Wilfred T. Kaleva.

Faculty of Science

Mathematics and Computing Department.

University of Goroka. Papua New Guinea

*Abstract.*

*The paper discusses the possible linkages between Ethnomathematics and language.
Research examples are given of the use of linguistic data, sources, terms
and methods of classification in identifying mathematical practices in socio-cultural
groups. It provides a case for collaborative research between linguists and
ethno mathematicians.*

**Introduction.**

The focus of this paper is on the study of mathematical
knowledge from traditional culture (ethnomathematics) and the identification
of this knowledge through the use of linguistic data, sources, terms and methods
of classification. The paper begins with a definition of ethnomathematics
and goes on to discuss the importance of linguistic data to ethnomathematics.
The discussions focus on the documentation of mathematical knowledge from
traditional culture through the use of documented forms of studied languages
and promotes the idea of “Bridging / moving boundaries” through
inter-faculty collaborative research. It also discusses the educational challenges
of incorporating mathematical knowledge from traditional culture in teaching.

**Definition of Ethnomathematics**

There are a variety of definitions of ethnomathematics
(for examples, see D’ Ambrosio, 1990: p.22; Howson and Wilson, 1986;
Presmeg, 1996; Frankenstein, 1990; Borba, 1990; Pompeu, 1992). This paper
adopts Barton’s (1996) definition, which states that:

*“Ethnomathematics is the field of study which examines
the way people from other cultures understand, articulate and use concepts
and practices which are from their culture and which the researcher describes
as mathematical”* (Barton, 1996: p.196).

This
writer also supports the view that “*ethnomathematics*” is
not the *mathematics* of particular groups of people but is an *academic
field of the study of mathematical ideas* *(knowledge), practices, activities*
which can be identified in socio-cultural contexts. Although it is not a mathematical
field of study, it is more like the study of anthropology or history. These
mathematical ideas, knowledge and activities are acquired and practiced by
various socio-cultural groups (in all cultures).

**Bishop’s fundamental activities.**

Bishop (1988: p.182) lists six fundamental mathematical activities, which he
suggests are common to all cultural groups and asserts that from these basic
notions “western mathematical knowledge” can be derived.

These fundamental activities are: *counting*, (which forms the basis of,
for example, number systems, algebraic representation, probabilities); *locating*,
(orientation, coordinates, bearings, angles, loci); measuring (comparing,
ordering, measurements, approximations); *designing*, (projections of
objects/shapes, geometric shapes, ratio); *playing,* (puzzles, paradoxes,
models, games, hypothetical reasoning) and *explaining*, (classification,
conventions, generalisations, symbolic explanations).

In the above activities, Bishop provides an alternative
way of thinking about mathematics. Although mathematics is usually regarded
as the abstract subject which one learns in schools, mathematics can also
be thought of in terms of the practices and activities such as counting, designing,
… etc that are listed above. In the case of traditional knowledge from
cultures, this framework (of fundamental activities) provides the tool that
can help to identify the mathematical knowledge practised in traditional societies.

**Examples of mathematics from traditional societies.**

Because of the limited scope of the paper only a few references are given of
the examples of mathematics from traditional societies. For example, see Gerdes,
1988 & 1990 (Mozambique weaving to illustrate mathematics and geometric
thinking and mathematics in traditional sand drawings); Keitel et al, 1989
(examples of mathematics from Africa & other parts of the world); Finau
& Stilman, 1995 (Tonga - Geometric skills in tapa designs); McMurchy-Pilkington,
1995 (Mathematical activities of Maori women in the Marae kitchen); Ascher,
1991 (Bushong sand figures and Malekula - Vanuatu - sand tracing to illustrate
graph theory, Inca strip patterns and Maori rafter patterns - geometry, Iroquoi
games - chance); Barton & Fairhall, 1995 (Examples of mathematics from
Maori designs, patterns and carvings); Lean, 1994 (counting systems of PNG
and the Oceania region) and Kaleva, 1995 (bamboo wall patterns from PNG -
number patterns).

**Importance of language to the teaching and learning of
mathematics**

*Conventional views about language as the medium of
instruction in teaching and learning mathematics.*

Conventional views about the link that exists between
language and mathematics are usually about the difficulties that are associated
with language as the medium of instruction. Some difficulties in learning
mathematics are language related. Language is therefore important to the teaching
and learning of mathematics. Macgregor & Moore (1991) note the following
reasons for the importance of language to mathematics:

- Language plays an important role in organising knowledge and thinking
logically.
- The learning of mathematics relies heavily on oral and written explanations
(or symbols).
- Most mathematical concepts are defined concepts, often about relationships
between things. They can’t be seen or touched or pointed to. Without
language they cannot be identified or described.
- Knowledge and language skills develop together.
- Verbalisation is essential for abstracting, generalising and categorising.
- Written and oral language skills are necessary for presenting the results
of mathematical investigations that form part of the school mathematics
curriculum.

I would like to add the following to the list.

- Proficiency in language is deemed as important to achievement in mathematics.
That is, it is assumed that problems or conflicts may arise if one is
not proficient in language, especially when the language of instruction
is in a second language. Much of the research in Papua New Guinea relates
to learning mathematics in a second language.
- Mathematics uses specialised terms and vocabularies.
- Language is an important cultural factor in learning mathematics.
- Mathematical knowledge from traditional cultures is identified through
the medium of language.

This paper will focus on points 3 & 4.

*Language as an important cultural factor in learning
mathematics.*

Language is an important cultural factor in learning mathematics, because the
learner possesses knowledge learnt in a different language (different to the
language of instruction), in a different cultural context with possibly different
interpretations of the mathematical concepts. It is of interest to note the
influence of cultural factors on the learning of mathematics.** **In writing
about issues dealing with linguistic and cultural influences on mathematics
learning, Saxe (1988) argued that culture constitutes a complex of intertwined
factors, one of which is the language background of children. In his study
of children in the Oksapmin area (West Sepik Province) of PNG, Saxe (1988)
observed children’s adaptation of the body parts system (a way of counting
in the local area where different parts of the body represented the numbers)
to do sums or arithmetic learnt in schools. Children used and adapted knowledge,
which are part of their home culture to that culture presented in school.
He suggests, “studying cultural supports for mathematics development
and how children utilise different backgrounds in coping with school mathematics
curriculum, can offer insights about the sources of language minority children’s
successes and failures in the mathematics classroom.” (Saxe, 1988: p.61).

The mathematical knowledge the learner has acquired out-of-school, in a traditional
cultural context makes up the entry characteristics of the learner. This mathematical
knowledge may consist of the child’s ability to count, design, locate,
measure, design, play and explain (Bishop’s, 1988, six fundamental activities)
and it is important to remember that this knowledge is gained in a cultural
context through the medium of language.

Clarkson (1991) makes the important point that “cultural
differences can affect mathematics learning in number of ways: clearly through
language impinging on the content of the curriculum directly, but also through
ways of behaving and knowing that are also embedded in the language”.
Lean (1994), who documented approximately two thousand counting systems in
the South Pacific), placed a lot of emphasis on the fact that counting systems
were a part of language and that language was embedded in culture. Ellerton
and Clarkson (1996: p.1017) quote Mousely, Clements and Ellerton who claim
that “one of the most fundamental aspects of all cultures is language”
They recognised the “… centrality of language factors in all aspects
of mathematics teaching and learning”.

Another important point to note is that language plays
an important role in defining socio-cultural groups.

*Identification of mathematical knowledge from traditional
cultures (societies) through the medium of language.*

The importance of language to mathematics and in particular,
ethnomathematics, is also in the fact that it is through language that we
can identify the mathematical knowledge and practices of various socio-cultural
groups. One key area of identifying the mathematical knowledge from traditional
cultures is through data gathered by anthropologists or linguists.

Traditional languages also have specialised terms and
vocabulary. Unfortunately, the knowledge about these specialised terms and
vocabulary is slowly disappearing from PNG societies. One can be proficient
in his or her own language but may lack the knowledge of these specialised
terms and vocabularies.

**Ethnomathematics research**

** **Bishop (1992) identifies three foci
of research in ethnomathematics. They are:

- Mathematical knowledge
from traditional societies (anthropology).
- Mathematical development from non-western cultures (history).
- Mathematical knowledge of different groups in society (social psychology).

A fourth area of research is the educational implication
research in ethnomathematics research.

**Ethnomathematics research in PNG and Oceania: A case ****for collaborative
research.**

The example used here falls into the category of anthropological
studies.

Lean’s (1994) PhD thesis entitled “Counting
systems of Papua New Guinea and Oceania” documents over 880 different
counting systems (from 1200 languages spoken in PNG and Oceania) in four volumes
of appendices. Most of the data collected are available from the Glen Lean
Ethnomathematics Centre at the University of Goroka in PNG. This study illustrates
the different ways of using documented forms of studied languages. Lean has
been referred to as an “Ethnomathematics anthropologist” (Bishop,
1995). I would like to refer to him as an “Ethnomathematics linguist”.
He was a lecturer at the Mathematics department at the University of Technology
in Lae, PNG. He read widely about linguistics and the many languages of PNG,
interviewed students and informants from all over the country, learnt several
of the languages, could speak “Kuanua” fluently and was initiated
into a Tolai clan in East New Britain. The result of his research is the most
extensive database on counting systems in PNG and the Oceania. In fact, some
would argue (as did Bishop, 1995: p.2) that it is “probably the more
extensive database of counting systems produced anywhere”.

In his keynote address at the History and Pedagogy of
Mathematics conference in Cairns, Australia, Bishop (1995) paid tribute to
Glen Lean and his research into counting systems in PNG and Oceania. I share
with you some of the points in the context of the sub-theme; culture, indigenous
knowledge and learning commission.

**How was he able to collect such enormous amount
of data?**

There were four primary sources of information (the
data was collected over a period of twenty years – 1968 to 1987):

- Field notes of interviews with informants living in villages that
Lean visited and those who travelled into Lae from remote villages.
There were a total of 64 records of interviews.
- Questionnaires from three populations; University of Technology students,
National high school students and Headmasters of community schools.
There are 2524 questionnaires from these sources.
- Unpublished material gathered during the Indigenous Mathematics Project
(IMP) during 1976 to 1979. There were a total of 238 of the IMP questionnaires.
- Unpublished primary source material derived from a survey word list
compiled by members of the Summer Institute of Linguistics (in PNG).
A search of these materials yielded 362 word lists which contained some
numeral data on various languages spoken in PNG.

The questionnaire technique used in the data collection
ensured that a wide range of languages was covered.

*Use of linguistic **data and sources*

Lean also had access to various sources of data, which
contributed to 600 items of data (ie. an item being, a set of numerals or
description of counting or tally systems for one language). In fact, the counting
systems of Oceania (Solomon Islands, Vanuatu, New Caledonia, Fiji and Rotuma,
Polynesia, Micronesia and Irian Jaya) were derived totally from secondary
published sources of data. Examples of secondary sources include such diverse
sources as: Somervile, (1897) Songs and specimens of the languages of New
Georgia.* Journal of Anthological Institute*; Ray, (1920) Languages of
the New Hebrides,* Journal of the Royal Society of New South Wales; *Raven-Hart,
(1953) A dialect of Yasawa Island, Fiji. *Journal of the Polynesian Society;
*Leenhardt, (1935) *Langues et dialects de l’Austro-Melanesie.
*Paris: Institut d’Ethnologie. A lot of these articles where obtained
from university libraries around the world (eg. Australian National University,
Cambridge University). Lean documented the counting systems of Oceania on
order to establish whether the diversity found in PNG was also found in the
neighbouring regions of Melanesia, Polynesia and Micronesia. Lean quoted from
linguistic sources, the most common being the *Language Atlas of the Pacific
Area, Part 1: New Guinea Area, Oceania, Australia *by Wurm and Hattori
(1981).

*Use of linguistic terms and methods of classification*

Lean classifies counting systems according to cycles
in the Austronesian (AN) and Non – Austronesian (NAN -Papuan) languages.

The counting systems were analysed into groups using
the classification system developed by Salzmann in Lean (1994). This classification
system uses the terms, *frame patterns, cyclic patterns* and* operative
patterns, *which are used together with unique *number morphs.* Consider
this example:

Numbers |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
16 |
19 |
20 |
25 |

Numerals in language |
1 |
2 |
3 |
4 |
4+1 |
4+2 |
4+3 |
2x4 |
(2x4) + 1 |
(2x4) + 2 |
(2x4) + 3 |
3x4 |
4x4 |
(4x4)+ 3 |
20 |
(20+4) + 1 |

In this language, they have distinct words for numbers
1 to 4, but five is made up of 4 and 1.

Distinct *number morphs*: 1 to 4, 20 i.e.
*Frame pattern *of the sequence is (1, 2, 3, 4, 20).

Cyclic structure: Sequence has a cycle of 4 and a super
ordinate cycle of 20. *Cyclic pattern *denoted by (4,20)

*Operative pattern: *8 is a complex number word
represented by the number sentence “8 = 2 x 4”, i.e. the number
sentence involves the multiplication (operation) of 2 and 4. The *operative
pattern* of 9 involves the multiplication of numbers 4 and 2 and the addition
of 1 to the result.

If the frame, cyclic and operative patterns are known
for a particular language, one can actually count up to large numbers in that
language, although it not always possible to identify these patterns in all
the counting systems.

*Numerical classifiers.*

According to Lean (1994) “In Languages like English,
the expression of quantity usually has the construction, ‘numeral +
noun’ as in three men or five houses. There are other languages however,
where the quantifying expressions contain not only a numeral or quantifier
and a noun but also an obligatory “classifier” which indicates
the specific class or category to which the noun belongs. In such languages
the universe of countable nouns is categorised into a number of classes, each
noun being assigned to a class according to some criteria”. It follows
this pattern: Quantifier (Q) + classifier (C) + Noun (N).

In the NAN languages in PNG, the existence of numerical
classification occurs in the Central (Nasioi, where it numbers over 100) and
Southern Bougainville. Other languages exhibiting classifiers are in the East
and West Sepik Provinces.

In the AN languages classification occurs most noticeably
in the Manus and Milne Bay provinces of PNG.

It is also interesting to note that not everything that
exists in society are counted and different things can be counted in different
ways.

*Language migration theories of the AN and NAN languages*

An interesting feature of Lean’s research was the
use of maps to denote the language groups, language clusters and their subgroups
and language migration maps in PNG and the Oceania region. Lean produced a
collection of maps. He had maps showing various language groups, clusters
and their subgroups. He had maps which showing the migration of languages
around the region. It was the use of migration maps that prompted him to look
at the diffusion theory of number development (Crump, 1990, Seidenberg, 1960).
Seidenberg’s theory, for example, states that several of the various
methods of counting that can be found in the indigenous cultures of the world
have a single centre of origin – ancient civilisations, no where else.
It began with the pure two-cycle system of counting, other methods of counting
evolved and then they spread to other parts of the world. He makes references
to genealogy of counting systems and order of diffusion. He agrees that 2
cycle and (5,20) cycle have primary status. But disagrees that 10-cycle system
evolved from a neo 2-cycle system.

Language and linguistic data can therefore be used to
identify mathematical practices in traditional societies.

**Educational ****challenges in mathematics education.**

The government of PNG embarked on a program of educational reform in 1994.
The main features of the reform are that elementary education (Year 1 –
3) is taught in the local vernacular, primary education is from year 4 –
8 and secondary education is from year 9 –12. The reform was initiated
following concerns about access to education and the relevance of the school
curriculum. Curriculum reform in all subject areas at both primary and secondary
level is currently taking place. One important question in the current curriculum
reform relates to the question posed by Thaman (2001) in her opening address
to this commission at this conference: What and whose knowledge is considered
as important?

**Challenges for the mathematics teachers.**

The challenge for mathematics educators is to provide answers to this important
question.

In PNG it would appear that the government has it easy to answer the above question.
There is a philosophy of education (Matane report in Kaleva, 1998) which encourages
a “culturally oriented” and “relevant” curriculum,
which could be interpreted to mean indigenous knowledge from local cultures.
However, in PNG where over 860 different languages are spoken, the question
becomes that much more difficult to answer. Whose (out of the 860 different
language groups is considered important? It is fair to assume that all are
equally important and that the situation should be seen as an opportunity
rather than a problem.

The situation in PNG is such that classes, at both the primary and the secondary
level consist of students from different language groups and in most cases,
the teacher would be from a different language group. These students posses
mathematical knowledge from their own cultures. If the intention is to teach
this mathematical knowledge from culture, then mathematics educators are faced
with the question: How do we teach the mathematical knowledge from cultures
in schools?

Ethnomathematics provides a *partial* solution to this question. It provides
examples of mathematical content, identified in the traditional PNG culture,
which can be used when teaching mathematics. The notion of *partiality*
is used because issues dealing with “relevant” curriculum are
multi-dimensional and depend very much on the interpretations of relevance
by those invested with powers to determine the contents of mathematics curriculum.
Perceptions of what mathematics should be included in the curriculum are also
deeply entrenched and so the ideal (to teach mathematics from traditional
culture) must be balanced with the reality (PNG wants to advance in a technological
society and mathematics is seen as the subject that contributes to that advancement).

The writer believes that using ethnomathematics as a solution involves two processes.
The first process involves the identification (“mathematicising”)
of the mathematics from traditional culture. Secondly, a decision needs to
be made on how to include the mathematics from traditional cultures in the
teaching of mathematics in schools.

The curricular implications of the decision reached may fall into any one of
the categories suggested by Bishop (1992). They are: the *culture free*
traditional view (assumes no cultural conflict); *assimilation* (learners
culture should be used as example); accommodation (learners culture should
influence education); *amalgamation* (cultures adults should share significantly
in educational decision and provision) and *appropriation* (community’s
culture takes over provision of education).

It can be recognised that the proposal to use mathematics from traditional culture
as examples in the teaching of mathematics fall into assimilation category.
For example, mathematics from culture can be used as “starting points”
when introducing topics in class. In reality, the starting point is the mathematics
prescribed in the official syllabus or curriculum but mathematics from culture
can be used as an example to help explain a concept.

The proposals for teachers to identify the mathematical knowledge their students
posses poses new challenges for the mathematics teacher. It changes the role
of the teacher from that of being the transmitter of knowledge who relied
on the textbooks and syllabi documents and adds on the responsibility of being
a researcher. Bishop, (1992) point out that ethnomathematics research needs
to be carried out in the communities concerned with due reference to languages
of these communities.

One other point needs to be made about the knowledge
derived from counting systems and language. The issue relates to ownership
of knowledge because it is a part of ones own language. Ownership empowers.
This view of knowledge is different to school mathematics, because it’s
content is determined by a centralised authority (courtesy of the centralised
curriculum system).

*Mathematics teacher education at the University of Goroka.*

Students taking the course “ Culture and language in Mathematics Education”
at the University of Goroka (premier secondary teacher educational institution
in PNG) do a project where students are required to gather as much information
about their counting systems and other mathematical practices in their traditional
societies. A questionnaire is used to identify the mathematical knowledge
from their traditional cultures. The aim of the project is to enable student
teachers to identify the mathematical knowledge the students posses. The classroom
approach advocated at the teacher education level is the assimilation model
(according to Bishop, 1992) where examples of mathematics from culture are
used as examples when teaching school mathematics.

Some preliminary results from the data show that:

- Many counting systems (language groups) do not have “equivalent
words or practices” for negative numbers or zero (although it could
be equated to “nothing”.
- There are cultural practices, which can have fractional ideas, but many
language groups do not have equivalent words for common fractions.
- Many language groups have a variety of mathematical practices such as
counting of days, measurement, patterns and designs, games and puzzles.

The data collected over three years by the students has been consistent. Students
also compare the data they have collected about their own counting systems
with data from the Glen Lean Ethnomathematics Centre. The data collected by
the students is crosschecked and added to the collection of counting systems
at the Center. In this years project, the students will identify mathematics
from various traditional PNG activities which will then be compiled into a
resource book for use by teachers.

**Conclusion.**

There is a strong case for collaborative research between
Linguists and Ethno mathematicians. The identification of mathematical knowledge
from traditional societies requires their collaborative efforts. The challenge
for mathematics educators in PNG is to identify and incorporate mathematical
knowledge from traditional cultures into their teaching.

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