This document discusses indigenous knowledges in Papua New Guinea related to positioning, measurement, and mathematics. It provides examples of indigenous practices for canoe making, string figures, graphs and calculus that demonstrate visuospatial reasoning and parallels to western mathematics. The document also discusses how indigenous languages conceptualize location, direction, and measurement differently than English. It advocates for recognizing and incorporating indigenous knowledges and practices into education through place-based learning, community partnerships, and modifying content and pedagogy to be culturally relevant.

2. Indigenous Knowledges in PNG
!! Recognising variables and relationships
visuospatially
"! Canoe making (Matang; Owens & Kaleva)
"! String figures (Vandenriessche; Sth Africa)
!! Indo-European mathematical equivalents
"! Graphs & calculus (Matang; UOG)
"! Knot theory & series of polynomes (Vandenriessche)
!! Position
"! Orthogonal axes versus place-based (Senft et al.;
Bishop; Owens)

3. Linguistic Studies of Position
!! Location & direction - deixis
"! Prepositions & postpositions (at, behind)
"! Locative adverbs (here, there)
"! Verbs (to go, to bring)
"! Directionals (to, into)
"! Presentatives (there is)
"! Spatial or dimensional adjectives (high, wide) (Senft et al)
!! Large variety across languages
!! Gestures
!! May be society with more man-made objects has fewer terms
"! as the object determines position e.g. lock and key
"! Context
!! Common
"! three positions: distant place, medial distance and close by
"! and/or three directions: at, going to or going from
given as postpositions and postdirectionals (suffixes)

4. Culture, Language and Measurement
!! Position may impact on measurement
"! Kilivila, PNG have vertical and horizontal but
horizontal may move from ground to arms
(Lawton, Wado)
!! May be few words but may have
qualifiers providing flexibility
!! Empty may indicate expectation
!! Comparison may be measured but
approximations are adequate
"! Anindilyakwa, Groote Island, Aust (Worsley)

5. Measurement practices
!! Vary with
"! activity,
"! relationships of people involved in the
activity & related activities determined
by culture,
"! tool availability,
"! body parts,
"! attribute being measured and
"! purpose for measuring
"! context such as environment/supply

6. Language and measurement
!! Lack of equivalence of English
adjectives and PNG language
!! General terms like size and use of
verbs for making bigger
!! Metaphors
!! Reduplication
!! Impact of other languages and
western education curriculum
!! Classification may have different
word for our one adjective

7. Policy recognition of Indigenous knowledges
!! Australia
"! Systemic change especially in funded
projects
"! Often project based rather than on-going
"! Involvement of community in decision
making and in the school at planning,
delivery and evaluation
"! Format of partnerships and means of
engagement considered
"! Consideration of available community
personnel
"! Open, trusting, authentic partnerships
"! Still need for more structural support

8. Australia cont. – Western NSW (Yunkaporta)
!! “building learning around deep understandings
of Aboriginal pedagogy;
!! Designing learning through intercultural
collaboration;
!! Allowing quality cultural context to emerge
through Aboriginal pedagogy and intercultural
collaboration”
!! Cultural metalanguage rather than western
linear logic; holistic; cultural connections from
different sources
!! Reflection and
!! representation

9. Smarter, stronger learning
communities
!! Pastto future to present
!! What do we do well now?
!! What would we like to change?
!! How might we change?

10. Moving Forward
!! Decolonising the curriculum
!! Employment of Indigenous staff
!! Reduction in turnover of other staff
!! Gaining knowledge through yarn-ups
!! Viewing child in a culture way and as part of
culture with Elders having a role in community,
in knowledge building, in school decision making
(Dubbo community)
!! Place-based, outdoor education with Elders
!! Democracy with Indigenous rights and respect
in meaningful, engaging and self-determined
education

11. Content (Frigo, literature review)
!! Value diverse cultural and linguistic heritages
!! Make explicit the differences between home and
school, value both equally
!! Make explicit links between community, home
and school based mathematics
!! Provide realistic and real-life classroom contexts
for mathematical activities
!! Plan with community
!! Modify content and pedagogy to reflect students’
interests and needs
!! Engage with suitable resources and materials

12. Examples
!! GarmaProject for Yolngu in Nth Territory
of Australia
"! complex relationship of position and every
other aspect of culture of Yolngu
"! Patterns of relationships and patterns of
school mathematics
!! Reform agenda in PNG
"! Elementary curriculum: Culture and
Mathematics emphasises principles like
counting words and their relationships,
principles and purposes for counting or ways
of measuring, units,

13. Matang’s Research
!! The use of the vernacular language and
mathematics resulted in stronger
arithmetic skills in their home language
and in English than those who used Tok
Pisin and English in elementary school.
!! Paraides also found vernacular language
resulted in good results in PNG.
!! Lipka & Adams found better mathematical
results for Yu’pik with culturally based
units of work being used

14. Value of appreciating Indigenous cultural
mathematics
!! Explicating visuospatial reasoning in
early childhood education
!! Alternatives resulting from cultural
linguistic contexts emphasise cultural
context
"! geocentric concepts before egocentric
concepts of position (Bali:Wassmann & Dasen; NT: Harris)
"! route descriptions with descriptive
landmarks or directions (British & Iranian: Spencer &
Darvizeh)
"! representations of space e.g. Australian
Aboriginal maps (Harris, Klich); Yupno, PNG
(Wassmann); Micronesian star maps and games
(Penn Museum)

15. Explicit content links
!! Tinatatuna counting (Paraides)
!! Kâte cutting bamboo (Matang)
!! Many groups using groupings of different
numbers to form a composite unit for counting
e.g. hand for 5 until 20 (Lean, Owens)
!! Combining numbers in different ways to
represent a number is common especially for
systems with a 2 cycle (and higher cycles)
(PNG) (Owens)
!! Body part tally systems like a number line for
comparing, adding and subtracting numbers
(PNG) (Owens; Kaleva)
!! Mathematics is a changing system in languages
which are rapidly changing (PNG) (Muke)

16. Appropriate Pedagogy
!! The outcomes and the journey are closely linked in
Indigenous pedagogy (Yunkaporta & McGinty)
!! Learner autonomy with a balance between self-
direction and social support (Dubbo community; Yunkaporta & McGinty)
!! High Indigenous expectations (not school
expectations) in terms of identity and engagement in
local knowledge protocols – major recognitive social
justice issue
!! Use of stories by teachers and students; modify
models or examples given by teachers to meet local
needs; recreate own visual versions of processes;
share with community
!! Build on cultural activities like weaving (Cherinda; University of
Goroka student projects; Owens); bridge and house construction to
develop mathematical concepts
!! Use artifacts e.g. bridges like mathematics trails that
establish concepts that link to the real environment

17. The traditional Cane Bridge on
Yalo River at Mambuanda
Village – Kagua, Southern
Highlands Province
by Kita Yambi, Kewabi
language area
When that is done, they tie
3 to 4 cross bars to make
immovable; long, strong
tied a special way to each
post equally distant at top
To start to build this kind of bridges, a and bottom; trees spread
group of man led by the village chiefs between for footpath and
usually take an initiative… bush materials tightened with bush rope.
such as bush canes, vines, ropes, long Hand bars and ropes to also
sticks from young trees, branches and support footpath like a
posts… left on the sides of the river…, they wave; ropes from top to
start by erecting four huge posts into the footpath so it does not
ground, two at each side of the river. move.

18. Prepares a series of examples e.g. on
angles based around the syllabus
Example 1:
Kota Yarisi was at the top of the posts
tightening the bars together during a
cane bridge making. Sualo Nakisi was on
the bridge, 10 meters away from the
posts on the opposite side of the river.
The distance from the bridge to the top of
the bars is about 5 meters. Kota Yarisi
looks down on Sualo Nakisi at an angle.
!! What is the angle of depression that Kota
Yarisi looks down at Sualo?

19. Constructing Telefol
Traditional Door
Board” (AMITUNG)
Telefomin, Sandaun Province
by Clement Oggi
three different traditional
colours to paint the
designs that have been
carved: maroon called
‘Baagaan’, white called
‘Buuguung’ and black
The height and the width are colour from carbon called
‘Amsiring’.
measured using bush ropes. A bush
rope is used to measure around the To make the paint shiny
and bright some grease
waist of the fattest man in the
pig [liquid pig fat] is
village … is used again to measure mixed according to the
the doorway of the door board. The correct proportion with
circle is drawn on the wood for easy the three traditional
carving. Then designs are carved colours.

20. Links to Grade 8 syllabus: Ratio
Anivatok and Naatum decided to paint their newly
constructed door board designs. They both had the three
traditional paints to paint their designs on the door board.
Before painting the door board, Naatum suggested that they
should mix the three traditional colours with some grease
pig before they can paint their door board. This is because
when the paints are mixed evenly with the grease pig, the
colours on the door board will be shiny and bright. And so
they both agreed and started to mix the paints with the
grease pig.
!! The three colours are made bright and colourful by mixing
each one of them with the grease pig in the ratio of 2:1
(two is to one). We are mixing all the three paints with the
grease pig in the ratio given above to produce the correct
brightness.
!! The ratio 2:1 is in its simplest terms because there is no
whole number that will divide exactly into both sides.
!! If Naatum and Anivatok decided to increase the ratio of
paints to grease pig up to 9:6.The ratio 9:6 is not in its
simplest terms. What is it in its simplest term?

21. Conclusion
!! Social justice issues and Indigenous rights in
schooling (O’Sullivan)
!! New perspectives on learning
!! Different mathematical relationships
!! Different views of mathematics in society
!! New ways of engaging students and teachers
!! Valuing and involving the expertise of Elders
!! Better ways of introducing western mathematics
!! Deeper understanding from alternative
perspectives, alternate mathematics, place-
based education, but globalised perspectives
!! Values education - valuing diversity