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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
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
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
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
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
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
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
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
(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
123
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
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
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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