Mutual Interrogation as an Ethnomathematical Approach

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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

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Outline

• Ethnomathematics from an indigenous persons
  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

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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

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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)

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Bartons Colonisation of Knowledge

• The 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)

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Skovsmoses and Vithals end of innocence

• Could 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)

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Dowlings 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)

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Dowlings 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)

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An Example Do they know its mathematics?

• The Weaving Patterns of the Northern Kankana-ey
  of Mountain Province (UPB Discipline of
  Mathematics Faculty, 1995)

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The Window Metaphor

• A shift in perspective
• Bartons 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)

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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)

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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

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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

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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 Aschers 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/Transf
ormation Example Lipka/Knijnik
Shifting Perspectives in the Study of Cultural
Practice
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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 ethnomathematicians
  mathematics.
• Entails more exploring of alternative conceptions
  and their effects in each knowledge system.

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Stone Walls and Water Flows
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The Rice Terraces of the Cordillera
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(No Transcript)
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The Stone walling Practice and Mathematics
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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

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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?

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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!

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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!

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Water Flows
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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 studys 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 Geoffs 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.

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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 Geoffs 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 . 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 Here, is the water
evaporation rate, is the seepage rate, is
the hungriness factor, 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, 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 farmers 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.
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• 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
  as the measure of dissatisfaction in every
  paddy. What we want is to minimise this
  dissatisfaction, not only in one paddy, but over
  the whole network.
• We can consider as a cost function.
  
• The objective is to drive to minimise
  dissatisfaction. But we can consider the average
  water level over a period of time. A better model
  is then given by
• Here, the integral
  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.
• is still the rate of flow of water
  going out from paddy i to paddy k.
• We can describe this situation in a diagram
  

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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.

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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.
• DAmbrosios lament Mathematics without ethics!

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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!