Mutual Interrogation as an Ethnomathematical Approach Willy V. Alangui University of the Philippines Baguio University of Auckland 3rd International Conference on Ethnomathematics Auckland, New Zealand 12-16 February 2006 Outline Ethnomathematics from an indigenous person‟s perspective The paradox in ethnomathematics, and issues of decontextualisation and colonisation The window metaphor: A shift in perspective Mutual interrogation as an approach to ethnomathematics Some examples from my research Why Ethnomathematics? Indigenous peoples‟ issues Ethnomathematics as a challenge to the influence and dominance of the west in the conception of mathematics and mathematical knowledge Ethnomathematics as site of IP struggle The Paradox of Ethnomathematics “How can anyone who is schooled in conventional Western mathematics „see‟ any form of mathematics other than that which resembles the conventional mathematics with which she/he is familiar?” (Millroy, 1992) Barton‟s Colonisation of Knowledge “[T]he concept of mathematics as a category of activity in any culture is a Western idea. Other cultures do not recognize „mathematics‟ as separate from some other aspects of their culture – it cannot be isolated out. How, then, does the union of all ethnomathematics come about? Does it include all the other parts of other culture which are regarded as inextricably linked? If not, then the Western idea of mathematics is being adopted, which is another expression of ideological colonialism.” Barton (1996) Skovsmose‟s and Vithal‟s end of innocence “[C]ould ethnomathematics itself become implicated in the formatting power of mathematics? Is there a possibility that ethnomathematics, in the very process of interpreting the activity of, say, basket weaving, invents new (mathematical) structures which then colonise and rearrange the reality of basket weaving?” Skovsmose and Vithal (1997) Dowling‟s myths The myth of reference: a division of the intellectual (mathematics) and the manual (cultural practice) “It is as if the mathematician casts a knowing gaze upon the non-mathematical world and describes it in mathematical terms. I want to claim that the myth is that the resulting descriptions and commentaries are about that which they appear to describe, that mathematics can refer to something other than itself.” Dowling (1998) Dowling‟s myths The myth of emancipation: a unification of the intellectual and the manual “Revealing the truly mathematical content of what might otherwise be regarded as primitive practices elevates the practices, and ultimately emancipates the practitioners… European mathematics constitutes recognition principles which are projected onto the other, so that mathematics can be „discovered‟ under its gaze. The myth announces that the mathematics was there already.” Dowling (1998) An Example: Do they know it‟s mathematics? The Weaving Patterns of the Northern Kankana-ey of Mountain Province (UPB Discipline of Mathematics Faculty, 1995) The Window Metaphor A shift in perspective Barton‟s Definition “Ethnomathematics is the field of study which examines the way people from other cultures understand, articulate and use concepts and practices which are from their culture and which the researcher describes as mathematical.” Barton (1996) The Window Metaphor Ethnomathematics is not mathematics. It may be thought of as a window with which to view mathematics. “What is being viewed through this window? What is the nature of the window? Who are the actors in this viewing process, and why are they looking through this window?” Barton (1996) Looking Out, looking In: Culture and Agency, Critique and Reflexivity Culture Anthropological view (bounded and discrete) Politicised view Aspectival view (overlapping, fluid, hybrid, a work in progress) Agency Critique and Reflexivity Mutual Interrogation Shifting of perspectives (transmutations) in Anthropology, the science of the local (Mendoza, 2001) Three stages in the way the West interrogates other cultures One-sided interrogation No interrogation; knowledge is contextual Mutual interrogation Standpoint: Superiority of conventional mathematics. Interrogation: One-sided. Mathematicians apply conventional mathematics to ‘explain’ the mathematical ideas in a particular culture. Motivation: Educational Curiosity Example: Weaving Patterns of UP Baguio Standpoint: The ‘other’ is just as ‘knowlegeable’ as ‘us’ mathematically although in a different way. Interrogation: No interrogation. There is no need to interrogate and compare because knowledge is strictly contextual. Motivation: Valuing other forms of knowledge. Example: Ascher’s work Standpoint: Cultural practice will tell the mathematician more about mathematics than what we can learn about the ‘other.’ Interrogation: Mutual. Cultural practice is used to interrogate conventional mathematical ideas and vice versa. Motivation: Shifting of perspectives/Transformation Example: Lipka/Knijnik Shifting Perspectives in the Study of Cultural Practice What is Mutual Interrogation? Mutual interrogation is the process of setting up two systems of knowledge in parallel to each other in order to illuminate their similarities and differences, and explore the potential of enhancing each other. Sets up a dialogue between cultural practice and mathematics Draws up parallels between the two practices, using elements in one system to ask questions of the other. Involves a series of reflection and questioning of assumptions about the ethnomathematician‟s mathematics. Entails more exploring of alternative conceptions and their effects in each knowledge system. Stone Walls and Water Flows The Rice Terraces of the Cordillera The Stone walling Practice and Mathematics Indicators of Suitability of Stone Walling for Mutual Interrogation with Mathematics Some parallelisms in the two practices Highly developed and systematised How practitioners are regarded by the community *Born with the skill *Wise person Positioning of Stones Why are individual stones with a longer dimension laid as headers instead of stretchers? To develop transverse strength through the wall (Conklin, 1980). Why are thin flat stones set up on their sides so that their longer cross- sectional dimension is placed more or less vertically? Ag-kagit ti bato Elders Lesser space for weeds to grow. More contact with other stones. Physicists Gravity and friction “We are missing something!” (Gio, Physicist) Ag-kagit ti bato: Stones clasping each other! GRIP Basic operative concept in the positioning of stones (ag-kagit, ag- innirot, ag-kinnagat) Sequence of stones that are gripping each other. Morabaraba game: A cow does not move on three legs! Water Flows The Mathematical Perspective on Water Flows How would a mathematician approach the problem of regulating water flows? This question arose as a result of this study‟s desire to set up a dialogue between cultural practice and mathematics. Fortunately, the Department of Mathematics of the University of Auckland has a number of members of the faculty who are recognised in the field of applied mathematics, and whose work revolved around modelling real-life phenomena. Dr. Geoff Nicholls was one of them. Geoff, a theoretical physicist cum practising mathematician, was a senior lecturer at the Department from 2002 to middle part of 2005. He taught modelling papers in both the undergraduate and graduate programmes of the Department. His more recent research interests were in linguistics and in spatial-genealogical models of bird song, population and statistical inference in archaeology. He has published extensively and has presented his research papers in numerous conferences around the world. Also, his office at the Department was directly across mine. We met twice to discuss two aspects of water irrigation in the research sites. The meetings became some sort of a dialogue between us. Geoff „interrogated‟ me about details of the practice, and „interrogated‟ back to make sure that I understood and contributed to the model that we were developing. In a way, I represented the voice of the farmers of Agawa and Gueday (having been able to document the practice), and Geoff represented the voice of the mathematician. One problem that was discussed was the regulation of the water level at each payeo, the other was that of maintaining a network of flows between the papayeo. There were two outcomes of these meetings. The first outcome was the formulation of initial models for the two practices mentioned above. The second was Geoff‟s ideas on what a model should look like and what indicators were there to determine whether the practitioners of a certain cultural practice go about mathematical modelling in some way. The following is a post-hoc reconstruction of how Geoff and I, both mathematicians, talked about the problem of water regulation in the papayeo. The account was based on the notes that I took during our conversation and whilst Geoff wrote his ideas on the board. The sequence of the discussion is rearranged to capture the way mathematicians would normally analyse a problem. How to model the water flow in a paddy The venue was at Geoff‟s office. He was thinking aloud and began writing on the board after I have explained to him the practice. As a mathematician myself, I understood what was going on: he was turning the problem into symbolic language, the first step in the process of abstraction. We can have to represent the actual water level in a paddy, and as the desired water level, or maybe we can call this optimal, or ideal? We need a critical level, let this be . im in w The variable is the water level. This is affected by several factors: evaporation, seepage, surface area, rate of flow of water going out from paddy i to paddy k. We then have the rate of change of water level with respect to time as: dwi ei Ai si Ai hi f i ,k f i 1,i rAi dt Here, ei (t ) is the water evaporation rate,si (t ) is the seepage rate, h i is ‘the hungriness’ factor, f i ,k is the rate of flow of water going out from paddy i to paddy k, r is the rain factor and is the surface area of the paddy. Also, 0 wi wim in f i ,k f i ,k o.w. We know that the rate can be controlled by the farmer by manipulating her/his own outlet. The negative sign means that the parameter contributes to a decrease in the level whilst the positive sign indicates a contribution to the increase in the water level. For example, represents water going out from a farmer‟s paddy i, the water going to paddy k, thus the its negative sign; on the other hand represents water into paddy i, coming from paddy i-1, thus its positive sign. Rain, represented by r, obviously contributes to the rise of water level, which explains its positive sign. How to model a network of water flows Satisfied with this initial model, we next turned our attention to the problem of regulating water flows in a network of paddies. Using the same notation as above, we have wi wi as the measure of ‘dissatisfaction’ in every paddy. What * to we want is N minimise this dissatisfaction, not only in one paddy, but over the whole network. We can consider w* w as a cost function. i 1 i i The objective is to drive f i to minimise dissatisfaction. But we can consider the average water level over a period of time. A better model is then given by: N 1 w i 1 * i T lastweek wi (t )dt 1 T lastweek Here, the integral wi (t )dt represents the average water level over a period of time T. Geoff now considered the network of papayeo. He continued by talking about something I was not longer familiar with: free chain. We have a finite number of paddies connected by the outlets. It is a free chain. So, consider a finite chain of N paddies that are connected by their respective water outlet. f i ,k (t ) is still the rate of flow of water going out from paddy i to paddy k. We can describe this situation in a diagram: f N , 0 (t ) f 0 , k (0) Rate at the first paddy/outlet Rate at the last paddy/outlet at time t=0. at time t. Geoff continued to think aloud: as a network, this problem is a complex one. Noting that these mathematical problems fall under control theory, he came up with several questions: What kinds of decisions are made for the system to work? Given that the system of networks works, what adjustments or decisions on f i ,k (t ) could have disastrous effects? If the existing network was stable, then the people have solved a complex mathematical problem! He sounded amused and excited, giving the impression that this was something significant if studied further. As for his questions, I suspected he was not waiting for the answers to come from me. Deficient model, ethically speaking… We missed something, again! Dagiti papayeo ditoy baba ti mangit- ited ti danum dita ngato It is ethically wrong to hold off water for her/his own payeo at the expense of the papayeo below it. Need to think of the effect on the whole network of flows. Water flows as a network of relationships of people Maintaining a desired water level is governed by a bigger “factor”, a cultural value or ethic, which is social responsibility. Social responsibility dictates how a farmer should deal with the problem of obtaining the desired water flow, on top of all the other variables that s/he takes into consideration. The network of water flows is a chain of social responsibility that goes up and down the papayeo. Ensuring that water flows from one payeo to the next is an expression of the value that people put in their relations with others. In a way, water flow is a metaphor for the relationships of the people in the community. D‟Ambrosio‟s lament: Mathematics without ethics! Conclusion Shifting of approach and perspective is needed to avoid critical issues of decontexualisation and recolonisation knowledge, and keep the integrity of cultural practice. Mutual interrogation as an ethnomathematical approach. THANK YOU!