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Interrogating Cultural Practice and Mathematics
Wilfredo Vidal Alangui
A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics Education,
The University of Auckland
2010
This thesis is dedicated to my parents,
Crispin and Carlina.
Interrogating Cultural Practice and Mathematics i
ABSTRACT
Ethnomathematical practice has been criticised for approaches that unwittingly privilege the mathematical gaze. The literature shows that uncritical investigations of mathematical knowledge in cultural practice may lead to ideological colonialism and knowledge decontextualisation. These concerns are legitimate and challenge the basic principles of ethnomathematics.
To address these concerns, a general methodological framework is developed in this study. The framework sets out generic questions to guide an ethnomathematical research study investigating a cultural practice. Mutual interrogation as an approach, and as a process of critical dialogue, was developed to allow the researcher to perform the ethnomathematical task of relating the structures of cultural practice to conventional mathematics whilst avoiding the dual dangers mentioned above.
Using ethnography, this study describes the knowledge embedded in stone walling and water management, which are two aspects of the rice terracing practice in the indigenous communities of Agawa and Gueday in Besao, Mt. Province, in the Cordillera region, northern Philippines. Exemplars of mutual interrogation between mathematical knowledge and the knowledge embedded in the two practices were then developed. Both mathematics and culture were used as frames of reference in the interrogation process.
This study suggests that the general methodological framework can be useful in guiding ethnomathematical research. It also demonstrates how the knowledge embedded in stone walling and water management may be used to interrogate conventional mathematical ideas. The study shows the potential of mutual interrogation in broadening the conception of mathematics, which is one of the goals of ethnomathematics.
ACKNOWLEDGEMENTS
I would like to express my thanks to my family, friends, and colleagues who have contributed to the completion of this thesis. I am grateful for the encouragement and support that I got from a lot of people, especially from the following:
• My supervisors Bill Barton and Ivan Reilly for their intellectual guidance, patience, and humour. My special thanks to Bill for his endless support and belief in me, and also for introducing me to the wonderful people of Brazil.
• The people of Agawa and Gueday, for sharing their knowledge to us and for welcoming me and John Rey as their own. Geoff Nicholls, Gio Malapit, and Cris Patacsil for our dialogues.
• My Igorot family in Auckland – Manong Steve and Manang Myriam for adopting me as their eldest, and to my extended siblings, Darrow and Sheldon who get displaced everytime I am around. My other whanau, Te Tuhi Robust, Roz, Helene and the kids, for making me a part of the family.
• Friends and colleagues at the Mathematics Education Unit who work very hard but always manage to find time together at Mission Bay. Thank you for making our stay at the UoA memorable. The same goes to the other colleagues at the Mathematics Department, The University of Auckland.
• Colleagues at the College of Science, University of the Philippines Baguio, and at the Department of Mathematics, for urging me to finish.
• Pip Neville-Barton, Kay and John Irwin, and Bob Scott for their friendship. Sir Paul Reeves, Chris Tremewan and Graham Smith for introducing me to New Zealand. Linda Tuhiwai-Smith, for her inspiration, and for giving me the crazy idea back in 1999 to contact a guy named Bill.
• Dear friends Shehenaz, Alan, Greg, Daniel, Bernd, Hannah, RR, Sel, Dech, Caster, Vicky, Yvonne, Raymond, Toots, Ron, Joseph, Phuong, Scott, Harley, Chris and Rob, housemates at No. 1 Domain Drive and Park Road Flats.
• My family for their love and prayers, and to John Rey who is very much a part of this journey.
Interrogating Cultural Practice and Mathematics iii
TABLE OF CONTENTS
ABSTRACT i
ACKNOWLEDGEMENTS ii
TABLE OF CONTENTS iii
LIST OF TABLES vi
LIST OF FIGURES vii
LIST OF VIGNETTES viii
CHAPTER ONE RETHINKING KNOWLEDGE,
EVOLVING IDENTITIES 1
1.1 Mathematics and its Discontents 1
1.2 Imperatives: Towards Transforming Mathematics 3 1.3 Rethinking Mathematical Knowledge: The Road to
Ethnomathematics 5
1.4 My Evolving Identities 6
1.5 Rationale of the Study 9
1.6 The Problem 10
1.7 The Aim of the Study 11
1.8 Overview of the Study 12
1.9 The Chapters 12
CHAPTER TWO ETHNOMATHETICS UNFOLDING 15
2.1 Sowing the Seeds 16
2.2 Ethnomathematics as a Slogan System 19
2.3 Critical Issues 25
2.4 Ethnomathematics: An Evolving Perspective 44
CHAPTER THREE ENGAGING CULTURE 47
3.1 Problematising Culture 47
3.2 The Anthropological Notion of Culture 48
3.3 Limitations of ‘Culture’ 49
3.4 Changing Views of Culture 53
3.5 Re-Viewing Culture: Challenges to the Theory of Ethnomathematics 58
CHAPTER FOUR FRAMING THE ETHNOMATHEMATICAL
RESEARCH PROCESS 61
4.1 Making Sense in an Unfamiliar World: A Methodological Framework 62
4.2 Utilising Ethnography 71
4.3 Ethnomathematical Research and the Challenges of Ethnography 73
4.4 Restructuring Experience 76 CHAPTER FIVE MUTUAL INTERROGATION 77 5.1 Looking Out, Looking In: Culture, Agency, Critique, Reflexivity and Dialogue 77
5.2 Transmutations in the Science of the Local 79
5.3 The Ethnomathematical Practice and its Transmutations 80 5.4 Defining Mutual Interrogation 86 CHAPTER SIX TALES FROM THE FIELD 89 6.1 Designing the Research: Myself as a Subjective Researcher 90 6.2 Into the Field: Initial Visits, Contact-Building and Establishing Camp 93 6.3 Making Marks, Writing Texts: My Research Strategies 95 6.4 Reflections about Fieldwork 103
CHAPTER SEVEN COMMUNITY LIFE AS THE CONTEXT OF RESEARCH 107
7.1 The Cordillera Region 108
7.2 Brief Historical Background 114
7.3 The Research Sites 119
CHAPTER EIGHT STONE WALLS AND WATER FLOWS IN AGAWA AND GUEDAY 130
8.1 The Papayeo of Agawa and Gueday 130
8.2 Menkabiti: The Indigenous Practice of Stone Walling 139
8.3 Binnanes: Water Management and Distribution 152
Interrogating Cultural Practice and Mathematics v CHAPTER NINE CULTURAL PRACTICE AND MATHEMATICS:
CRITICAL DIALOGUES 162
9.1 Stone Walls: Mathematics as Frame of Reference 163
9.2 Water Flows: Culture as Frame of Reference 176
9.3 Insights from the Critical Dialogues 181
CHAPTER TEN TOWARDS A VIBRANT THEORY OF ETHNOMATHEMATICS 184
10.1 The Methodological Framework for Ethnomathematical Research and the Use of Ethnography 185
10.2 Mutual Interrogation as a Process of Critical Dialogue 186
10.3 Limitations of the Study and Recommendations 192
10.4 Final Reflection 193
REFERENCES 194
APPENDIX A 209
APPENDIX B 211
LIST OF TABLES
Table 4.1 Framework for Ethnomathematical Research 70
Table 6.1 Respondents Classified according to Gender 97
Table 6.2 Summary of Participants’ Coding System 97
Table 6.3 Distribution of Respondents Based on Type of Interview 98
Table 7.1 Population in Agawa and Gueday by Gender, 2000 120
Table 8.1 Parts of a Typical Payeo in Agawa and Gueday 148
Table 8.2 Areas Serviced by the Tampo-oc Irrigation 153
Interrogating Cultural Practice and Mathematics vii
LIST OF FIGURES
Figure 5.1 Transmutations in the Ethnomathematical Study
of Cultural Practice 83
Figure 7.1 Map of Southeast Asia 108
Figure 7.2 Maps of the Philippines, the Cordillera Administrative Region (CAR) and Mountain Province 109
Figure 7.3 A Boy at a Dap-ay in Gueday 122
Figure 7.4 The Stone Calendar of Gueday 125
Figure 7.5 The Bridge that separates Gueday and Agawa 129
Figure 8.1 The Map of Besao 131
Figure 8.2 The Papayeo of Agawa and Gueday 138
Figure 8.3 Repairing a Stone Wall in Gueday 143
Figure 8.4 A Newly-Built Stone Wall in Gueday 144
Figure 8.5 Sketch of a High Wall with a Secondary Support 145
Figure 8.6 Sketch of a High Wall with Secondary Terraces 145
Figure 8.7 A Newly-Repaired Payeo in Gueday that was Terraced 146
Figure 8.8 The Tampo-oc Irrigation System 154
Figure 8.9 A Cluster of Payeo between Agawa and Gueday 159
LIST OF VIGNETTES
Vignette 7.1 The house at Nabanig 121
Vignette 7.2 Obaya 129
