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Closing the gap.

In discussing the widespread misunderstandings of Australian Aboriginal number systems, John Harris (1990, p.137) recently suggested that "Aboriginal people [want their children]...to learn the three R's [sic] and to grow up Aboriginal". In essence this means that most Aboriginal people recognise that for their children to be 'successful' in a Western society, they need to understand and be fluent or have facility with Western mathematics and standard Australian English, while at the same time embracing the Aboriginal culture and world view.

Both at national and state level there continues to be a concern about the size of the 'gap' between the results of non-Aboriginal and Aboriginal students on the national benchmark tests in numeracy. In Western Australia the present gap between the percentage of Aboriginal children meeting the national benchmark and non-Aboriginal children meeting the benchmark currently ranges from 31% in Year 3 to 44% in Year 7. Governments respond usually by throwing more and more money at the problem--trying to 'close the gap' often with little success.

My personal view is that this concern and the responses it evokes may be thought of as being almost condescending and even arrogant. Indeed, why should we as a Western society, expect Aboriginal children to perform as well as non-Aboriginal children on tests that test understanding of Western mathematics? This discussion paper is prompted by the difference between Aboriginal and non-Aboriginal numeracy results on benchmark tests. The paper draws on some of the relevant literature, in an attempt to understand the sources and significance of the difference and the place of mathematics in Aboriginal cultures generally. The paper rests on the assumption that before we can close the gap, we must understand its nature and origins.

Aboriginal ways of knowing about mathematics

The term 'mathematics' may be vague from an Aboriginal and Torres Strait Islander perspective. Steen (1988) defines mathematics not as the science of space and number but as the science of patterns: 'The mathematician seeks patterns in number, in space, in science, in computers and in imagination. Applications of mathematics use these patterns to "explain" and predict natural phenomena' (p. 616). Christie (1996) states that:

   Mathematics is not a language, nor is it an object. It is a
   practice: the unseen work done by individuals and groups making
   sense of their lives, their territories, their histories, and
   economies through particular discourses which involve naming,
   ordering, recursion and valuing.


It is widely agreed in Australia that mathematics can enhance our understanding of the world and the quality of our participation in society. The mathematics of the Western technological world however, is filtered by the people belonging to this cultural group. Researchers have suggested that:

Presenting this compartmentalised de-contextualised body of Western knowledge to learners with a different world view scheme invites failure for both the learner and the teacher (Jones et al., 1995, p. 2).

Similarly, Harris (1989, p. 91) describes this practice in terms of the "wide difference between teachers and pupils in their understanding of the nature of reality and the way they organise the world to find meaning in it".

Ideally perhaps, in order to minimise this difference, a 'two-way' school will have Aboriginal teachers teaching Aboriginal children.

In reality, we know that this is rare and even unlikely--at least in the near future--and so the next best models are teachers from the non-Aboriginal cultures working alongside Aboriginal paraprofessionals, or non-Aboriginal teachers who are educated in the Aboriginal "ways of knowing'. That is, at best, making an attempt to understand the way that Aboriginal children organise the world and find meaning in it. Clearly, because of the Aboriginal 'ways of knowing', the mathematics an Aboriginal child learns will always be a different mathematics from that taught by the non-Aboriginal teacher.

We view the world through our own cultural filter. Pam Harris (1991, p.13) states that "A world view is the way individuals and groups see the world and how they understand that they should behave within it". She goes on to say (p. 14) that:

   ... there are some parts of reality which have been organised to a
   much higher level of abstraction in Aboriginal culture than in
   Western cultures, and there are some parts of reality which in
   Western industrialised culture have been organised to a much higher
   degree than in Australian Aboriginal cultures. This is part of the
   difference in world views. Each culture develops more vocabulary
   and higher degrees of abstraction for those parts of reality which
   they consider most important.


It is important to note that this observation does not imply that the culture that is more organised is 'better' in any way or that those less organised in some way are defective; it merely makes the cultures different. Sutherland (1999) states that:

   Whereas Western mathematics appears to be dominated by abstraction
   from reality, Aboriginal mathematical understanding centres on the
   functionality of concepts--irrelevant abstractions are not
   considered important.


Thus an Aboriginal child may have a word or language for 'five turtles' or 'five stones' but not a word for 'five', for example. The concept of 'fiveness' is abstract but may be embedded in Western culture through the activities of that culture.

In commenting on the relationship between a traditional Aboriginal world view and Western mathematics, Michael Christie (in Harris, 1991, p.13) states that:

   ... the Aboriginal world-view provides for the unity and coherence
   of people, nature, land, and time. In a world which has not been
   broken down by science and (Western) mathematics, the spiritual
   unities which transcend Western analysis remain primary.


Pam Harris (1991, p.13) suggests that Aboriginal world views "... are characterised by a very personal view of the universe in which humans are seen as united with nature rather than separate from it." By contrast, the world view dominant among non-Aboriginal Australians is marked by the separation of humans from nature which allows for the ". exploitation of nature so that it serves the perceived needs and desires of humanity. This is the world view that gives rise to the mathematics which is taught in school." (p.11) As a result of this, she contends that if Aboriginal people from a traditional Aboriginal community were to set up their own mathematics curriculum (uninfluenced by white Australian traditions), they may give highest priorities to kinship relations and to the space strand.

Harris (1990) has explained that Aboriginal world views stem from spiritual and religious beliefs, in contrast to Westernised cultures, which have their roots in science. Watson & Chambers (1989, p. 31) state that the Aboriginal people of Australia use a genealogical pattern to make sense of their world: "By this we mean ordered ways of naming and construing the relationships of natural things according to perceived ancestral or familial linkages." Whereas non-Aboriginal people use number patterns based on counting and measurement, the patterns used by Aboriginal people are based on relationships between people. Number patterns are also used by Aboriginal people as a form of ordering, just as familial patterns are used by non-Aboriginal people; the difference is found in the importance placed on these. They go on to suggest that, for Aboriginal people:

   ... the genealogical pattern explains all relationships
   in both the social world and the world of
   nature. Furthermore ... the number system is
   involved in Aboriginal life only secondarily. Little
   hangs on the functioning of number, which is
   to say number does not carry the deterministic
   weight nor the aura of objectivity and inevitability
   that it carries in non-Aboriginal Australia.
   In other words, in all Aboriginal-Australian
   communities the genealogical recursion, not
   the numbering system, patterns both knowledge
   and behaviour and carries the productive
   processes of the social order (p. 32).


This is difficult for a non-Aboriginal person to understand. I will draw on some of the discussion presented by Watson & Chambers (1989) in an attempt to clarify some of the understandings associated with the Aboriginal ways of knowing or world view.

The Western number system presents a sort of 'grid' that can be placed over the physical, economic and social landscape in an attempt to order and make sense of the world. The number names enable notions of equivalence and hierarchy that 'fit' with the ideas of economics and competition that dominate Western society. The patterns of Western lifestyles are strongly determined by cycles; routines pervade and organise daily, monthly and yearly cycles. A spatial grid covers the planet and enables every square centimetre to be described and labelled, thus allowing the landscape to be managed and controlled. The gathering, organising and summarising of data enables predictions about weather or natural disasters. Units used for measurement and the numbers attached to these when counted are powerful elements of trade in negotiation--an important part of Western society.

Contrast this with the way the world is organised by many of the Aboriginal people in Australia. The Yolngu people of Arnhem Land, for example, may give a name to a stranger among them with whom they expect to have prolonged personal contact. The name is given in order to 'locate' the person in their system of genealogical pattern. On being named, the person then receives continued instruction about their relationships with other people from the group, for example: "He is your brother--greet him", or "He is your son-in-law --stay away". The names not only imply emotional involvement, as in Western society, but also carry with them an acknowledgement that they are 'someone'. As Watson and Chambers put it: "To be a 'real' entity in Yolngu life, a person or place must be named, and thus located within the genealogical order." (1989, p. 36).

A stranger from another people group must have their location from another group translated into their own group, and once the relation between the stranger and one person from the Yolngu has been determined, then the relationship with all other Yolngu is known. "When a non-Aboriginal person is recognised in this way he or she will have established, in one moment, formal family relations and responsibilities involving several hundreds of people and more" (Watson & Chambers, 1989, p. 36) and is expected to learn these complex, formal connections as do Yolngu children, who have been instructed in them from a very early age. The Yolngu people use this system to make connections to all parts of nature and across all time. Thus, the Yolngu live 'close to nature' meaning they have a "precise mathematically articulated relationship" (Watson & Chambers, 1989, p. 36) with nature.

This 'gurrutu' system of organising used by the Yolngu people is basically a series of names or labels with a recurring pattern. It enables the formal articulation of all types of relationships. Like number names, these gurrutu names form a pattern of language, the basic form of which requires alternation across three generations. The Yolngu divide the universe into two moieties based on the fact that every person has two direct ancestors, a mother and a father. In the Yolngu world, every named thing belongs to one of these two moieties. Thus the gurrutu cycle operates in two dimensions: the moieties and the generations. This system imposes order on the relationships between individuals and groups to each other as well as to the land and to all things in the Yolngu world, determining for example, potential marriage partners, designating parcels of land belonging to individuals and groups. "At a general level it is a formally articulated system of beholdenness; it orders degrees and types of indebtedness". (Watson and Chambers, 1989, p.37). Thus, in a mathematical sense, every relation has a sort of negative value in contrast to the notions of value in Western society where everything has value in a positive sense based on the way the number system works.

In their comprehensive discussion of the Yolngu gurrutu, Watson and Chambers (1989, p.38) describe how "...genealogical relatedness, formally articulated in a recursive system of names, constitutes an encompassing pattern of Aboriginal life." Individuals play roles in hierarchies which are woven together in an orderly 'grid' which works in such a way as to emphasise co-operation instead of competition. Thus it enables both the social and the natural worlds to be ordered.

Jones et al. (1995) contrasted world views of Aboriginal and Western cultures under headings of Life and Living, Time, Activity, and Structure of Society. Their helpful analysis has been summarised in Table 1.

Tables such as these often result in stereotyping and hence it is important to understand that many Aboriginal people do not necessarily have an Aboriginal world view and, likewise many non-Aboriginal people may have a world view that is, or is similar to, what is categorised as 'Aboriginal' above. In reality, it should be appreciated that most people, both Aboriginal and non-Aboriginal, lie somewhere on a continuum between these two extremes.

Educational implications

Implications for pedagogy have already been drawn from some of the differences in world views indicated in the table above. There are however, other more specific implications for the teaching of mathematics; in particular Western culture--and hence Western mathematics--places considerable emphasis on quantity and hence, measurement. Since the ability to quantify rests heavily on the ability to count and hence understand number, it is apparent that the capacity to be able to succeed with Western mathematics relies heavily on understanding concepts in number. This is quite different from the situation in Aboriginal culture, so that, significantly, Aboriginal children may need to be shown or provided with a reason for needing to know these quantitative concepts, since they may not exist in the cultural or economic environment in which they live.

As a result of these differences, the teaching of Western mathematics in schools and classrooms can be problematic for many Aboriginal children. Real understanding of an abstraction comes through application and relevance; this is true for all learning. But it is often difficult to provide relevance and application for abstractions for non-Aboriginal children, let alone from outside this group.

There is an inevitable tension in the desire of Aboriginal people to have Western mathematical understandings and knowledge while at the same time not having familiar cultural contexts through which non-Aboriginal teachers can make the abstractions relevant and meaningful. For example, it seems pointless to teach skills such as time-stables or reading a clock to Aboriginal children if these can not be made immediately relevant and useful for them.

In effect what this means is that in attempting to teach Aboriginal children the abstractions of Western mathematics it may be essential to also teach the reason for wanting to know and the context in which this knowledge becomes meaningful and relevant!

In teaching Aboriginal children to measure exact quantities using a measuring cup for example, a teacher may need to first give the students a piece of cake, and then say, "We will now learn how to make a cake like this; it will mean that we have to measure the ingredients exactly with a measuring cup and this is how we do this". Alternately, they could all taste some cake in which too much salt or not enough sugar has been used and the teacher may say, "We will learn how to measure the ingredients so that our cake will not taste like this". Without such contextual support, an Aboriginal child may see reading measurements from a measuring cup as a pointless exercise; the further abstraction of the 'degree of accuracy' required may be meaningless.

This of course, may also be true for a lot of children from non-Aboriginal cultures. It is more likely, however, that non-Aboriginal children will be familiar with the notion of measuring accurately, having seen the behaviour modelled within the home and possibly even taking part in similar activities whilst being at home. As a result the child knows it is a useful skill to have for the present and future and this provides both relevance and meaning

In his work with Milingimbi Aboriginal people in the Northern Territory, Kepert (1993) found that their world view was one of relationships and not number, shapes or dimensions. As a result, contexts which related to the Aboriginal people and their culture or relationships provided an immediate context. For example, he used information provided by the students about the jobs held by people in the community to build a makeshift pie-graph. With this information in visual form it was then an easy matter for him to lead into a discussion about fraction and percentage. Similarly, Aboriginal teachers were able to understand the concept of 'one quarter' once it was related to the passage of time between sunrise and midday--a quarter of a day. This however, is more about making connections to what is known than about relevance.

Table 1 clearly indicates that family and relationships, places and the land, and participation in living and life activities are of paramount importance in Aboriginal cultures. This is true for most Aboriginal communities no matter where they are located, whether remote, rural or urban. An effective teacher will draw heavily on this fact, using these as the contexts for mathematics teaching and learning wherever possible.

Conclusions

It can be seen then that there is a clear distinction between mathematics and Western mathematics. Aboriginal people have their own mathematics, consisting of the techniques and structures they use in order to make sense of and order their world. That this mathematics is successful is unquestionable: it has operated and worked efficiently for thousands of years in Aboriginal society.

Is it no wonder then, that when Aboriginal children come to school and begin instruction in Western mathematics they are automatically being shown that the mathematics they have learned within their culture and by their people is deficit. They feel devalued and may begin to lose respect for their culture and society. They are likely to feel ashamed when they cannot grasp the basics of this new way of ordering the world. They may not even understand why this new way of ordering is necessary, especially if there is no immediate relevance or practical use for it. In short, they may be learning a whole set of facts which connect in no way to their experience and which they may never see modelled by their society or family, in contrast to the experiences of most non-Aboriginal children.

The under-valuing of Aboriginal mathematics and other aspects of Aboriginal culture and tradition is likely to be a major factor contributing to underachievement of Aboriginal children in our schools. I personally believe that we need to learn to understand and value Aboriginal mathematics and in so doing, present the mathematics taught in schools as 'Western mathematics', the mathematics used by Western society to make sense of their world. It should not be presented as the mathematics used by all progressive and technologically advanced societies which seems to be the way mathematics is currently taught and delivered in most of our schools.

In closing, I ask the question: were non-Aboriginal people to sit a 'numeracy test' developed by Aboriginal people about knowledge of Aboriginal mathematics, would they pass or would there be a gap between the results of the non-Aboriginal people and the Aboriginal people? Perhaps we should be celebrating the 'gap' and acknowledging the unfairness of expecting Aboriginal children to be as efficient with Western mathematics as with their own cultural mathematics. Perhaps test questions might even have a stem such as, "What would be the answer to this question if you used Western mathematics to solve it?"

Published in Vol. 58, No. 2, 2002, Barry Kissane (Ed.)

References

Christie, M. (1996). Yolnguphilosophy and intercultural education. Unpublished paper presented at Northern Territory University.

Harris, J. (1990). Counting: What we can learn from white myths about Aboriginal numbers. Australian Journal of Early Childhood, 15(1).

Harris, P. (1989). Teaching the space strand in Aboriginal school. Mathematics in Aboriginal schools project: 4. Northern Territory Department of Education, Curriculum Development Centre.

Harris, P. (1991). Mathematics in a cultural context: Aboriginal perspectives on space, time and money. Geelong: Deakin University.

Jones, K., Kershaw, L. & Sparrow, L. (1995). Aboriginal children learning mathematics: Issues in primary mathematics education. Perth: Edith Cowan University.

Kepert, B. (1993). Mathematics for Aboriginal students who have a different world view. In T. Herrington (Ed.), New Horizons: New Challenges (pp. 210-20). Adelaide: AAMT.

Sutherland, L. (1999). Aboriginality and Mathematics. Cross Section, 11(4).

Steen, L. A. (1998). The science of patterns. Science, 240, 29, 616. Watson, H. & Chambers, D. W. (1989). Singing the land, signing the land. Geelong: Deakin University Press.

Table 1: Contrasting world views of Aboriginal and Western cultures
(after Jones et al., 1995)

World       Aboriginal                  Western
view

Life and    * Life always exists and    * Life is transient
  living      exists in spirit form     * Past is separate from
            * Discusions always           present
              intertwine land, life,    * Discussions separate
              religion, kinshio, and      land, religion
              therefore quality is        relationships as
              important                   autonomous ideas, and
                                          therefore quantity is
                                          important
Time        * Cyclic                    * Linear
            * Passage of time is        * Time is divided into
              related to events           units to be measurered
              --nature, place and         as quantities
              participants are          * Events are related to
              important                   progress--when they
                                          occurred is important
Activity    * Concerned with            * Activity is concerned
              harmonious interactions     with setting goals and
              with people and the         ways of achieving goals
              environment               * Completing a task is
            * Maintaining meaningful      important
              relationships is          * Emphasis is on
              important                   measuring individual
            * Emphasis is on shared       achievement
              achievement and working
              together
Structure   * Living and working are    * Work is related to
  of          synonymous;                 earning money to live
  society     participation is          * School is about
              important                   learning how to gain
            * Learning is ritualistic     work and become
              e.g., participation is      employable
              important in schooling    * Impersonal
            * Personal relationships      relationship with
              with those in authority     those in authority


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Title Annotation:Section 2
Author:Perso, Thelma
Publication:Australian Mathematics Teacher
Article Type:Essay
Geographic Code:8AUST
Date:Sep 22, 2016
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