Dossier: History of Mathematical Education COMPLEMENTARY EDUCATION SPACES OF VENEZUELAN MATHEMATICS EDUCATORS. CASE: VENEZUELAN SCHOOL FOR THE TEACHING OF MATHEMATICS-EVEM

Qualitative study on mathematical educators complementary education spaces, considering the Venezuelan School for Mathematics Teaching. Referents: Social history (Bernal); Epistemology of Science (Toulmin); Scientific Field (Bourdieu); Systemic vision of Mathematics Education in Venezuela (Beyer). Finding: EFC-EMVs are dialogic, interdisciplinary, generate knowledge about teaching-learning of mathematics, for the Venezuelan educational reality.


INTRODUCTION
One of the most important tasks that the Mathematics Education (ME), seen as a field for the production of knowledge related to the mathematics teaching and learning processes, should assume is the professional development of the mathematics educators. Therefore, it should impulse formative processes that empower such development. The education of the mathematics educators plays a fundamental role in the students' math learning, and consequently in the probability that the citizens reach a high-quality education in mathematics (ADLER, 2008).
As it is stated by León, Beyer, Serres and Iglesias (2012) in their report presented in Costa Rica, the education of the mathematics educators in Venezuela exists in different types and levels: an initial education (undergraduate studies) and a continuous education (graduate studies). In addition to these two types of education, we have considered in this research a third type that we name Complementary Education. This kind of education corresponds to the studies of not formal matter and short time that are procured by the students who expect to become mathematics teachers (the ones in initial education), the mathematics teachers who carry out specialization, Master and Doctoral studies (who are in an advanced education), and the educators who teach mathematics in the different levels of the educational system. All of them are motivated by their necessities and opportunities to update and improve academically. This third type of education is found in academic events and congresses devoted to the teaching of mathematics in local, regional, national and international levels.
These aspects invite us to the following reflection: to have a high quality teacher education, all the education types mentioned above must be very interrelated, that is to say, that all of them must converge in common goals: Educate a professionally competent educator; to foster a coherent learning and to help the professional development of the educators who teach mathematics in the country. In this process the Complementary Education Spaces portray a great importance because they represent spaces of synergistic convergence (BELISARIO, 2015) for both teachers of initial education and the teachers in continuous education in the search of improvements for their professional performance through the different activities that take place in those spaces.
Then, our interest is to offer an answer to the main question of this study: what are the characteristics of those Complementary Education Spaces?
The Evem is one of the education spaces that the Venezuelan mathematical educators count on and which constitutes a scenario for various of the aspects of the SBa (BELISARIO, 2015). This is an annual event that has taken place since 1997 until current days which is carried out in the Facultad de Ciencias Básicas of the Universidad de Los Andes in Merida State (Venezuela).
Future and active mathematics educators from almost all of the states of the country participate in the courses, workshops and conferences of this event.
Among those who participate repeatedly in the Evem there are some of the reference actors (TOULMIN, 1977) of the Venezuelan Mathematics Education.
The courses are taught by guest professors who should produce a book that serves of support material for the development of the corresponding course.
The Evem is considered a useful case to characterize the

Complementary Education Spaces of the Venezuelan Mathematics Educators
(EFC-EMV in Spanish) due to its stability (an uninterrupted implementation since 1997), the kind of participants (future and active educators), the bibliographic production associated (books), and the repeated participation of reference actors. In order to confirm this conjecture, a study was carried out to see the details in depth, beginning, development and actual situation of the  (FLECK, 1986), the notion of Scientific Field, (BOURDIEU, 1994), the conditioning factors of the development of disciplinary field (MALIZIA, 2013), the concept of Epistemic Synergy Of The Venezuelan Mathematics Education (SEEMV) (BELISARIO, 2015), and the Conceptual Evolutionism theory (TOULMIN, 1977).
The social character of the sciences proposed by Bernal (1968), in which he makes reference to the role sciences can exert in a society willing to serve itself of its capacity to produce social progresses; as well as Barros's position of development as disciplinary field of history (2011). This author recognizes ten important dimensions to characterize a scientific discipline. The system of ten dimensions presented by this author shows a set of criteria applied to History, as field of knowledge, to certify it as scientific discipline which is adaptable to other alternative disciplines. Due to these disciplines are social facts represented by a history, these perspectives helped us to characterize the historic trajectory of the Evem as a Space of Complementary Education for the Venezuelan Mathematics educators.
It is important to point out that the approaches of historic and social nature used to explain the creation of disciplinary fields are different. Among them the vision of Fleck (1986)  In other words, according with Fleck, science is a social process done by communities of researchers who constitute thought collectives that develop a specific thought style which links concepts, theories and methods developed by specific sectors and individuals of the thought collective.
From Bourdieu (1997), we assume the notion of scientific field that he defines as "[…] the place of a competitive struggle that has the scientific authority monopoly as specific challenge" (p. 12). Bourdieu (op. cit.), attributes to history a big importance as a explanatory factor of the social phenomena. The social implies a double existence, suppressed in what is considered: the field as the objective (the social made a thing) and the habitus as the subjective (the social included in the body), and from the dialectic relationship between both areas the social practices begin. The objective structures cultivate the habitus which is "a set of established and long-lasting dispositions that determine our way of acting, feeling and thinking." (BOURDIEU, 1994, p. 173). This allows the agents to handle dynamics inside the field established by a structure of relationships. The capital is made of everything that can be used in the field to obtain a benefit (BOURDIEU, 1994, p. 173); thus, the capital is a product of the field and it does not exist out of it. The different kinds of capital create different effects in other fields; the fields are defined by the power relationships that the capital exerts, and by the actions of the subjects to maintain and acquire more capital. The central elements of this theory are: the fields, the capital and the habitus.
It is worth noting that the habitus of Bourdieu's field theory can be assimilated with the institutional forums and the groups of reference of Toulmin's Conceptual Evolution Theory (1977) that was adopted here with the purpose of supporting the notion of Complementary Education Spaces which characterization and dynamics are the main objective presented in this article. Toulmin (op. cit.) asserts that the purposes, concepts and theories that result from the solution of disciplinary problems, as well as the methods and the dominant philosophical approaches in the scientific thinking, are changeable characteristics in the disciplines. Meanwhile the concrete problems of the discipline are the starting point of such characteristics. Consequently, this author points out that the proto-specific elements: ideas and conceptual notions are progressively accepted thanks to what he denominates "groups of reference" and "forums of dissemination." The former are made of all those individual or collective actors that carry the ideas, concepts and general expressions of theoretical nature that constitute the ideational core of the disciplines in formative process. Meanwhile, the latter constitute spaces that promote social interactions between the future professionals of the new emergent scientific discipline (MALIZIA, 2013, p. 18).

Regarding the notion of Epistemic Synergy of the Venezuelan
Mathematics Education (SEEMV) developed by Belisario (2015), it can be said that it constitutes an extension of Beyer's systemic vision (2001a; 2001b) about the mathematics education in Venezuela. According to Beyer (op. cit.) the ME is a system constituted by the organizations, the graduate programs, the investigations, publications and the events (local, regional, national and international). According to this author, these elements considered systematically constitute what he calls the Venezuelan system of Mathematics Education, and they are sufficient to draw the teaching of mathematics in the country. Although, Belisario (2015) extended it, developing the Epistemic Synergy of the Venezuelan Mathematics Education that consists of A relationship of bilateral mediation that is established between two or more individuals or entities, especially when they work or do something of interest in the context of this disciplinary field, such as contributing in the construction of its history and providing evidences for the certification of its scientific condition (BELISARIO, 2015, p. 123).
It is important to point out that we consider Mathematics Education as a scientific field that joins social and diversified activities related to the teaching and learning of mathematics through the events of the area. Moreover, these conditions have favored the permanent transformation that the ME has experienced as a scientific discipline and it will continue experiencing depending on the development of the ideas and concepts of mathematics itself and other sciences.

METHOD
The considerations by González and Villegas (2009) were taken into account to construct a suitable strategy that worked as a base for the recollection of the necessary information to answer the research question of this study. These authors consider that each problem of investigation requires and demands a specific approach. They conceive this moment of the investigation "as a strategic disposition of resources (material, financial, conceptual, human, etc.) that serve to collect the necessary information to answer the interrogatives of the investigation" (p. 103). The construction of such strategy requires the adaptation to a research paradigm associated to a methodological perspective that supports the design of the investigation. It implies the definition of techniques and the adaptation or construction of instruments ad hoc. For the study here reported it was adopted a design of case study. The details are presented below.

RESEARCH PARADIGM
The approach of this investigation was qualitative. This makes possible approaching the phenomenon with the capacity to discover, to interpret and to understand it.

METHODOLOGICAL PERSPECTIVE
A historic perspective of descriptive-interpretative nature was assumed.
Historic, since a timeline of the Evem was reconstructed to identify characters, dates and key situations that define its trajectory. Descriptive, since the characteristics of the school, their qualities and attributes, were identified, pointed out and explained. It is interpretative because besides the collection of information, the meaning of the event was added in the context of the disciplinary development of the Venezuelan Mathematics Education.

RESEARCH DESIGN
Regarding the design, the investigation constituted an instrumental, unique, critical accessible and analytical case study. It is instrumental because the Evem was assumed as an illustrative case which study gives enough and suitable elements to characterize theoretically the Complementary Education Spaces of the Venezuelan mathematics educators. Furthermore, it is unique since the Evem is the only case considered for the study. Moreover, it is critical because the study was not limited to a mere description of the characteristics of the Evem, but it was also accompanied of reflections that place it in a major framework with the disciplinary development of the Mathematics Education in Venezuela. Also, it is accessible due to the direct relationship that the researcher maintains with the Evem because, through the time, she has performed diverse roles in it since her first participation as assistant until becoming a guest speaker. Finally, it is analytical because the direct participation of the researcher in the Evem allowed her to access to the information obtained in primary sources. This factor facilitated the conformation of a corpus which content analysis provided high quality information that was used to answer to the interrogative of this investigation.

CORPUS OF STUDY
Two types of sources were used to construct the Corpus of the present study: documental and testimonial. First, some efforts were made to obtain the biggest amount of documents that contained the direct information related with the Evem. Then, the testimonies of the people linked with the school were procured parallel and simultaneously. Therefore, non-structured interviews, face to face meetings and conversations were made with people who are associated with the Evem; this is the way the experiential information about the School was obtained. Information that could have been impossible to obtain through other ways. The following criteria was followed to the selection of the key informants: (1) they should have knowledge of the history of the Evem; (2) they should have participated in any of the editions of the school; (3) they should be available and willing to provide information about the Evem.
Following this criteria, eight (8) key informants were selected, see table 1. The interviews were made following an initial script, which was extended and modified depending on the interview, elaborated according with the selection criteria and the kind of participation of the person interviewed. Another testimonial information was provided by the researcher who, from her own memories, recovered her personal experiences associated with her participation in the Evem. She wrote a detailed, exhaustive and meticulous narration of the different roles and participations that she has had in the Evem.
This allowed her to make explicit the pertinence of the investigation considering her "personal epistemological place" conceived by González (2003) as the position from which knowledge is produced; which is associated with the life story, personal and professional formation, as well as the vital experiences as a human being (GONZÁLEZ, 2003, p. 140).
The narration based on the perspective by Lindón (1999)

CORPUS CONTENT ANALYSIS
After organizing the constitutive information of the Corpus-build from the interviews, the narration of the researcher and the theoretical frameworkthe analysis of the content was carried out in two dimensions: quantitative and qualitative. This analysis was based in a source triangulation process as it is seen in the Figure 1. The quantitative information is related to measurable aspects such as: amount of participants, courses, workshops, conferences, released books in the Evem (Table 2), taking into consideration the period 2000-2016 since there is no available information about the period 1997-1999.

Books 115
Source: research data.
The Evem has been a space for meeting between mathematics educators (professors and students of mathematics and other alike careers) from different regions of the country who look for the update and improvement of their insights and didactics; as well as their personal growth to develop general and specific competences of their educational labor through the academic activities promoted in the School.

ANALYSIS OF THE QUALITATIVE INFORMATION
The qualitative information is associated with the names of people, topics studied in the different editions of the Evem, theoretical and methodological approaches from the books used to support the courses, the content of the conferences and many others. The analysis was done in two phases.

PHASE I
This phase consisted on the primary analysis of the Evem taking into account the categories that Belisario (2015)

PHASE II
The key informant interviews were analyzed during the second phase

COMPLEMENTARY EDUCATION SPACES OF THE VENEZUELAN MATHEMATICS EDUCATORS (EFC-EMV): A CONCEPTUAL CHARACTERIZATION PROPOSAL
From the analysis of the collected information about the Evem it was contextualized the EFC-EMV in the following way:

EFC-EMV CHARACTERISTICS
1. It offers diverse modalities, places, update moments and practical improvement to the participants. 7. It counts on the frequent participation of reference actors of the mathematics education of local, regional, national and even international contexts.
8. It promotes the production of didactic material that supports the academic activities carried out such as books, brochures, bulletins, and some others.
9. It deals with the academic necessities of the participants promptly and flexibly during the event.

BASIC PRINCIPLES OF THE EFC-EMV
The interests about some of the topics of the discipline, and who interact to acquire, spread and share their insights. 5. Self-regulating its own development: it constitutes a formative assessment process that generates information that allows the constant revision of a high quality education and social pertinence.
This is how the EFC-EMV displays a knowledge production dynamics that permits the self-regulation of the educator. It allows the educator to revise and prepare himself for the innovations that occur in his environment. It becomes a professional self-improvement process.
6. Counting on social support: every EFC-EMV should count on a group of human and material resources provided by the institutes, organization and persons related with the mathematics education, all of this with the intention of generating a harmonic climate between the people who seek for the improvement of their pedagogical tasks.
7. Promoting the social interaction: in an EFC-EMV the exchange between the assistants and other reference actors must be promoted in order to achieve the expected goals.
8. Producing support material: it is indispensable to design, publish and apply didactic resources that can be used as support material that prevail to the disposition of the mathematics educator's community. Furthermore, they should be adapted to the individual and collective context and based on the reflection of the education process.

AN APPROXIMATION TO THE CONCEPTUAL DEFINITION OF COMPLEMENTARY EDUCATION SPACE OF THE VENEZUELAN MATHEMATICS EDUCATORS (EFC-EMV)
Three types of arguments were used to construct the definition of EFC-EMV: theoretical, historical and social.
From the theoretical viewpoint, we assumed the notion of scientific field developed by Pierre Bourdieu (2005), and the notion of conceptual evolutionism of Stephen Toulmin (1977), specifically his notion of "