complexity of fractals originates from following some simple recursive rules. Often a simple recursive formula can generate