Mathematics Teacher Preparation in Central America and the Caribbean

Front matter

SPRINGER BRIEFS IN EDUCATION

Angel Ruiz Editor

Mathematics leacher Preparation in Central America and the Caribbean

Foreword

The International Commission on Mathematical Instruction (ICMI) has a substantive interest in ensuring that its resources and influence extend to any country that is able to mobilize the elements of its mathematics education enterprise to make productive use of this connection. In line with this goal, ICMI in conjunction with the International Mathematical Union (IMU), and with the support of UNESCO and ICSU (International Council for Science), promotes the Capacity & Networking Project (CANP). It aims to enhance mathematics education in developing countries by supporting the educational capacity of those responsible for mathematics teachers, and to create sustained regional networks of teachers, mathematics educators and mathematicians, linking them to international support.

CANP consists of a program that is being carried out since 2011 in different developing world regions: each program has, at its center, a two-week workshop of about forty participants, half from the host country and half from regional neighbors, who interact with experts in mathematics, math education, and school policy coming from different parts of the world. It is primarily aimed at mathematics teacher educators, but each event includes also mathematicians, researchers, policymakers, and key teachers.

The Capacity & Networking Project is a major international initiative in the mathematical sciences in the developing world to help exchange information, share the state of the art research, enhance mathematics education and build a sustainable network for policymakers, scholars and practitioners across those targeted regions.

The program builds on existing activities in the region and does not seek to reproduce or compete with existing development programs.

At the moment when this book is printed (2016) five CANP workshops have been held: CANP-1 in sub-Saharan Africa (2011), CANP-2 in Central America and Caribbean Area (2012), CANP-3 in South East Asia (2013), CANP-4 in East Africa (2014), CANP-5 in Andean Region and Paraguay (2016).

The main goal of a CANP consists in building capacity in mathematics education and creating a sustainable regional network in the countries, which participate to the workshop, with a common goal of improving mathematics education in vi Foreword the region. The initial two-week workshop is an occasion for launching the network and for collecting and sharing information about the situation of mathematics teaching in the region. For this, before the workshop each group of participants from a country prepares a report about the state of the art in their own country: the reports are presented, compared and discussed during the meeting. After that, they are further elaborated according the results of the discussions and constitute a final report for that CANP.

They constitute interesting documents about mathematics education in the regions touched by the different CANPs, and give a piece of information not always accessible in an easy way. For this reason ICMI decided to launch a new series of books with an international publisher, Springer, in order to make accessible non-expensive format reports to an international audience of informed policymakers and scientists.

The present volume is the second in the series of CANPs reports: it is the result of a huge work of elaboration of the original documents presented in Spanish at CANP-2 workshop, held from August 6 to 17, 2012 in San José, Costa Rica. The event involved 66 participants from Central America and the Caribbean Region and concerned the initial and continuing formation of mathematics teachers in those countries. It was organized in a splendid way thanks to the wonderful work both of the International Program Committee, and of the Local Organising Committee, and especially of Angel Ruiz, vice-president of ICMI: as liaison person with ICMI he participated to the scientific design of this CANP and with his team took care of all its organizational aspects. In fact CANP-2 included lectures given by outstanding mathematicians and math educators, regional presentations, workshops, round-table discussions, panel presentations, and other parallel activities. Many hours were devoted to the discussion of the regional reports, which are the germs from which this book was originated, and to the creation of a Mathematics Education Network (REDUMATE: Red de Educación Matemática de América Central y El Caribe —www.redumate.org).

Angel Ruiz is also the editor of the volume: with all the other authors, he made a huge effort to have the different articles written in a suitable and uniform way. They illustrate in an updated form the initial and continuing preparation of mathematics teachers in Colombia, Costa Rica, Dominican Republic, and Venezuela.

I thank all those who have made possible the existence of this book: the editors, the authors, the excellent translator of the reports from Spanish, Patrick (Rick) Scott, the publisher, and particularly the participants to the CANP-2 event and to its follow-ups. I do think that making accessible its content to math educators, teachers, and policymakers also outside Latin America represents a useful tool for approaching the problems of mathematics education within a global landscape, but without forgetting the specific cultural and social needs of specific developing regions, in this case Central America and Caribbean area.

Foreword vii

It is my strong hope that with the publication of these CANP books, we will have a wide updated picture of mathematics education needs and problems from relevant parts of the developing world. This will help to avoid the dangers of the alienation generated by the loss of the variety of cultural richness existing in the different regions of the world.

Torino Ferdinando Arzarello September 2016 President of the International Commission on Mathematical Instruction

Acknowledgments

The following persons and institutions were very important in the preparation of this book:

E. Guacaneme, G. Obando-Zapata, J. Villa.Ochoa, D. Garzón, Y. Morales, I. Cruz, S. González, N. León and W. Beyer, the authors of the chapters of this book. The research, organization and intellectual condensation they realized made possible CANP 2 and this publication that is instrumental for the progress of mathematics education in the Central American and Caribbean region.

Patrick (Rick) Scott, Vice President of the Inter-American Committee on Mathematics Education. He produced the English translation of the different sections and chapters of the book.

Jhony A. Villa-Ochoa (Colombia) and Edison de Faria (Costa Rica). They carried out the first review and editing process of the national reports from the four countries that are the base of this book.

International Commission on Mathematical Instruction (ICMI). It was the main force in conveying and supporting the realization of the Capacity and Networking Project (CANP 2) and the publication of this book.

International Mathematical Union (IMU). Its support was essential to the organization of CANP2.

International Council for Science (ICSU). This institution provided important support to CANP 2.

Inter-American Committee on Mathematics Education ([ACME). Through this international academic network it was possible to organize CANP 2 and provide special support for the development of the Mathematics Education Network for Central America and the Caribbean, and the elaboration of the national reports that nurtured the this publication.

x Acknowledgments

Mathematics Education Network for Central America and the Caribbean (REDUMATE). These chapters were possible by the support provided by this new organization that emerged from CANP 2.

Mathematics Education Reform in Costa Rica Project, Ministry of Public Education of Costa Rica. It provided the local organization of CANP 2 in Costa Rica and was supportive of the edition of all the materials included in this publication.

Cuadernos de Investigación y Formación en Educación Matemática (Journal of Research and Teacher Preparation in Mathematics Education). Center of Research and Teacher Preparation in Mathematics Education, University of Costa Rica. It published the original national reports (written in Spanish) that emerged from CANP 2.

Contents

1 Mathematics Teacher Preparation in Central America Angel Ruiz

2 Colombia: Mathematics Education and the Preparation Edgar Alberto Guacaneme-Suárez, Gilberto Obando-Zapata, Diego Garzón and Jhony Alexander Villa- Ochoa

Yuri Morales-López

Chapter 1 Mathematics Teacher Preparation

in Central America and the Caribbean.

An Introduction

Angel Ruiz

Abstract This chapter provides a summary of the state of Mathematics teacher’s preparation in Central America and the Caribbean based on four papers presented at a workshop of the International Commission on Mathematical Instruction: Capacity and Networking Project, held in Costa Rica in August 2012. The countries considered here are Colombia, Costa Rica, Dominican Republic and Venezuela. First, a description of the conditions of this region in various international comparative tests of Mathematics is established, as a prelude to offer elements of each country about the general structure of their education systems and the main features of their curricula in school Mathematics; then the initial preparation and professional development of teachers are studied. Finally, graduate programs and research in Mathematics Education are analyzed and, to conclude, the main challenges that these countries face in the current scenario are indicated. Throughout all this work, comparative elements between the four countries are given in the dimensions studied.

Keywords Teacher preparation - Mathematics - Mathematics education + Central America - The Caribbean

The second workshop of the Capacity and Networking Project (CANP 2) of the International Commission on Mathematical Instruction (CMI—www.mathunion. org/icmi/home) was held in San José, Costa Rica from August 6 to 17, 2012. This event brought together mathematics educators, mathematicians, university administrators, and elementary and secondary teachers from Central American and the Caribbean. Financial support was received from the International Mathematical Union (IMU) and from the International Council for Science (ICSU). It was organized with assistance from the Inter-American Committee on Mathematics

A. Ruiz (DX) Inter-American Committee on Mathematics Education,

Mathematics Education Reform in Costa Rica Project, San José, Costa Rica e-mail: ruizz.angel @ gmail.com

URL: http://www.angelruizz.com

© Springer International Publishing Switzerland 2017 1 A. Ruiz (ed.), Mathematics Teacher Preparation in Central America

Education—IACME (the multinational affiliate of ICMI in the Americas, www. ciaem-iacme.org.). Local arrangements were the responsibility of Mathematics Education Reform in Costa Rica Project (www.reformamatematica.net). The creation of the Mathematics Education Network (REDUMATE—www.redumate.org)' was one of the most important outcomes of the event.

National reports on the status of initial and continuing mathematics teacher preparation were presented during the event. These national reports became important references in establishing collaborative actions related to the teaching and learning of mathematics in the region.? This book presents summaries of the reports: a synthesis of initial and continuing preparation for the teaching of mathematics in Colombia, Costa Rica, the Dominican Republic and Venezuela. These reports have served as a starting point for comparative analyses, showing similarities and differences while highlighting the various perspectives.

The reports consider various dimensions:

. The structure of the educational systems.

. The school curriculum for mathematics.

. Initial teacher preparation.

. The professional development programs for in-service teachers. . Graduate programs.

. Research in mathematics education.

Du UND Ke

To assist the reader, some sketches of these dimensions that will be further developed below are presented. But to begin, it is convenient to offer an orientation to the region.

Central America and the Caribbean

The countries of this region of Latin America have certain characteristics in common: all are part of the Caribbean Basin; there is a shared European heritage (predominantly Hispanic) with ethnic and cultural contributions from pre-Colombian, African and Asian communities; educational achievement is not reaching the levels needed to meet development goals; there are often conditions of poverty that are among the highest in the Americas (Fig. 1.1).

The situation with respect to the teaching and learning of mathematics in Central America and the Caribbean should be considered in a larger context. One image of its reality is provided by international comparative testing.

LA study on these and other multinational mathematics education organizations can be found in Ruiz (2013).

“The complete national reports were published in Spanish in the journal Cuadernos de Investigación y Formación en Educación Matemática published in Costa Rica (Mathematics Education Reports 2013).

Fig. 1.1 Central America and the Caribbean. Source Free vector map of Middle America political with shaded relief.

The achievement in Latin America on PISA, the international assessment from the Organization of Economic Cooperation and Development (OECD) (that is given to 15 year-old students) is systematically among the lowest of participating countries. The following table shows the 15 countries with the lowest scores on the 2012 PISA mathematics test. More than half of them are from Latin America (Table 1.1).

The average scores for countries participating from Latin America was approximately 397, almost 100 points lower than the OECD average and 215 points lower than Shanghai. Fully 63 % of Latin American youth scored under Level 2, which is considered to be the level necessary to function adequately in the modern world in which we are living (and that is 40 % more than was the OECD average). Less than 1 % scored at the highest two levels. Even if you do not accept all of the criteria and methodology used by PISA, these results show very weak achievement in school mathematics which presents these countries with the need to design very serious actions to improve education. Also, within the region there are significant differences, for examples there is a 55-point difference between the highest (Chile) and the lowest (Perú).

There has also been an effort on the part of UNESCO’s Latin American Laboratory on the Evaluation of the Quality of Education to measure achievement in the third and sixth grades in schools in the region. Their two latest studies have been the “Second Regional Comparative and Explanatory Study” (SERCE) in 2006

Table 1.1 The 15 countries with the lowest achievement levels on PISA 2012

Country Average Percentage of students Percentage of students score on with the lowest scores with high scores (level PISA 2012 (lower than level 2) (%) 5 or 6) (%)

Malaysia 421 52 1.3

Montenegro 410 57 1.0

Costa Rica 407 60 0.6

Argentina” 388 67 0.3

Colombia 376 74 0.3

Indonesia 375 76 0.3

Average for the 397 63 0.8

Latin American

countries

OECD average 494 23 12.6

Shanghai-China 613 4 55.4

“Latin America. Many nations in this region did not participate, including the Dominican Republic and Venezuela Source OECD (2014)

and the Third Regional Comparative and Explanatory Study” (TERCE) in 2013. Some results follow (Table 1.2).

These data show that the countries of the Caribbean Basin that have participated in these tests (without including México) have consistently scored below the rest of Latin America. Latin America as a region on international comparative tests has had low achievement levels with respect to the rest of the world, but Central America and the Caribbean is even weaker. On three of the tests the difference between Chile (with the highest scores) and the Dominican Republic (with the lowest scores) is more than 130 points.

The purpose of CANP 2 was to study the conditions related to mathematics education in Central America and the Caribbean, and search for elements to promote development. And the objective of this book is to offer to the international mathematics education community for the first time an academic summary of some dimensions of the development of the teaching and learning of mathematics in this specific region.

Why are only Colombia, Costa Rica, the Dominican Republic and Venezuela in this book? Although other countries in the region were invited to participate, for various academic and socioeconomic situations they chose not to. This work should

Table 1.2 Results from SERCE and TERCE (UNESCO) in Latin America: 2006, 2013

Third grade Sixth grade SERCE |TERCE |SERCE | TERCE

Argentina 505 530 Brazil 505 520 Chile 529 581 Colombia* 499 515 Costa Rica? 538 535 Ecuador 473 513 Guatemala” 457 488 Honduras” 480 México 532 566 Nicaragua” 474 462 Panamá* 463 461 Paraguay 486 456 Perú 474 527 Dominican Republic* 396 448 416 437 Uruguay 539 551 578 567 Overall average 491 521 492 509 Average for countries from Central 471 502 471 483 American and the Caribbean (not including

México)

*Countries from Central America and the Caribbean Venezuela did not participate in these studies Source OREALC-UNESCO (2014)

Table 1.3 Area and population of Colombia, Costa Rica, the Dominican Republic and Venezuela

Country Area Approximate population in millions in 2016 Colombia 1,142,903 km? and territorial waters 48 988,000 km?

Costa Rica 51,100 km? and territorial waters 5 589,000 km?

Dominican 48,442 km? and territorial waters 10 Republic 138,000 km?

Venezuela 916,445 km? and territorial waters 31 670,000 km?

be seen as a first approximation. Without a doubt, it will be important to replicate this study in other countries in Central America and the Antilles.

It is important to note that the four countries in this study are particularly diverse geographically and demographically. Below are data on surface area and population that should help to situate the reader (Table 1.3).

Colombia is the largest with the greatest population, followed by Venezuela. The Dominican Republic has an area similar to Costa Rica, but twice the population.

Colombia has an area 20 times that of the Dominican Republic and Costa Rica, and a population 10 times that of Costa Rica. Colombia and Venezuela share a long border. Costa Rica and the Dominican Republic share no borders with the rest of these countries, but the latter does share an island with Haiti. All these countries were a part of the Spanish Empire, although with diverse levels of importance. Costa Rica was the most “peripheral”. All four experienced distinct processes of independence from Spain. They have all had distinct relationships with the main power in the Americas, the United States. For example, the Dominican Republic was occupied various times by the United States while Costa Rica has always enjoyed a close relationship with the country to the north. Politically, all are representative democracies, but historically they have lived quite different conditions. Their levels of economic, social and educational development are distinct which indicates the need for care in analysing these realities.

The Structure of the Educational Systems

The following table provides a visualization of the educational structure of the four countries with respect to the school years from first to twelfth (Table 1.4).

In all these countries there are, of course, pre-school opportunities and diverse higher education systems. It can be said that currently there is a shared educational structure, although the names given to the different levels can vary a bit.

Why was it decided to focus on the initial and continuing preparation of mathematics teachers? Because, although it is not the only factor that should be

Table 1.4 The structure of the educational systems in Colombia, Costa Rica, the Dominican Republic and Venezuela: Years 1 to 12

Country Colombia Elementary Secondary Upper Secondary Grades 1-5 Grades 6-9 Grades 10-11 Costa Rica Elementary Secondary Grades 1-6 Grades 7-12"

Dominican Republic Basic* Media”

Grades 1-8 Grades 9-12 Elementary? Secondary?

Grades 1-6 Grades 7-12 Venezuela Elementary Secondary Grades 1-6 Grades 7-12"

“The structure of the educational system in the Dominican Republic when this book was originally drafted

PThe structure in the Dominican Republic beginning in 2014

“Most secondary schools have a total of 11 years while technical secondary schools have a total of 12 Source Mathematics Education Reports (2013)

analyzed, it is a crucial dimension for understanding what happens in mathematics education, as well as a powerful route for improving the mathematical capacities of the population.

To address issues related to initial and continuing preparation of teachers in these countries several common topics will be considered:

Institutions that offer initial teacher preparation.

Teacher preparation for elementary (grades 1 to 5 or 6) and secondary (grades 6 or 7 to 11 or 12). e Components of teacher preparation: content, general pedagogical, content-based pedagogy, student teaching, other subjects.

Areas of mathematics in the programs.

Institutions that provide professional development.

Professional development modalities.

Institutions that offer graduate programs.

Maturity of research in mathematics education.

The School Mathematics Curriculum

The preparation of mathematics teachers should be considered in the larger school curricular context. Venezuela, Costa Rica and the Dominican Republic each have a mandatory, official, national curriculum. In Colombia, however, there are general orientations that are followed in the different regions of the country, but an identical national curriculum for all does not exist.

With respect to mathematics, all these countries experienced in their own way the “Modern Mathematics” reform, which had emerged as a possible solution to an important problem for mathematics education: closing the gap between the mathematical practice of professional university research mathematicians, and the mathematics in elementary and secondary schools. Using the language of sets and with perspectives taken from university mathematics there was a desire to integrate mathematics as a single discipline. It was proposed that modern symbolism be adopted, that the use of graphs be given greater importance, that much of traditional algebra be eliminated, and that something extremely serious be modified and practically eliminated: traditional Euclidean Geometry. A famous war cry of some of the reformers was “Euclid must go” (J. Dieudonné). The reform began in Europe (especially France) and the United States. Later it was expanded to Latin America and other latitudes. Textbooks and curricular changes were the main mechanisms to drive the reform. The reform failed to achieve its initial objectives and was rejected by many educators, students, and even parents. Nevertheless, the actions and ideas that the reform promoted were dominant for almost 30 years. In different ways all of these countries backed away from the reform in the 1980s and 1990s as they were influenced by constructivism, systems theory or curricular models based on competencies.

During the 1970s and well into the 1980s much of the school mathematics curriculum in Colombia was determined by “Modern Mathematics”. It is perhaps noteworthy that the /st Inter-American Conference on Mathematics Education (ACME I) was held in Bogotá, Colombia, in December of 1961. Its main purpose was to promote the development of the modern mathematics reform in the Americas (Ruiz and Barrantes 2011). Afterwards Colombia experienced the influence of “systems theory” until the second half of the 1990s. At the time they began to emphasize the construction of knowledge and the development of thinking using “problem situations”, interdisciplinarity, action research, and a perspective based on competencies. In Colombia there are two seminal documents that orient school mathematics: Curricular Guidelines for Mathematics and Basic Standards for Mathematical Competencies, published in 1998 and 2006, respectively. These should be considered complementary. The lack of a specified national curriculum is being debated in the country.

In Venezuela the influence of modern mathematics was also felt. In 1965 the Pedagogical Institute (a very influential institution in that country) also assumed that reform model in the first three and that same year national mathematics programs with that orientation were officially approved. Venezuelans participated in the first three Inter-American Conferences on Mathematics Education and organized the fourth. This wave continued until 1980 when a national education law broke with modern mathematics and adopted the international perspective of “Back to the Basics”.

The modern mathematics reform also influenced mathematics programs in the Dominican Republic where a translation of the books of the School Mathematics Study Group (SMSG) from United States was used in teacher preparation, and later on in some high schools.

In Costa Rica, as in the other countries considered, modern mathematics dominated the school curriculum beginning in 1964 when new programs with this orientation were officially approved. The universities adopted teacher preparation processes based on the new orientation. It was not until the middle of the 1990s that the modern mathematics model was officially replaced, although it had already been largely abandoned in practice. The next strong influence was constructivism (although the approach was very general), and many dimensions of behaviorism continued to exist (particularly with respect to assessment). In 2012 there was an authentic revolution in mathematics education. A new national curriculum was approved based on problem solving with an emphasis on real contexts, and that introduced an innovative vision of the use of competencies.

Initial Teacher Preparation

In this work the types of institutions where teachers are prepared, the degrees that are given, the extent to which teacher preparation is related to the national school curriculum, as well as the curricular components of teacher education, are of interest. Identifying the specific pedagogy of mathematics (mathematics teaching methods courses) is of particular relevance, as it is an indication of the extent to which mathematic education in the country has developed as an independent discipline.

In some of the cases presented here, teacher preparation is designed for secondary teaching, but also serves for teaching mathematics in institutions of higher education (in programs that require mathematics), but not for doing research in mathematics.

In these countries initial teacher preparation is provided in universities and other higher education institutions (such as “normal schools” in the Dominican Republic). In each country the names of the undergraduate degrees are different (or the same names may refer to different degrees).

In Colombia, initial teacher preparation was in normal schools, then in higher normal schools, and eventually the “normal” programs were passed to schools of education in the universities. Since the middle of the first decade of the 21st century, teacher preparation has been considered to be “technical” rather than “professional”. With respect to secondary education, since the 1990s individuals with undergraduate degrees (“licenciados”) have taught mathematics in secondary schools. By the end of the last century, a basic undergraduate degree with a major in Mathematics (Licenciatura Básica con Enfasis en Matemáticas) has been offered for lower secondary teachers and an undergraduate degree in Mathematics (Licenciatura en Matemáticas) for teachers in upper secondary schools. However, in practice those with the degree in Mathematics work at all levels of secondary schools, as do those with degrees in other fields. Elementary school teachers have either received a very general teaching degree or a degree in some other field.

In the last 15 years, the initial teacher preparation programs in Colombia have moved from an emphasis on mathematics to an emphasis on pedagogy, which has led to the expansion of mathematics education as a field. The main emphasis to date has been to follow a curricular model based on competencies.

In general, initial teacher preparation programs in Colombia include mathematics, curriculum, mathematics pedagogy, general pedagogy, and elements of communication to support actions in the classroom. There is also a course in either physics or computer science. The mathematics courses include Arithmetic and Algebra, Calculus, Geometry, Probability and Statistics. With respect to curriculum and pedagogy, the courses are related to the national Curriculum Guidelines for Mathematics or to research results in mathematics education.

Since 1996, in the case of Venezuela, initial teacher preparation from elementary through secondary has been structured with four dimensions: general, pedagogical, specialized and professional practice. For secondary schools the initial preparation is in public universities. For elementary schools, both public and private universities provide programs. There are a variety of degrees offered to future secondary teachers: Mathematics Teacher; and Bachelor’s Degrees in Education with a major in Mathematics, in Mathematics, in Mathematics and Physics, in Mathematics Teaching or in Mathematics and Computer Science. The programs are for four or five years. The mathematics courses include Geometry, Calculus and Analysis, Algebra, Probability and Statistics. Except for one institution there are no courses specifically on mathematics pedagogy. Student teaching experiences vary widely across the country.

Elementary teachers that teach mathematics in Venezuela are prepared as generalists. They usually have three mathematics courses. Two of them attempt to relate to work in the classroom and the other (Geometry) emphasizes strengthening the logical, deductive and spatial reasoning of teachers.

In Venezuela there is a deep divide between the state educational agencies and the institutions that prepare teachers. In particular, there is not a close and consistent correlation between the official national curriculum and the programs for initial preparation (there is almost no mention of the school curriculum in the courses). The majority of the characteristics of the programs for initial preparation were set in the 1990s and have not been changed very much. A relevant detail is that there is a shortage of secondary mathematics teachers.

In the Dominican Republic most of the initial teacher preparation programs for grade 1-8 are in normal schools and universities under the coordination of the National Teacher Institute for Preparation and Professional Development (Instituto Nacional de Formación y Capacitación del Magisterio). Recently there has been a great demand for teachers particularly for grade 1-8. The teacher education programs were divided into grades 1—4 and grades 5-8. In those programs mathematics courses were no more than 10 % of the total and mathematics teaching methods courses were almost non-existent. For upper secondary education (grades 9-12) initial teacher preparation which is called “Secondary Education with a major in Physics and Mathematics” is structured with the usual dimensions: mathematics, general pedagogy, mathematical pedagogy, general education, etc. There are also courses in physics given the double major. The mathematics courses that are usually present are Algebra, Trigonometry, Higher Algebra, Statistics and Calculus. Mathematics teaching methods are usually confined to one course associated with student teaching. In the Dominican Republic the student teaching experience varies greatly from institution to institution.

In Costa Rica the public and private universities are charged with providing initial teacher preparation for both elementary and secondary teachers. Elementary teachers receive a bachelor’s degree (four years in the public universities) or a licentiate’s degree (5 or more years in the public universities). The program prepares generalists with at most two or three mathematics courses; in some private universities courses from other disciplines replace the mathematics courses. For secondary teaching the initial preparation can be a three-year Teaching degree often called a “profesorado”. This degree is considered to be a lateral exit from the bachelor’s or licentiate degrees. These three-year programs for preparing secondary mathematics teachers have courses in mathematics, general teaching methods, mathematics pedagogy, general education and student teaching. Beginning near the end of the first decade of the 21st century the public universities have made efforts to increase the time dedicated to mathematics pedagogy. As of 2016 this process is still being developed with different levels of success. The mathematics courses include abstract algebra, linear algebra, calculus and analysis, geometry and topology, probability and statistics, and number theory. Various private universities offer initial teacher preparation programs. Although their programs are fairly similar to those in the public universities, they usually require one or two years less. In Costa Rica the programs for initial teacher preparation, particularly at the secondary level, have had to make changes related to the new school mathematics curriculum adopted in 2012.