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Linkages between Ethnomathematics and Language: A case for collaborative research.

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:

  1. Language plays an important role in organising knowledge and thinking logically.
  2. The learning of mathematics relies heavily on oral and written explanations (or symbols).
  3. 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.
  4. Knowledge and language skills develop together.
  5. Verbalisation is essential for abstracting, generalising and categorising.
  6. 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.

  1. 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.
  2. Mathematics uses specialised terms and vocabularies.
  3. Language is an important cultural factor in learning mathematics.
  4. 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:

  1. Mathematical knowledge from traditional societies (anthropology).
  2. Mathematical development from non-western cultures (history).
  3. 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):

  1. 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.
  2. Questionnaires from three populations; University of Technology students, National high school students and Headmasters of community schools. There are 2524 questionnaires from these sources.
  3. Unpublished material gathered during the Indigenous Mathematics Project (IMP) during 1976 to 1979. There were a total of 238 of the IMP questionnaires.
  4. 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:

  1. Many counting systems (language groups) do not have “equivalent words or practices” for negative numbers or zero (although it could be equated to “nothing”.
  2. There are cultural practices, which can have fractional ideas, but many language groups do not have equivalent words for common fractions.
  3. 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.

References.

Ascher, M. (1991). Ethnomathematics : A Multicultural View of Mathematical Ideas. Pacific Grove California.: Brooks / Cole publishing company.

Barton, B. (1996). Ethnomathematics : Exploring Cultural Diversity in Mathematics. Unpublished Doctor of Philosophy thesis : University of Auckland.

Barton, B., & Fairhall, U. (1995). Is mathematics a Trojan Horse ? In Mathematics in Maori Education. A collection of papers for the History and Pedagogy of Mathematics Conference. Cairns. Australia : School of Mathematical and Information Sciences, University of Auckland.

Bishop, A. J. (1988). Mathematics Education and Culture. Dordrecht. Netherlands : Kluwer Academic publishers.

Bishop, A. J (1992). Cultural conflicts in mathematics, science and technology education: The curricular issues. Paper presented at Monash University.

Bishop, A.J. (1995). What do we learn from the counting systems research of Dr. Glendon Lean. Key note address to the History and Pedagogy of mathematics conference. Cairns. Australia.

Borba, M. (1990). Ethnomathematics and Education. For the Learning of Mathematics, 10(1), 39-43.

Clarkson, P. C. (1991). Bilingualism and mathematics learning. Melbourne, Australia: Deakin University.

D’Ambrosio, U. (1990). The role of mathematics in building a just society. For the Learning of Mathematics, 10(3), 20-23.

Ellerton, N. F., & Clarkson, P. C. (1996). Language factors in mathematics teaching. In A. J. Bishop et al. (Eds.), International Handbook of Mathematics Education. Dordrecht. The Netherlands : Kluwer Academic Publishers.

Finau, K. & Stilman, G. (1995). Geometric skills behind the Tongan tapa designs. Paper presented at the history and pedagogy of mathematics conference. Cairns. Australia.

Frankenstein, M. (1990). Critical Mathematical literacy. In R. Noss, A. Brown, P. Dowling, P. Drake, M. Harris, C. Hoyles & S. Mellin-Olsen, (Eds.), Political Dimensions of Mathematics Education : Action and Critique. Proceedings of the First International Conference. London : Institute of Education, University of London.

Gerdes, P. (1988). On Culture, Geometrical Thinking and Mathematics Education. In A. Bishop, Mathematics Education and Culture. Dordrecht. Netherlands : Kluwer Academic publishers.

Gerdes, P. (1990). On mathematical elements in the Tchokwe “Sona” tradition. For the Learning of Mathematics, 10, 31-3 4.

Howson, A., & Wilson, B. (1986). School Mathematics in the 1990s. U.K : Cambridge University press.

Kaleva, W. (1998). The cultural dimension of the mathematics curriculum in PNG: Teacher beliefs and practice. Unpublished Doctorate of Philosophy thesis. Monash University . Australia.

Keitel, C., Damerow, P., & Bishop, A. J. (1989). (Eds.), Maths, Education and Society. Document 35. Paris : Division of Science, Technical and Environment Education, UNESCO.

Lean, G. (1994). Counting Systems of Papua New Guinea. Unpublished Doctor of Philosophy dissertation. PNG University of Technology. Lae. Papua New Guinea.

Macgregor, M., Moore, R. (1991). Teaching mathematics in a multicultural classroom. University of Melbourne: School of Science and Mathematics Education.

McMurchy-Pilkington, C. (1995). Mathematical activities of Maori women in a Marae kitchen. Paper presented at the history and pedagogy of mathematics conference. Cairns. Australia.

Pompeu, G. (1992). Bringing Ethnomathematics into the School Curriculum. Unpublished Doctoral Dissertation. Cambridge University (U.K).

Presmeg, N. C. (1996). Ethnomathematics and Academic Mathematics : The Didactic Interface. Paper presented at the Eighth International Congress of Mathematical Education. Seville, Spain.

Saxe, G. (1988). Linking Language with Mathematics Achievement : Problems and Prospects. In R. Cocking and J. Mestre (Eds), Linguistic and Cultural Influences on Mathematics Learning. Hillsdale. New Jersey : Lawrence Erlbaum Associates publishers.

Thaman, K. (2001). Towards culturally inclusive teacher education with specific reference to Oceania. Paper presented at the 11 th World congress of Comparative Education Society. Korean National University of Education. South Korea.

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