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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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?
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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
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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
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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-
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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.
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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.
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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,
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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