Introduction

The Ethnomathematics Programme (^{D’Ambrosio, 2008}, ²⁰¹²) brings together researchers interested in developing diverse mathematical ideas, concepts, and actions by different cultural groups, to building a holistic view of mathematical knowledge, thus perceiving the influence of these concepts and actions in human social construction. Nonetheless, the motivations of interest are different depending on the context, so they have different purposes. Therefore, the views and ways to approach research reflect the diversity of contexts and are continuously evolving to adapt to the socio-cultural reality (^{Albanese et al., 2017}). Today, many ethnomathematics researchers are working to find answers to questions such as: what are the implications of the Ethnomathematics Programme for (mathematics) Education? Touching on more general questions such as: how to integrate the needs of increasingly multicultural contexts with globalisation positions? (^{D’Ambrosio, 2006}), or, however, in a neo-colonialist context, for whom should ethnomathematics research be positioned?

The current agenda of the Ethnomathematics Programme contributes to the construction of a fairer society (^{D’Ambrosio, 2018}) through the reallocation of the human intellectual construction in comparison with the different primary concepts of survival-such as counting, measuring, inferring, analysing, etcetera-identified in Western culture (^{Restivo, 1994}) as mathematical knowledge. This construction is based on respect for the dignity of life and valuing the encounters of diversity (^{D’Ambrosio, 2008}). One of the current challenges of the Ethnomathematics Programme is to formulate alternatives to the formal mathematics education system towards an education that transforms society (^{D’Ambrosio, 2012}).

Communitarian education (^{Mesquita et al., 2014, p. 12} ) is a concrete possibility in this regard. It already exists within a community (^{Coppe & Mesquita, 2015}), as it is the process that enables that community to establish a shared knowledge base for survival. This conception of education has its roots in Paulo ^{Freire’s popular education (1970)}, which proposed, through the emancipation of local knowledge, different from that imposed in formal education, the construction of a political consciousness to fight against the culture of silence (^{Freire, 1970}). The concept of the culture of silence and its corollary, the politics of silencing, correspond to a set of ways of being, thinking and expressing oneself that are the consequence of a social structure of secular domination in Latin America, which are also manifested in developing countries, and in the marginalised communities of countries considered developed, conditioning both oppressors and oppressed. The culture of silence is one of the dimensions of the banking concept of education (^{Freire, 1970}), which is revealed in the low voice and lack of communication that characterise a society without dialogue.

A critical research practice, when focused on communitarian education, encourages community members to become aware of the educational processes intrinsic to their survival, concretising communitarian education as a praxis of living in the community with respect for its spatiotemporal diversity. This process enables researchers to be with and within the community, to know the environment from the community’s own perspective and thus to identify local knowledge collaboratively with its members.

The local knowledge that emerges is transdisciplinary, in the sense that ^{D’Ambrosio (2008)} gives to the concept of ethnomathematics¹¹, as the ways and techniques (-tics) of understanding, explaining and dealing with (matema-) the diverse natural, cultural and socio-economic dimensions of reality (ethno-). The learning of this knowledge is based on a form of organisation, here understood as communitarian. The trivium curriculum proposes to organise the learning process around three instruments: literacy (communicative instruments), matheracy (analytical instruments), and technoracy (material instruments).

In their work, ^{Mesquita et al. (2011)} include a discussion of the three concepts provided by ^{D’Ambrosio (2008)} in the trivium curriculum, based on local learning practice. For the authors, the concept of literacy-which, according to ^{D’Ambrosio (2008)}, is the ability to process information, such as the use of written and spoken language, signs and gestures, codes and numbers-points to a new meaning of reading. Literacy requires, for example, reading a film, social media content, or even reading political actions, governance proposals and the social relations surrounding the reader’s context. Nowadays, reading also includes skills of interpretation of graphs, charts and other forms of informing the individual, thus including comprehension and expression with language condensed into codes.

According to the same authors, materacy, which is defined and understood by ^{D’Ambrosio (2008)} as the ability to infer, propose hypotheses and draw conclusions from data, is a first step towards an intellectual stance, invisible in our school systems… Materacy is closer to the way mathematics was present in both classical Greek and indigenous cultures. The focus was not simply on counting and measuring, but on divination and philosophy. Materacy entails a deeper reflection on man/woman and society and should not be restricted to the elite, as it was in the past.

Following ^{Mesquita et al. (2011)}, technoracy is the critical familiarity with techniques and technologies. Technology, here, is not linked to computer technology development, but to each instrument that can collaborate in the materialisation of actions on human relations. Today, with technology focusing more on the digital than on the analogue, the operational aspects are, in most cases, inaccessible to a large part of the still excluded population.

Communitarian (mathematics) education combined with Critical Ethnography (^{Gérin-Lajoie, 2009}; ^{Thomas, 1993}) establishes a bottom-up approach that foresees the creation of spaces for interaction between people with diverse experiences and life stories to collectively discuss their problems and define their own way of acting in society. The integration of approaches and points of view provides solutions that improve society as a whole and can be re-proposed, with appropriate adaptations, in other contexts.

This critical and participatory approach is a possible solution to some of the tensions in the Ethnomathematics Programme (^{Pais, 2010}; ^{Parra-Sánchez, 2017}) that problematise the relationships established between researcher and researchee, and between ethnomathematical currencies, reproducing colonial power schemes. Other researchers have proposed different ways of confronting these tensions sharing the same direction, i.e., the search for more symmetry in this relationship: the investigation of the practice itself (^{Mesquita et al., 2011}), the mutual interrogation (^{Adam et al., 2010}), the dichotomy between recognition and the search for (ethno) mathematics (^{Albanese et al., 2017}), the notion of Propio as a decolonial tool (^{Parra-Sánchez, 2018}) and the dialogical perspective as a balance between the emic and the etic (^{Rosa & Orey, 2018}).

Research approach

Among the various research interest focuses, we propose here to analyse the practice of profit sharing by fishermen and its relationship with mathematics and social geometry from a critical and participatory methodology based on community education.

In the following, we will briefly review the main concepts that we will need later on for our analysis. On the one hand, we will describe the mathematical notions involved in this practice from an educational perspective. On the other hand, we will clarify, based on previous definitions, what we mean by social geometry.

The sharing of profit between different people is a situation of sharing which-among many others (measurement, quotient, division) (^{Llinares & Sánchez, 1988}; ^{Mancera, 1992}; ^{Pinxten et al., 1983})- gives meaning to the mathematical concepts of fraction and ratio. There are different types of distribution: the partitioning of the whole or part/whole situation occurs when a certain number of congruent parts are made from a whole or total -which we will henceforth indicate as equal in order to approach the common language used by fishermen-; from these, some parts are taken, that constitute a portion of the total. Uniform sharing, also called equitable sharing, establishes that a whole/total is shared out so that each partition or portion is equivalent in number (if the whole is discrete) or quantity (if it is a continuous whole) to another partition or portion. Proportional sharing occurs when a whole is shared out in such a way that each partition or portion maintains a certain relationship with the others, which can be additive or multiplicative (^{Godino, 2004}).

Now, the mathematical representation of each portion of the división is usually made with a fraction a/b, where the denominator, b, indicates the number of parts into which the whole is divided and the numerator, a, indicates the number of parts that make up a portion or partition. However, to indicate the multiplicative relationship that exists between two portions when the division is proportional, we use the ratio, which we indicate here with the notation c:d, which means that for every c parts that make up a portion, the other is made up of d parts (^{Godino, 2004}).

The term social geometry was coined at the end of the 19th century by the German sociologist George Simmel, who studied social relations, processes and boundaries as fundamental categories of sociological thought. This is the origin of a movement among sociologists who approached the study of social space by investigating problems related to social movement and social dimensions using a geometric-mathematical language (^{Ferreira, 1955}).

In more recent times, in the architectural field, the term social geometry has been used to indicate the geometry of the social interaction of people, which conditions the shape of the spaces to be built or constructed (^{Fernández-Álvarez, 2006}).

^{Le Bras (2000)} conceives social geometry as the study of the mathematical models that govern the distribution in space of peoples and individuals. This process is guided by two opposing forces -attractive and repulsive-which have caused humankind (as communities or as individuals) to follow tendencies to disperse and tendencies to associate throughout the ages, reaching alternating stages of equilibrium and crisis.

It is the latter that has inspired our own conception. In the context of the present ethnomathematical study, we are interested in analysing social geometry, in the sense of how mathematical practices reflect social relations and their materialisation in the social space within the fishing community and, vice versa, how social relations and their positioning in the social space influence the mathematical practices of the community. Furthermore, we will see how these relationships are transformed by technological innovation that revolutionises the practice of fishing, which also strongly conditions mathematical practices.

Methodology: critical ethnography

The methodology used places special emphasis on the participation of the community in the processes of data collection and analysis, which are carried out through critical ethnography (^{Gérin-Lajoie, 2009}; ^{Thomas, 1993}), due to the flexibility that this provides when opening up to dialogue with the community.

This methodology allows the integration of different cultural constructs of the group under study with the observer’s vision of the culture, to accompany the processes of identity production, in which research becomes an immersion of the researcher in the experience of the chosen group and vice versa. This process implies the participation of key actors of the group under study in the dynamics of the academy, such as data collection, report production, data analysis and dissemination of the results, in this case through the teaching work.

This process involves the integration of communitarian education (^{Mesquita et al., 2014}) as a research methodology. However, in this case it has been more of an inter-communitarian education, where a two-way dialogue has been generated between the local fishing community and the academic community.

All of this makes it possible to problematise the educational and social challenges with the community, blurring the roles of researcher/researchee, and allowing for a critical analysis of the academic community’s approach to education and its limitations in collaborating with the current issues of the dichotomy between local and global needs (^{Mesquita, 2017}).

Learning spaces

Spaces of encounter (^{Mesquita, 2016}; ^{Mesquita et al., 2011}), the fulcrum of communitarian education, are key elements for the dynamics of dialogue. In this research, we distinguished two moments during which the encounters took place in different spaces. Firstly, one of the researchers was immersed in the spaces of the fishing community of the Costa de Caparica - Portugal, in the winter of 2017-2018. This was made posible by the creation of the Ocean Literacy Observatory¹² (olo) research laboratory, which brings together members of the academy, technicians, and local communities. Secondly, the participation of fishermen in academic activities has been organised, so that their discourse becomes the starting point for the reflection of a class on Ethnomathematics in a master’s degree at the University of Granada for two consecutive years at the beginning of 2019 and 2020.

Source: Photo taken during the immersion in the fishing community, September 2017.

Figure 1 In the learning space constituted by the alveolos of the fisherman Mr. Mário Raimundo. 

It is worth noting that this community of fishermen had already worked closely with the olo coordinator in previous projects: d.a.r. à Costa - Tr@nsFormArte and Urban Boundaries¹³ - in building a mutual interest in the major issues of development and sustainability of local artisanal fisheries, through meetings in countless spaces of encounter over the last 18 years. Here, a bond of belonging to local causes, both fishing and academic, was created, thus building up a unified body of emancipatory, libertarian, and counter-colonialist actions. All this facilitated the entry and active participation of the fishermen in the two moments of this research, fostering their interest and desire to participate into this research and its context, to be with a new researcher, overcoming, through a social and slow pedagogy, also the barriers of linguistic diversity; to be with master’s students in another country, share knowledge and get to know other visions on artisanal fishing in different parts of the world; after all, the master’s students came not only from Spain, but also from Venezuela, Colombia, and Brazil.

During the first moment, we went daily to the alveolos, a building near the Costa de Caparica (Portugal) beach where each fisherman is assigned a small enclosed space to store his tools and work on the nets (Figure 1), and to the beach (Figure 2), to carry out non-participant observation, participant observation (the researcher came to help the fishermen in their work of repairing nets), and individual in-depth unstructured interviews with several fishermen (^{Geertz, 1973}) about working with the nets and about fishing. All this information has been compiled in the field notes of one of the researchers.

Source: Photo taken during the immersion in the fishing community, November, 2017.

Figure 2 In the learning space constituted by the beach, observing the repair of nets by Mr. Mário Pedro. 

Finally, the olo Room, still in Costa de Caparica, is a space conceived and created by the olo study and research group and financed by the mare Research Centre, and is located in a building near the alveoli where the fishermen keep their tools and repair their nets, and in the so-called fishermen’s neighbourhood, where they live. It is precisely because it is so close to the area where the fishermen live and work that the encounter dynamics and opportunities for dialogue are greatly facilitated.

This room has been the setting for the collective elaboration with the fishermen of an ethnographic report (^{Angrosino, 2012}) that arose from the sharing with three fishermen of the field notes resulting from the encounters described above. These three fishermen were chosen by the fishing community itself to represent it, as they are in charge of the two active associations of traditional fishing arts that exist nowadays on the Costa da Caparica.

Such analysis arises from the proposal to take the systematisation of academic work to the collective practice within the community, since in the critical ethnographic process all those involved discuss the field notes in their meetings, point by point.

The report is the result of a collective and shared reflexive analysis during which the main role of the researchers has been to organise the information, while the fishermen commented among themselves, clarifying and thinking together about the implications of the visibility of the issues raised for understanding the establishment of social -and power- relations within their own community, in other words, analysing the data themselves (Figure 3). Importantly, this process has enabled the triangulation of information from different sources and lends communal validity (^{Moral, 2006, p. 156} ) not only to the information collected, but also to the analysis and interpretation of the data itself, which is a characteristic of critical participatory ethnography.

Source: Photo taken during the immersion in the fishing community, January 2018.

Figure 3 In the learning space constituted by the olo room, working together on the ethnographic report with Mr. Mário Raimundo and Mr. Lídio 

The dynamics of encounter in a critical ethnography is based on the cyclical socialisation between different communities, in this case the academic and the fishing communities, which have a common interest. In this study, the academic community aims to uncover the cloak of invisibility that hangs over the local-traditional knowledge of the fishing community, which has requested collaboration in order for its voice to be heard in local political decisions. Here, the purpose was to create a more participatory space in the local society, to guarantee some of their basic cultural traditions for their survival, such as the making of the fishing net and the knowledge of fish production -preparing to fish, fishing and working economically on their fish.

In this sense, the dynamics proposed in the encounters was based on dialogue and knowledge transfer from the communities to which they belong, in order to co-construct a common local knowledge -the starting point for the building up of a bottom-up collaborative work.

In the second moment, the academic space consisted of a university classroom of a master’s class at a Spanish university where the fishermen organised a workshop on net construction (Figure 4), bringing their work tools to class and teaching the students how to weave a net cloth. What interests us most for the subsequent discussion, they gave a presentation on the history of the fishing community of the Costa de Caparica and the distribution of profit among the fishermen of a company, taking into account its variations over the last decades. The meeting with the groups of master’s students was a new opportunity for the fishermen themselves to re-elaborate their knowledge and the implications of mathematical practices for the social fabric of their cultural group. The activity was videotaped for the participation of some virtual students in non-synchronous mode.

Source: Screenshot taken from the recording, January 2020.

Figure 4 In the learning space constituted by the classroom of the master’s class taught by the fishermen, showing the nets to the students. 

In this second moment, the academic community received the fishing community -the opposite movement to the first moment. This Exchange promotes intercultural dialogic interaction, which makes it possible to make visible and value the knowledge of the different communities that participate in the encounters. In addition, the fishing community manages to elaborate a translation of its own knowledge systematised and expressed in a language accessible to the academic community.

These intercultural intellectual encounters allowed to re-signify the processes of community mathematics education that objectify justice. This praxis exemplifies a position of the Ethnomathematics Programme in which the promotion of the encounter, recognition, valorisation, and dialogic integration of any knowledge is valued, building a new common, efficient, effective, and relevant knowledge for the solution of local symptoms of global problems. Thus, the Ethnomathematics Programme aims to understand and work on the relational positioning of cultural knowledge, how it is constructed and how it evolves, which shows the degree of justice between the parties involved in the learning processes.

From these spaces of encounter, we selected fragments that allude to the way in which the fishermen communicate their knowledge about the reading of their reality (literacy), analyse the relationships that are established in the social space in real life and their decisions and actions (matheracy), as well as the specific techniques -tools and strategies, here mainly of calculation- created to express and materialise their knowledge in their work (technoracy). Below, we present a possible analysis through these concepts, remembering that they are transversal and are identified here in a specific context, always bearing in mind their close relationship.

Selected data of interest on profit sharing

Although Arte-Xávega¹¹⁴ is legally considered a trawl fishing art, in practice it is carried out with an encircling net to pull ashore, sometimes referred to as a cod-end seine. This means that the Arte-Xávega net is thrown into the sea from a boat, leaving one end of the network cable to land and returning the end of the other cape to land, which then allows the net to be pulled ashore -formerly by hand, nowadays by using tractors- with the fish. The fishermen gather in companies, that is, groups of people who fish together with the same boat and net. Each fisher performs certain tasks and his/her financial reward for his/her work depends on the type of tasks he/she performs.

Below, we have selected, from the data collected throughout the research -i.e., from the ethnographic report co-constructed with the fishermen and from the transcription of the master’s session organised by the fishermen, the most exemplary fragments describing the distribution of profit in some fishing companies at different historical moments, using the fishermen’s own way of communicating it (literacy-their own reading of reality expressed through a specific language).

Extract 1

In the 50’s and 60’s, the sharing of the profit among the fishermen of a company was as follows: from what was earned from the sale of the fish, first the expenses were taken out, which were then the oil for the lanterns, the taxes and the faena (this is the money charged for the task of setting the net on the boat before going fishing, which the calador did with two other men and earned extra money for it), and it is divided into four parts. One part, i.e., a quarter of the profit, goes to the company owner. The remaining three parts are divided as follows:

The unsold fish was divided into equal shares for everyone, a little pile for each one, and in the few cases where there was a difference between species, a lot was drawn.

(From the report made jointly with the fishermen, February 2018).

It should be noted that the company’s scribe is in charge of selling the fish and then making the accounts for the distribution of the profit.

Extract 2

Mr. Lídio: The scribe knew how many fishermen went on the boat that day, how many stayed on land, and how many children went to Arte- Xávega and he was in charge of doing the accounts. (Transcription and own translation of Mr. Lídio’s explanation in the master’s class, January, 2020)

An example of profit sharing in a company in the 70’s-80’s, when a single tractor was used to move the ship, is now presented.

Extract 3

At that time, the total profit, excluding expenses, i.e., petrol for the tractor, the engine and oil for the lanterns, was divided as follows:

Today’s profit sharing reflects the amount of expenses they have to incur for the operation and maintenance of tractors and the ship’s engine.

Extract 4

From the profit, after deducting the tax that is kept by the lota¹⁵, the fuel for the ship’s engine and tractors (which is sometimes counted in tractor working hours) is deducted. After that, the distribution depends on the company.

Each tractor takes more or less 5 parts (the tractors are usually 3 or 4: two tractors to pull the net, one for the boat and one to take the fish to the lota, being in some companies these last two tasks carried out by the same tractor).

The trafo (maintenance of engine, boat and net) takes another 5 parts, or sometimes 4 parts for the boat and net and 4 parts for the engine.

The men who go on the boat get more (2 or 3 or 4 parts, skippers usually get one more part) than those who stay on land (1 or 2 parts, those who drive the tractors may get a little more than the others on land), depending also on the constancy with which they go to the beach [i.e. participate in the fishing tasks].

Nowadays the work is not as physically hard as it used to be; the hardest tasks are to pick out the fish when the net comes ashore, and to set up and repair the nets.

(From the report made jointly with the fishermen, February 2018).

Results and participatory discussion...

During the research process the fishermen became researchers and analysed their own practices (here, the profit sharing) from a mathematical point of view and, at the same time, addressing the connections with the social geometry of their community. Below, we share the discussion over these questions, combining our observations with fragments of the discourses, which emerged from the different moments of collective analysis with the fishermen.

Conclusions

In this research we wanted to share the mathematical practices of the fishing community of Arte-Xávega in Costa da Caparica (Portugal). We highlight the intellectual tools of the community of artisanal fishermen, as we approach, in the extracts presented, both the way in which fishermen read and communicate reality (literacy) and analyse (matheracy) the relationships between their social position and its effects on profit, and the way in which they use specific calculation techniques as tools and strategies (technoracy) to adapt and apply their knowledge to their work.

In this context, the search for and construction of spaces of encounter between the fishing community and the academic community in which to carry out a reconstruction of the local community education process, has allowed us to co-construct a dialogue that has made possible collective reflection on the mathematics involved in the practice of profit sharing, as well as on social geometry. The later, we recall once again, consists of the analysis of the relationships established in the social spaces that are expressed in some mathematical practice which, in turn, reflects the social fabric of a community.

We also highlight how this experience of critical ethnography combined with community education is part of a set of proposals that problematise the relations between researcher and researchee, aiming for greater symmetry in the power relations in the research process itself, each of them responding to the specific demands of the context in which it is adopted.

Collectively with the fishermen we managed to analyse how social transformations, among which we highlight access to education -particularly their mathematical empowerment- and technological innovations have altered the relationships between fishermen, as can be seen in a modification of the mathematical practice of profit sharing. Moreover, it has become clear, in a very concrete context, how mathematics plays a leading role in the negation or search for a fairer society. On the one hand, mathematical knowledge is seen as a tool of power in the hands of the few who handle it (the owners and the scribes); on the other hand, greater awareness of mathematical techniques is taken as an instrument of empowerment for fishermen who demand a more equitable redistribution of profit and greater transparency when it comes to making expenses visible.

Finally, we emphasise how, through the experience and collaboration of fishermen, also in the educational field, we have left a seed of change that is projected into the future in the awareness of teachers (master’s students), key actors in the educational field (^{Albanese et al., 2017}). In fact, sharing this experience within the framework of a teacher training course has provided the occasion, not only for a deeper reflection of the fishermen, but also to generate an interesting debate on the role of formal and informal education in relation to cultural practices, particularly in mathematics. The possibility of addressing, in school and academic learning spaces, real mathematical situations faced by the community is a concrete proposal for incorporating community education, generally informal, into formal education.

Although we do not go into this aspect in depth here, we would still like to leave -by way of conclusion and as a call to continue valuing these processes- the comment of a student in which the value of mathematics (education) for social change can be glimpsed.

Student: [The fishermen] gained political and social awareness, taking an active role in a new construction of cultural practices that promoted equality. New mathematical operations, new ways of measuring, comparing and sharing would bring about this change, in which mathematics appears as a tool to improve the quality of life and the dignity of human relations. (From the reflections of a student in the Ethnomathematics course, February, 2020)