Why are we interested in gender studies in mathematics education?

Mathematics education is concerned with the systematic study of the phenomena that characterize the teaching of mathematics and its continuous improvement (^{Cantoral & Farfán, 2001}). The national and international results of standardized tests such as Planea (^{Secretaría de Educación Pública [sep], 2017}; ²⁰¹⁹) FOR Mexico AND PISA (^{OECD, 2019}; 2020) for all the countries belonging to the Organization for Economic Cooperation and Development (OECD) show that for certain sectors of the population the traditional school mathematics discourse has not met the specific characteristics and needs for them to fully develop their potential. Among these, we can find groups with different characteristics: ethnic, economic, or social background; different drives and interests; attitudes and skills (both intellectual and physical) developed at different levels; and indeed, one of the most notorious and neglected difference: the students' gender. Each one of these characteristics represents the diversity found in the classrooms. This is why it is important to bear them in mind to achieve one of the most important objectives in the educational system: to make inclusive, egalitarian, comprehensive, and quality education. This could make that every person participated and enjoyed the mathematical culture, so deeply rooted in their own lives (^{Cantoral et al., 2014}).

Throughout time, the teaching of mathematics has been a concern of all societies on account of its importance for the development of other areas of knowledge; and at present, for science, technology, engineering and mathematics (SIEM). These disciplines have brought about progress in many fields, such as health, infrastructure, renewable energies, to mention a few. Moreover, it is considered that mathematics is key to prepare students of any gender to have access to these undergraduate programs which will be in high demand in the future, especially because they are the foundation for inclusive and fair sustainable development (Unesco, 2019).

Nonetheless, it is precisely in these areas where women have less participation, regardless of their country of origin. Worldwide, there is a trend when choosing undergraduate programs in higher education. Women are more prone to choose education, health sciences, social sciences, humanities and arts, while men will be more inclined to choose majors oriented to technology, architecture, and engineering. Thus, there are clear differences between disciplines since only very few women are present in majors related to technology, mathematics, and engineering (Unesco, 2019). In Mexico, we analyze more in depth these specificities in the matriculation in higher education and identify that the same trend holds true for graduate programs although the percentage of women pursuing a doctoral degree in these areas decreases considerably. For example, in programs related to mathematics, women's registration diminishes by 30% (^{Simón, 2018}).

On account of women's little participation in the STEM areas, the Unesco (2019) has paid special attention to ensure the egalitarian access to these areas for girls and women. From the gender perspective, it is considered that their participation will enrich and broaden the outlook in these areas since different perspectives add creativity, reduce potential biases, and generate more knowledge and solid solutions, all of which, according to Unesco perspective, promote scientific excellence. In addition to our loss of women's talent, it entails a great loss for society (^{Farfán & Simón, 2016}; Unesco, 2019) and significant consequences for them, since it has been identified that women's economic, work, academic, and personal development is biased due to an androcentric culture.

The gender studies have brought to the table how such segregation is a product of the gaps that are present throughout life within the family circle, social life and, of course, within the school system, where the gender stereotypes are reproduced and lead to categorize certain professions as essentially "masculine" and others as "feminine" and that they work as determining factors when choosing a major (^{Aros et al., 2017}).

The mathematics classroom stands for one of these spaces since the interactions and representations that occur in it (family and school relationships, pedagogic interventions, interactions within the classroom, the students' interests and drives, school achievement and performance) are regulated by the stereotype built over mathematics as an androcentric paradigm of knowledge (^{Farfán & Simón, 2016}).

On account of this situation and because of other phenomena described in numerous research studies, promoting equality between women and men will allow the empowerment of the former as an integral part of the international public policies of the United Nations through the objectives for sustainable development (^{ONU, 2019}).

Of the seventeen objectives for sustainable development, the fifth corresponds to gender equality and it has six goals, three of which correspond to the analysis made from the educational field: 1) end discrimination against women and girls; 2) promote women's empowerment through the use of instrumental technology, and 3) approve and enforce policies and laws focusing on gender equality.

Objective 4, "Quality Education," is also concerned with the goals referred to as gender equality. In addition to considering knowledge on gender equality as fundamental to promote sustainable development, it proposes eliminating gender disparities in the access to all levels of teaching and college education and offering the physical conditions needed to attend the differences in gender within safe, non-violent, inclusive, discrimination-free, and effective learning environments.

The main theoretical lines from mathematics education in Mexico that have dealt with this problematic are the socio-epistemological and the one of social representations; both meet in a particular point: placing the learner in the center, including their cultural baggage, knowledge, history, experience, and community; in other words, their social representations.

Throughout this article, we will show how from the mathematics education we have analyzed the binomial gender and mathematics, taking as a reference the first research studies on the topic worldwide (^{Fennema, 1974}). Indeed, we take into account the outlook from Latin America, considering the research done by Rosa María ^{González (2010)}, a pioneer in dealing with these topics in Mexico and that done by research groups within our discipline that emerge from the Department of Educational Mathematics of the Center for Research and Advanced Studies of the Instituto Politécnico Nacional.

The feminist epistemology and its relationship with our standpoint

The feminist epistemology has reflected, among other things, about how gender has an influence on the methods, theories, and organizational structures of science as well as on the way in which it reproduces androcentric patterns and social prejudice of gender, especially how gender influences the person they meet, the way in which they research, ask, and justify (^{Harding, 1998}). To illustrate, we can mention those research studies done at the end of the 19^th century and at the turn of the 20^th century that assured that the form, size, and structure of a female brain made them unfit to develop certain kind of intellectual abilities, and thus, it became par for the course for them to only do house chores (^{Maffia, 2007}). Other social and ethnic groups went through the same kind of biological biases: slaved people, black people, and jews (^{Museo Memoria y Tolerancia, 2020}). This kind of scientific results show how social prejudice of gender -passed on by the person that knows- reflects on the object of study chosen, the methods used, the conclusions drawn, etc. In Patricia Castañeda's words, "There is no objectivity without subjectivity" when it comes to the value load that the researcher brings along as a subject of gender (^{Castañeda, 2016}).

In this kind of analysis, it is possible to mention the first research studies done about the relationship between gender and learning of mathematics, even those that pretended to leave the biological differences aside. Within these, we can refer to the one done by David ^{Tzuriel & Gila Egozi in Israel (2010)}, who ascertained that women had some backwardness in the development of the spatial ability, as a result of the lack of spatial experiences lived by the girls outside school, but that it could be compensated with a period of remediation or training at a young age to prevent the gender differences. In other words, from our perspective, we believe that this research from an androcentric perspective disdains the skills women develop in daily activities. Other studies, such as ^{Bethencourt & Torres (1987)}, drew the conclusion that the differences in mathematical achievement in standardized tests were due to the differences of gender while being raised and their social adaptation in school; furthermore, it depended on the type of mathematical task required. Even though these two studies do not attribute the gender gap to a biological factor, they do rest value to that which is traditionally feminine as a space that also allows the construction of mathematical knowledge.

On account of this kind of results, one could think that the remedy for the evil that afflicts women in regards to low academic performance is to make them feel more confident, encourage them in different ways for them to develop their joy and skills in mathematics and to bring them close to the female scientists who would be their role models as well as to reduce the obstacles that stop women from being interested in this area. Nevertheless, the feminist epistemology has also concluded that women are excluded from the spaces where knowledge is built; her epistemic authority is denied, her feminine ways and styles of knowledge are belittled, and women's interests and activities are made invisible (^{Blázquez, 2012}).

Since the classroom is a space that promotes the construction of knowledge, these situations start coming up even at an early age (3-5 years) (^{Flores & Guevara, 2018}) and are strengthened each step of the way throughout the educational system.

A similar phenomenon can be seen in the mathematics classroom. In its role of transmitter of a hegemonic culture, the school reproduces -in both its structure and its socializing media- the practices, beliefs, representations, and social mandates that serve for sexual difference; in other words, the attributions according to gender (^{Lamas, 2000}) throughout the hidden curriculum. Thus, the activities that are traditionally done by women are devalued and excluded from any kind of analysis in the classroom, which diminishes importance and value when considered as contexts that could lead to the construction of knowledge. Evidence of this can be found in textbooks since it has been recorded that the activities that are prepared with women are mostly in the cases occurring within the private circles even after several analyses and interventions made. Likewise, since it is done within the contexts of knowledge production, the qualities attributed to the feminine, such as emotions, subjectivity, singularity, or narrativity are less appreciated in their epistemic value (^{Maffia, 2007}) compared to others such as rationality, the capacity for logic, abstraction, universalization and objectivity, all associated to the masculine and highly valued in the mathematics classroom.

Some research in educational mathematics has identified the uses of mathematical knowledge by ethnic, cultural, and economic groups whose ways of building knowledge had not been considered in the traditional curriculum (^{Cervantes & Reyes, 2016}; ^{Cordero et al., 2014}; ^{Covian & Romo 2014}; ^{Yojcom, 2013}). We have not identified any research study that analyzes, from the gender perspective, the uses of mathematical knowledge in activities and professions that are traditionally developed by women.

Doing so, on one hand, would give us evidence of women's roles as constructors of mathematical knowledge, and on the other hand, it would allow us to describe how women build mathematical knowledge in spaces which are traditionally feminine.

The socio-epistemological theory has provided evidence of how from a patriarchal culture with an androcentric viewpoint (male, white, with access to private property, heterosexual, westerners, etc.), the so-called mathematics school discourse has been built. It is characterized for being a hegemonic and utilitarian reasoning system that lacks a reference framework; it is a discourse that imposes meanings, arguments, and procedures all of which are centered on mathematical objects and men. It privileges argumentation supported by characteristics of what is masculine; and competition in the classroom and the most valued forms of knowledge, such as mathematics, keep the structure of power from where women and other social groups are excluded (^{Cantoral & Soto, 2014}).

What questions have we wondered about from the mathematics education perspective?

Gender studies in mathematics education in Mexico have been done from different theoretical schools for more than fifteen years. The reflections around the topic arose from empirical evidence of research studies that showed that by making a differentiation by gender, new categories for analysis emerged. The research studies done in the discipline took those works that had shed light on the gender gaps in the learning of mathematics as a primary source. By the seventies, Fennema had highlighted elements that we are still analyzing: the gender gap in the achievement and performance in the mathematics class, and how these differences increased when starting middle school (aged 12-15), the low self-concept of the female students as learners of mathematics, and the explanation of this gap attributed to sociocultural factors associated to gender (^{Fennema, 1974}; 1979; 1990, cited in ^{Espinosa, 2010}).

More recent research studies have pointed out the differences in mathematics performance between female and male students of different levels, and they gave account of the great variety of sociocultural factors to which these differences are attributed (^{Eccles, 1989}; ^{Vale, 2008}). Some report the factors that place female students' academic performance below that of their male classmates; for example, beliefs and conceptions (^{Andrews & Hatch, 2000}); motivation (^{Middleton & Spanias, 1999}); some cognitive and attitude-towards-math variables (^{McGraw et al., 2006}; ^{Pierce et al., 2007}); and self-confidence to work on mathematics (^{Eccles, 1989}; ^{Jacobs et al., 2002}). These factors were later identified in the Mexican population; however, we were pursuing the objective of advancing in the knowledge of how to face them, seeking to go deeper in the questions we were wondering about.

Pertinence of the theoretical frameworks in mathematics education

We pursue leaving evidence of the coexistence and complementarity of the theoretical frameworks that support the research studies referenced in this article. First, we will say that both the feminist theory and the social representation theory are born at the same time of social change as alternatives to hegemonic models within socio-epistemology (^{Arruda, 2010}). Even though the socio-epistemological theory is younger, its history within mathematics education is also accompanied by the need to construct a theory characterized by understanding the phenomenon of social construction of mathematical knowledge and its institutional diffusion (^{Cantoral, 2013})

At a theoretical level, the socio-epistemological, the feminist, and the social representation theories have their onset at the construction of human realities from social, cultural, and historical contexts. Ontologically, the three theories understand the human being as a producer of senses and focus their analysis on the symbolic productions, meanings, language, through which human beings construct the world in which they live.

From the feminist theory, the woman's condition is a historical creation whose content is the set of circumstances, qualities and essential characteristics that define a woman as a sociocultural generic being (^{Lagarde, 1990}).

From the social representation theory, the woman represents above all declarative knowledge that fixes its characteristics and meaning in the social contexts and by means of understanding the everyday acts since it is there where the real symbols, values, costumes, and traditions for the social group crystalize (^{Berger & Luckman, 1972}). It is in the same everyday life where it is defined that being a woman is a generic category and at the same time the parameters of a woman within society are determined (Flores, 2001).

The socio-epistemological theory supports the idea that everybody, men and women, should be able to use and enjoy the mathematical knowledge in their lives (^{Cantoral, 2013}); however, it has been documented that specially women have not reached the opportunity or the freedom to build mathematical knowledge or enjoyed putting it into play (^{Farfán & Simón, 2016}). Such research seeks to characterize how the school mathematics discourse excludes women, and in this way set the keys that will contribute to strengthen them as constructors of mathematical knowledge (^{Simón, 2018}).

In these theories, another meeting point is the method. The socio-epistemology conceives that in the construction of mathematical knowledge not only do the adaptation of theoretical explanations and empirical evidence intervene, but it also enhances the fundamental role played by the historical, cultural, and institutional stages in which the human activity develops when knowledge is produced. Therefore, different pre-existing research tools and techniques in several areas of social sciences have been useful to explain the dynamics of situated knowledge (^{Cantoral, 2013}).

Social representations consider that the complexity of the represented phenomenon legitimizes combining approaches or theoretical perspectives that are articulated and assume different approaches that favor a more in-depth and multidimensional close-up (^{Rodríguez, 2009}); and they allow to be constructed through symbolic elements, mainly verbal or written, endowed with meaning and social and personal sense.

In the case of the feminist theory, when talking about method and methodology, "they are often referred to viewpoints and the epistemo-logical focus rather than to procedures and research techniques" (^{Kaplan, 1995}, p. 89). Nevertheless, in practice, a great variety of research methods are used. In these, subjectivity has increasingly become a more sought focus of approximation in correspondence with that emerging trend in the set of human and social sciences that propose communication with all female and male actors as well as self-awareness of their own identity (^{Borderías, 2009}). It is important to mention that a characteristic in the evolution of the research done from this viewpoint is based on re-taking women's experience as an empirical and theoretical resource, in words of Castañeda, thinking of them and organizing research in relation to them (2016). In a way, it is a criterion shared by the three theoretical approaches analyzed, since they focus on the person and on their condition as builders of mathematical knowledge.

Another element of closeness among the three theories is the vindicating essence of the subjects and the irrevocable posture for social change. In the case of feminism, both concerns drove the movement. Just as expressive and constructive of the social, the social representation is not only a means of knowledge, but it is also an instrument of action. Its particularity lies on the possibility of deconstructing and reconstructing; thus, it becomes a methodological tool that serves as a trigger for reflection, interpretation, and change. Likewise, just as the social representations, the other two perspectives serve as a reconstructive methodology that seeks achieving more just, equitable, and egalitarian relationships.

Research development and evolution from these theoretical perspectives

Social, mathematical, and gender representations

In regards to mathematics, gender and social representations, the gender differences have been approached in different aspects of school mathematics in grade school (female and male students, 6-15 years of age). ^{Flores' work (2007)}, which analyzes the gender social representations of the mathematics faculty in regards to their students' learning, shows that the expectations and beliefs of female and male teachers have an influence on the pedagogical interactions set with the students in the everyday life in the classroom, and thus, condition the women's mathematics learning. The work highlights the relations of power, the kind of language, the attention span, the number and kind of questions towards female and male students. The combination of all these elements benefits the mathematical learning of male students, but excludes female students.

^{Ursini's longitudinal research (2010)} analyzes the gender differences in the social representation of mathematics learning. In the study are shown the changes in students' attitudes towards mathematics when they are in middle school. The statistical analysis reveals that female students have a more negative attitude towards the courses in comparison with male students. The author's theoretical framework reveals that the attitudes are a sociocultural construct highlighting the gender differences in relation to their intellectual, cognitive and behavioral capabilities to be successful in mathematics; in other words, stereotypes are strengthened. While male students are oriented by the social consensus to develop more aptitude, female students face the social representation of having less intellectual capability, and thus, they conclude they need to be more focused and obedient, and make a greater effort to reach positive or equivalent results. This is how gender stereotypes are intertwined with self-perception, printing a gender bias in the learning processes; and it is only through deconstruction and reconstruction that these socially imposed stereotypes can be overcome.

^{Mosquera & González (2015)} pose as the main concern the social representation of gender contained in the mathematics textbooks used in school. Such representations contribute to reproduce, reflect, and strengthen conceptions that go around these gender inequalities. Their work acknowledges the impact of textbooks as learning tools, since their contents have an influence on legitimizing the social representations of gender that build up in the present and future generations that go through the school system. The authors report three gender representations: a) androcentricity (masculine prominence in the analyzed school textbooks); b) separate environments, public and private space (tendency to differentiate stages, characteristics, and roles according to sex, attributing what is public to male students and what is private to female students); and c) gender stereotypes (referred to the resources used to illustrate people with the tendency to differentiate men and women.) This study recommends illustrating books bearing in mind the presence of humankind in its diversity and in conditions of equality with the end of reflecting a society with equal opportunities for women and for men.

In a study with elementary school teachers, ^{Rodríguez & Ursini (2008)} proposed identifying the gender differences in the social representation of the teaching staff in regards to teaching mathematics with a multimedia resource. Thirty teachers participated (22 female teachers and 8 male teachers), and they used multi-methodology. Almost all of the teachers used the resource in an ongoing fashion and declared having good command of it. However, classroom auditing of its use revealed contradictions in this discourse. Only the female teachers acknowledged the need of teacher training to handle this resource. The female teachers showed solidarity when exploring and sharing their findings with colleagues. These last two results and our work in "professional teacher training" have led us to think about the need to analyze how the characteristics of these female teachers allowed them to be more flexible to reflect on their teaching practice.

From the feminist theory and the social representation theory, ^{Rodríguez & Ursini (2014)} analyze the relationship between the gender identity processes, the representational construction of mathematics, and the successful performance of female students in this course, using a qualitative and interpretive methodology, combining techniques used in anthropology and social psychology. Twenty elementary school students were followed up during three school years. Data triangulation allows to describe the gender identities that favor the construction of social representations of school mathematics that characterize the female students with successful performance. Throughout time, positive changes were seen in some students when building up an awareness of being women and being mathematics learners, in addition to being better prepared to face the math assignments. By analyzing the in-depth interviews, this study shows evidence of the elements that clear the construction of generic identities. The most successful female students of mathematics are those who are raised in environments where gender equality is promoted more frequently and in families that foster the development of abilities, knowledge, capabilities, values, and affection in the same way for women and for men.

These research studies invite us to open up the possibilities in face of the emerging categories and gender constructions, not with the intention of promoting pigeonholes to categorize people, but to acknowledge and enable human diversity without hierarchies.

From our theoretical position, we consider that the studies about differences do not give us the elements for an intervention at the public policy level nor instruments to build ad hoc designs that enable us to have interventions in the classroom. We moved from centeredness and we are interested in systematizing and make visible all the ways of feminine construction that will give us a hypothesis for the redesign of the school mathematics discourse.

Research prospective: where recent findings point at

From the perspective of the presented theories, it is worthwhile to wonder: How does a woman think about mathematics? We have no answer for the time being, but we can assure that because her culture, knowledge, and history have been disrupted by the social attributions made to her gender, the way in which a woman thinks about mathematics cannot have the same bases as that of a man's. The same situation can be present in each individual in the classroom; their history, their culture, the knowledge of their community, their experiences and their social representations will be specific to each one. However, there are things that are shared and can be considered when it is sought to intervene in the classroom.

The results bring about another matter of concern: Is it possible to consider the design of a learning situation with gender perspective? Research studies based on the theory of social representations have allowed to show evidence of the advantage of building didactic environments where the female students can legitimize their identity as mathematical communicators; this condition modifies their participation, self-concept, and self-confidence as learners (^{Rodríguez & Ursini, 2014}). Other findings allowed us to identify that the learning situations proposed from the social epistemology led women to participate with more motivation and interest in the mathematical activity when they were valued and when their experience, arguments, and reasoning were included as part of the knowledge construction (^{Farfán & Simón, 2016}; ^{Carranza, 2019}). By then, it seemed that our problem had been solved; however, the analysis and putting into practice our designs have taken us to identify that more research is needed about how the school mathematics discourse leaves women out.

We believe that the next step is to increase the value of what is feminine, in such a way that the female students can bring to the game their experiences and arguments when working on mathematics; and at the same time that what is traditionally feminine can be valued by providing contexts that allow construction of knowledge. We need in-depth research about how to create learning environments that are inclusive, egalitarian, and fair to our girls at a very early age. We also need broader knowledge about how women put into play their mathematical knowledge, how it was done in the past and how it is done at present, inside and outside the classroom, in everyday and in crafted activities. This way, we will have more elements to consider when the activities are designed to participate in the classroom.