Abstract

Mexican mathematics educational researchers have made important contributions to what we now know about the learning and teaching of mathematics. However the scope of these contributions is often limited to spanish speakers. In this article, I will give an overview of the research developed in Mexico, which is often not widely disseminated in English language journals or academic societies. One particular theoretical contribution, Local Theoretical Models, relevant to the teaching and learning of mathematics education is discussed. Also, I will mention Mexican contributions to the Mathematical problem solving for teaching and learning mathematics in higher education.

Keywords: research, mathematics, education, Mexico, problem-solving, local-theoretical-models

1. Introduction: Educational context of Mexico

Mexico is the 5th largest country in the Americas by total area. With an estimated population of over 113 million, it is the world’s 11th most populous country and the most populous Spanish-speaking country [1, 2]. It was the first Latin American member of the Organisation for Economic Co-operation and Development (OECD) in 1994, and it is now considered an upper-middle income country by the World Bank [3] although 7.2% of the adult population remains illiterate [4].

In Mexico, three phases of education encompass schooling from pre-kindergarten (PreK) to higher education: 1) Basic education: Preescolar (Pre-K), Primaria (grades 1-6) and Secundaria (grades 7-9); 2) High school education: Preparatoria (grades 10-12); and 3) Higher education: Universidad (Undergraduate education) and Postgrado (Postgraduate education). Mandatory education in Mexico is from grade 1 through to grade 9. Statistics for the school year 2011–2012, according to National Institute of Statistics and Geography (INEGI[1]) [4], show that approximately 33 million students attended public (and private) education across the educational system, although students typically leave school at grade 8. Further details are presented in Table 1.

1.1. Educational regulation and curriculum design

Since 1921 the federal government, through the Secretariat of Public Education (SEP[2]), has regulated education in Mexico, established the national curricula for basic education, and designed or approved the corresponding material (textbooks, lesson guides, library material, etc).

On the other hand, high school and higher education in Mexico involve diverse federal, state, and university regulated schooling systems, which do not share a curriculum. Each system is based on particular needs and objectives. The options available to students, in high school education, include technical (professional technicians) or general orientations (preparation for university). For example, the National Autonomous University of Mexico (UNAM[3]) and the National Polytechnic Institute (IPN[4]), both located in Mexico City, are two large higher education institutions that have their own 10-12 educational options.

1. 2. Mathematics curricula, standards and achievement

The required mathematics courses are held from basic through high school education every year (see Table 2 for details). For basic education, since 2006 [5], the official curriculum encompasses four standards that frame the mathematical syllabus: 1) Number sense and algebraic thinking, 2) Shape, space and measurement, 3) Data analysis; and 4) Attitudes toward the study of mathematics. Here it is important to point out that the mathematical syllabus identifies problem solving activities as key aspects in the student’s mathematics learning [6]. Meanwhile, for high school education, the varying curricula establish courses of mathematics at higher level, including different areas such as: Algebra, Analytic Geometry, Trigonometry, Differential and Integral Calculus, and Probability and Statistics.

The majority of mathematics teachers, in basic education, are trained at Normal Schools (this idea was adopted in Mexico from France in the beginning of the 20^th century) or what are referred to as ‘teachers colleges’ in other countries. On the other hand, the majority of mathematics teachers in high school education are trained at universities. However, in recent years a great amount of people, who have ended their undergraduate studies not related to mathematics, have become teachers of mathematics at different levels even when they are not qualified, and this is due to the lack of jobs in Mexico. This issue of untrained mathematics teachers diminishes the quality of mathematics courses across the educational system.

Mexico suffers the lowest levels of numeracy amongst OCED countries. For example, the Program for International Student Assessment (PISA) results from 2009 shows that Mexico performs compared with Australia, New Zealand, USA, Canada, Spain, Chile [7]. The low student performance in mathematics remains a constant, even at the higher education level [8].

The SEP, concerned with the low student performance in mathematics, has sought to improve mathematics teaching and learning in basic and high school education. Beginning in 1970, SEP has initiated various educational projects to address the mathematics education including (1) implementing new technological resources for teaching and learning, (2) creating study programs for training researchers and teachers, and (3) creating interdisciplinary groups of research for understanding cognitive and pedagogical processes that influence the learning and teaching of mathematics.

As a result, there are currently many groups of academics, in different institutions across the country researching in the area of mathematics education not only in basic and secondary education but at the high school and university levels. In fact, Mexico has a long-standing tradition in research in mathematics education as one of the first countries worldwide to have research groups specifically dedicated in this area [9].

2. The origins of mathematics education research in Mexico

2.1. Mathematics education research in Mexico City

Technically, mathematics education research in Mexico started in 1970 when the government launched a major educational reform. In response, a group of mathematicians at the Centre of Research and Advanced Studies (Cinvestav[5]) were asked to develop a new curriculum of mathematics, together with textbooks for basiceducation. The overarching focus of these early Cinvestav researchers was that of teachers’ mathematical content knowledge with the belief that ‘teachers need to know more mathematics’ [10]. However, some researchers became concerned with teachers’ mathematical pedagogical content knowledge and problems teachers encountered in the teaching of mathematics [10]. The Cinvestav research group also began researching in the history of mathematics and relating it to current trends in curriculum design [10]. Publications from this group began in 1980’s.

Some of the initial aims of the group were the following [11, p. 20]:

(1) General research on the learning of mathematics and the methods of teaching it.

(2) Experimentation, reviews and corrections of the new primary school mandatory textbooks.

(3) Study of the real needs of primary-school teachers and development of different types of auxiliary materials.

(4) Structuring of a mathematics curriculum for the teacher-training schools.

(5) Study of the problems faced by secondary-school teachers, in particular those in public schools, with emphasis in the development of materials suitable for those levels, both for teachers and for students.

(6) The structuring of an undergraduate degree focusing on the teaching of mathematics with the aim of training secondary-school teachers specialized in the teaching of mathematics.

(7) Designing Master’s and doctoral programs with the same aims.

(8) The development of popularization materials.

Despite its origins to address basic education level, much of the research of the Cinvestav focused on the upper educational levels, due in part to the experience of the first members of the group [12]. Gradually, the research interests of the group developed, and research expanded to include all levels from Pre-K to university with much broader areas of research. Currently, the Department of Mathematics Education[6], at the Cinvestav, explores the following research areas:

· Arithmetic and algebraic thinking;

· Advanced mathematical thinking and the teaching of calculus and analysis;

· Geometrical thinking;

· The teaching and learning of statistics and probability;

· The history and epistemology of mathematics;

· Theoretical foundations;

· The social construction of mathematical thought;

· Cognition;

· Technology based learning environments;

· Problem solving;

· Gender studies;

· Teacher training and assessment.

Others important research groups, located in Mexico City, can be found at the Universidad Pedagógica Nacional (UPN), UNAM, the Instituto Tecnológico Auntónomo de México (ITAM), the Universidad Autónoma Metropolitana (UAM), and the Universidad Ibero Americana (UIA).

While responding to local and national challenges in training mathematics educators and preparing students with the appropriate mathematical knowledge, Mexican researchers have made several theoretical contributions of international relevance to the mathematics education community. Two such contributions are discussed below. The first, a theory for learning algebra developed by Mexicans researchers at Cinvestav that is applicable to the higher education community. I do not intend to present an extensive exposition, instead focusing on the crucial principals of this theory (for an extensive presentation, see [13, 14]). Next, I present Mexican contributions to the Mathematical problem solving and the use of computational tools for teaching and learning mathematics in higher education.

2.2. Contribution of Mexican researchers

2.2.1. Local Theoretical Models

In 1990’s, Eugenio Filloy developed the idea of Local Theoretical Models (LTM) for learning algebra. Filloy [13, p. 30-31] defines LTM according to the following four characteristics:

(1) a LTM consists of a set of assumptions about a concept or system;

(2) a LTM describes a type of object or system by attributing an internal structure to it, which when taken as reference will explain several of the object’s or system’s properties;

(3) a LTM is considered an approximation, which is useful for certain purposes;

(4) a LTM is often formulated and developed and perhaps even named on the basis of an analogy between the object or system that it describes and some other object or different system.

Considering the above, symbolic algebra is considered a language and there exists the interest, as has been previously stated, in also studying the relationship between the latter language and prior or intermediate language levels, thus incorporating the notion of a Mathematical Sign System (MSS) in the broad sense of the term [15, 16]. The MSS has to serve as a tool to analyze the texts produced by students when they are taught mathematics in school systems as well as to analyze historical mathematical texts, taken as monuments, petrifactions of human action, or processes of cognition belonging to an episteme [13].

According to Rojano [17, pp. 39-40], the MSSs are:

sign systems in which a socially agreed possibility exists to generate signic functions. As a result, the definition has scope to include cases in which functional relations (signic) have been established for use of didactic devices within a teaching situation and in which usage thereof may be intentionally temporary. In other words, also considered or included are sign systems or sign system strata produced by students in order to give meaning to what is presented to them within a teaching model, even when said systems are governed by correspondences that have not been socially established, but that are rather idiosyncratic. The notion of MSS plays an essential role in (locally) defining the components that make up the LTM and that deal with different types of: i) teaching models; ii) models for cognitive processes; iii) formal competency models, which simulates the performance of an ideal user’s of a MSS; and iv) communication models, in order to describe the rules of communicative competency, text formation and decodification, and contextual and circumstantial disambiguation’.

Filloy, Rojano and Puig [13, pp. 31-32] claim that the LTM can fulfil the same functions as theories: they can be used for purposes of explanation, prediction, calculation, systematization, derivation of principles, and so on. They also provide explanations but these explanations are based on assumptions that may be simplified, and this condition must be borne in mind when one compares them with theories.

2.2.2. Mathematical problem solving and the use of computational tools for teaching and learning

Mathematical problem solving (MPS) has been widely recognized as a framework to analyze learning mathematical processes in which it plays two roles. On one hand, it guides performing research in mathematics education (see for example [18]) and on the other hand, it supports the development of curricular proposals (see for example [19] from the USA and [5] from Mexico).

Mexican researchers have been contributing to this area in the last two decades conducting substantial research on the implementation of MPS and the development of computational tools for teaching and learning mathematics at different levels from basic to higher education. Some of the most representative researchers on MPS, working at different universities and research institutions, are Manuel Santos-Trigo, Fernando Barrera-Mora, Aarón Reyes-Rodriguez and Hugo Espinosa-Perez.

As examples of research carried out on MPS and computational tools for teaching and learning and published in english see [6, 20, 21]. In particular, research to illustrate and contrast students’ approaches based on paper and pencil and compared with the use of dynamic software, Santos-Trigo [6, p. 531] uses the next:

Problem.You are given two intersecting straight lines and a point P marked on one of them, Figure. 1, below. Show how to construct, using straightedge and compass, a circle that is tangent to both lines and that has the point P as its point of tangency to one of the lines [18, p.15].

Figure 1. Two intersecting lines and a point on one of them

Santos-Trigo [6, p. 535] claims that

[...] the students’ use of dynamic software seems to favour the construction of dynamic representations of mathematical objects or problems. As a consequence, students are likely to develop a set of heuristics that involve measuring particular mathematical attributes (segment lengths, angles, areas, perimeters, etc.), dragging objects with a geometric configuration, and the appropriate use of the Cartesian system to detect, explore, and support mathematical relations or conjectures.

3. Mathematics Education research in other areas of Mexico

Even though most of the research takes place at institutions located in the capital Mexico City, other research groups of mathematics education are forming in the country. Researchers’ contributions at different universities (e.g.the Autonomous Universities of Guerrero, Querétaro, Morelos, Zacatecas, Sonora, Yucatán, Chiapas, Tamaulipas, Quintana Roo, Nuevo León; Michoacana University, Veracruzana University, Pedagogical University of Zacatecas and San Luis Potosí, Technologic Institute of Advanced Studies of Monterrey, among others) have been growing continuously and are developing local graduate programs focusing mainly on teachers’ continuous education.

Recently, some new groups dedicated to mathematics education research are arising at institutions such as at the Latin-American Institute of Educational Research (ILCE[7]), the National Institute of Educational Assessment (INEE[8]), and the Centre of Studies on Assessment (CEE[9]).

4. Final remarks

Even though research has been growing and spreading across the country there are many challenges that the Mexican community has to face. As I have pointed out before, the SEP has sought to improve mathematics teaching and learning in basic education through various educational projects, however the low student performance in mathematics remains a constant, even at the higher educationlevel.

Another issue is that, although research results have been taken into account for the development of new curricula, there is still a need of promoting more opportunities of interaction between the policy-makers and the research community. A major problem is the fact that young researchers have many difficulties for finding jobs and positions within research centres and universities. Young researchers are needed to make innovations, to foster new ideas and impact on the dynamics of the community. However, many talents are lost when young people leave the field for positions in the job-market that are not related to mathematics education [9].

On the other hand, another major issue in Mexico is corruption. As Stephen D. Morris has pointed out: “Scandals, anecdotes, official reports, the rhetoric of politicians, surveys, scholarly analyses, and even popular legend all indicate that corruption pervades Mexico, spanning the country, the layers of government, and the years’’ [22, p. 1]. The Corruption Perception Index (CPI), published by Transparency International (TI) in 2012, ranks Mexico in the position 105 with a score of 34, on a scale from 0 (highly corrupt) to 100 (very clean) [23]. According to TI’s 2010/11 Global Corruption Barometer, the institutions perceived to be most affected by corruption in Mexico are police and political parties [24], the latter is closely related to the educational institution SEP. One example is the case of the head of Mexico’s powerful teachers’ union who was recently arrested for allegedly embezzling some $200 million (money that belonged to the teachers union and to the SEP) [25]. It's clear that corruption is a major threat, not only in Mexico but all around the world. It generates popular anger in communities, and undermines countries and institutions.

Finally, a pressing issue is the training of qualified mathematics teachers. Although there are several programs implemented by various institutions throughout Mexico to improve the performance of teachers of mathematics, there is still a huge lack of on-going professional development for teachers to update and inform their mathematical knowledge and pedagogical practices.

Regardless of the many challenges that Mexico faces, the Mexican mathematics education community is better known to a wider national and international audience thanks to their contributions that have broad relevance beyond the borders of Mexico or the confides of the Spanish language.

References

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[1] Instituto Nacional de Estadística y Geografía: http://www.inegi.org.mx/

[2]Secretaría de Educación Pública.

[3]Universidad Nacional Autónoma de México.

[4]Instituto Politécnico Nacional.

[5] Centro de Investigación y de Estudios Avanzados: http://www.cinvestav.mx/

[6] Departamento de Matemática Educativa: http://www.matedu.cinvestav.mx/

[7]Instituto Latinoamericano de Investigación Educativa.

[8]Instituto Nacional de Evaluación Educativa.

[9] Centro de Estudios sobre Evaluación.