13
ORIGINAL ARTICLE The history of mathematics education in Brazil Joa ˜o Bosco Pitombeira de Carvalho Bruno Alves Dassie Accepted: 6 June 2012 / Published online: 20 June 2012 Ó FIZ Karlsruhe 2012 Abstract This paper describes the broad lines of the development of mathematics education in Brazil since 1500, emphasizing the development of secondary mathematics education. We divide this history into seven major periods, based on the political and cultural development of Brazilian society, and stress the characteristics of each period. Keywords History of mathematics education Á History of education 1 Introduction In this paper we try to unravel the development of math- ematics education in Brazil, from the sixteenth to the twentieth century. In the last few years, this history has been actively pursued, but there are as yet no general surveys of the field. This paper is a contribution toward filling this gap, with the hope that it will encourage others to offer their own interpretations. A great part of our discussion will focus on secondary mathematics education, whose history has been more studied in the last decades and which displays character- istics very tied to the evolution of Brazilian society as a whole. It is also easier to study, since it has almost always been regulated by the central government, and so has had a comparatively more homogeneous structure than elemen- tary school mathematics education .Also, secondary edu- cation has always been a thorn in all discussions about educational policies in Brazil. It has been viewed, at dif- ferent times, as preparation for post-secondary studies, as professional education for middle level careers, or as a way to prepare all citizens, male or female, for full participation in society. This last view has been the trend for the last 20 or so years, and vindicates the view of many important Brazilian educators (Teixeira 1936, 1969, 1994). So, the study of the evolution of secondary education has much to tell about Brazilian society. Also, as part of political and cultural characteristics of Brazilian society, primary level education was institutionalized much later than secondary education, only in 1946, in contrast with secondary edu- cation, which started being organized in 1837. We begin with a brief sketch of Brazilian history, as a frame for our discussion. A comprehensive view of this history may be found in Fausto (1994, 1999) and Skidmore (1999). Next, we propose a division of the history of mathematics education in Brazil into seven periods, fol- lowed by a discussion of their characteristics. Lastly, we attempt some broad interpretations of the subject. It is impossible to give due credit to all authors. We present our apologies for this and welcome comments, information on unmentioned research, and comparison of viewpoints. Also, many situations painted here as black and white do actually have many shades of color in between, but this is unavoidable in such a general presentation. 1.1 A brief sketch of Brazilian history Brazil, like other European ex-colonies, received its corpus of school mathematics ready-made from its ‘‘mother J. B. P. de Carvalho (&) Programa de po ´s-graduac ¸a ˜o em ensino de Matema ´tica, Instituto de Matema ´tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] B. A. Dassie Faculdade de Educac ¸a ˜o, Universidade Federal Fluminense, Nitero ´i, Brazil e-mail: [email protected] 123 ZDM Mathematics Education (2012) 44:499–511 DOI 10.1007/s11858-012-0439-5

The history of mathematics education in Brazil

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Page 1: The history of mathematics education in Brazil

ORIGINAL ARTICLE

The history of mathematics education in Brazil

Joao Bosco Pitombeira de Carvalho •

Bruno Alves Dassie

Accepted: 6 June 2012 / Published online: 20 June 2012

� FIZ Karlsruhe 2012

Abstract This paper describes the broad lines of the

development of mathematics education in Brazil since 1500,

emphasizing the development of secondary mathematics

education. We divide this history into seven major periods,

based on the political and cultural development of Brazilian

society, and stress the characteristics of each period.

Keywords History of mathematics education �History of education

1 Introduction

In this paper we try to unravel the development of math-

ematics education in Brazil, from the sixteenth to the

twentieth century. In the last few years, this history has

been actively pursued, but there are as yet no general

surveys of the field. This paper is a contribution toward

filling this gap, with the hope that it will encourage others

to offer their own interpretations.

A great part of our discussion will focus on secondary

mathematics education, whose history has been more

studied in the last decades and which displays character-

istics very tied to the evolution of Brazilian society as a

whole. It is also easier to study, since it has almost always

been regulated by the central government, and so has had a

comparatively more homogeneous structure than elemen-

tary school mathematics education .Also, secondary edu-

cation has always been a thorn in all discussions about

educational policies in Brazil. It has been viewed, at dif-

ferent times, as preparation for post-secondary studies, as

professional education for middle level careers, or as a way

to prepare all citizens, male or female, for full participation

in society. This last view has been the trend for the last 20

or so years, and vindicates the view of many important

Brazilian educators (Teixeira 1936, 1969, 1994). So, the

study of the evolution of secondary education has much to

tell about Brazilian society. Also, as part of political and

cultural characteristics of Brazilian society, primary level

education was institutionalized much later than secondary

education, only in 1946, in contrast with secondary edu-

cation, which started being organized in 1837.

We begin with a brief sketch of Brazilian history, as a

frame for our discussion. A comprehensive view of this

history may be found in Fausto (1994, 1999) and Skidmore

(1999). Next, we propose a division of the history of

mathematics education in Brazil into seven periods, fol-

lowed by a discussion of their characteristics. Lastly, we

attempt some broad interpretations of the subject.

It is impossible to give due credit to all authors. We present

our apologies for this and welcome comments, information

on unmentioned research, and comparison of viewpoints.

Also, many situations painted here as black and white do

actually have many shades of color in between, but this is

unavoidable in such a general presentation.

1.1 A brief sketch of Brazilian history

Brazil, like other European ex-colonies, received its corpus

of school mathematics ready-made from its ‘‘mother

J. B. P. de Carvalho (&)

Programa de pos-graduacao em ensino de Matematica,

Instituto de Matematica, Universidade Federal do

Rio de Janeiro, Rio de Janeiro, Brazil

e-mail: [email protected]

B. A. Dassie

Faculdade de Educacao, Universidade Federal Fluminense,

Niteroi, Brazil

e-mail: [email protected]

123

ZDM Mathematics Education (2012) 44:499–511

DOI 10.1007/s11858-012-0439-5

Page 2: The history of mathematics education in Brazil

country’’. This should be taken into account when dealing

with problems of periodization, and we propose one based

on the broad lines of the country’s history and education.

Brazil was discovered by Portugal in 1500, and

remained a Portuguese colony till 1808, under strict control

from the mother country. In 1808, the Portuguese gov-

ernment, fleeing from Napoleon’s armies, crossed the

ocean and settled in Brazil, which became the seat of the

Portuguese empire. This fact had deep consequences for

Brazilian society, including its culture, science, and edu-

cation. In 1822, Brazil became independent from Portugal,

under the rule of Pedro I, the son of the Portuguese king.

He was succeeded by his son, Pedro II, who ruled from

1841 through 1889.

In 1889 the country became a republic, following which

the 1920s and 1930s were a period of great social changes,

with a coup d’etat by Getulio Vargas in 1937, who

remained in power till 1945. The republic was reinstituted

in 1945, and lasted till 1964, when the military took over.

They withdrew gradually from power, and in 1985 a

civilian president was elected.

2 The major periods of mathematics education

in Brazil

Since mathematics education—as a matter of fact, all of

education—depends on the society in which it is considered,

the divisions we propose are based on important political or

educational events that had considerable influence on the

development of mathematics education in Brazil, as will be

explained in the discussion of each period.

2.1 First period (1500–1808): the period of Jesuit

dominance

2.1.1 Background

In Portugal, as in many European countries, from the

Renaissance on, there was a growing need of mathematical

expertise, which was provided in technical and professional

schools. Since Portugal was a Catholic country, and followed

Spain as a stronghold of the Catholic counter-reformation, the

Jesuits played an important role in its educational system.

There were three Jesuit colleges in Portugal, at Lisbon,

Coimbra, and Evora. Starting from 1590, in Lisbon, at the

Colegio de Santo Antao of the Jesuits, we find the ‘‘Aula da

Esfera’’, of cosmography and theoretical navigation, which

was not restricted to the regular students and could be attended

by noblemen and other persons interested in the nautical arts.1

2.1.2 The beginnings of mathematics education in Brazil

The Jesuits came to Brazil in 1549, with the first ‘‘Gov-

ernador Geral’’ (Governor General), and set up reading and

writing schools for the Indians, as part of their efforts to

convert them to the Catholic religion. Later on, these

schools were opened to Europeans and native non-Indian

Brazilians. Around the end of the sixteenth century, the

Jesuits made a clear option for secondary and post-sec-

ondary education. This contrasted strongly with the view of

Manoel da Nobrega (1517–1570), who was the first Pro-

vincial of the Jesuits in Brazil and believed that the Jesuits’

efforts should concentrate on the teaching of children only

in elementary reading and writing schools. The Jesuitsre-

structured schools along the lines of the Ratio studiorum

and so created schools for the teaching of the ‘‘liberal arts’’

to the sons of the landed gentry who were expected to go to

Coimbra, in Portugal, to study law and theology, or to

Montpellier, in France, to study medicine.

Very little is known about the teaching and learning of

mathematics in the Jesuit schools in Brazil. Leite (2006,

p. 163), one of the standard sources on the subject, states:

The teaching of mathematics in Brazil started, of

course, as it should, that is, with lessons on numbers,

or the first operations, a teaching which was gradually

improved, and in 1605, in the schools [colegios] in

Bahia and Rio de Janeiro, there is mention of arith-

metic lectures.2

Very early on, after its separation from Spain in 1640,

Portugal started sending military engineers to Brazil,

to build the forts and fortresses along the coast. In

1699 Portugal created fortification schools in Salvador,

Maranhao, and Rio de Janeiro. This was followed, in 1738,

by the establishment in Rio de Janeiro of an artillery and

fortification school (Aula de artilharia e fortificacoes).

In 1759, the Marquis of Pombal3 expelled the Jesuits

from the Portuguese empire. Fortunately, some other reli-

gious orders had schools in Brazil, and this allowed the

survival of organized teaching (Azevedo and de 1963,

p. 255). Also, some Jesuits went on providing schooling,

either as lay priests or affiliated to other religious orders.

To fill the void created by the expulsion of the Jesuits,

Pombal created the aulas regias, which consisted of lec-

tures on a specific subject; from 1772 on they were

financed by a new tax, the subsıdio literario. These aulas

1 See Baldini (2004) for a detailed exposition of the teaching of

mathematics at Santo Antao.

2 All translations from Portuguese to English were made by the

authors of this paper.3 Sebastiao Jose de Carvalho e Melo, Marquis of Pombal

(1699–1782), was State Secretary to Dom Jose I, from 1750 through

1777. He is one of the most controversial and influential figures in

Portuguese history. See, for example, Maxwell (1995).

500 J. B. P. de Carvalho, B. A. Dassie

123

Page 3: The history of mathematics education in Brazil

took care of almost all secondary level education and were

the first public education schools in Brazil (Cardoso 2002).

Initially, in 1759, Pombal authorized the creation of

aulas of Greek, Latin, and rhetoric. Nevertheless, slowly,

along the years, some of the aulas introduced a new ped-

agogical line, and were dedicated to mathematics, modern

languages, and drawing (Silva 1969, pp. 188–189).

According to the same author, these changes pointed

towards an increasing characteristic of secondary education

in Brazil: its eclectic character, without the ‘‘hegemonic

imposition of an educational frame’’ (Silva 1969, p. 190).

At the end of this period, in 1800, we witness the cre-

ation of a Catholic seminary at Olinda, Pernambuco, set up

following the ideas of the enlightenment (Alves 1993).

This seminary offered a regular course and stressed

the importance of the sciences, but very little is known

about the mathematics taught (Azevedo and de 1963,

pp. 326–327).

As befits a colonial society based on slavery, education

was viewed strictly as a provider of the professionals

needed for administration, trade, defense and minor roles in

society, and as a means of preparing the male offspring of

the colonial elite for their duties and roles. So, formal

education, including mathematics education, was almost

completely for men. Women learned, at most, to write

and read. The lack of formal education of women up to

the nineteenth century was often commented on by for-

eign visitors (Freyre 1946, 1980, 1986; Machado 1953;

Vasconcelos 2004).

2.2 Second period (1808–1837): mathematics

education in a new setting

The transfer of the Portuguese Crown to Brazil had far-

reaching consequences. Soon after his arrival in 1808,

D. Joao VI created schools of medicine in Salvador and in

Rio de Janeiro (1808); a course on economic sciences in

Rio de Janeiro (1808); and, more importantly for us, the

Real Academia de Guardas-Marinha, to prepare naval

officers for the defense of the long unprotected coast, and

the Academia Real Militar da Corte (1810), to prepare

army officers. Later, this school became the Escola Cen-

tral(1858) and, still later on, the Escola Politecnica. Very

soon the Academia Real Militar opened its doors to non-

military students. From 1858 on, it offered three courses of

instruction: a theoretical course on mathematics, physics,

and the natural sciences; a course of military engineering;

and a course of civil engineering.

To prepare students for these courses it was required to

provide them with mathematical knowledge. This was done

mainly in the aulas regias, isolated, dedicated to a single

subject and which lasted till 1834. And of course mathe-

matics books were needed. As early as May 1808, D. Joao

VI established the Impressao Regia in Rio de Janeiro,

which began immediately to publish translations of math-

ematics and science books into Portuguese, for example

Legendre’s Elements de Geometrie, which came out in

1809, Monge’s Geometrie descriptive, and Euler’s Vol-

lstandige Anleitung zur Algebra Silva and da (2009).4

The creation and expansion of post-secondary profes-

sional schools forced, inevitably, the improvement of math-

ematics teaching at the lower levels. Propaedeutic courses in

mathematics were needed to prepare the future students of the

engineering, medical, and military schools. Spix and Mar-

tius,5who visited Brazilbetween 1817 and 1820, mention that

in Rio de Janeiro, the seat of the Portuguese empire, one could

find ‘‘several good institutions for young people. The rich hire

private tutors to prepare their sons [of the rich] for Coimbra

University’’ (Cardoso 2002, p. 181). Some of these private

schools, most of them owned by foreigners, provided good

secondary education for those that could afford it. Primary

education was very lax, and the rich prepared their children

for secondary school at home. The very rich had, sometimes,

‘‘live-in’’ tutors. But this should not hide the fact that there

was not an organized and effective public education system

and that education was a privilege of a very few. By and large,

the government favored post-secondary professional educa-

tion, required for the pressing administrative, technical, and

military needs of the country.

In 1826 we witness the creation of the Liceu Provincial

de Pernambuco, whose very name reflects French influ-

ence. Ten years later, in 1836, a similar institution was

created in Salvador, the Liceu da Bahia, followed the very

next year by the establishment of Colegio Pedro II in

Rio Janeiro. According to Silva (1969, p. 191), these two

last-mentioned institutions arose from the union and

organization of already existing aulas, and they reflect a

long-standing characteristic of secondary education in

Brazil: the attempt to reconcile the classical tradition of

education based on the humanities and the Latin language,

and the need for new scientific studies and for modern

languages, required for post-secondary studies.

On the other hand, Cardoso (2002, p. 57) states that we

see in Brazilian secondary education, from the very first

attempts in 1837 to set up an organized system, a two-fold

character: first, a propaedeutic one, of preparation for

4 We believe that the exceptionality of this situation has not been

sufficiently studied, at leastregarding mathematics. We witness the

translation and publication of very up-to-date mathematics books in a

definitely backward society, just a few years after their appearance in

Europe.5 Johann Baptist von Spix (1781–1826), a German naturalist, visited

Brazil from 1817 to 1820, with Carl Friedrich Philipp von Martius

(1794–1868), a German botanist and explorer. They visited several of

the southern and eastern provinces of Brazil and went up part of the

Amazon River.

The history of mathematics education in Brazil 501

123

Page 4: The history of mathematics education in Brazil

further studies; and, secondly, its wide reach, in an attempt

to preserve part of the old humanistic educational tradition

inherited from the Jesuits. As a consequence, there was

never a really unified curriculum, with clear educational

goals. This can be best seen in the curriculum of Colegio

Pedro II, which included, for many years, Greek, Latin,

Rhetoric, French, English, and elements of Geography,

History, Philosophy, Zoology, Mineralogy, Algebra,

Geometry, and Astronomy!

A factor that contributed to the inclusion of mathematics

in the cultural formation of all students was the decision to

incorporate geometry in the entrance examination to the

two law schools, established in 1827 in the cities of Sao

Paulo and Olinda, the latter in the province of Pernambuco,

near Recife. Since many young men, particularly the sons

of the landed gentry, entered law schools, this inclusion

had a far-reaching effect (Valente 1999, p. 79).

As a sign of the need to organize a system of secondary

education, Cardoso remarks that, in 1821, pressure from the

city of Rio de Janeiro succeeded in the setting up of a pro-

fessional school, in the building of a religious seminary. Its

curriculum included drawing and geometry (Cardoso 2002,

p. 196). This was the seed from which was born, 16 years later,

Colegio Pedro II, an epoch-making institution, as we shall see.

In 1834 there was an amendment to the 1824 Consti-

tution, granting the provinces more autonomy. This inclu-

ded a very important step towards the organization of a

secondary education system: the aulas regias were abol-

ished, and primary and secondary education became the

responsibility of the provincial governments with central

government being responsible for post-secondary educa-

tion. This ended, at least in principle, the widespread dis-

order in the educational system of the country. In 1837, the

Seminario de Sao Joaquim was transformed into a regular

secondary school, called Colegiode Pedro II. The period

from 1837 till the end of the monarchy in 1889 will be

discussed in the next section.

Adapting the views of Vasconcelos (2004, p. 34), one

can say that, during the eighteenth and nineteenth centu-

ries, there coexisted the following kinds of schooling:

• Public education, mostly non-seriated, supported by the

state or by institutions subjected to it […].

• Education, mostly non-seriated, provided by religious

establishments.

• Private teaching, mostly non-seriated, in private insti-

tutions or in the teachers’ homes […].

• Home teaching, in the students’ homes, by teachers

hired by parents.

In the 1820s and 1830s, with the establishment of sev-

eral liceus, among them Colegio Pedro II, organized public

teaching slowly increased its participation in the educa-

tional activities in Brazil.

Who were the teachers of mathematics during this per-

iod? First of all, after the Jesuits were expelled in 1759,

some other religious orders (the Franciscans, for example),

already had schools in their convents that were able to

provide more or less organized teaching. As already

mentioned, some Jesuits became lay priests and went on

teaching. Also, some professionals (the military, lawyers,

doctors) and foreigners taught or earned their often meager

living solely by teaching.

Practically anyone could set up a ‘‘reading and writing’’

classroom. In small towns and villages, sometimes the

local priest took care of this. The big estates of the landed

gentry often had a resident chaplain, who served as tutor

for the owner’s children (Freyre 1986). To become a tea-

cher paid by the government one had to pass an exam,

given by a ‘‘banca’’, which issued a teaching permit. Car-

doso (2002) was able to collect actual sheets of these

exams, for the period 1797–1807, which consisted of two

tests: one in arithmetic and the other in written spelling.

Similarly, Soares (2007) studied who were the teachers of

mathematics from 1759 through 1879. She located many

exam sheets of the candidates and studied their mathe-

matical content, which was indeed very elementary.

2.3 Third period (1837–1889): the struggle

between systematization and fragmentation

It is generally accepted that the establishment of Colegio

de Pedro II, or simply Colegio Pedro II, was a decisive

event for the organization of secondary education in Bra-

zil.6 For the first time, there was a public institution with a

clearly defined curriculum. Students of secondary educa-

tion establishments that adopted Pedro II’s curriculum did

not need to pass further examinations to enroll in post-

secondary professional schools, a fact that led to a pro-

gressive consolidation of a very stable secondary school

mathematics corpus.

Miorim stresses the fact that with the establishment of

Colegio Pedro II there was, for the first time:

(…) a gradual and integral plan of studies for secondary

school, in which the students were promoted year by

year, and not, as before, by subjects and earned, at the

end of the course, a bachelor’s degree, which enabled

them [from 1843 on] to enroll in post secondary

establishments, without the need to undergo entrance

examinations. In this plan of studies, that followed

6 The standard reference for the history of secondary education

during the second empire is Haidar (1972).

502 J. B. P. de Carvalho, B. A. Dassie

123

Page 5: The history of mathematics education in Brazil

French models, the emphasis were the classics and the

humanities. (Miorim 1988, p. 87).

How was mathematics distributed along the school years

in Pedro II? Fig. 1, adapted from Beltrame (2000), shows

this distribution from 1838 through 1932. One sees that the

duration of the course varies from a minimum of 5 to a

maximum of 8 years, while mathematics was not always

taught in all school years.

Changes in the length of secondary education and in

the distribution of mathematics courses through the school

years were a consequence of educational reforms, care-

fully detailed by Haidar (1972).7A hypothesis to explain

the distribution of mathematics in the several programs is

the existence of the ‘‘isolated exams’’ (exames parcelados,

exames avulsos). These enabled students to present

themselves, whenever they felt prepared, to an isolated

exam on one of the areas of school mathematics: arith-

metic, algebra, and geometry (including trigonome-

try).8So, it did not matter at which period of the

curriculum mathematics was taught. We further remark

that lecture attendance was not required, and that the

student could even request to take the exams without

being enrolled in any school. In addition, he could change

schools whenever he wished.

The first geometry textbook adopted at Pedro II was

Lacroix’s9 Elementos de Geometria, translated from

the French by Manuel Ferreira de Araujo Guimaraes.10

From 1856 until 1869, the geometry (Elementos de

Geometria e Trigonometria), algebra (Elementos de

algebra), and arithmetic (Elementos de Aritmetica) text-

books were those written by Christiano Benedicto Ottoni.

According to Beltrame (2000), until 1869 his textbooks

were the only ones used at Pedro II. In 1870, he started

losing this position, but his geometry and trigonometry

texts were used until 1880.

In the period 1856–1892 all textbooks used at Pedro II

were written by Brazilians. In 1892, we note the appear-

ance of an arithmetic textbook written by a Portuguese

author, the Tratado elementar de arithmetica, by Jose

Adelino Serrasqueiro,11widely used both in Portugal and

Brazil.

The mathematics curricula of Pedro II had a great

influence in the establishment of a very stable corpus of

secondary school mathematics in Brazil. The fact that the

different subjects of secondary school mathematics were

studied separately, with their own textbooks, should not

obscure the fact that one had, indeed, a corpus of secondary

school mathematics, mainly of a propaedeutic nature as

preparation for post-secondary studies in the professional

schools already established in Brazil in this period: law,

medicine, and engineering.

As far as subjects are concerned, we can roughly

describe the corpus of secondary school mathematics,

instituted during the period 1837–1889, as follows:

Arithmetic—operations with numbers (including roots);

‘‘complex numbers’’12; proportions; ratio13; arithmetic

and geometric progressions; logarithms; commercial

mathematics (simple and compound interest, ‘‘rule of

three’’, etc.); the decimal metric system.14

Algebra—algebra as generalized arithmetic; linear and

quadratic equations with applications.

1º ano 2º ano 3º ano 4º ano 5º ano 6º ano 7º ano 8º ano1838–1840 1841–1855 1856–1857 1858–1861 1862–1869 1870–1876 1877–1878 1879–1897

1898 1899–1914 1915–1918 1919–1925 1926–1928 1929–1930 1931–1932

With mathematics Without mathematics

Fig. 1 Distribution of mathematics courses in school years at Pedro

II, 1838–1932

7 The graph shows the year each reform was actually implemented,

not the year it was approved by the Brazilian Congress.8 Usually, the presentation of mathematics in the programs followed

this order.9 Sylvestre Francois Lacroix (1765–1843) was a French mathema-

tician who authored many books for secondary and post-secondary

education. For his activities as a mathematics author see, for example,

Schubring (2003a).

10 Guimaraes deserves high praise for the diffusion, in Brazil and

Portugal, of good up-to-date mathematics books. Among his trans-

lations, one should mention Legendre’s Elementos de Geometria, one

of the first books printed by the Impressao Regia, in Brazil.11 Jose Adelino Serrasqueiro published a mathematics textbook

collection for secondary schools, influenced by Joseph Louis Francois

Bertrand. His works were innovative, in Portugal, due to the fact that

they had lists of exercises at the end of the sections, just like

Bertrand’s books.12 In nineteenth century school mathematics in Brazil, a ‘‘complex

number’’ was a ‘‘number’’ with more than one unit: degrees and

seconds; pounds, shillings, and pence; yards, feet, and inches, etc.13 The study of proportions and ratio was purely arithmetical. It had

nothing to do with Euclid’s treatment of these concepts.14 The decimal metricsystem wasofficiallyadopted in Brazil, by law,

in 1862, but only in 1872 did this law really become effective.

The history of mathematics education in Brazil 503

123

Page 6: The history of mathematics education in Brazil

Geometry (elementary plane and space geometry)—

relative position of straight lines; triangles; quadrilater-

als; polygons; the circle; similarity of plane figures;

relative position of straight lines and planes; polyhedra;

spheres, cones, and cylinders; areas and volumes.

Trigonometry—proofs of formulae; construction of

trigonometric tables; the theory of triangles.

One very important development during this period was

the creation of ‘‘normal schools’’, which prepared teachers

for the primary schools. The first one was created in 1835

in Niteroi, a town next to the city of Rio de Janeiro, and

was one of the first normal schools in the Americas, pre-

ceded only by the Escuela Normal de Ensenanza Mutua of

Uaxaca, in Mexico, established in 1824. One should note

that only in 1880 do we see a normal school in the city of

Rio de Janeiro (the Escola Normal da Corte). After this,

several of the provinces opened their own normal schools,

which played a very important role in the preparation of

elementary school teachers and in later mathematics edu-

cation reforms.

In this period, the educational system begins to take a

more organized shape. We have the professional and

military schools, Colegio Pedro II and similar establish-

ments in the provinces, private schools for the children of

the rich (often run by Englishmen or Germans, for boys,

and by French, for girls), and the growth in numbers of

the normal schools. All these contributed to the

improvement of the mathematical level of general edu-

cation, and to the establishment of a new profession, or, at

least, activity: the mathematics teacher. This is studied,

for example, among others, in Soares (2007). During this

period, the country followed the French model for sec-

ondary and post-secondary education with the liceus,

inspired by the French lycees, and the several profes-

sional schools. On the other hand, English and American

influences are felt in primary education (Lorenz and

Vechia 2005; Neves 2006, 2007, 2008, 2009; Gomes

2011; Valente 2012).

Haydar (1972) and Zotti (2005) stress that the great

problem in secondary education during this period and in

the next, up to 1930, was the provision that students could

go to Pedro II or to other secondary schools in the prov-

inces and take examinations in each subject till they had

completed the requirements to enter post-secondary

establishments. These ‘‘partial exams’’ foiled repeated

attempts to make students complete the full secondary

school course. Only in 1931 was regular attendance made

unequivocally mandatory in secondary school education.

One should also note that there was no evidence in this

period of the idea of ‘‘mathematics for all’’, that is, of basic

common mathematical knowledge as part of the education

of all citizens.

2.4 Fourth period (1889–1930): mathematics education

in the First Republic

The monarchy was abolished in 1889. In 1890, the new

regime created the Ministerio da Instrucao Publica, Cor-

reiose Telegrafos (Ministry for Public Instruction, Postal

and Telegraph Services), which was headed by Benjamin

Constant, a leading republican and a staunch follower of

Augusto Comte’s positivism.15Slowly, the country mod-

ernized itself. The evolution from an agrarian society based

on slavery to an urban industrialized society based on free

labor generated many tensions, which came to a head in the

1920s (Nagle 1974).

In 1889 we witness the first republican educational

reform, when Benjamin Constant was Minister of Educa-

tion. It was attacked both by positivists, who claimed it did

not follow Comte’s pedagogical ideas (Comte 1851, 1986;

Dias 2002b; Rocha and da 2006), and by the defenders of a

humanistic education, because the reform introduced sci-

entific subjects into the curriculum. Since the traditional

subjects were not abolished, secondary education assumed

a truly encyclopedic scope.16 From 1889 through 1898 the

ideas introduced by Benjamin Constant’s reform were

systematically eroded (Llopis 2008).

With the Republic, Pedro II became the Gymnasio

Nacional and, starting with its 1898 programs, teaching

was divided into two simultaneous courses: one of 6 years,

the ‘‘curso propedeutico ou realista’’, which enabled stu-

dents to enter post-secondary establishments; and the other,

of 7 years, the ‘‘curso classico ou humanista’’. The dif-

ference between the two courses was the presence of Greek

and Latin in the seventh year of the second course. Stu-

dents entered the Gymnasio aged 10 or 11 years.

During the imperial period few changes had been made

in secondary school mathematics. The first years of the

republican government showed an attempt at significant

reforms, with the introduction of new subjects, namely

analytic geometry, differential and integral calculus, higher

algebra, and Monge’s descriptive geometry. In geometry,

one notices the appearance of the conic sections in the

programs. The function concept appeared for the first time

in the programs of Pedro II: in 1892, it was studied in the

first year; and in 1893 in the second year.

15 Benjamin Constant Botelho de Magalhaes (1836–1891), an

engineer, was a very influential teacher at several institutions during

his career. At the military academy he ‘‘preached’’ Comte’s ideas and

he was one of the strongest defenders of the republican ideas. Later,

before he became Minister of Education, he left the orthodox group of

Brazilian positivists.16 Law n. 981, of 8 November 1890. It stipulated that secondary

schooling at Pedro II comprised twenty subjects, ranging from

classical Greek to mineralogy and music!

504 J. B. P. de Carvalho, B. A. Dassie

123

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The presence of the calculus in the secondary school

curriculum was short-lived.17In 1895 it did not figure in the

programs which, later on, for example in 1915, did not

even include the concept of a function. This remained

unchanged until 1929, when the program for the sixth year,

the complementary course for students planning to pass the

entrance examination to the Escola Polytechnica, included

a first course in differential calculus.

The enrollment requirements of post-secondary institu-

tions continued to have great influence in shaping sec-

ondary school mathematics. They defined what kind of

mathematics was taught, and to what extent.18 The very

competitive examinations for the more prestigious engi-

neering schools (the Escolas Politecnicas at Rio de Janeiro

and Sao Paulo, among others) dictated in practice what

kind of mathematics was taught at Pedro II and other

schools. The propaedeutic character of the curriculum had

its clearest formulation in the program for 1928, which

stated that in the fifth year ‘‘one should follow, as much as

it is feasible, the program for the entrance examination to

the Escola Politecnica’’ (Beltrame 2000). Besides, the

survival of the ‘‘isolated exams’’ made unimportant and

irrelevant in which years of the curriculum mathematics

was taught.

According to Beltrame (2000), for a long period, namely

from 1837 to 1929, there was a great stability in secondary

school mathematics, the only exception being the attempt

at changes made by Benjamin Constant. The same opinion

is stated by Pfromm Neto (1974, p. 81) regarding mathe-

matics teaching in the first half of the twentieth century.

The decade of the 1920s saw many changes in Brazilian

society. Industry was growing, the new urban professional

groups demanded a role in the political process, immigra-

tion greatly changed the profile of the working classes,

particularly in Sao Paulo, and foreign educational, artistic,

political, and philosophical ideas were more and more

discussed (Carvalho 2006). It is in this setting that we

witness major educational changes, particularly in the

teaching of secondary school mathematics.

This reform movement had many roots. One should

mention the modern educational ideas of Dewey which, in

Brazil, found expression in the so-called ‘‘Escola Nova’’

(New School) movement, and the echoes of the first

international mathematics education reform at the begin-

ning of the twentieth century, from which resulted the

creation of IMUK, the Internationale Mathematische

Unterrichtskommission—Commission internationale de

l’enseignement mathematique (Miorim 1988; Schubring

2000, 2003b; Valente 2003). Also, as has been brought to

light by recent research (Dassie 2008; Dassie and Carvalho

2010a, b), the escolas normais were an important factor in

the reform movement. In the background there was the

perception by many intellectuals, educators, and a few

politicians that modernizing efforts required deep changes

in education. Some of these educators had already imple-

mented educational reforms in several states of Brazil (the

former provinces) and were to play important roles in the

forthcoming wide reforms (Azevedo 1963).

Influenced by the ideas of the first international reform

movement of the teaching of mathematics, and also

reflecting the educational questionings of the 1920s, and

drawing on his teaching experience in a normal school,

Euclides Roxo who was director of Pedro II from 1925 to

1935 succeeded, in 1929, in approving a reform which was

incorporated, ‘‘in toto’’, in the national reform enacted by

the federal government in 1931, the ‘‘Reforma Campos’’.

This reform will be discussed in more detail in the next

section.

As shown by Rocha (2001), Roxo’s reform had two

dimensions: a structural one and a pedagogical one.

Structurally, this reform instituted the teaching of mathe-

matics in all years of the curriculum at Pedro II, tried to

integrate the several subjects of secondary school mathe-

matics, and started the practice, which lasts to this day, of a

single mathematics textbook for each school year. Like all

past reforms of secondary education, this one was specific

to Pedro II. Even though since 1911 enrollment in post-

secondary institutions could be achieved only through

entrance examinations, the ‘‘vestibulares’’, Pedro II’s pro-

grams remained very influential. As of 1931, with the

Campos reform, the Faculty of Pedro II was made

responsible for issuing the national mandatory programs.

From the pedagogical point of view, Roxo states that he

was influenced by Felix Klein’s ideas, mentioned exten-

sively in the preface of the book written for the new pro-

gram, the Curso de Matematica Elementar, vol I (Roxo

1929). These ideas are repeated in Roxo (1937, 1940).

Quoting Klein, Roxo states that one should19

1. Make the psychological point of view essentially

dominant […]

2. In the choice of topics to be taught take into account

the applications of mathematics to other subjects17 For its comings and goings, see, for example, Carvalho (Carv-

alho1996) and Silva (1996).18 For a very long period, this has been a stranglehold on attempts to

change high school mathematics. With a recent measure taken by the

Federal Government, namely the creation of a national exam that can

be used as entrance examination to post-secondary institutions, this

situation has started to change.

19 Roxo reproduces, in his writings, whole paragraphs of Klein’s

Elementarmathematik vom hoheren Standpunkt aus, without indicat-

ing he is using Klein’s very words.

The history of mathematics education in Brazil 505

123

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3. Subordinate the teaching of mathematics to the goals

of modern education […]

2.5 Fifth period (1931–1961): upheaval, reform,

consolidation, and accommodation

From 1837 through 1930 there was a seesaw between two

tendencies: on one side, the attempt to make students fol-

low a regular course, lasting several years, with a well

defined curriculum; on the other side, the permission to let

them prepare themselves attending aulas or private lectures

and to present themselves at Pedro II or other authorized

establishments just for examinations on each school

subject.

In the social and educational settings described in the

last section, and with the purpose of changing the goals of

secondary education, which should lose its almost exclu-

sively propaedeutic character, the federal government

enacted the ‘‘Reforma Francisco Campos’’, with national

scope.20 Secondary schooling was divided into two parts: a

fundamental course of five years; and a complementary

course of 2 years. According to Dallabrida (2009), this

reform strove for the modernization of secondary education

as part of the urbanization of society. The rigid school

culture imposed by the reform aimed to produce well dis-

ciplined secondary students.

The 1931 reform, enacted by a very authoritarian gov-

ernment, extinguished the ‘‘exame sparcelados’’, required

that students attend at least three quarters of the lectures,

restricted changes of schools by students to the end of the

school year, and instituted periodic student evaluations,

capped by final exams. The unification of the several

branches of school mathematics under a single heading

‘‘mathematics’’ made isolated exams in particular areas of

mathematics impossible and was a significant step towards

mathematics for all—basic common mathematical knowl-

edge as part of the education of all citizens. Also, an

increasing number of students enrolled in secondary level

schools.

Rocha (2001), among others, studied the antecedents of

this reform, which were the changes in the programs

instituted by Roxo in 1929 at Pedro II. If we compare the

program and the methodological instructions of the Cam-

pos reform with what had already been done at Pedro II, we

see that Roxo’s proposals were adopted ‘‘in toto’’. In par-

ticular, mathematics started being taught in each year of the

fundamental course.21 There was no substantial change in

the mathematical topics in the curriculum. The great

change was in their distribution, and in their presentation,

which followed Roxo’s innovative ideas, as made clear by

the methodological instructions for the new programs

(Rocha 2001).

Several voices raised strong and strident objections to

the new mathematics program instituted by the ‘‘Reforma

Campos’’. Among these, we should mention Arlindo Vie-

ira, director of a prestigious Catholic school in Rio de

Janeiro, and who was a staunch defender of the teaching of

the humanities; Joaquim Ignacio de Almeida Lisboa, a

colleague of Roxo at Pedro II, who took the side of the

traditional teaching of mathematics; and the army, repre-

sented by the military school in Rio de Janeiro.22 These

critiques were published in newspapers, in which we can

find a long, bitter, and passionate exchange of ideas

between Roxo and Almeida Lisboa.23

Vieira (Vieira 1936a, b) criticized the encyclopedic

nature of the mathematics programs, and preached a return

to the humanities. His critiques were part of a campaign he

led against the ‘‘decadence’’ of secondary education.

Almeida Lisboa had already fought against Roxo’s (1929)

reform of mathematics teaching at Pedro II, and continued

his attacks of the new programs after the reform was

instituted. His objections were against the low mathemat-

ical level of the new programs and the practical inductive

approach to geometry in the first years of the curriculum.

The military based their very strong critiques on Augusto

Comte’s positivism (Silva 1999). They were against the

‘‘error’’ of teaching simultaneously arithmetic, algebra, and

geometry.

All these reactions were taken into account by Campos’

successor, Gustavo Capanema, who became Minister for

Education in 1934 and was responsible for the next great

reform, in 1942, which was studied by Dassie (2001),

among others.

Dassie (2008) brought a new and fundamental view

point to the study of the reforms of the 1930s and 1940s.

He showed that there form ideas did not come only from

Klein and other mathematicians, as claimed by Roxo, but

were preceded by a much wider reform movement in the

normal schools, which prepared teachers for primary level

education. It is very significant that important educators at

the time, such as Fernando de Azevedo, Francisco Campos,

and Carneiro Leao, established normal school reforms in

their States.

After years of consultations with the several parties

concerned with secondary education—educators, the

church, the military, mathematics teachers—as shown by

20 Law 19.890, of 18 April 1931, with its consolidation by Law

21.241, of 4 April 1932.21 Compare with Fig. 1, which shows the distribution of mathematics

in Pedro II through the years 1838–1932.

22 A detailed discussion of these objections can be read in Carvalho

(2003, 1996), Dassie (2001), and Rocha (2001).23 Later on, Roxo synthesized his points of view in this virulent

exchange of ideas with Almeida Lisboa in Roxo (1937).

506 J. B. P. de Carvalho, B. A. Dassie

123

Page 9: The history of mathematics education in Brazil

(Dassie 2001), the ‘‘Lei Organica do Ensino Secundario’’,

known usually as Capanema’s reform, was enacted on 9

April 1942, during Varga’s dictatorship. It instituted the

division of secondary education into two parts. The first,

the curso ginasial, was mandatory for all students. In the

second, the curso colegial, the student had two choices:

one emphasizing the sciences, the curso cientıfico; the

other, the humanities, the curso classico.

In its final form, the mathematics programs instituted by

Capanema show the great stability of the corpus of school

mathematics in Brazil. Some of the ideas defended by

Euclides Roxo since the late 1920s were preserved:

mathematics was present in all secondary school years, and

there were no more separated schoolbooks for each area of

school mathematics. On the other hand, the ‘‘integration’’

of the several fields of school mathematics retreated.

Both Campos’ and Capanema’s reforms are a significant

step towards mathematics for all, basic common mathe-

matical knowledge as part of the education of all citizens, a

goal continuously pursued since their time, notwithstand-

ing the pressure imposed by the entrance examinations to

post-secondary education to preserve the purely propae-

deutic character of secondary level mathematics education.

The mathematics programs for the curso ginasial

remained very stable in the period we are studying—

1930–1961—while the ones for the curso colegial suffered

a slow evolution, characterized by the steady decrease of

Euclidean geometry, the disappearance of ‘‘theoretical

arithmetic’’, and the introduction of the function concept

only in the third year, the last. This evolution was fully

completed by 1951, when there was an adjustment of the

original programs instituted in 1941, which became much

lighter, as indicated above.

The programs instituted by Capanema’s reform

remained mandatory till 1961, when the first National Law

for Education was approved.24 The debates on this law

generated a long and bitter ideological battle, from 1948 to

1961, centering on the role of the State in education with,

on one side, the defenders of a free lay public school

system against, on the other side, the advocates of religious

private schools. With this law, the mandatory national

curricula were abolished and great curricular liberty was

allowed.

This liberty was, in practice, limited by some basic

factors. In the first place, the weight of tradition, the fact

that there was already an established corpus of school

mathematics. In the second place, the propaedeutic char-

acteristic of secondary school mathematics, which was

viewed as preparation for the entrance examinations

(vestibulares) to post-secondary institutions, either profes-

sional schools or universities.

We now turn to the 1960s, when attempts were made at

significant changes in mathematics education, with the

arrival of the ‘‘new math’’ movement.

2.6 Sixth period (1961–1988): decentralization,

experimentation, advances, and setbacks

At last, we reach the period of the latest big change in

mathematics education in Brazil, when the ‘‘new math’’

movement was introduced into the country. This has been

studied in detail by Soares (2001), among others.25

This period was characterized by educational decen-

tralization. It also saw the diffusion in Brazil of the ‘‘new

math’’ movement, which subsequently lost its momentum

in the mid-1970s. The period closed with the new Brazilian

constitution which introduced changes in the educational

system of the country. Also, from the mid-1980s to the

early 1990s, there was a big federal program for mathe-

matics and science education, which promoted the growth

and strengthening of the budding community of mathe-

matics education researchers. Groups of educators estab-

lished educational institutions26 and undertook the task of

writing or translating books and providing in-service

courses for mathematics teachers. In Sao Paulo, the group

headed by Oswaldo Sangiorgi (GEEM, created in 1961)

had a particularly impressive performance: he or his

companions translated the School Mathematics Study

Group (SMSG) texts, wrote their own materials, and were

responsible for the wide dissemination of the new ideas

(Soares 2001, pp. 133–134). Papy, among others, was

invited to Rio de Janeiro, and materials based on his books

were still used in the late 1990s in a very traditional and

demanding Catholic school in Rio de Janeiro.

The flexibility allowed by the new law was used par-

ticularly in the first eight years of schooling. As far as

secondary mathematics education was concerned, there

were not many changes, probably due to the already

mentioned propaedeutic character of this school level for

further studies. The main changes were the addition of

some very simple facts about sets and the introduction of

some notions of logic.

Two important consequences of this movement were, in

the first place, the creation of the groups already mentioned

(at least two of them, GEPEM and GEEMPA, still exist),

which originated reflections and research on the teaching

24 Lei de Diretrizes Bases da Educacao Nacional, Lei 4.024, de

dezembro de 1961.

25 See also Burigo (1989), D’Ambrozio (1987), Matos and Valente

(2007), and Burigo et al. (2008).26 For example, GEEMPA (Grupo de Estudos de Ensino daMatematica de Porto Alegre) in Porto Alegre; GEPEM (Grupo dePesquisas e Estudos em Educacao Matematica) in Rio de Janeiro; and

GEEM (Grupo de Estudos do Ensino de Matematica) in Sao Paulo.

The history of mathematics education in Brazil 507

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and learning of mathematics; and in the second place,

many teachers were motivated to study not only mathe-

matics but also its learning and teaching. The ideas of Jean

Piaget, George and Frederique Papy, Hans Freudenthal,

Tamas Varga, and Zoltan Dienes became known in Brazil

mainly because of the interest aroused by the movement. In

this way, it contributed indirectly to the awareness by

Brazilians of the international community of research in

mathematics education.

2.7 Seventh period: from 1988 to the present

In 1988 a new Constitution was enacted. Eight years later,

the second national law of education was approved. This

instituted the ‘‘ensino fundamental’’ (elementary and mid-

dle school education), lasting 8 years, augmented in 2010

to 9 years, and the ‘‘ensino medio’’ (secondary school

education), lasting 3 years. The new law of education, like

the first one, did not impose a mandatory national curric-

ulum. The Ministry of Education issued, instead, a series of

guidelines, that, though not mandatory, have had a great

influence on curricula (Brasil 1997, 1998, 2002, 2006).

The guidelines for the ‘‘ensino fundamental’’, the first

eight years of schooling, were long and detailed. Those for

the secondary school, by contrast, were short and very

general. In 2002, a supplementary document (PCN ? En-

sino Medio: orientacoes complementares ao Parametros

Curriculares Nacionais) was published, which is more

detailed and gives specific orientation on how to deal with

the abilities one ought to develop and their relationship to

mathematics. Secondary school mathematics was divided

into the following major themes: algebra, numbers and

functions, geometry and measurements, and data analysis.

This document addressed questions related to methodology

and didactical approaches and stressed the importance of

problem solving.

In this period, universal primary education has been

almost attained and secondary education has made great

strides. However, the vast increase of primary and sec-

ondary school attendance requires many teachers, who

often do not have the necessary qualifications.

One should also mention the establishment, in 1988, of

the Sociedade Brasileira de Educacao Matematica

(SBEM), the Brazilian mathematics education society, that

seeks to promote research and act as a bridge between

teachers in the classrooms and researchers in mathematics

education.

During this period, several graduate programs in math-

ematics education have been established and the number of

researchers in the field has increased continuously. The

seeds for this development may be found in the groups

interested in the ‘‘new math’’ movement which we men-

tioned in the preceding section.

3 Concluding remarks

The history of mathematics education in Brazil has several

interesting characteristics. It is a case study of transmission

and reception of concepts and ideas, as explained by

Schubring (2000), understood as dynamical processes, not

as mechanical actions like transferring water from one vase

to another.

At first, Brazil received a corpus of school mathemat-

ics27 from Portugal, which, in turn, had received its corpus

from other countries, mainly France. So we have a two-fold

transmission movement. Once this corpus had taken roots

in Brazil, there were transmission movements directly from

France, in the ninteenth century, when authors of mathe-

matics textbooks for Colegio Pedro II adapted French

textbooks. Also, the use of French authors in post-sec-

ondary establishments influenced secondary school math-

ematics. Slowly, as we near the twentieth century, we can

see French influence ebbing away, superseded by English

and American ideas.28 In the 1920 s and 1930 s, we can

also note some German influence on school mathematics,

with the diffusion of Felix Klein’s ideas on mathematics

education.29

Next, in the 1950s and 1960s, with the arrival in Brazil

of the ‘‘new math’’ movement, we see both American and

European influences, sometimes competing for primacy.

The signing of broad educational agreements between

Brazil and the United States certainly had a part in the

influence wielded by American groups and organizations.

Recently, the guidelines issued by the Ministry of Educa-

tion show a strong debt to similar documents prepared by

the American NCTM (National Council of Teachers of

Mathematics).

We would like to stress the greatest ability of secondary

mathematics education in Brazil if we look at its evolution

from 1837 on. From 1837 till 1889 the curricula saw few

changes and the main issue was to make school attendance

mandatory. The government made several attemptsto

eliminate the practice of the ‘‘exames avulsos’’ (isolated

exams), which made it possible to complete the require-

ments for entrance to post-secondary education by taking

the necessary exams without having attended a regular

secondary school course. This problem was definitively

solved only in 1931, with the ‘‘Reforma Francisco

Campos’’.

27 School mathematics is viewed here in a very broad sense,

embracing all levels of mathematics instruction.28 This is a general trend in Brazilian culture. See, for example,

Freyre (1986).29 It remains to be studied whether this is part of a general trend at the

time, as discussed, for example, in Schubring (2003c). One should

note, for example, the German influence on the Brazilian military in

this period.

508 J. B. P. de Carvalho, B. A. Dassie

123

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In the first years of the Republic, the curriculum was

subjected to significant changes which were slowly put

aside, and it reverted to its previous form. With the

Reforma Campos (1931) we witness a great rupture with

a century-old tradition in secondary mathematics educa-

tion. This reform imposed a national mandatory curric-

ulum, mathematics was taught in all school years, and

the teaching of arithmetic, algebra, and geometry

(including trigonometry) in different years was abolished.

The Reforma Capanema (1942) extended the Reforma

Campos by organizing elementary education and dividing

secondary education into two main branches, one for the

sciences and the other for the humanities.

Kruger (2003) compared the fates of two reforms, one in

France, imposed from above in 1902, and the other in

Germany, which was ‘‘from below’’, the ‘‘Meran reform’’.

The French reform floundered, while the German one

flourished. The history of mathematics education in Brazil

also provides a case study of imposed reforms that suc-

ceeded, possibly due to the dictatorial nature of Vargas’

regime.

From 1942 through 1961 we witness another stable

period in mathematics education. In secondary mathemat-

ics education, we see a continuous reduction of curriculum

content, with Euclidean geometry being dislodged by

coordinate geometry. In 1961, the first national law for

education abolished mandatory national curricula, and the

‘‘new math’’ movement had considerable influence in ele-

mentary school mathematics. Slowly, however, it ebbed

away, leaving few traces in the curriculum.

In the 1990s, instead of trying to impose a mandatory

national curriculum the government issued a series of

guidelines for both elementary and secondary school. In

mathematics, the guidelines emphasized problem solving

and that mathematics should not be just a tool to enter

post-secondary establishments, but an essential part

of everyone’s preparation for an active role in a mod-

ern, complex society. With almost universal elementary

school attendance and the great expansion of the

secondary school system, the country now comes

close to this goal of providing mathematics education for

all.

The reforms decreed in the 1930s and 1940s shaped

mathematics textbooks for a long period. There was not a

‘‘winning collection’’ which imposed a textbook model.

Prior to the reform established by Euclides Roxo at Colegio

Pedro II there were already textbooks which followed the

ideas later adopted by Roxo in the textbook collection he

wrote for the new programs (Dassie 2008). The main

shapers of the textbooks and, to some extent, of classroom

practices were the official detailed mandatory official

programs with their pedagogical instructions. This can be

seen if we follow the periodic program revisions issued by

the Ministry of Education and the corresponding changes

in the textbooks.

From 1837, when the Colegio Pedro II was created, till

1961, there is a very close parallel between the textbooks

for secondary education and the official programs. This can

be seen if we follow the debates on the teaching of math-

ematics by the faculty of Colegio Pedro II (Tavares 2002).

Also, from 1931 till 1961, when there were mandatory

national curricula, mathematics textbooks had to comply

with the official programs. So, even though mathematical

textbooks are not the main subject of the history of math-

ematics education, one can find in them helpful elements

for the study of this history.30 In general, textbooks provide

information on how the several mathematical topics were

presented, but not if they were actually taught or not, and,

most important, whether they were learned. One should

look ‘‘inside’’ the textbooks, because their style of subject

presentation has much to tell about conceptions of mathe-

matics and its teaching, the fashions of presentation in each

period, and how these changed along the years.31Along

these lines, we need research on mathematics textbooks out

of the mainstream, and so not often mentioned in the

literature.

Research is sorely needed on other sources besides

textbooks. In Sao Paulo, the ‘‘Grupo de Pesquisa de

Historia da Educacao Matematica no Brasil’’ (GHEMAT),

established in 2000, has done pioneering work, organizing

the personal archives of important mathematical educators

such as Euclides Roxo, Ubiratan D’ambrozio, and Oswaldo

Sangiorgi. In addition, it has set up several databases,

among them one with actual sheets of exams taken by

students, and another one with scanned parts of many

textbooks.

Institutional history has been pursued by, among others,

Dias (2002a), who has written about the development of

institutions in his State, Bahia. Meanwhile, the archives of

several important institutions, such as the Liceu Pernam-

bucano in Recife, remain unexplored. Schubring (2003c),

among others, has studied the very well organized German

schools in southern Brazil, which were abolished by the

Vargas regime.

Even though the introduction, expansion and ebbing of

the ‘‘new math’’ movement in Brazil has been studied by

several authors,32 new studies are needed, in particular to

interpret the subject as a case study in the transmission and

modification of ideas, along the lines of, for example,

30 See, for example, the elements mentioned in Genette (2002).31 These remarks do not aim at belittling the study of mathematical

textbooks, which is important and fruitful, but only to stress that one

should not rely solely on them.32 See, for example, Burigo (1989), D’Ambrozio (1987), and Soares

(2001).

The history of mathematics education in Brazil 509

123

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Schubring (2000). In particular, the articulation of the

movement in Brazil with competing foreign leaders

deserves further study.

Acknowledgments We thank both Gert Schubring for many fruitful

discussions and the reviewers, whose suggestions contributed much to

the improvement of this paper.

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