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

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 Unpublished doctoral dissertation, University of Auckland, New Zealand.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.

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

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

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.

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

Construction of a Tudung Saji

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)

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.

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

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…

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 ”

Starting Point = 6 Strands

Starting Point = 5 Strands

Starting Point = 7 Strands

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.

Framework Transformation – Phase 3

3-peak tudung saji

2-peak tudung saji

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.

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

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

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

Graphical Representations

DARK BLUE LIGHT BLUE

2 STRANDS: (0 = RY; 1 = YR)

000 011

101 110

111 100

010 001

3 STRANDS: RRY, RYR & YRR

BLUE PINK GREEN

4 STRANDS: RRRY, RRYR, RYRR & YRRR

DARK BLUE LIGHT BLUE PINK RED

5 STRANDS: RRRRY, RRRYR, RRYRR, RYRRR & YRRRR

DARK BLUE LIGHT BLUE PINK RED GREEN

6 STRANDS: RRRRRY, RRRRYR, RRRYRR, RRYRRR, RYRRRR & YRRRRR

RED PURPLE BROWN GREEN BLUE

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

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.

Thank You