weaving mathematics and culture: mutual interrogation as a methodological approach
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WEAVING MATHEMATICS AND CULTURE: MUTUAL INTERROGATION AS A METHODOLOGICAL APPROACH - NOOR AISHIKIN ADAMTRANSCRIPT
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WEAVING MATHEMATICS AND CULTURE: MUTUAL INTERROGATION AS A METHODOLOGICAL APPROACH
NOOR AISHIKIN ADAM [email protected]
Supervisor: PROF. BILL BARTON University of Auckland, New Zealand
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Mutual Interrogation •! A methodological process for ethnomathematical research
•! Definition of mutual interrogation:
“ A process of setting up two systems of knowledge in parallel to each
other to illuminate their similarities and differences, and explore the
potential of enhancing and transforming each other ” (Alangui, 2010)
•! Proposed as a way of avoiding:
ideological colonialism (imposition of mathematical concepts and
structures onto cultural knowledge);
knowledge decontextualisation (taking of knowledge and practice out
of cultural context to highlight ‘inherent’ mathematical values)
•! Barton’s QRS system: A system of meanings that occur when a group of people attempt to manage quantities, form relationships and
represent space within their own surroundings (Barton, 1999)
Alangui, W. (2010). Stone walls and water flows: Interrogating cultural practice and mathematics. Unpublished doctoral dissertation, University of Auckland, New Zealand.
Barton, B. (1999). Ethnomathematics and philosophy. Zentralblatt fur Didaktik der Mathematik (ZDM), 31(2), 54-58.
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The Approach •! Carried out through a process of critical dialogue between cultural
knowledge and mathematics (via the practitioners)
•! The researcher (ie. ethnomathematician):
i)! Facilitates the interactions between practitioners (by representing one knowledge system to the other);
ii)! Critically reflects on his or her assumptions and beliefs about mathematics;
iii)! Experiences perceptual shifts about mathematics;
iv)! Explores alternative conceptions;
v)! Disseminates outcome of dialogue to mathematical communities
•! Internal and external aspects of mutual interrogation
•! May lead to a broadening or transformation in conventional mathematical ideas, as well as contemporary development of cultural practice
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The Study on Weaving •! Primary aim: To test the efficacy of mutual interrogation and
facilitate its employment as a methodology in ethnomathematical research
•! Dialogue between Malay food cover weavers and mathematicians (Malaysia & NZ)
•! Researcher as mediator of dialogue
•! Three phases of fieldwork - Phase 1: March - June 2008
Phase 2: Nov 2008 - Jan 2009
Phase 3: Oct – Dec 2009
•! Ethnographic techniques - participant observation, interviews,
audio & video recording, fieldnotes
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Research Objectives 1. i) To undertake participant observation with weavers in order to
understand weaving processes and conceptual frameworks;
ii) To trial and develop extension to existing weaving in order to understand weaving limitations and possibilities;
2. i) To document mathematical responses of mathematicians to Malay weaving;
ii) To develop conventional mathematics that relates to weaving in order to formalise weaving limitations and possibilities;
3. To facilitate an exchange of ideas between weavers and mathematicians and investigate the extent to which each others’ concepts can enhance their own perspectives and practices.
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Malay Food Cover (Tudung Saji)
•! Samples produced in the states of Terengganu and Melaka
(east coast and west coast of Malaysia, respectively)
•! Weaving technique: triaxial or hexagonal weave (interlacing of
three strands in three directions)
•! Cone-shaped framework - pentagonal hole surrounded by
hexagonal holes
•! 11 different sizes (diameter: 2” - 32”), 20 basic patterns/designs,
at least 10 combined patterns
•! Focus of investigation: weaving technique, structural
construction, pattern formation
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Construction of a Tudung Saji
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5-6 Connections in Patterns
Lima Buah Negeri (Five States) Tebeng Layar (Spread Sails) Bintang Tabur (Scattered Stars)
Kahwin Merdeka (Free Union) Kapal Layar (Sailboat) Pati Sekawan (Flock of Pigeons)
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The Dialogue PHASE 1 PHASE 2 PHASE 3
WEAVERS MATHEMATICIANS WEAVERS MATHEMATICIANS WEAVERS
Starting point: 5 strands = peaked
6 strands = flat
Why not 3, 4 or 7 strands?
Possible to form a peak from 3 or 4 strands
What about 7 strands? (Saddle-shaped peak)
Wave-like structure
Not many patterns can be achieved with
left-turning peak.
All of the patterns can be created regardless of
the turning at the peak.
Agreed. However, the handedness determines
the turning of motifs.
Why are there discontinuities in certain
patterns?
Natural occurrences:-
i)! arrangement of uneven number of coloured strands at the peak
ii)! overlapping of
strands.
Why conical, and not any other shape?
The shape is ‘high and rounded’ to allow hot
steam to travel upward.
Where did the idea of food covers originate?
A fusion between triaxial latticework of Chinese
hats and Malay basket weaving.
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Structural Changes – Phase 2
4-strand peak vs 5-strand peak:
3-strand peak:
Weavers’ views on 4-strand
and 3-strand peaks: -! unsuitable
-! impractical -! uneconomical
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Weaving Template
Developed to:
a) reproduce existing patterns and create fictitious ones
b) classify two-colour patterns, R (red) and Y (yellow)
- blocks of 2 to 6 strands
eg. Blocks of 2 strands (RY)
A: RYRYRY…
B: RYRYRY…
C: RYRYRY…
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Fictitious Patterns
A weaver’s comment:
“ Template weaving does not resemble actual tudung saji
weaving, therefore it is impossible to determine whether
the generated patterns could be replicated ”
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Starting Point = 6 Strands
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Starting Point = 5 Strands
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Starting Point = 7 Strands
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Structural Changes – Phase 3 7 strands as starting point:
“ Hidden mathematical ideas can be uncovered through a
reconstruction of past knowledge. In order to understand the reasons
behind the form of the product, it is necessary to learn the production
techniques and vary the form at each stage of the process. This
method would lead to an observation of its practicality and the
possibility of the form being the optimal or only solution of a production
problem ” (Gerdes, 1994)
Gerdes, P. (1994). On mathematics in the history of Sub-Saharan Africa. Historia Mathematica, 21, 345 - 376.
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Framework Transformation – Phase 3
3-peak tudung saji
2-peak tudung saji
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Dialogue Analysis •! Both weavers and mathematicians were highly engaged in the dialogue
•! The interactions had succeeded in uncovering several perspectives that
concerned both parties
- structural changes in framework construction
- development of new ideas in tudung saji weaving
- mathematicians gained insights on the mathematical ideas embedded
in weaving
•! Power relations in the dialogue – imbalance in the interrogation process
“ Cultural is not only the result of interactions with the natural and social
environment, but also subjected to interactions with the power relations
both among and within cultural groups ”
(Vithal & Skovsmose, 1997)
Vithal, R. & Skovsmose, O. (1997). The end of innocence: A critique of ‘ethnomathematics’. Educational Studies in Mathematics, 34(2), 131-157.
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Pattern Classification – Blocks of 2 Strands
ORDER ORIENTATION
000
011
101
110
000
111
100
010
001
111
( 0 = RY…; 1 = YR… ) = 8 orders of arrangement
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Pattern Classification – Blocks of 3 Strands
ORDER ORIENTATION ORDER ORIENTATION ORDER ORIENTATION
000
111
222
012
120
201
021
102
210
010
121
202
020
101
212
011
122
200
002
110
221
001
112
220
022
100
211
( 0 = RRY…; 1 = RYR…; 2 = YRR… ) = 27 orders
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Ordering of Two-colour Strand Blocks
NO. OF STRAND ORDERING TOTAL
2 0 = RY; 1 = RY 8
3 0 = RRY; 1 = RYR; 2 = YRR 27
4 (i)! 0 = RRRY; 1 = RRYR; 2 = RYRR; 3 = YRRR
(ii) 0 = RRYY; 1 = RYYR; 2 = YYRR; 3 = YRRY
64
5 (i)! 0 = RRRRY; 1 = RRRYR; 2 = RRYRR; 3 = RYRRR;
4 = YRRRR
(ii)! 0 = RRRYY; 1 = RRYYR; 2 = RYYRR; 3 = YYRRR;
4 = YRRRY
125
6 (i)! 0 = RRRRRY; 1 = RRRRYR; 2 = RRRYRR;
3 = RRYRRR; 4 = RYRRRR; 5 = YRRRRR
(ii)! 0 = RRRRYY; 1 = RRRYYR; 2 = RRYYRR;
3 = RYYRRR; 4 = YYRRRR; 5 = YRRRRY
216
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Graphical Representations
DARK BLUE LIGHT BLUE
2 STRANDS: (0 = RY; 1 = YR)
000 011
101 110
111 100
010 001
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3 STRANDS: RRY, RYR & YRR
BLUE PINK GREEN
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4 STRANDS: RRRY, RRYR, RYRR & YRRR
DARK BLUE LIGHT BLUE PINK RED
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5 STRANDS: RRRRY, RRRYR, RRYRR, RYRRR & YRRRR
DARK BLUE LIGHT BLUE PINK RED GREEN
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6 STRANDS: RRRRRY, RRRRYR, RRRYRR, RRYRRR, RYRRRR & YRRRRR
RED PURPLE BROWN GREEN BLUE
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Underlying Group Structure
Strand Blocks Triangle Hexagon
2-strand 2 0
3-strand 4 1
4-strand 6 2
5-strand 8 3
6-strand 10 4
n-strand 2n - 2 n - 2
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How Effective is Mutual Interrogation? •! Ensures that the voices of practitioners who take part in the dialogue are
heard, thus addresses the following criticism:
“ Ethnomathematics attempts to interpret practices found in different
cultural groups in terms of mathematical concepts and models, ie. to
identify and establish thinking abstractions that underpin these practices.
They are seen as providing cultural affirmation and entry into
mathematical abstractions themselves. But for whom? By and large we
do not hear the voices of the people whose practices are thus
interpreted. ” (Vithal & Skovsmose, 1997)
•! Perceptual shifts occur in both the researcher and practitioners, leading
to alternative conceptions (eg. contribution of ideas by mathematicians
in refining the weaving template and analysis of patterns)
•! Mediator plays a major role in the process
Vithal, R. & Skovsmose, O. (1997). The End of Innocence: A Critique of ‘Ethnomathematics’. Educational Studies in Mathematics 34(2), 131 – 157.
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Thank You