Mathematics And Its Teaching In The Southern Americas: With An Introduction By Ubiratan D'ambrosio

with An Introduction by Ubiratan D'Ambrosio

Hᅢᄅctor Rosario, Patrick Scott, Bruce Vogeli

• Publisher: WSPC
• Year: 2014
• ISBN:
  9789814590587
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This anthology presents a comprehensive review of mathematics and its teaching in the following nations in South America, Central America, and the Caribbean: Argentina, Bolivia, Brazil, Chile, Colombia, Costa Rica, Cuba, Guyana, Haiti, Honduras, México, Panamá, Paraguay, Perú, Puerto Rico, Trinidad and Tobago, and Venezuela. The last summary of mathematics education encompassing countries from the Southern Americas appeared in 1966. Progress in the field during five decades has remained unexamined until now.

Contents:

• ARGENTINA: A Review of Mathematics Education through Mathematical Problems at the Secondary Level (Betina Duarte)
• BOLIVIA: An Approach to Mathematics Education in the Plurinational State (A Pari)
• BRAZIL: History and Trends in Mathematics Education (Beatriz S D'Ambrosio, Juliana Martins, and Viviane de Oliveira Santos)
• CHILE: The Context and Pedagogy of Mathematics Teaching and Learning (Eliana D Rojas and Fidel Oteiza)
• COLOMBIA: The Role of Mathematics in the Making of a Nation (Hernando J Echeverri and Angela M Restrepo)
• COSTA RICA: History and Perspectives on Mathematics and Mathematics Education (Ángel Ruiz)
• CUBA: Mathematics and Its Teaching (Otilio B Mederos Anoceto, Miguel A Jiménez Pozo, and José M Sigarreta)
• GUYANA: The Mathematical Growth of an Emerging Nation (Mahendra Singh and Lenox Allicock)
• HAITI: History of Mathematics Education (Jean W Richard)
• HONDURAS: Origins, Development, and Challenges in the Teaching of Mathematics (Marvin Roberto Mendoza Valencia)
• MÉXICO: The History and Development of a Nation and Its Influence on the Development of Mathematics and Mathematics Education (Eduardo Mancera and Alicia Ávila)
• PANAMÁ Towards the First World through Mathematics (Euclides Samaniego, Nicolás A Samaniego, and Benigna Fernández)
• PARAGUAY: A Review of the History of Mathematics and Mathematics Education (Gabriela Gómez Pasquali)
• PERÚ A Look at the History of Mathematics and Mathematics Education (César Carranza Saravia and Uldarico Malaspina Jurado)
• PUERTO RICO: The Forging of a National Identity in Mathematics Education (Héctor Rosario, Daniel McGee, Jorge M López, Ana H Quintero, and Omar A Hernández)
• TRINIDAD and TOBAGO: Mathematics Education in the Twin Island Republic (Shereen Alima Khan and Vimala Judy Kamalodeen)
• VENEZUELA: Signs for the Historical Reconstruction of Its Mathematics Education (Fredy Enrique González)

Readership: Graduates and professionals in mathematics education; education planners.
Key Features:

• Featured introduction by Professor Ubiratàn D'Ambrosio of Brazil — the most prestigious of Latin American mathematics educators
• Insights into the impact of political changes of mathematics education in Cuba, Venezuela, Brazil etc.
• Historical references, not available elsewhere, are covered in this book

Chapter 1

ARGENTINA: A Review of Mathematics Education through Mathematical Problems at the Secondary Level

Betina Duarte

Abstract: We propose a review of the teaching of mathematics in Argentina based on the presence and evolution of problem-solving in textbooks for the secondary school. We also take into account the changes introduced by educational policies in curricular documents. The period selected (1950–2000) has significant influence on the current situation of teaching.

Keywords: History of mathematics education; mathematics education in Argentina; mathematics education; math problem solving; secondary school; math textbooks; curricular documents.

Historical Overview

While we have chosen a timeframe to discuss mathematics education in Argentina, we believe it is necessary and appropriate to provide a brief historical background to contextualize our proposed analysis.

We can surmise that the colonial education promoted and sustained largely by religious orders arriving to Río de la Plata during the mid-sixteenth century, established the first ideas about the teaching of mathematics.¹ Among the various groups who came to this region were the Jesuits who had greater presence through the establishment of institutions dedicated to education: education or nursing homes, colleges and universities. One of those, created in 1613 under the name Colegio Máximo, with the aim of granting bachelor’s and doctoral degrees, and train teachers, was next transformed into the Universidad de Córdoba del Tucumán in 1621. Its teaching structure emulated the University of Salamanca. It was administered by Jesuits until they were expelled in 1767. This was the first university in Argentina, now known as National University of Córdoba.

Buenaventura Suárez, a priest, astronomer, geographer, and mathematician (1679–1750), established an astronomical observatory. His calculations were used to select the exact locations of the thirty Jesuit missions in Paraguay (Stacco, 2011).

The same order, settled in what is now the center of our country, founded in 1662 the San Ignacio School in Buenos Aires, a great intellectual and cultural center. Similarly in 1685 they created the Monserrat School, an institution that prepared students for college. For many years it trained the future leaders of the region. Both schools — today depending on their respective national universities — contributed to the formation of those who gestated the emancipatory project of our country.

In 1773, the Viceroyalty propelled the creation of a school of applied mathematics by French engineer Joseph Sourryères of Souillac. The shortage of students forced the establishment to reorient their activities. Shortly after, around 1799, and thanks to the initiative of Manuel Belgrano, it transformed into the Nautical Academy, which included the teaching of arithmetic, geometry, spherical and rectilinear trigonometry, cosmography and differential and integral calculus.

His first 26 students used the texts of Etienne Bezout, Cours de mathématiques à l’usage du Corps Royal de l’artillery (Paris 1770–1772) and Benito Bails, Principles of Mathematics (Madrid, 1789–1798).

Manuel Belgrano was a central figure to ensure the presence of mathematics since the late viceroyalty to the early period of emancipation in the region. In this second stage, the teaching of mathematics was thought necessary for the instruction of the army because it was “the more useful science for a military man” and the more efficient means to form “smart military in the art of defense” (Arata and Mariño, 2013). This included the teaching of arithmetic, plane geometry, rectilinear trigonometry and practical geometry. Instruction was carried out in the School of Mathematics established in September 1810 (Gaceta de Buenos Aires, 1910).

The University of Buenos Aires, founded in 1821, had an Exact Sciences Department since its inception. Under the initiative of Felipe Senillosa, of Spanish origin and trained at the Academy of Engineering of Alcalá de Henares, Lagrange’s trigonometry texts, Lacroix’s Arithmetic, De Monge’s Descriptive Geometry and Poisson’s Principles of Mechanics were taught. This evidences the early influence that the French school of mathematics had at the beginning of the University of Buenos Aires, now the largest public university in Argentina. In 1872, driven by members of the Faculty of Exact Sciences, the Argentina Scientific Society was created to “promote, in particular the study of the mathematical, physical and natural sciences” (Stacco, 2011).

Before the end of the nineteenth century, three new universities were created: the Tucumán University (1875), La Plata University (1882), and Santa Fe University (1889). Discussion on a possible educational model, the role of the State, teaching contents, the conditions to be met by teachers, teaching methods and funding sources, were developed largely in the Pedagogical Congress of 1882 and crystallized by the 1420 Law, also known as Láinez Law, in 1884.

So far we have tried to present some historical milestones of education in Argentina, particularly in mathematics education, to give a context to our proposal. These facts do not deny the earlier presence of pueblos originarios (native peoples) in our region who, in one way or another, constructed mathematical notions. There was a mathematical knowledge developed by those who inhabited these regions that were neglected or destroyed by the effect of the arrival of the Spaniards. In order to make clear some kind of testimony to these cultures, it is appropriate to say that, for example, the Mapuche people, from the Patagonia region, “developed an oral decimal number system” to meet their need to count and, therefore, limited to numbers less than 10,000 (Belloli, 2009). Also, the north territories populated by Kolla and Aymara communities, significantly influenced by the Inca culture, used yupanas to count and quipu to record information. The numbering system of these communities was also decimal. The geometry was present in their clothing designs and the construction of their homes.

Foundational Ideas

The teaching of problem-solving in mathematics has attracted the attention of various educational communities in Argentina: teachers, educators, researchers, school heads, government officials and their respective technical teams, and textbook publishers. Some of them are also interested in problem-solving as a way to teach mathematics. These are two different matters — different approaches to the same topic, namely, how math problems are conceived in the classroom. The two views indicate different conceptions about the teaching of mathematics.

Some of the questions regarding the first topic we have indicated center around the problems themselves. How is the solving of problems taught? What is the goal? Based on what mathematical topics? Does mathematics begin and end with the solving of problems?

The questions that arise when considering the possibility of teaching mathematics on the basis of problem-solving brings to light the need to think about classroom management and the link between the problems and the mathematical knowledge one intends to teach. How do math problems exist within a secondary school classroom? Who presents them? Who solves them? How is a class organized around the solving of a problem? What part of the student’s problem-solving task should be left to the individual, and what part should be dealt with in the collective space of the classroom? How can interaction among students be generated while problems are being solved?

To answer whether problem-solving can be a way of teaching mathematics, we must accept that problem solving has certainly been and continues to be a powerful driver of mathematical output. One has only to look at the colossal production that resulted from Fermat’s marginal note on his copy of Diophantus’ Arithmetica. We can also point to the 23 problems posed by David Hilbert at the International Congress of Mathematicians in Paris at the beginning of the twentieth century (Stacco, 2011).

Nevertheless, turning away from the production of mathematics to the teaching of mathematics, when research in math education establishes the existence of a strong link between the mathematics teaching and learning process and the mathematical production of students, an impulse is given to the idea of considering problem-solving as a process capable of being interwoven in the learning process. Research on this topic (G. Brousseau, M. Artigue, R. Charnay, J. Kilpatrick and A. Schoenfeld, among others) has had repercussions in Argentina.

The contributions of these researchers confirm the need for problems to be considered in interaction with the subject that resolves them. In addition, they promote the idea that problem-solving gives meaning to the mathematical notions in play. We will take these approaches as a starting-point for the study of the presence of problems in the textbooks we have selected.

Our proposal in this chapter consists of reviewing some of the history of the presence of problem-solving in the teaching of mathematics in Argentina. To do so, we use two points of reference: (1) textbooks and teaching manuals — books that have defined secondary school teaching in the country for many generations of Argentines, and (2) curricular educational policy documents. We have considered it advisable to restrict the period to be considered. Hence, we propose reviewing two eras: the period between the 1950s and 1970s inclusive, and the period from the return of democracy until the end of the 1990s. We highlight certain historical landmarks that represent significant changes in the way mathematics has been taught, as well as changes in the customs and traditions of the publishers of such texts.

An Anachronistic Example

We have chosen to include an example from outside the timeframe we have adopted, as well as from outside the secondary school system that concerns us, because of its originality. In 1901, a book written by a “normal professor”² named Alfredo Grosso under the title of Arithmetic Exercises and Problems for Use by Standard Schools, dedicated a chapter to methodological considerations regarding “studying together”. He points out to students the advantages of sharing the task of problem-solving with peers, based on the description of the studying process of two hypothetical children (from the fourth grade of primary school)³:

When it came to problem-solving, they acted as follows: Each one, separately, solved the problems in his own manner, and once solved, wrote the result on the blackboard; if the answers were the same, they checked their calculations to verify whether their solutions, despite arriving at the same result, were the same or not. If the results differed, each explained his answer, discussing it until arriving at a solution they considered to be exact. If there was no agreement, they submitted the problem with their own solution (Grosso, 1907).

At first sight we find the validity and currency of such words stressing the importance of interaction among peers and the clarifying of strategies by the students. Images of classrooms in the early twentieth century show students organized in a manner (the classic wooden bench pattern) that is not conducive to group study.⁴ Such a structure would only begin to be seen at the beginning of the 1970s. At the start of the twentieth century, classrooms were designed for individual work by students at all levels of learning.

A second reading of the paragraph in question allows us to infer that the problems to which the author refers have a “single” correct, numerical answer, which serves as a starting point for consideration of procedures or “solutions” in the form of calculations based on what is indicated next, reinforcing the idea that mathematics is about performing calculations. It therefore states, albeit implicitly, that (a) mathematical problems have “a result or solution”, (b) such a result derives from numerical calculations, and (c) desirable solutions in mathematics are precise, exact, and unique.

These inferences seem to describe the concept of mathematics that has been transmitted through the teaching of mathematics, at least since the start of the twentieth century. In this chapter we document the scope of this approach.

First Stage: Textbooks from the 1950s to the 1980s

Publisher Kapelusz brought out the first edition of a series of books for all secondary school years in their different categories (regular high school, commercial and technical) in 1940, all of them written by Celina Repetto,⁵ Marcela Linskens⁶ and Hilda Fesquet.⁷ A dozen mathematics textbooks were written and re-written. In the early 1950s these textbooks were widely used in the country’s various types of secondary schools. Some ran into over 20 editions through the end of the 1970s.⁸ In the mid-1960s, they incorporated the proposals for change formulated for Latin America⁹ with the introduction of the teaching of the new math.

The tables of contents of these books all have something in common. At the end of each chapter came an identical heading in bold print: Applied Exercises and Problems.

The term “exercises” is applied principally to the performance of mathematical operations that can result in simplification (w...

Table of contents

• Cover
• Halftitle
• Editions
• Title
• Copyright
• Content
• Preface
• Introdution
• Chapter 1 ARGENTINA: A Review of Mathematics Education through Mathematical Problems at the Secondary Level
• Chapter 2 BOLIVIA: An Approach to Mathematics Education in the Plurinational State
• Chapter 3 BRAZIL: History and Trends in Mathematics Education
• Chapter 4 CHILE: The Context and Pedagogy of Mathematics Teaching and Learning
• Chapter 5 COLOMBIA: The Role of Mathematics in the Making of a Nation
• Chapter 6 COSTA RICA: History and Perspectives on Mathematics and Mathematics Education
• Chapter 7 CUBA: Mathematics and Its Teaching
• Chapter 8 GUYANA: The Mathematical Growth of an Emerging Nation
• Chapter 9 HAITI: History of Mathematics Education
• Chapter 10 HONDURAS: Origins, Development, and Challenges in the Teaching of Mathematics
• Chapter 11 MÉXICO: The History and Development of a Nation and Its Influence on the Development of Mathematics and Mathematics Education
• Chapter 12 PANAMÁ: Towards the First World through Mathematics
• Chapter 13 PARAGUAY: A Review of the History of Mathematics and Mathematics Education
• Chapter 14 PERÚ: A Look at the History of Mathematics and Mathematics Education
• Chapter 15 PUERTO RICO: The Forging of a National Identity in Mathematics Education
• Chapter 16 TRINIDAD and TOBAGO: Mathematics Education in the Twin Island Republic
• Chapter 17 VENEZUELA: Signs for the Historical Reconstruction of Its Mathematics Education
• Epilogue A Half Century of Progress
• Index