Due to the epistemological changes involved in ethnomathematics, there have been seminars, courses, and workshops out for teacher education with the common objective of raising awareness, among teachers, on the nature of mathematics as a social and cultural product. Some examples are:

Oliveras (1996) used microprojects on Andalusian handicrafts in early teacher training courses to raise awareness about their conceptions on the nature of mathematical knowledge, producing changes in their mathematics conceptions and their learning environments.

In addition, Gerdes (1998) reflected on the need to raise awareness about the social and cultural bases of mathematics in teacher training. In the Mozambican context, it pointed to the importance of future teachers learning to recognize the mathematical roots in their own and in other African cultures.

Whereas, Presmeg (1998) focused on the above points in his courses for future US teachers, leading them to affirm, value, and take advantage of the points of view concerning cultural diversity and the relation of mathematics to social reality in their professional lives.

More recently, Gavarrete (2012), in a course with indigenous Costa Rican teachers, promoted relativistic conceptions about mathematics that integrate traditional academic knowledge with indigenous cultural knowledge.

Naresh (2015) mentioned a transforming learning experience during a Phd course where she embrace a culturally relevant conception of what mathematics is.

Finally, Nolan and Graham (2018) challenged the teachers’ perception of what mathematics is during a course on cultural-responsive pedagogy.

But even though all of this authors suggest a direct or indirect influence on the participants’ epistemological conceptions, they do not present a structured analysis of this influence, that is why we present the following proposal of approaches and related categories as innovative.

Following the precedent set by these authors, a workshop was developed for pre-service teachers at the University of Buenos Aires as part of a broader Phd project of the first author. The workshop had the purpose of fostering participants’ reflections mainly about the social and cultural aspects on the nature of mathematical knowledge.

The objective of this research is to describe and analyze their conceptions about the nature of mathematics after the proposed workshop.

For this, we identified some approaches that are characteristic of the ethnomathematical perspective which were used later to analyze the participants’ conceptions. These approaches took shape during a complex process, partly inductive from the data analysis and partly deductive, deepening the study of the theoretical contributions made by some prominent researchers of ethnomathematics. Finally, we grouped the participants into profiles according to the incorporation of the above-mentioned approaches.

D’Ambrosio's definition of ethnomathematics (2008) as the styles, arts, and techniques (Ticas) of explaining, understanding, managing, and connecting with (Matema) the natural and social environment (Ethno), points to mathematical knowledge as an instrument for understanding and managing reality. Such knowledge develops as a consequence of the need to make decisions and arises from representations and reality models that relate to the perception of time and space. Then, mathematics is a way of knowledge used to act in the context that surrounds us for controlling it and eventually modifying it.

In this practical approach, elements of the conceptual and cognitive dimensions of D'Ambrosio (2008) converge, referring to mathematics respectively as an instrument to represent and model reality, and as organizational structures of experience.

Regarding the educational context, this practical approach implies developing the relation of mathematics with reality, valuing its origin in concrete practices of daily life, as well as its use for environment manipulation. Methodologically, direct experience and research are promoted (D’AMBROSIO, 2008), as well as experiencing mathematical attitudes, contact, and participation in mathematical inquiry (OLIVERAS, 1996; PRESMEG, 1998) to develop skills critical to the environment. These considerations are widely recognized when speaking of sciences in general, but hardly accepted when it comes to mathematics.

Barton (2012) proposes to interpret mathematics as a system of meanings through which a group of people gives meaning to quantity, relationships, and space. Communities that share a common vision of reality and agree on common codes build these systems. This perspective assesses the importance of language that allows communication within the group and is based on the theoretical reflections of Barton (2012) and Knijnik (2012). These authors coincide in referencing Wittgenstein and his language games as a key philosopher to look at mathematics as a system, as a socially-shared language which exploration has to focus on its operation and use.

In this social approach, elements of the political dimension of D'Ambrosio (2008) also come into play given that factors of power affect communication and the achievement of consensus.

The implications for the educational context are linked to social constructivism, that is, to a methodology that values the construction of knowledge with work in small groups, reaching agreement on a shared interpretation of experiences (OLIVERAS, 1996; PRESMEG, 1998).

Ethnomathematics establishes a deep relationship between mathematics and culture. If the location of mathematics is the collective mind of the human species, the origin of mathematics is situated in culture, the body of behavior and thought of mankind (WHITE, 1988 quoted in GAVARRETE, 2012).

A key element of this position is the assumption that there are several mathematics and their differences depend precisely on the different cultures in which they are constituted. From the philosophical perspective, it is accepted as a form of relativism that allows the coexistence of different mathematics. D'Ambrosio (2008) argues mainly for a historical vision, meaning that it admits an evolution of mathematics due to the development in the time of cultures. However, Barton (2012) insists on the need of philosophically justifying that there are different mathematics simultaneously, which are not subordinate to each other, meaning that there is no development hierarchy between them.

Therefore, in this cultural approach, D'Ambrosio's (2008) historical dimension - with the clarifications of the previous paragraph – and D'Ambrosio's (2008) epistemological dimension come together.

For the educational context, these reflections lead to proposals of working with cultural groups that maintain different visions of mathematics, addressing the variety of their contents, thoughts, and contributions (SHIRLEY, 2001). In this context, the aspects of our own cultural background or of a culture close to the student are investigated (BISHOP, 1999; GAVARRETE, 2012; OLIVERAS, 1996). As far as fieldwork is concerned, the importance of learning about sociocultural factors is highlighted - context, relationships, needs (GERDES, 1998).

The research geographical context is the city of Buenos Aires, which was selected for two reasons: 1) the workshop focuses on Argentinean braids crafts, and 2) the Argentinean legislative guidelines in the educational field are based on principles that are shared with ethnomathematics (ALBANESE; SANTILLÁN; OLIVERAS, 2014).

Argentina is a country deeply linked to its country tradition. Likewise, the country has shown great interest in valuing its cultural heritage and folklore (DE GUARDIA, 2013) through educational actions as well. The handicraft chosen has great cultural significance for its diffuse use in the northern and central regions where the braids are used to work with animals and in characteristic decorations (ALBANESE; OLIVERAS; PERALES, 2014).

The workshop that we carried out followed the line proposed by Oliveras (1996), which deals with the study of a cultural feature – a trait or element of a determined culture or microculture that contains some mathematical potential, to later design mathematical tasks – and the idea of investigating the mathematics enclosed in the ethno-modeling of reality done by cultural groups (ROSA; OREY, 2012).

The cultural feature that we chose was a handicraft from the region of Salta, Argentina; the mathematics enclosed in these ethnomodels has been investigated in previous research (ALBANESE; OLIVERAS; PERALES, 2014). The ethnographic research in the artisanal environment has provided a mathematical ethnomodel of braiding that presents great possibilities for its educational use (OLIVERAS; ALBANESE, 2012).

The workshop is designed as a session of a pre-service teachers’ course, and was validated with a previous pilot experience. Specifically, this took place in June 2013, in two four-hour sessions each. There was a group of 14 participants from the optional course “Workshop of Mathematical Modeling and Production” in the Mathematics Teachers degree at the University of Buenos Aires. Here we present a brief description of the workshop, but a more detailed one can be found in Albanese and Perales (2017).

In the workshop, we used an investigative methodology, where the participants constructed and agreed on a creative modeling for the making of a simple four-wire braid that involves the construction of the mathematical concept in an oriented graph. In a later moment, participants worked with the model created and used by the artisans (the ethnomodel). The course teachers acted spontaneously as guides and mediators in the debates.

The workshop was organized in three stages (see Frame 2 below): (A) initial debate with presentation of ethnomathematics through the sharing of observations on selected fragments of several authors previously read by the participants (BARTON, 1996; BISHOP, 1999; D'AMBROSIO, 2008; GERDES, 1998); (B) representation and elaboration of braids; (C) final debate about the work done and answers to an open questionnaire on the work's epistemological implications (such questions are detailed below in paragraph 4.3). The central stage (B), focused on the craft activity, was composed of four phases.

Each phase was characterized by a different interaction between participants:

Phase 1: Participants were provided with the necessary tools for braiding. Then, they were asked to individually describe the process; first graphically-ironically and then with words.

Phase 2: A creative representation was made aimed at the construction of a model. That is, from the individual description previously made, the participants reach a consensus within the group on an iconic and then a symbolic representation that synthesizes and describes the process of braiding.

Phase 3: In the sharing of the agreed representations, the participants reflect on the decisions made by each group (advantages and limitations). Finally, the artisanal four-wire braiding modeling that involves the mathematical concept of an oriented graph was presented (first graph in Figure 1).

Phase 4: the participants were offered a handmade modeling of three eight-wire braids (last three graphs in Figure 1). Pattern recognition in the graphs that represented them allowed the participants to later create graphs of 16-wire braids.

The approaches mentioned above were reflected in the phases. Since conceptions are formed through experience (COONEY, 2001), participants experienced the practical, social, and cultural approaches. Phases 1 and 4 emphasized the practical approach of mathematics as a tool for understanding reality and organizing experiences. In Phase 2, emphasis was placed on the social approach of consensus and sharing of mathematical productions within the community. Whereas in Phase 3, the cultural approach was emphasized in the different ways of thinking mathematically and the artisan culture being recognized as a culture creating knowledge.

The research is an exploratory study presented in a qualitative framework due to the affinity of this methodology with the ethnomathematical purposes, as well as for the interest of investigating people's conceptions in the educational context (WOODS, 1987).

The data were collected through audiovisual recordings, transcripts, field notes of the researcher who runs the activity, evaluation sheets of the two course teachers, worksheets of all participants, and the open questionnaire. This variety of data, characteristic of a qualitative research, is fundamental to ensure triangulation and, therefore, research validity.

We performed a data qualitative content analysis (CABRERA, 2009) with the support of the program MAXQDA7, and we show below the results of the debates transcripts and the answers to the final questionnaire.

Descriptive codes are inductive. The higher order categories that group codes determined by practical, social, and cultural approaches were obtained through an inductive-deductive cyclical process. Their observation and first outline emerged during a pre-analysis or speculative analysis (WOODS, 1987), and its refinement was contrasted to the theoretical foundations mentioned above which were also organized according to these approaches.

We initially hypothesized a strong presence of the practical category, then an intermediate presence of the social category and finally a minor presence of the cultural category, as a sign of a progression in the reflection process about the sociocultural characterization of mathematics.

Figure 2 shows the results of the initial shared analysis with the observations made after the readings on ethnomathematics. The participants’ codes preserve their anonymity.

Six participants acknowledged as ethnomathematics studies the different mathematics that arise in culture (cultural category), but all referred to the proposed text without expressing their own opinions. Some emphasized the relation of mathematics to real situations (practical category) by mentioning examples of their personal experience. One person named interactionism (social category) as a collective construction of knowledge (concept absent in the proposed readings). We emphasize that only two participants made observations that explicitly related several approaches and, in both cases, happened in relation to an educational problem. One person observed that different guilds (doctors and psychologists) use different mathematics according to the contextual needs of their profession. Another showed that the social construction of mathematics is linked to situations that arise in concrete contexts and suggested to take advantage of them to stimulate students’ interest in what mathematics are useful for.

We emphasize that only a few participants, who had already come up with concerns from previous experiences, made deep observations and expressed doubts, criticisms, or personal comments to the proposed documents.

Figure 3 shows that the final debate was much richer.

Here, codes related to the emergence of doubts show the clash between participants’ previous understanding and the approaches introduced with the workshop (COONEY, 2001). One teacher strongly advocated focusing on school mathematics. Several participants (8/14), in contrast to this comment, considered that mathematics are somewhat more about the school-academic conception and goes beyond concepts. Someone explicitly asked the question: what is math? This issue brought forward the problem, highlighted by some (5/14), of who decides whether in a given practice there is mathematics. Furthermore, they highlighted the question of what authority determines it, whether the community itself or an external observer. These same questions appear in the theoretical literature of ethnomathematics and were answered in many ways (ROSA; OREY, 2007; BARTON, 2008; ALBANESE; PERALES, 2014).

Finally, two observations emphasized mathematics as an instrument to systematize experience and solve real problems (practical category). Whereas two others insisted that ideas arise in interaction and agreement is needed (social category). There were no comments on the cultural category, despite being the most present in the initial debate. The cultural approach seems to have been perceived but not internalized by the participants.

In Frame 3, we summarize the inductive codes obtained in the analysis of the responses to the final questionnaire, distributed by item and category.

As an example of the coding process followed, we show the process for the second item.

These two participants demonstrated the role of mathematics as the generator of models to understand and control situations of everyday life (e.g., inventing new braid patterns). We assigned them the code “Modeling of daily patterns” and placed them into the practical category.

These two participants went a little further, both recognizing that mathematics is a tool to model reality but also insisting on detecting internal relationships and patterns. We encoded these observations as “patterns” and we also considered them in the practical category.

The next one solved its dilemma about the nature of mathematics in the historical sense, looking at the transformation of mathematics over time. We associated this observation with the idea that mathematics is a historical product and we assigned it to the social category.

These last two observations showed the existence of different ways of doing mathematics with respect to the academic and different points of view of the same problem (elaboration of diverse notations), respectively. In addition, they showed the particularity in the interpretation of the same representation depending on the observer. The other mathematics answers were related to the existence of different mathematics and we assigned them to the cultural approach.

We now synthesize in Figure 4 the overall results of the group of participants using inductive codes developed according to each item of the questionnaire. It is worth mentioning that the category indicated with the dotted frame cells is not considered in this analysis, it is actually related to the static view of the conceptions about the nature of mathematics mentioned in the theoretical framework (section 2).

As evidence of the social and cultural approaches, we grouped the participants into profiles according to their conceptions considering that more complex and articulated conceptions are desirable in pre-service teachers.

The proposed profiles are described in Frame 4.

We then assigned and described the participants’ location in these profiles:

Profile 1: Two participants presented this profile (TA and EL). In their answers, they manifested only codes of the practical category. We highlight the case of EL who seems to reject the social and cultural approaches, in fact absent in his questionnaire, despite the initial concern about the educational implications related to the social approach. His intervention in the debate was in favor of recognition just of academic mathematics.

Profile 2: Six participants present in the final classification categories related to the doubts and hardly outlined the social or cultural approaches in the questionnaire answers. It is important to remember that doubt is a fundamental element to generate reflection and, eventually, changes (COONEY, 2001).

Profile 3: Three participants (MAR, JE, KA) in the debate recognized that mathematics go beyond academics and they present observations related to social and/or cultural categories in two responses to the questionnaire and -except one- presented both categories. Here the manifested doubts reflect a deeper reflection than the previous profile.

Profile 4: Three participants made observations on social and cultural approaches in three or more responses. During the debate, they generated the dilemma about who decides what is, or is not, mathematics and this showed a good internalization of the relativist positions. In the answers to the questionnaire, a high articulation of the three approaches was detected; in addition, the categories were well interrelated. Two of these answers came from people (VE, DI) whose experience and reflection were very deep according to their trajectory. However, it is important to highlight the case of GA that in the final debate and in the questionnaire answers insisted on the different points of view determined by culture and environment.