LOCAL MATHEMATICS EDUCATION:The implementation of local mathematical practices into the

mathematics curriculum

Karen François, José Ricardo e Souza Mafra, Maria Cecilia Fantinato and Eric Vandendriessche1

ABSTRACTIn this article we explore the possibility of implementing local mathematical practices in the mathematics curriculum. This research relies on two theoretical frameworks that focus on the value of culture in relation to the study of science and scientific practices in relation to the learning process. The analysis is based on two empirical long-term ethnographical investigations in different places and with different peoples. The field research was carried out in the Northern Ambrymese society (Ambrym Island, Vanuatu, South Pacific) and in the region of Aritapera, a rural area near the city of Santarém, state of Pará, North of Brazil. The local activities investigated in this empirical research can be described as, respectively, string figure-making and handcrafted cuias. The peoples involved are the Ambrymese society (South Pacific) and the craftswomen from the city of Santarém, organized since 2003 in the Santarém Riverside Craftswomen Association (ASARISAN). Based on the comparative analysis and exploration which will improve our understanding of the implementation of local mathematical practices in the mathematics curriculum, we will also show the added value of these implementations and we will provide some guidelines for best practices.

KEYWORDS: Ethnography, South Pacific, North of Brazil, local mathematical practices, local mathematics curriculum, practical turn, social turn, learning theory.

INTRODUCTION

Research on local mathematical practices and the implementation of these practices in the formal mathematics curriculum relies on two theoretical traditions. The first of which is the practical turn within the philosophy of sciences; the second, the social turn in learning theory. Both traditions share a similar concern for the fusion of both scientific practices and learning processes in a given sociocultural environment. In this introduction we shall conduct a brief overview of the way in which both traditions evolved over the past decades and how the research we are reporting on is a continuation of these traditions.

Looking back at the recent history of the philosophy of sciences, one of the most striking revolutions in the field of research is the work of Thomas Kuhn (1962) with his The Structure of Scientific Revolutions, where he describes the way in which a scientific theory evolves over

1 Prof. Dr. Karen François, Brussels University (Vrije Universiteit Brussel), Pleinlaan 2, 1050 Brussel, Belgium <Karen.Francois@vub.ac.be >

Prof. Dr. José Ricardo e Souza Mafra, Federal University of West Pará.68040-030 (PAC) Marechal Rondon Avenue, Santarém/PA, Brazil <jose.mafra@ufopa.edu.br>

Prof. Dr. Maria Cecilia Fantinato, Federal University Fluminense (UFF), Rua Prof. Marcos Valdemar de Freitas Reis, s/n, Bloco D, Campus do Gragoatá Niterói/RJ, 24210-201, Brazil <mcfantinato@gmail.com> Dr. Eric Vandendriessche, Paris Diderot University, Sorbonne Paris Cité, Science-Philosophy-History, UMR 7219, CNRS, F-75205 Paris, France <eric.vandendriessche@univ-paris-diderot.fr>

1

time. Kuhn (1962, 1969) refers to several historical cases (e.g. the Copernican revolution) to substantiate his thesis that scientific theories develop along a continuous line including different stages (from a pre-scientific program, to a scientific program, normal science, then a period of inconsistencies and finally a revolution leading to the rise of a new scientific paradigm). Although Kuhn is acknowledged as a prominent philosopher who emphasized the relevance of studying how scientific theories come to the fore and how they are then replaced by newer theories, refers to his precursors Ludwik Fleck (1935) and Alexander Koyre (1961) who had already studied scientific practices in the early twentieth century. Fleck (1935) studied the way in which scientific facts are collected depending on the particular mindset of a research community. He studied the perception of Syphilis and how it evolved from a mythical, empirical, then pathological concept towards an etiological one. But his ideas were too well ahead of the times to gain a wide acceptance. It was Kuhn who referred to his work in his preface as follows: “I have come across Ludwik Fleck’s practically unknown monograph, Entstehung und Entwicklung einer wissenschaftlichen Tatsache (Basel, 1935), which is an essay that anticipates many of my own ideas. (Kuhn [1962] 1970: vi-vii). Kuhn’s work was revolutionary, although his perspective still remained that of an internalist view on sciences. With Robert Merton’s work (1973), the field of study was broadened by taking into account the external sociological processes that are connected to scientific practices. In his Sociology of Science, Merton (1973) developed his CUDOS system with which he describes the socio-economic and political factors that come to play in the development of scientific practices. This work was previously published in 1942 as Science and Technology in a Democratic Order, but became more widespread with its publication in The Sociology of Science (Storer, 1973), a collection of 23 papers published by Merton. The acronym CUDOS stands for norms of scientific practices, i.e. Communism, Universalism, Disinterestedness, and Organized Skepticism. Notably, these norms or standards are of great importance today in the context of the ongoing debate on scientific integrity.

It took quite a few decades before the study of sciences evolved into the study of scientific practices. With Pickering’s work (1992) we have a collection of articles that emphasize the study of scientific practices from a broad perspective, including anthropological, sociological, ethnographical and critical standpoints.

It is within this tradition of studying practices rather than studying the final product of sciences that ethnomathematics could breathe and develop its research area. Mathematics, considered to be the most robust and universal science, also became of interest for the study of its practices. The relation between mathematics and culture was one of the first investigations (Powell & Frankenstein, 1997) undertaken, although culture was perceived as non-western culture. The topic also became of interest in the study of western mathematical practices (Larvor, 2016a, 2016b) with a large research community (e.g. Association of the Philosophy of Mathematical Practice (APMP) founded in 2010).

The second aspect of our article focuses on the implementation of local mathematical practices in the formal mathematics curriculum. The theoretical background of this research topic relies on the social turn within learning theory and the emphasis on the fusion of the learning process into a sociocultural environment. Social sciences have developed during the last decades three main learning theories, each of them with a specific focus on the learning process (François & Pinxten, upcoming). Whereas behaviorism focuses on the input-output mechanisms ignoring the black box in between, genetic psychology mainly focuses on the black box and what is going on in the student’s mind. The socio-cultural theory is taking into account the environment in which the learning process is taking place. The social turn within

2

the learning theory became of interest in the field of mathematics education (Bishop 1985) mostly as a reaction to the huge dropping out of pupils with a specific socio-cultural background. Even in countries with high scores in international comparative research (which doesn’t imply our support of these tests) one can easily conclude that migration and language backgrounds are an important determinant of failure in mathematics (OECD, 2014).

Socio-cultural theory elaborates on the learning theory of the Russian Lew Vygotsky (1978) and his concept of a zone of proximal learning (ZPL). The concept can be understood as the cognitive field of the pupil, which can be spotted at the fringe of the background knowledge and the out-of-school worldview. It is the zone of learning where the pupil will be able to connect insightfully to new knowledge because of the intrinsic relation between background knowledge and new inputs. Background and out-of-school knowledge are integrated in formal learning as a stepping-stone for acquiring new knowledge, new meanings and new mental frames. The concept of ZPL was an inspiration for later developments of socio-cultural learning theory and such central concepts as the pupils’ background and foreground (Renuka & Skovsmose, 1997; Skovsmose, 2005).

Lave & Wenger (1993) elaborated on the ZPL concept and developed a more specific notion of legitimated peripheral participation (LPP). They emphasize that the learning process is always a situated learning that considers the student holistically. The student is an agent who is active within a specific world context and all these aspects are mutually constitutive for the learning process. Learning is not perceived as the reception of factual knowledge or robust information. It is a social activity that takes place within a community, at first legitimately peripheral. Later on it increases gradually in engagement and complexity. Participation in social practices is the fundamental form of learning. It implies more than connecting the immediate context to the instruction. It is even more important “to consider how shared cultural systems of meaning and political-economic structuring are interrelated, in general and that they help to constitute meaning within communities of practice.” (Lave & Wenger, 1993, p. 54). This concept of LPP will be an important tool to analyze our data on informal learning practices in the local communities and to understand the importance of implementing them into the formal schooling and learning environment. François & Pinxten (upcoming) state that:

Indeed, when we agree that children are raised throughout the world in the particular meaning production processes and mental frames of their particular environment, it is important and sensible (i.e., possibly beneficial) to take into account and even actively study the contents and the learning strategies of the out-of-school knowledge and skills the child possesses and uses when first coming into contact with what we–western educators–call mathematics education. (p. 6)

We have to investigate the background knowledge that children actually bring to school and how we can introduce their learning context in the teaching of mathematics at school. Blindness and ignorance concerning the local culture, local practices and knowledge may well explain the gap between success and failure, in a formal mathematics classroom. In the following sections of our empirical investigations, we will examine the possibilities of implementing local practices within the formal curriculum. Throughout our empirical journey we became aware that besides closing the socio-cultural gap, there would be additional benefits not only for the pupil, but for the local community as well.

Empirical investigation

3

In this section we will look into empirical cases and the way we developed the data gathering during the field research. Both investigated cases are situated within their own theoretical background, specific circumstances and local practices.

1. String figure making from VanuatuFor over a century, string figure-making practices have been observed by anthropologists in many regions of the world, especially within “oral tradition” societies (Paterson, 1949; Maude, 1978; Braunstein, 1992). Some mathematicians have also regarded string figure-making as a worthy topic within their discipline. At the beginning of the 20 th century, Cambridge mathematician W. W. Rouse Ball (1850-1925) devoted a chapter to string figures in his popular book on mathematical recreations (1911) which is—to our knowledge—the first attempt made by a mathematician to demonstrate the connection between mathematics and string figure-making (Vandendriessche, 2014a). Thereafter, a few mathematicians have developed mathematical and modeling tools in order to formalize this practice (Amir-Moez, 1965; Storer, 1988; Yamada, Burdiato, Itoh & Seki, 1997).

In the Republic of Vanuatu, a large archipelago in the South Pacific, string figure making was first documented in the 1920s by anthropologist L. A. Dickey (1928). As noticed while carrying out fieldwork since 2006, this activity is still practiced today in that part of the world. In these islands, making a string figure requires making a loop by knotting the ends of an approximately two-meter-long string—which is made with a thin slice of a pandanus tree leaf. The activity then consists in applying a succession of operations to the string, using mostly the fingers, and sometimes the wrists, mouth or feet. This succession of operations, which is generally performed by an individual and sometimes by two individuals working together, is intended to generate a final figure, whose name refers to a particular being or thing.

Figure 1: Ambrymese string figures, Mata performing mel (a nut), Ambrym, Vanuatu. ©Vandendriessche

Our ethnomathematical project2 (still in progress) is based on a long term ethnography (around the village of Fona, Northern Ambrym, Vanuatu) aiming at collecting various types of data: 1) the procedures leading to the various figures (using an original symbolic writing system for noting/recording them) 2) the vernacular (technical) terminology linked with the studied practices 3) the oral texts and/or discourses which are sometimes associated with the latter.

2 This project is currently carried out as part of the collective research project “Encoding and Transmitting Knowledge with a String: a comparative study of the cultural uses of mathematical practices in string-figure making (Oceania, North & South America)” (2016-2020), coordinated by Eric Vandendriessche, and supported by the French National Research Agency (ANR).

4

This in-depth collection has been mainly collated through semi-structured interviews with the practitioners considered by the other members of the community as the more knowledgeable in string figure-making. At a second stage, the collected data are analyzed—and put in perspective with other ethnographical sources—in order to comparatively analyze the mathematical dimension of the latter procedural activity in their relationships with other forms of knowledge in a given society (Vandendriessche, 2014b).

The process of making a string figure can be regarded as a series of “simple movements” analyzable as “elementary operations”, insofar as the making of any string figure can be described by referring to a certain number of these operations. A string figure can thus be seen as the result of a “procedure” (or “algorithm”) consisting of a succession of elementary operations. Most of these operations can be defined as “geometric” operations whose purpose is to modify one configuration/state of the string in order to transform it into another (Vandendriessche, 2015).

Figure 2: Elementary operations. Left: “picking up” a string. Right: “twisting” a loop

The activity of creating new string figure procedures (as it occurred in the past3) can be regarded as mathematical at different levels. Their production very likely required an intellectual task of selecting the elementary operations and organizing them in procedures and “sub-procedures” (i.e. ordered sets of elementary operations either iterated within a given procedure or repeated identically within several different string figure algorithms of the same corpus). String figures thus appear as the result of genuine algorithms. Based on an algorithmic practice, the production of string figure algorithms is also of a “geometrical” and “topological” order, insofar as it is based on investigations into complex spatial configurations, aiming at displaying either a 2-dimensional or a 3-dimensional figure. Several recurrent phenomena confirm this point: the concept of ‘iteration’ (iteration of a pattern or a sub-procedure) and the concept of transformation (of the final figure ‘geometry’ i.e. combination of motifs) are ubiquitous in this practice. Finally, some Ambrymese string figures suggest that practitioners have elaborated some procedures by altering one or several operations involved in the making of another string figure4.

3 In Northern Ambrym, as well as the other societies where we carried out fieldwork, many people know how to perform numerous string figures, and we often had the opportunity to work with real “experts”. Nevertheless, we have never met anyone with the ability or even the desire to invent a new string figure. Therefore, we can only speculate about which methods were carried out by the actors to create new string figure algorithms.4 The concepts of ‘transformation’, ‘iteration’, and ‘alteration’ related to string figure-making can be found in many societies from Oceania, North and South America. However the occurrence as well as the modalities of their use may vary significantly from one society to another (Vandendriessche, 2015).

5

Figure 3: Iteration and variation. Left: string figure meya (red fruit), resulting from the iteration of a same pattern. Right: variation on meya, named Titi man liseseo (“The devil’s breasts”- represented by two hanging loops) © Vandendriessche

2. Handcrafted gourds from the North of BrazilThe Northern region of Brazil has a large diverseness of social practices reflected in various cultural and aesthetic manifestations, such as: braids, embroidery, lace and ornaments. These practices are very firmly linked to routine, day-to-day actions in the communities scattered along the Brazilian Amazon. Among these sociocultural practices brought to light in northern Brazil, the processes of ocher preparation and the fashioning of ornamental items made from gourds5—one of the objects of the research that comprises this proposal—provides us with a unique opportunity for discussion. These elaborations, produced by artisans living in the western region of Pará, display a uniqueness in peculiarities and multiplicities of (artefacts?) evidence, which are closely related to the cultural dynamics characteristic of these riverside populations.

The handcrafting of cuias provides a unique and permanent feature in the lives of its practitioners, a part, therefore, of the local culture, way of life and natural dynamics of the populations residing there. We accompanied a group of craftswomen residing in five (05) distinct locations in the region of Aritapera, a rural area near the city of Santarem. These activities reveal several6 particular and idiosyncratic expressions in daily life, drawing on their traditions, social relations and the very nature of their existence, in the production of cuias.

In 2003, the craftswomen of Aritepera founded the Association of the Riverside of Santarem Artisans-ASARIAN, for the production of cuias and the popularization of their craftsmanship.The study (Mafra & Fantinato, 2016) of the riverside communities in the region of Aritapera took place in 2002. These communities are located on the banks of the Amazon River, on wetlands three hours by boat from the city of Santarem.Naturally, they suffer changes in geographical handicaps stemming the flow of this hydrological region. The period of flood/ebb of the Amazon River, alternates between periods of flood, during the first half of the year, and periods of drought, during the second half. In this context, the riverside populations’ experience is a part of this dynamic of seasonality, interacting well with their many subsistence activities: fishing, agriculture, grazing, domestic life and craft trading.

5 The gourd (bowl) is a fruit originated from calabash tree (traditional tree of the Amazon Region) and that it presents diverse formats and dimensions, may take the dimension of a sphere, oval or slightly flat. It is primarily used to drink the tacacá (stock or soup typical of the Amazon region, of Indian origin), but, with the passing of time many other artifacts were being created, using the raw material.6 The region of Aritapera composes a set of communities. These communities are: Enseada do Aritapera, Centro do Aritapera, Carapanatuba, Cabeça d’Onça e Surubim-Açu.

6

Among the many crafts practiced in this region, the production of cuias is an ancient traditional women’s custom. Production takes place with greater frequency during the flood season (Maduro, 2013). This is due to the fact that most of their subsistence activities, soil management and conservation, planting, harvesting and other similar activities are quite restricted during the flood season, limiting economic activities and jeopardizing their livelihood. The collection of cuias—calabash gourds harvested from the Crescentia cujete tree that are then cut, scraped, dyed and etched—has no impact on the region’s ecological balance. The crafted cuias have many different uses in these community’s daily life: drinking vessels, food and water storing, bathing, bailing water from canoes as well as ornaments and decorations, etc. The production developed by the artisans provides an indication of typologies identified and reflected in the items produced.

In this research, the study involved a detailed description of the group’s activities, the procedures used, instrumental techniques and intrinsic characteristics they practice. The methodological procedures used involved elements of ethnographic research, looking for a dense description of events related in research (Geertz, 2008). The investigation implies the the researcher’s immersion in the environment in which the object of study lies (Bogdan & Biklen, 1994), making the researcher-observer the main instrument in the collection and subsequent analysis of the data. It was elaborated at different times: Dec. 2014, Jan., Feb. and Aug. 2015. The data collection techniques used were a field diary, note taking, interviews, photography and filming.

The central focus of investigation was to gain an understanding of the techniques and processes involved in the preparation of sections and records on the curved surface of cuias. Produced from mechanisms and strategies created in the environment, the technical aspects of the craftswomen’s work showed a very strong with the interaction with the group in moments that require their cooperation, indicating an obvious variation in the know-how of the craftswomen. When scratching, various materials are used, such as knifepoints, penknives, punches, among others (see, for example, fig. 4).

Figure 4: Preparation of the gourds for scratching. Source: record held by the authors, with ASARISAN’s permission.

The standards and records evidenced show a diversity so obvious in relation to the multiplicity of records. There are basically two types of standards of records sections in cuias: the floral and the tapajonic (Fig. 5).

7

   

Figure 5: Some patterns of items recorded in cuias. Source: record held by the authors, with ASARISAN’s permission.

These standards provide indications of how these records reflect elements of meanings, such as the flora and fauna present in the region, bringing with it visions and reflections of the labor activity and know-how of the craftswomen.

INFORMAL LEARNING

Based on the empirical investigations and the abundant data gathered from the field, we analyzed the data from the perspective of the informal learning process. In the following subsections we will report on the results of our analyses as a first step in our research project, where we will look for possible connections between the informal and formal learning systems.

1. Knowledge transmission through string figuresIn Vanuatu, there are no less than 120 different vernacular languages, corresponding to different cultural areas in the archipelago7. As Eric noticed while doing fieldwork in different Ni-Vanuatu societies (North Ambrym, South Malekula, South Santo, etc.), a large number of string figures can be found in various linguistic areas, whereas a few string figures seem to be more locally practiced8. However, there are significant linguistic variations (from one area to another) related to string figure making, in the names given to the activity as well as to the final figures, and in the use of technical expressions.In Northern Ambrymese society, string figure making is locally termed using the vernacular expression “tu en awa” (literally “to write with a string”), suggesting that this activity is perceived in this society as an encoding of information. Some other vernacular expressions are used by practitioners to refer to the (basic) movements involved in string figure making. In particular, the (elementary) operations implemented to the string are designated through action verbs; the subjects being the finger names (for instance, pokolam hu pokokiki, the thumb picks up the little finger i.e. implicitly a string running from the little finger). A few (short) ‘sub-procedures’ (ordered sequences of a small number of elementary operations) are also named in the Northern Ambrymese language. All of these technical expressions are used 7 In the Republic of Vanuatu (ex New Hebrides, a French & English Condominium until its independence in 1980), English, French, and the pidgin language ‘Bichlamar’ are the three official languages. Children generally learn Bichlamar before schooling, in parallel to their mother-tong, and begin learning English or French in elementary school.8 For instance, the Ambrymese string figure meya (red fruit, cf. Fig. 3) is known in many other Ni-Vanuatu societies (and also more generally in Melanesia: I personally recorded it in the Trobriand Islands where it is called vivi (a nut) – see http://www.rehseis.cnrs.fr/www/vandendriessche/kaninikula/OpeningsA/OpeningA/sg-vivi/26-vivi/26-vivi.html) By contrast, the variation on the latter string figure, named titi man liseseo (The devil’s breasts, cf. Fig. 3) by Northern Ambrymese, has not been collected—as far as we know—in other islands so far.

8

in instances of transmission from one person to another, although not consistently. The “tu en awa” procedures are indeed taught or shown most often without any technical comments. However, the existence of these expressions is an indicator of the perception by the actors of orderly sequences of operations, suggesting a local perception of the notion of ‘elementary operations’ and ‘sub-procedures’ revealed by ethnomathematical analysis. Furthermore, vernacular terms explicitly express the property of ‘symmetry’ (shared by a number of these figures) and the ‘iteration’ of a pattern or a sub-procedure.

In Ambrymese society, the practice of string figure-making is said to be a feminine activity. However, it is indeed a shared knowledge, almost everyone being able to perform a few of these figures. We had indeed the opportunity to work with a few male practitioners who are well acquainted with the subject. However, we definitely met many more women interested in this activity. Some of them are genuine “experts”—as they are able to do all the procedures of the corpus known in the village (including the most complicated ones); they are also the ones able to perform string figure procedures slowly, by clearly dissociating each operation, allowing transmission. Finally, it seems that string figure making is mostly transmitted to children by their mother and/or grandmother.

The long “sub-procedures” (made with more than three elementary operations) are nameless for the Ambrymese practitioners. However, they spontaneously relate one string figure procedure with another when both “tu en awa” procedures share an ‘ordered set of basic movements’. The practitioners do so by pointing out that from a given stage of a string figure procedure, “you do it in the same manner as you do it in another procedure”. While working with Ambrymese children, I often noticed their ability in making such links between string figure procedures. This suggest that ‘sub-procedures’ play—for these practitioners—a major role in the process of memorizing the making of string figures.

In Ambrym (and more generally in Vanuatu, and even in Melanesia), the practice of string figure-making is—or was—meant to record, memorize and/or express a particular knowledge of mythology, cosmology, geography, social rules, and ritual prescriptions (Vandendriessche, 2014b, 2015). For instance, the Ambrymese string figure named bulbul algon (literally “canoe lizard”) is related to the story of Yaulon, one this society’s mythical heroes. Bulbul designates this hero’s canoe, while algon (lizard) recalls the symbol of one of the seven grades of the chieftainship system, whose conception is attributed to this local mythological hero.

In this society, string figures are preferably performed during the yam harvest (from February to July), while their usage is prohibited outside this period, the making of such figures being perceived as having a negative impact on the growing of the plant’s stem winding around the stake: it would favor the entanglement of the stem, slowing down the plant’s growth. In Ambrymese society, the practice of string figure-making can thus be analyzed as a method for the organization and the transmission of knowledge (mythological, cosmological, sociological, geographical, etc.), involving the use of (ethno-)mathematical concepts. When learning how to make string figures, Ambrymese children become acquainted with a technical activity (with a geometric and algorithmic character), requiring dexterity and concentration, and, at the same time, they develop their knowledge of their cultural environment.

2. Knowledge transmission through cuias’ craftswomen cultural practices

9

In this topic, we are bringing in some concepts on informal education studies, in relation to the empirical investigation on cuias’ craftswomen cultural practices. We agree with Dasen (2004), when he writes:

All societies, either from the North or the South, have their peculiar ways of transmitting their culture from generation to generation, away from formal education represented by schools [..] good knowledge about informal education can help schools adapt these cultural contexts, they are immersed in, to the environment realities they are placed in, in much better ways (Dasen, 2004, p. 23).

The Aritapera craftswomen group can be defined as a community of practice (Wenger, 1998), since they share the same knowledge and practices repertoire. They get together searching for common solutions and their collective activity sustains a feeling of identity for the group. The concept of legitimated peripheral participation (LPP) (Lave & Wenger, 1993) provides an important tool to approach learning constituted as a social practice within this community. The apprentices joining the craftswomen’s group begin by gradually performing carving and ornamental tasks, guided by the craftswomen through a well-defined progressive teaching/transmission. The apprentices actually learn the kinds of patterns considered to be the easiest ones to handle. According to Chamoux (1978), the apprentices can learn through impregnation, a process that presupposes two conditions:

Firstly, it grounds on a corporal training common to every single member of the village group: gestures, postures, material perception ways, language […] This training is connected to what is generally called group culture. Secondly, it implies in observation repetition of different techniques and gestures experimentation (Chamoux, 1978, p. 63).

In case one of these two conditions isn’t fulfilled, osmosis doesn’t happen by itself, and a master is required to allow the conveyance of such know-how (Chamoux,1978).Throughout the fieldwork, we managed to observe an interaction situation between two craftswomen, Lélia and Marinalva, where the first assumed the part of the master. Marinalva was working with a large knife, building the support base for a fruit bowl, adjusting the piece gradually into a spherical shape. Then, Lélia took the piece Marinalva was producing and carried out a test placing a fruit bowl in the hollow semispheric shape on the piece Marinalva was working on to check the balance of the piece. She showed Marinalva, who was attentively watching her friend and master, and the set, while Lélia pointed to where the piece needed a few other adjustments. After the demonstration, Lélia handed both pieces to Marinalva, who applied the large knife to the areas in need of modifications, while she looked at her friend\master. Right after verbal confirmation, she resumed her scraping activity with the large knife. Minutes later, Lélia asked Marinalva to hand her the piece that had just been fixed and carried out the same balance test, saying:

L: Look Marinalva, look.M: yes.L: there is just this one more little thing here to fix, but reduce from here, can you see?

Marinalva gets hold of the large knife to sharpen another which she was working with and remarks:

M: or we break it or we fix it (04/08/15).And Marinalva carried on her scraping work with the large knife.

10

We noticed variations in the acquirement and transmission of the skills required for the decoration of cuias in Aritapera’s craftswomen community. The floral pattern’s informal processes are taught by mothers and grandmothers. Tapajonic patterns were unheard of before the establishment of the ASARISAN and CNFCP\ IPHAN organizations, who compiled a printed catalogue of these indigenous patterns, and organized courses and workshops to favor their acculturation by the community. Using the tapajonic models assimilated during the workshops, the elder craftswomen elaborated numerous variations based on tapajonic patterns. They gradually shared this knowledge with the younger craftswomen. Angeli said, as she pointed out two elderly members of their group: “Both of them attended a workshop, I didn’t, but they taught me what they had learned” (05/08/15).

Therefore, the distinctions characterizing formal and informal education are not dichotomous, they offer a wide array of contextual nuances. Based on Greenfield and Lave’s research (1979) De Vargas (2009) refers to the presence of an informal\formal continuum. The traditional art of fashioning cuias using floral motifs was enriched by exogenous patters acquired in a more formal teaching environment. In turn, the women assimilated their knowledge and practice of tapajonic carvings with their own, and so, diversified the repertoire of indigenous patterns. Therefore the craftsmanship used in designing cuias is a dynamic process.

FROM INFORMAL LEARNING TO FORMAL LEARNING

Based on social learning theory and more specifically on the concepts of ZPL as developed by Vygotsky (1978) and later on elaborated on by Lave & Wenger (2013) as outlined in our introduction, we can now move on to the field data to analyze how local practices can be of interest for the local community. We will detail the added value of implementing local mathematical practices in the school curriculum. From social learning theory we have evidence that pertains to the pupil’s background information (Renuka & Skovsmose, 1997; Skovsmose, 2005), which is essential for the learning process. Pupils attending classes need to be apprehended in their entirety, as agents who develop extra-curricular skills, and who live and interact in a complex environment (Lave & Wenger, 1993), where they share common cultural systems and social rules. Most of which are handed down through informal learning systems by the community. In the next subsections we will analyze our two empirical cases.

Traditional knowledge and the Vanuatu National CurriculumAbout a decade ago, the Republic of Vanuatu began an evaluation in order to elaborate a National Curriculum taking into account the various local cultures and the different vernacular languages, in order to “nurture” Ni-Vanuatu children and students

with this rich background which provides the prior knowledge and skills that teachers can build upon. This applies to all levels of schooling from Kindergarten to Year 13 and tertiary institutions. This is how we should make Vanuatu’s curriculum more relevant and contextualized. [...] Our multicultural society is challenging as teachers need to cater for the diverse needs of children and students from many backgrounds and languages and ensure that we conserve and sustain our cultural heritage and languages (VNCS, 2010: 19-20).

11

In that perspective, the Vanuatu National Curriculum Statement (VNCS, 2010) recognizes the value of traditional knowledge and practices—such as string figure-making, mat making, and sand drawing9 in particular—for calling upon the latter practices in formal education. The VNCS claims moreover that:

Teachers should usually begin from what children know and then introduce unfamiliar knowledge, skills and attitudes. Teachers need to breathe life into the curriculum and demonstrate its relevance to children and students by using local examples whenever possible (VNCS, 2010: 50).

At the same time, it is implicitly recognized that some of these traditional practices (“weaving, carving, sand drawing and string games [...] all custom art forms” that “children should be familiar with”) are sometimes in decline, and prompts the members of each community to “assist the children in learning about these art forms and make simple mats and other objects” (VNCS, 2010: 52).

It is in this context of the revision of the National Vanuatu Curriculum (still in progress) that the “Vanuatu Cultural Centre”, the local institution working for the preservation and the promotion of different aspects of Vanuatu’s culture,—drawing on our ongoing research on string figure-making—has given its (mandatory) assent for this project in ethnomathematics, provided it leads to pedagogical applications. Beyond this institutional incentive, another motivation for undertaking such educational research is that the communities involved in the project (and the Northern Ambrymese society in particular) welcome with interest the idea of using their traditional practices in the curriculum. Whereas the purpose of ethnomathematical (theoretical) research is clearly not generally not perceived as vital as we think it is, indeed, its educational valorization makes sense for these people. Aware of these practices’ decline in their community, and asserting that young people are no longer interested in traditional knowledge, they consider this valorization as a way of preserving their local culture.

In the final stage of the latter project, pedagogical materials (related to the outcomes of the present research and in collaboration with Ni-Vanuatu educators) will be elaborated in an attempt to help local teachers in experimenting with the use of (culturally-related) mathematical string figure-making practices ‘as such’, in and of themselves. A pedagogical mathematical sequence (say in Northern Ambrymese 6th year classrooms) related to string figure-making could start by the collection of the string figures (and their vernacular names) that the pupils do remember. They should be prompted to use their local vernacular language for expressing the various operations involved in the making of these figures, as well as the symmetries and the iteration of patterns. The set of string figures thus collected could be then completed with other “tu en awa” procedures known to the elders of the community.

Previous studies on string figures’ value in mathematics education suggest that practicing string figure-making may develop vital skills necessary for practicing mathematics—such as concentration, self-evaluation, spatial relation consciousness, or conducting step by step ordered sequences of instructions (Moore 1988; Murphy 1998). Beyond this analogy between string figure-making and mathematical practice, the pedagogical sequence might continue through an in-depth analysis of a set of procedures, bringing to light the (elementary) operations involved, their impact on the string configurations and their organization in sub-

9 Sand drawing consists in drawing a continuous line with the finger, either in the sand or on dusty ground,—generally drawn through the framework of a grid made of perpendicular lines, without retracing any part of the drawing (François & Vandendriessche, 2015; Vandendriessche, Mafra, Fantinato & François, upcoming).

12

procedures. Furthermore, the teacher might induce his pupils to reflect on how a string figure has been sometimes transformed into another, and how the iteration of an ordered set of operations may allow the iteration of a given motif (cf. meya, Fig. 3. above). Finally, it can be seen that different (Ambrymese) string figures sometimes differ on one—and only one—elementary operation. This remarkable property actually implies a methodology to create new string figures (Ball, 1911; Murphy, 1998): by altering a few operations within some “tu en awa” procedures, the pupils would become creators of new string figures themselves.

This pedagogical (ethno-)mathematical sequence on string figure-making will not be isolated from the other forms of local knowledge imbedded within this activity. One way to do so, would be to include this sequence in an interdisciplinary pedagogical project, bringing into the classroom the cultural and cognitive complexity of this practice.

A possible interest by the community of AritaperaOne of the possibilities for conducting the research developed previously shows a probable link with mechanisms of mathematical instrumentation. The knowledge abstracted from the craftswomen’s work, suggest very strongly cognitive operators involving mental calculations benchmarks of metrics and estimates of references. These characteristics can be transposed to educational actions, from the organization of pedagogical actions, aiming at the development of skills and abilities related to measurement, counting aspects and geometric topology.These possible multiple relations can be established on the basis of the aspects of strategic actions used in those symbolic elements and space are integrated with the social and cultural context in which the craftswomen live. For example, the incisions and records of markings indicated in the cuias allow a structured work, aiming the dimensioning of mathematical forms and conceptual elements proper to the school curriculum, such as plane geometry, spatial measures and arithmetics.

These interactions then project a set of elements and propositional factors of actions concerning the purposes of ethnomathematics, in relation to the educational context. In the case of know-how and knowledge to be promoted, in Aritapera’s communities, these characteristic concepts are as important for students in the initial process of schooling, as well as for students who are progressing from secondary education to higher education.Understanding that such conceptual elements may be developed in these environments, in articulation with other disciplines, thus permitting a possibility of integration between interdisciplinary knowledge, able to show at school, the multiples possibilities for resolving day-to-day situations. Chools could become positive elements of initiatives, in which the school space is interconnected to the various aspects. Within the reality of mathematics teaching.

Such an approach can be forwarded around the interlocution, discussion and understanding of different forms of procedure for resolution of a same activity, be it developed from informal learning or following a formal method, like, for example, the axiomatic method used in geometry. This perspective of analysis is based on the principle that it is possible to understand the mathematical processes that are interconnected, and/or articulated with a sense, with a reason to exist, which can gradually favor the ability for abstraction and developing the capacity of reasoning, creativity and cognitive processing.

For example, we can cite—the title of future researches—a systematic study of standards and metrics used by craftswomen and their similarities, in terms of strategies used in school contexts, for students.

13

To witness the artisans' work on curved surfaces, using in their construction strategies and cognitive elements, it is possible to realize a proximity with geometric principles, leading to apprenticeships, which impresses us with their skills and their dexterity in handling the instruments and the perfection in the preparation of sections. Signs of possible explanations allow a multiplicity of possible specific strategies—from the perspective of a lay reader or of students—which suggests alternative possibilities of understanding and knowledge of different strategies, representations, techniques and processes that can show resolutions and explanations of the same problem—situation or immediate need.

Here in this moment of discussion, the possible interest—by the community of Aritapera—in the implementation of local mathematical practices and their relationship with informal learning by entering them in the official mathematics curriculum, taught to the community’s students, will depend on the extent to which, for the community’s members these practices assert their meanings, and that teachers working in the community’s schools understand these possible implementations. From the point of view of the curriculum, it becomes essential to validate this knowledge as an element of apprenticeships and areas included in the school environment, so that a balance between these dialogs may be achieved.

The artistic possibilities have been investigated and the presence of techniques, strategies and materials evidenced in this research lead us to think about pedagogical futures and related to ethnomathematics. Guidelines designed and projected in teacher training courses can be set in the surroundings of riverside communities, neighboring city of Santarem/PA and directly related to the basis of the processes of creation of cuias.

The artisans understand that these educational referrals are significant, in a contribution to their traditions and keeping sociocultural practices alive. Thus, the school takes on a central role as a comprehensive incentive in accordance with this proposal—for the local community to implement the handcraft practices evidenced in the communities, related to the craftwork with the cuias. Such referrals show the school as a living and powerful instrument, thus contributing to the maintenance of their traditions.

DISCUSSION

Based on the two empirical long-term ethnographical investigations of Northern Ambrymese society, Vanuatu, South Pacific, and from the region of Aritapera, state of Pará in Northern Brazil we will discuss some interesting points on the local practices themselves and of the possibilities and the value of implementing local practices in the formal curriculum.

Concerning the local practices themselves we have to emphasize that local practices are a part of the daily way of life and that they are embedded in their cultural tradition, in the natural environment and in the communities’ social networks.

These practices as well as the use of the artifacts they produce are connected to both, the very basic daily life and the deeper cultural tradition. Daily life uses are identified in the making and using of cuias, e.g. drinking, cooking, storing, and collecting. String figure making is also practised by children of the Ambrymese community to play without any deeper meaning given to this practice. Connections to the broader cultural tradition are analyzed in the case of string finger making, e.g. knowledge transfer, storytelling –related to mythical heroes of that

14

society, and encoding information. Also in the case of cuias making, the relation to the broader cultural tradition was identified, e.g. transmission of design, aesthetics, representations of the natural environment, taking care of the ecological balance of the region.

Furthermore local practices are enmeshed with their natural environment. Both cases we studied have a deep connection with nature, with a circular timeline, with seasons and/or with river flooding. Local practices are performed during a specific time of year. String figures are performed during the yam harvest (February–July). They are not practiced during the growing of the plant’s stem because of its the negative effects on growth. Cuias making is related to the floods cycles of the Amazon. Production increases during the first half of the year, when the river is swelling.

Local practices are also part of the social interactions, social relations and of the informal learning processes in these communities. Which brings us to a discussion on the learning processes in relation to local practices in the communities we engaged with.

A first observation is the continuum between informal and formal learning. As we have found in the region of Aritapera, the informal learning process became partly formalized by the founding of the Association of the Riverside of Santarem Artisans (ASARISAN). Since then, the production is increasing thanks to a more systematic procedure. New designs were introduced at the initiative of the local community. So called indigenous designs –which are quite popular and of commercial value—were transmitted to the community by setting up formal learning situations. New formally implemented designs became part of the community’s repertoire and novel applications were created. This case makes clear that out of school practices are dynamic in nature and they are performed along an informal-formal learning continuum. From this continuous practice we can bridge the gap between the out-of-school learning and the formal curriculum. Data analysis provided evidence that people from the ASARISAN community became interested in the implementation of cuias making and carving in the formal school curriculum. Negotiations between community members and teachers will be held to underscore the surplus values of both, the pupil’s individual learning process of the pupil and the value of cultural and traditional knowledge transmission in order to preserve community values. Our study of string figure making can provide us with a good practice.

A second observation is that out-of-school practices can be of interest in the formal mathematics curriculum and for the formal school system. Our case of string figure making, with an emphasis on the mathematical analyses of the procedures (e.g. symmetry, iteration) and the comparative analysis with other forms of knowledge in the community, shows how a local practice can be analyzed as a method for the organization and the transmission of knowledge. Implementing these practices in mathematics and school curriculum acquaint pupils with a technical/mathematical (e.g. geometric, algorithmic character) activity. At the same time, pupils develop knowledge of their own cultural environment. This was one of the reasons for the Vanuatu National Curriculum Statement (VNCS, 2010) to include string figure making (amongst other traditional practices) in the curriculum. They value local traditional practices and knowledge as a means of preserving their culture.

Both observations relate to socio-cultural learning theory and to the concept of zone of proximal learning (ZPL) as well as the concept of legitimated peripheral participation (LPP). The local practices we studied—string figure making and cuias making and carving—are performed in a situated context, they are connected to their daily life experiences and are

15

related to broader cultural traditions and transmission of knowledge. Even when implemented in a formal school curriculum they remain connected to the pupils’ background and their out-of-school worldview. Pupils will be able to connect insightfully to new (mathematical) knowledge because of the intrinsic relation between local practices and the mathematical procedures that underlie the local practices.

It is interesting to observe the fact that new pedagogical materials shall never be isolated from the other forms of local knowledge they are imbedded in. This clears a path for interdisciplinary activities and the integration of mathematical knowledge in a lively and meaningful context. A situated learning process considers the student as a whole person as is the case in the out-of-school transmission of the local practices we studied.

Future research should be carried out and we are convinced that collaboration at the level of the analysis and the interpretation of data of empirical long term ethnographical investigations from different places and different peoples, is an added value in studying the implementation of local practices in formal (mathematics) curricula.

REFERENCES

Amir-Moez, A. R. (1965). Mathematics and String Figures. Edwards Broth. Ann Arbor, Michigan.

Ball, W. W. R. (1911). Mathematical Recreations and Essays (fifth ed). London: Macmillan and Co.

Bishop A.J. (1985) The Social Psychology of Mathematics Education. In: Streefland L. (Ed.) Proceedings of the 9th PME International Conference, Volume 2. Northern Illinois University, NA. Illinois.

Bogdan R.C. & Biklen S.K. (1994) Investigação qualitativa em educação. Portugal: Porto.Braunstein, J. (1992). Figuras juegos de hilo de los Indios Maka. Hacia una Nueva Carta

Etnica del Gran Chaco, III, 24-81. Chamoux, M.N. (1978) La transmission des savoir-faire: un objet pour l’eth-nologie des

techniques? Techniques et culture. Bulletin de l’équipe de recherche 191, 3, 46-83.Dasen, P.R. (2004). Education informelle et processus d’apprentissage. In P.R. Dasen & A.

Akkari. Pédagogies et Pédagogues du Sud (pp.23-52). Paris: L’Harmattan. De Vargas, S. (2009) Estratégias não-escolares de ensino-aprendizagem e formação de

professores da EJA. In M. C. C. B. Fantinato (Ed.) Etnomatemática: novos desafios teóricos e pedagógicos (pp.193-201) Niterói: Editora da UFF.

Dickey, L. A. (1928). String figures from Hawaii, including some from New Hebrides and Gilbert Islands (Reprint Edition 1985, Germantown, New York Ed., Vol. Bulletin n˚54). Honolulu, Hawaii: Bishop Museum.

Fleck, L. [1935] (1981). Genesis and Development of a Scientific Fact. The University of Chicago Press, Chicago. Translation by F. Bradley and T.J. Trenn of the original work ‘Entstehung und Entwicklung einer wissenschaftlichen Tatsache: Einführung in die Lehre vom Denkstil und Denkkollektiv’. Basel: Benno Schabe & Co.

François, K. & Vandendriessche, E. (2016). Reassembling mathematical practices: A philosophico-anthropological approach, Revista Latinoamericana de Etnomatemática, 9 (2), 144-167.

François, K. & Pinxten, R. (upcoming). Multimathemacy? Time to reset. Paper submitted for the MES-9 Mathematics Education and Society, 9th International Conference, Greece, Volos, 7–12 April 2017.

16

Geertz, C. (2008). A Interpretação das culturas. 1.Ed. 13ª reimpr. Rio de Janeiro: LTC.Greenfield, P. & Lave, J. (1979) Aspects cognitifs de l´éducation non scolaire. Recherche,

Pédagogie et Culture, 44, 16-35.Koyré, A. (1961). La révolution astronomique. Copernic, Kepler, Borelli. Histoire de la

pensée III. Ecole pratique des hautes études. Sorbonne. Hermann, Paris.Kuhn, T.S. [1962] (1970). The Structure of Scientific Revolutions. Second Edition, Enlarged.

The University of Chicago Press, Chicago.Kuhn, T.S. [1969] (1970). Postscript. In Kuhn, Thomas S. [1962] (1970) The Structure of

Scientific Revolutions. Second Edition, Enlarged (pp. 174-210). Chicago: The University of Chicago Press.

Kuhn, T.S. [1976] (1981). Foreword. In Fleck, Ludwik [1935] (1981) Genesis and Development of a Scientific Fact. The University of Chicago Press, Chicago. Translation by Fred Bradley and Thaddeus J. Trenn of the original work ‘Entstehung und Entwicklung einer wissenschaftlichen Tatsache: Einführung in die Lehre vom Denkstil und Denkkollektiv’ (pp. vii-xi). Basel: Benno Schabe & Co.

Larvor, B. (2016a). Editorial introduction. In B. Larvor (Ed.), Mathematical Cultures. The London Meetings 2012-2014 (pp.1-6). Birkhäuser: Trends in the History of Science Series.

Larvor, B. (2016b). What are mathematical cultures? In Ju, S. et al. (Eds.), Cultures of Mathematics and Logic (pp.1-22). Birkhäuser.

Lave, J. & Wenger, E. (1993). Situated learning: legitimate peripheral participation. Cambridge: Cambridge University Press.

Mafra, J. R. & Fantinato, M. C. (2016). Artesãs de Aritapera/PA: técnicas e processos em uma perspetiva Etnomatemática. Revista Latinoamericana de Etnomatemática, 9(2), 180-201.

Maude, H. (1978). Solomon Island String Figures. Canberra: Homa Press.Maduro, R.G.A. (2013) A cuia nossa de cada dia. In A. M. S. Santos & L. G. Carvalho (Eds.)

Terra, água, mulheres & cuias: Aritapera, Santarém, Pará, Amazônia (pp. 32-38) Santarém: UFOPA.

Merton, R. (1973). The Normative Structure of Science. In N.W. Storer, (Ed.), The Sociology of Science. Theoretical and Empirical Investigations (pp. 267-278). Chicago: The University of Chicago Press.

Moore, C. G (1988). The Implication of String Figures for American Indian Mathematics Education. Journal of American Indian Education, 28 (1), 16-26.

Murphy, J. (1998). Using String Figures to Teach Math Skills. Part 2: The Ten Men System. Bulletin of the International String Figure Association, Vol. 5, 159-209.

OECD. (2014). PISA 2012 Technical report, Paris: OECD.Paterson, B. T.T. (1949). Eskimo String Figures and their Origin. Acta arctica, 3, 1-98.Pickering, P. (Ed.) (1992). Science as Practice and Culture. Chicago: The University of

Chicago Press.Powell, A.B.; Frankenstein, M. (Eds.) (1997). Ethnomathematics, Challenging Eurocentrism

in Mathematics Education. State University of New York Press, Albany.Skovsmose, O. (2005) Travelling Through Education. Uncertainty, Mathematics,

Responsibility. Rotterdam: Sense Publisher.Storer, N.W. (Ed.) (1973) The Sociology of Science. Theoretical and Empirical Investigations.

Chicago: The University of Chicago Press.Storer, T. (1988). String Figures, Bulletin of the String Figures Association, 16(1), 1-212.Vandendriessche, E. (2014a). W. W. Rouse Ball and the Mathematics of String Figures,

Historia Mathematica, 41, 438-462.

17

Vandendriessche, E. (2014b). Cultural and cognitive aspects of string figure-making in the Trobriand Islands. Journal de la société des Océanistes, 138-139, 209-224.

Vandendiessche, E. (2015). String figures as mathematics? An Anthropological Approach to String Figure-making in Oral Tradition Societies. Studies in History and Philosophy of Science, 36. Switzerland: Springer.

Vandendriessche, E.; Mafra, J.R.S; Fantinato, M.C. & François, K. (upcoming). How Local Are Local People? Beyond Exoticism. Paper submitted for the MES-9 Mathematics Education and Society, 9th International Conference, Greece, Volos, 7–12 April 2017.

VNCS (2010). Vanuatu National Curriculum Statement. Port Vila: Ministry of Education.Vithal, R. & Skovsmose, O. (1997) The End of Innocence: A Critique of Ethnomathematics.

Educational Studies in Mathematics. 34(2), 131–157.Vygotsky, L. (1978). Mind in society: The development of higher psychological processes.

Cambridge, MA: Harvard University Press.Wenger, E. (1998). Communities of Practice: Learning, Meaning and Identity. New York:

Cambridge University Press. Yamada M., Burdiato R., Itoh H. et Seki H. (1997). Topology of Cat’s Cradle Diagrams and

its Characterization using Knots Polynomials. Transaction of Information Processing Society of Japan, 38(8), 1573-1582.

18