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					       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
 Ethnomathematics from an indigenous
  person‟s perspective
 The paradox in ethnomathematics, and
  issues of decontextualisation and
 The window metaphor: A shift in
 Mutual interrogation as an approach to
 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
 Ethnomathematics as site of IP
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
 “[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
                                          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
 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
   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
   Mutual interrogation
      Standpoint: Superiority of conventional
      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
              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
                      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
    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
    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
Indicators of Suitability of Stone Walling
for Mutual Interrogation with Mathematics

 Some parallelisms in the two
   Highly developed and systematised
   How practitioners are regarded by the
    *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!
 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

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

       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
                                                              i 1
                                                                     i   
                                                                             T   
                                                                                            wi (t )dt
                     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
 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
 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
 D‟Ambrosio‟s lament: Mathematics without ethics!
 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!

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