WEAVING MATHEMATICS AND CULTURE: MUTUAL INTERROGATION AS A METHODOLOGICAL APPROACH - NOOR AISHIKIN ADAM

1. WEAVING MATHEMATICS AND CULTURE:
MUTUAL INTERROGATION AS A
METHODOLOGICAL APPROACH
NOOR AISHIKIN ADAM
aishikin@math.auckland.ac.nz
Supervisor: PROF. BILL BARTON
University of Auckland, New Zealand

2. 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 Unpublished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.

3. 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

4. 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

5. 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.

6. 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

7. Construction of a Tudung Saji

8. 5-6 Connections in Patterns
Lima Buah Negeri (Five States) Tebeng Layar (Spread Sails) Bintang Tabur (Scattered Stars)
Kahwin Merdeka (Free Union) Pati Sekawan (Flock of Pigeons) Kapal Layar (Sailboat)

9. The Dialogue
PHASE 1 PHASE 2 PHASE 3
WEAVERS MATHEMATICIANS WEAVERS MATHEMATICIANS WEAVERS
Starting point: Why not 3, 4 or 7 Possible to form a peak What about 7 strands? Wave-like structure
5 strands = peaked strands? from 3 or 4 strands (Saddle-shaped peak)
6 strands = flat
Not many patterns All of the patterns can Agreed. However, the
can be achieved with be created regardless of handedness determines
left-turning peak. the turning at the peak. the turning of motifs.
Why are there Natural occurrences:-
discontinuities in certain i)! arrangement of
patterns? uneven number of
coloured strands at
the peak
ii)! overlapping of
strands.
Why conical, and not The shape is ‘high and
any other shape? rounded’ to allow hot
steam to travel upward.
Where did the idea of A fusion between triaxial
food covers originate? latticework of Chinese
hats and Malay basket
weaving.

10. 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

11. 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…

12. 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 ”

13. Starting Point = 6 Strands

14. Starting Point = 5 Strands

15. Starting Point = 7 Strands

16. 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.

17. Framework Transformation – Phase 3
2-peak tudung saji
3-peak tudung saji

18. 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.

19. Pattern Classification – Blocks of 2 Strands
( 0 = RY…; 1 = YR… ) = 8 orders of arrangement
ORDER ORIENTATION
000
011
101
110
000
111
100
010
001
111

20. Pattern Classification – Blocks of 3 Strands
( 0 = RRY…; 1 = RYR…; 2 = YRR… ) = 27 orders
ORDER ORIENTATION ORDER ORIENTATION ORDER ORIENTATION
000 012 021
111 120 102
222 201 210
010 020 011
121 101 122
202 212 200
002 001 022
110 112 100
221 220 211

21. 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 64
(ii) 0 = RRYY; 1 = RYYR; 2 = YYRR; 3 = YRRY
5 (i)! 0 = RRRRY; 1 = RRRYR; 2 = RRYRR; 3 = RYRRR; 125
4 = YRRRR
(ii)! 0 = RRRYY; 1 = RRYYR; 2 = RYYRR; 3 = YYRRR;
4 = YRRRY
6 (i)! 0 = RRRRRY; 1 = RRRRYR; 2 = RRRYRR; 216
3 = RRYRRR; 4 = RYRRRR; 5 = YRRRRR
(ii)! 0 = RRRRYY; 1 = RRRYYR; 2 = RRYYRR;
3 = RYYRRR; 4 = YYRRRR; 5 = YRRRRY

22. Graphical Representations
2 STRANDS: (0 = RY; 1 = YR)
000 111
011 100
101 010
110 001
DARK BLUE LIGHT BLUE

23. 3 STRANDS: RRY, RYR & YRR
BLUE PINK GREEN

24. 4 STRANDS: RRRY, RRYR, RYRR & YRRR
DARK BLUE LIGHT BLUE PINK RED

25. 5 STRANDS: RRRRY, RRRYR, RRYRR, RYRRR & YRRRR
DARK BLUE LIGHT BLUE PINK RED GREEN

26. 6 STRANDS: RRRRRY, RRRRYR, RRRYRR, RRYRRR, RYRRRR & YRRRRR
RED PURPLE BROWN GREEN BLUE

27. 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

28. 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.

29. Thank You