Revista Brasileira de Educação do Campo
The Brazilian Scientific Journal of Rural Education
ARTIGO/ARTICLE/ARTÍCULO
DOI: http://dx.doi.org/10.20873/uft.2525-4863.2018v3n2p519-2
Tocantinópolis
v. 3
n. 2
p. 519-548
may/aug.
2018
ISSN: 2525-4863
519
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Mathematical education of young and adults: pedagogical
implications of historical-cultural theory
José Carlos Miguel
1
1
Universidade Estadual Paulista Júlio de Mesquita Filho - UNESP. Departamento de Didática/Programa de Pós-Graduação em
Educação. Faculdade de Filosofia e Ciências. Avenida Hygino Muzzi Filho 737, Mirante. Marília - SP. Brasil.
Author for correspondence: jocarmi@terra.com.br
ABSTRACT. The present study addresses some pedagogical
implications of historical-cultural theory for the exploration of
mathematical ideas in the field of youth and adult education
(EJA). Starting from an analysis of the state of the art regarding
the difficulties of teachers and students for teaching and learning
of Mathematics throughout the schooling process indicates
elements to the debate that return to the explanation of the
problems listed and to refer a process of constitution of
mathematical learning subjects within the scope of the EJA. It is
a bibliographical and documentary research, besides the analysis
of usual mathematical situations in EJA classes, whose results
show the difficulties of the basic school culture to overcome
didactic actions still strongly marked by the association of
models. It points to the theoretical construct of the historical-
cultural perspective as a perspective for the realization of a
broad process of production of meanings and negotiation of
meanings of teaching and learning of Mathematics in the EJA.
Keywords: Youth and Adult Education, EJA, Mathematical
Education, Formation of Concepts, Production of Meanings,
Negotiation of Mathematical Meanings.
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
Tocantinópolis
v. 3
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p. 519-548
may/aug.
2018
ISSN: 2525-4863
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Educação matemática de jovens e adultos: implicações
pedagógicas da teoria histórico-cultural
RESUMO. O presente estudo aborda algumas implicações
pedagógicas da teoria histórico-cultural para a exploração de
ideias matemáticas no âmbito da educação de jovens e adultos
(EJA). Partindo de uma análise sobre o estado da arte no que se
refere às dificuldades de professores e alunos para o ensino e a
aprendizagem da Matemática ao longo do processo de
escolarização indica elementos ao debate que se voltam à
explicação dos problemas elencados e para encaminhamento de
um processo de constituição de sujeitos de aprendizagem
matemática no âmbito da EJA. Trata-se de pesquisa
bibliográfica e documental, além da análise de situações
matemáticas usuais em aulas de EJA, cujos resultados mostram
as dificuldades da cultura escolar básica para a superação de
ações didáticas ainda fortemente marcadas pela associação de
modelos. Aponta para o constructo teórico da perspectiva
histórico-cultural como perspectiva para a efetivação de um
amplo processo de produção de sentidos e de negociação de
significados de ensino e de aprendizagem da Matemática na
EJA.
Palavras-chave: Educação de Jovens e Adultos, EJA, Educação
Matemática, Formação de Conceitos, Produção de Sentidos,
Negociação de Significados Matemáticos.
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
Tocantinópolis
v. 3
n. 2
p. 519-548
may/aug.
2018
ISSN: 2525-4863
521
Educación matemática de jóvenes y adultos: implicaciones
pedagógicas de La teoría histórico-cultural
RESUMEN. El presente estudio aborda algunas implicaciones
pedagógicas de la teoría histórico-cultural para la exploración de
ideas matemáticas en el ámbito de la educación de jóvenes y
adultos (EJA). A partir de un análisis sobre el estado del arte en
lo que se refiere a las dificultades de profesores y alumnos para
la enseñanza y el aprendizaje de las Matemáticas a lo largo del
proceso de escolarización indica elementos al debate que se
vuelven a la explicación de los problemas enumerados y para
encaminamiento de un proceso el proceso de constitución de
sujetos de aprendizaje matemático en el marco de la EJA. Se
trata de una investigación bibliográfica y documental, además
del análisis de situaciones matemáticas usuales en clases de
EJA, cuyos resultados muestran las dificultades de la cultura
escolar básica para la superación de acciones didácticas aún
fuertemente marcadas por la asociación de modelos. Se apunta
al constructo teórico de la perspectiva histórico-cultural como
perspectiva para la efectividad de un amplio proceso de
producción de sentidos y de negociación de significados de
enseñanza y de aprendizaje de las Matemáticas en la EJA.
Palabras clave: Educación de Jóvenes y Adultos, EJA,
Educación Matemática, Formación de Conceptos, Producción de
Sentidos, Negociación de Significados Matemáticos.
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
Tocantinópolis
v. 3
n. 2
p. 519-548
may/aug.
2018
ISSN: 2525-4863
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Introduction
The historical development of
mathematical education as a theoretical
field shows us, among other relevant
formulations, that the attempts to explain
the difficulties with the learning of
Mathematics go through the ideas of
inadequate working conditions in the
school, inadequate teacher training (Ponte,
2003; (Danyluk, 1993; Oliveira & Moreira,
2010), and every aspect of the teaching of
Mathematics (Danyluk, 1993, Oliveira &
Moreira, 2010), problems of student
assimilation, school devaluation,
inadequate teaching programs, of this
problem deserves consideration and plays a
role for students' performance in
mathematical learning.
The difficulties of mathematical
learning of young people and adults are
due in general to the school culture whose
methodological procedure is still marked
by the association of models, that is, a
didactic conduct in which, if the student
observes well the teacher does, he must
learn, prevailing the utilitarian view and
the Platonic view of Mathematics as we
can conclude based on Chacón (2003).
By this way of understanding the
teaching and learning of Mathematics,
interesting logical-mathematical relations
for the development of theoretical thought
within this science, present in the social
relations of working, playing, playing and
interacting in the context of the EJA are
little explored mathematically or even
neglected.
Observation and follow-up of
Mathematics classes at all levels, but in the
education of youths and adults, in
particular, reveal a certain distance
between the evolution of mathematical
thinking, clearly marked by
contextualization and the attempt to solve
problems that are for humanity throughout
its historical trajectory. One loses sight of
the fact that mathematics education is a
social practice of an interdisciplinary
nature, and therefore obliged to dialogue
with other social practices, somehow
disregarding in the methodological form of
its diffusion the need to stick more to
psychological and sociocultural bases than
to systematic ones.
In spite of the acknowledged efforts
to overcome the problem in the formative
contexts, the organization of educational
programs and public policies for education,
this way of understanding the constitution
of mathematical thinking, clearly marked
by the attachment to the formal
systematization, and the form of its
diffusion in the school still marks,
impregnates and determines the relation
between content and form in this area of
knowledge. Strictly speaking, it is
necessary to understand how students think
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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2018
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from the analysis of their supposed errors
(Cury, 2007).
The most notorious didactic
behaviors in schools indicate that since
mathematics is a hypothetical-deductive
science, it must predominate in its
diffusion, from the first steps in the
schooling process, the explication of its
logical-formal chain and, therefore,
students are required a level of abstraction
and formalization that is beyond their
capacity for understanding. In the case of
the students of the EJA there is another
contradiction: they mentally make
interesting mental calculations, which
usually do not know how to register in
writing and, as a rule, in the school they
seek the appropriation of formal models
sometimes distant from their ways of
thinking.
This tendency to exaggerated
formalism in the teaching of Mathematics,
a tradition that is due to the inadequate
understanding of the formal Euclidean
model (Imenes, 1987) by teachers, crosses
practically all the themes of this area of
knowledge and has reduced the approach
of the mathematical notions to a axiomatic
treatment that consists much more of
seeking the algebraic formulation of this
idea by the attachment to logical-formal
reasoning than to an attempt to know and
interpret the properties involved as
fundamental concepts for the
understanding of significant phenomena of
students' lives.
In our view, the competence to deal
with mathematical ideas is presented to the
subjects before their inclusion in school
and should be emphasized throughout
schooling, starting in the literacy process
by understanding that Mathematics is an
important support component to reading
and writing processes. From birth the
person establishes relationships with the
environment, developing, structuring and
perfecting the intelligence through the
development of basic structures of thought,
ie, topological, algebraic and order.
It seems to us that the EJA school
has little explored such relationships that
are fundamental for the development of
theoretical thinking and already in the first
school experiences, therefore, teachers
need to pay attention to the need to favor
this construction since the EJA student
brings for the school wide range of
experiences of appropriation of
mathematical ideas, although not
systematized from the formal point of
view.
The relative disregard for the
development of the aforementioned
thought structures and the school way of
dealing with the diffusion of mathematical
thinking neglects the fact that the good
performance of pupils in the first years of
formal schooling and their successful stay
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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in school implies the careful work of
stimulation of the senses, coordination,
attention and direction of the construction
of a symbolic language guided by activities
that favor the development of theoretical
thinking, a fundamental aspect for the
effectiveness of the formation of
mathematical concepts.
Data from the INAF, National
Indicator of Functional Literacy, reveal
over the last two decades the need to
incorporate mathematical skills into the
constitution of functional literacy
indicators in order to reflect the diversity
and progressive sophistication of reading
and writing demands subjects must meet to
be considered functionally literate in
contemporary society.
Nevertheless, what has been seen
over the last two decades is a mismatch
between such needs imposed by the mode
of production defined in the context of the
proliferation of technologies, and
especially of microelectronics. Thus,
... the INAF 2004 results indicate that
only 23% of the Brazilian young and
adult population is able to adopt and
control a strategy to solve a problem
involving the execution of a series of
operations. Only this portion is also
capable of solving problems
involving proportional calculus. Even
more disturbing is the revelation that
only in this group are the subjects
who demonstrate certain familiarity
with graphical representations such
as maps, tables and graphs. (Ação
Educativa, 2004, p. 8-9)
i
.
Despite the efforts of technical staff
of education secretariats and education and
training agencies, these indicators remain
virtually unchanged. Why does it happen?
It is our hypothesis that in the
multisecular tradition of a schoolized
approach to mathematical knowledge, the
traditional form of diffusion of
mathematical facts does not give due
importance to the experiences developed
by the students from a very early stage of
sensory exploration of the physical
environment, interpreting the environment
in which they live, knowing and
transforming the relations present in it. As
a consequence, Mathematics teaching
programs are much more concerned with
activities related to language,
symbolization and quantification, failing to
explore the development of the capacity
for logical reasoning possible in a
pedagogical work with the subject in
question and that perpasses the practical
activities of daily life, play activities and
mathematical experiences, via mental
calculation and estimation, so recognized
in the context of the EJA.
That said, two distinctive marks of
the didactic activity regarding the teaching
of Mathematics in the EJA in the Brazilian
context are evidenced. On the one hand, in
the set of teachers who overestimate the
role of mental calculation and estimation in
mathematics teaching, this stage of basic
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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2018
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training should emphasize the process of
appropriation of reading and writing. But
to what extent, the articulation and the
diffusion of mathematical ideas cannot be
formed from the processes of reading and
producing texts? Of what Mathematics
should we speak regarding the teaching of
youth and adults of the EJA? A quick foray
into the reality of school will show that
these issues are not very clear. On the other
hand, there is a current clearly affected by
the mercantilist perspective of education,
reinforcing the EJA students' rush to learn,
and putting into practice a pedagogical
action that is part of the formal
systematization of mathematical models,
neglecting the specificities of the
intellectual development.
Both positions prove to be
inadequate, judging by the main indicators
of the evaluation of the results obtained by
the students in the continuity of the
schooling process. It is a problem that,
despite the various invariants that compose
it, has a position marked by the formation
of the educator in its constitution. In
discussing the role of research as an
element of teacher's professional culture,
Ponte (2003) states that
The valuation of a research culture
among teachers does not depend only
on a more or less voluntaristic action
at the individual level. It
presupposes, on the contrary, a
fundamental role of the collective
instances where the professors carry
out their professional activity,
emphasizing the schools, the
pedagogic movements and the
associative structures. One of the
major obstacles to the affirmation of
a research culture in teachers is the
old opposition between theory and
practice. In this opposition, theory is
something fanciful, unsuitable for the
interpretation of reality, useless or
even pernicious. Practice is the realm
of normality and the inevitable,
where all problems always find
external justification (whether
students, caregivers, explainers, lack
of working conditions or Ministry
policy). It is a bizarre conception of
theory and practice. In fact, theory
and practice are two sides of the
same coin. They always coexist.
Where there is a theory there is a
practice and where there is a practice
there is a theory. What is needed is
whether the theory serves or does not
serve and whether the practice is
commendable or problematic. (Ponte,
2003, p. 18-19)
ii
.
The citation is long, but it is
illuminating for our discussion as the
teaching practices in Mathematics teaching
and the EJA need to be placed at a level
that contemplates the recent advances of
the research in Education, especially, as far
as the sociocultural contributions of
mathematical learning.
Overcoming in the EJA the
utilitarian and merely instrumental
conception of mathematical knowledge
requires thinking the formation of an
epistemologically curious teacher, willing
to reflect on the meaning of mathematical
knowledge, how it is constituted and to
insert students in a process of meaning
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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production and negotiation of learning
meanings through establishment of a
dialogic relationship:
The relationship between thought and
word is a living process: thought is
born through words. A word devoid
of thought is a dead thing, and a
thought not expressed by words
remains a shadow. The relationship
between them is not, however,
something already formed and
constant: it appears throughout
development and also changes. ...
The word was not the beginning -
action already existed before it: the
word is the end of development, the
crowning of action. (Vygotsky, 1991,
p. 131)
iii
.
Based on the author's thinking, it is
necessary to consider that the development
of language skills is carried out in
conjunction with mathematical activities,
which has consequences for the planning
of actions in schools. In addition to the
scientific and technological dimensions,
Mathematics is consolidated as a
component of the general culture of the
citizen that can be observed in everyday
language, in the press, in laws, in
advertising, in games, in games and in
many other everyday situations.
Thus, the present study aims to
analyze the theoretical and methodological
assumptions of the pedagogical action to
be developed in the education of youths
and adults with a view to sustaining the
process of formation of concepts in
Mathematics. For that, it discusses the
theoretical foundations involved in the
construction of the fundamental
vocabulary of Mathematics and its
implications for the teaching practice in the
EJA in order to contribute to mathematical
literacy in the first years of elementary
school.
It is a bibliographic and documentary
research that starts from the assumption
that the EJA student is an active being who
thinks, perceives things, facts and objects;
elaborates mental images; establishes and
formulates relationships; operating
mentally and formulating concepts. It is a
theoretical-conceptual construction that
results directly from our work as
coordinator in the context of the UNESP
Program for Youth and Adult Education,
PEJA, developed since the year 2000 and
the Institutional Program for the Initiation
to Teaching, PIBID-EJA, post in practice
starting in 2009, both projects being aimed
at the initial and continuous training of
EJA educators and the articulation between
teaching, research and university extension
in this area of knowledge.
The developmental systematic of
both programs involves intervention
processes in the school reality and
permanent reflection on the school context
of the EJA. It is in the debate about the
difficulties that the teachers in initial or
continuous formation face that the
possibilities of change in the form of
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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2018
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methodological treatment of the diffusion
of the mathematical knowledge are
obtained. In this way, the present article
discusses issues that daily apprehend in
this process of action-reflection-action.
We also believe that mathematical
knowledge can be taught, but that its
appropriation must be based on the
relations that the subject establishes
between objects, facts and events.
Knowledge, therefore, imposes on the
subject who learns interaction and
exchange with others, and in particular
with the object of knowledge. The
appropriation of the mathematical fact is
simultaneously collective, active and
individual action.
Finally, we consider that the
understanding of these relationships is a
central element for the formulation of
proposals to address the difficulties faced
by teachers and students in the teaching
and learning process of Mathematics and
for the consolidation of pedagogical
principles aimed at an adequate
epistemological formulation with a view to
the organization of the Mathematics
curriculum as an organic, articulate and
flexible whole, highlighting its relations
with the formation of concepts in the area.
In this sense, in addition to the symbolic
registration, the pedagogical work in
Mathematics should contribute to the
development of reasoning skills that begins
with the support of oral language and, over
time, incorporating more elaborate texts
and representations.
We start from the belief that it is only
from our own experience that the
appropriation of mathematical knowledge
is facilitated. Only a methodology based on
the subtlety of one's own reasoning can
lead to more abstract propositions and to
the use of formal, logical, and deductive
reasoning typical of mathematics.
Based on the assumptions of the
Historical-Cultural Theory, it is our
conviction that in organizing the teaching
of Mathematics according to the needs of
the students of the EJA, it is imperative to
consider that the teaching activities
organized by the teachers must focus on
the development of the human personality.
In order to do so, the activities of
EJA students are directly linked to the
process of appropriation of the bases of
mathematical concepts, such as the control
of different quantities, quantities, space
and form in their direct relation with the
world and with things, of a physical nature
or symbolic.
These activities are crucial for the
appropriation of mathematical concepts
involving the topological, algebraic, and
orderly relationships so evident in people's
daily actions such as orientation, location
and notion of boundary, numbering,
problem solving, and occupation and
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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exploration of space, effective in the
relations of people with the physical
environment, with objects, with other
people and, finally, with the world that
surrounds them.
The production of meanings and
meanings in the teaching and learning of
Mathematics
A discussion about meanings and
meanings in the teaching and learning of
Mathematics in EJA is fundamental
because teachers and students find
themselves in their daily activities with
problems of great complexity, both
teaching and learning, affections to the
dimensioning of what they do daily. In
fact, the central element of the issue is the
lack of attribution of meaning to what is
done in the educational process.
Bishop (1999) points to three levels
of culture that in their perspective
determine the production of mathematical
meanings and meanings: technical, formal
and informal. According to the author, the
technical culture of Mathematics involves
the set of symbols and arguments used by
mathematicians in their formulations.
Formal culture relates to systematically
organized mathematical concepts. On the
other hand, informal culture considers the
particular mathematical knowledge of an
individual or social group.
To illustrate, let us take as examples
mathematical situations of addition or
subtraction of natural numbers effected
mentally by EJA students. Almost always,
in the heuristics they develop, they make
an analogy to the use of money. In
addition, we note that in expressing the
heuristics they put into practice in mental
calculus they reveal decomposition
calculations, generally beginning with the
higher order of the numeral, based on how
they speak or interpret amounts of money:
200 + 40 + 9
+ 100 + 30 +5_
300 + 70+14
300 + 70 + 10 + 4
300 + 80 + 4
384
600 + 40 + 2 500 + 140 + 2 500 + 130 + 12
- 300 + 90 + 7_ - 300 + 90 + 7 - 300 + 90 + 7_
200 + 40 + 5
245
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Note that the idea of decomposition
can also be maintained in multiplication
and division, and in the latter it is also used
to estimate:
Obviously, it is the school's role to
lead pupils to the development of the usual
arithmetic algorithms, both for their social
use and for the possibility of generalization
that it holds, but one cannot ignore the
knowledge they bring to school as a
starting point for didactic-pedagogical
action.
For Leontiev, the psychology of man
is linked to the activity of concrete
individuals included in the system of
relations of society. Activity cannot be
considered unrelated to social relations, for
in this way it does not exist. The author
explains this by stating: "Man finds in
society not only the external conditions
that must accommodate his activity, but
also the social conditions contain the
motives and ends of his activity, his
procedures and means." (Leontiev, 1978,
p. 68).
I turn to Leontiev to establish that the
activity of knowing involves figurative,
operative, and connotative aspects.
Figurative knowledge relates to the
external real and does not require the
establishment of relationships. The subject
forms an isolated mental image in which
colors, sizes and shapes stand out. It is in
this way that initially the students of the
EJA can recognize the notes of the money
or determined urban bus line. It is a kind of
knowledge that focuses on memorization,
repetition and tricks. Thus a pupil can see
or utter numeral 209, but not know what
209 has to do with 208 or with numeral
210.
In turn, connotative knowledge
relates to the formation of concepts, of
meanings. It goes beyond the figurative
knowledge, involving the establishment of
relations, raising hypotheses and drawing
conclusions. Thus, as an example, the car
ceases to be just a four-wheeled vehicle,
body and seats, as a means of
transportation, and is also seen as a
medium for production and services on a
larger scale, qualitatively different from a
20 + 2 20 + 2
X 20 + 3_ X 20 +3_
6 60 + 6
+ 60 400 + 40____
40 400 + 100 + 6
400__________ 506
400 + 100 + 6 = 506
1000 + 100 + 2 29
- 870 30
130 + 100 + 2 + 8_
200 + 30 + 2 38
-200 + 30 + 2_
0
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scooter or of a stroller. The subject
apprehends the real, giving meaning to it,
using the concepts elaborated and using the
objects according to their meaning.
In the operational context,
knowledge is the interaction of the learning
subject with other people and reality,
characterized by active thinking, in order
to overcome conflicts and contradictions
arising from interaction.
All psychic activity, then, is a
reflection of the practical activity,
transporting to the subjective activity all
the activity with objects realized in the
cultural world, objective. Obviously, this
transport does not occur mechanically, but
implies the active participation of the
subject, a process called historical-cultural
theory as objectification, always
determined by the social relations in which
the subject is involved.
In this sense, Vygotsky (1995)
explains that the formation of concepts
does not take place mechanically, as a
simple overlapping of photos taken from
reality. There is a whole elaboration on the
part of the subject in the constitution of the
natural thought, that occurs in the exact
moment in which it attributes sense to that
all moment of experience.
For the author, psychic functions are
internalized social relationships, that is,
they are originally constituted in social
processes. Therefore, the mediating
function of the meaning of words
constitutes a system of reversible signs
allowing the two main functions of
language, interrelated, communicative and
representative, articulating thought. These
functions are related to the processes of
contextualization and decontextualization,
giving meaning and value to the meanings
of words. It should be noted, then, that the
author highlights both the nature and its
constitution, from and by the socio-cultural
reality, which attributes to the concept of
semiotic mediation a social and historical
density.
In fact, it is
... access to objects necessarily
passes through semiotic
representation. Moreover, this
explains why the evolution of
mathematical knowledge leads to the
development and diversification of
registers of representation. (Duval,
2003, p. 21)
iv
.
It is essential, however, to recognize
that, in fact, the issue is not restricted to the
problem of registers of representation,
given the fact that students recognize a
certain numeral but do not know the
relation between it and its predecessor or
successor. However, Duval's contribution
is relevant as regards the strict context of
symbolic registers, that is, as regards the
forms of representation of mathematical
concepts, since the semiotic
representations are external and conscious
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of the individual, that is, they constitute the
explicit understanding of a given object,
and such perceptions are formed in
different ways and is a problem of
development.
Thus, if the variety of forms of
registers and representations indicates,
marks, and characterizes, to a certain
extent, the functionality of human thought,
it does not seem to us sufficient to
expressly express our understanding of the
object under study. And it is precisely at
this point that the school sins because it
places weight precisely in the formal
systematization, read strictly in the
symbolic representation, neglecting aspects
related to the historical evolution of
mathematical ideas, the production of
meanings of learning and the negotiation
of meanings mathematicians.
Thus, it is necessary to consider the
complementarity of functions between
thought and language and the fact that,
strictly speaking, it is the semiotic function
that enables thought, which is supported by
Vygotsky (1995) of mental representations,
is associated with the internalization of
semiotic representations initiated by the
mother tongue.
The difference in understanding the
problem is subtle, but fundamental. While
for Vygotsky (1995) the representative
function is a function of language, together
with communicative function, in the other
way of thinking, although it is agreed that
representation is the result of semiotic
activity, such a function appears as
autonomous, from which emerges
representative intelligence.
Hence language, an articulated
system of signs, historically and socially
constructed, involves relatively stable, but
changeable, instituted meanings, forming
itself in the polysemy of words. It is in the
context of interlocution that such meanings
assume their concrete meaning.
Therefore, to understand the speech
of the other is not enough to understand his
words, it is necessary to understand how
the other thinks and the motivation that
mobilizes him.
For Vygotsky (1995), the function of
representation is what defines the sign, that
is, it is the specific function of sign
systems, such as language. Therefore,
sensory perception, the primordial
manifestation of knowledge of the real, is
already delineated semiotically, although
the subject of it is not fully aware. It is
famous the following passage of text of the
author on the question:
A special aspect of the human
species - which comes at a very early
age - is the perception of real objects.
This is something that does not find
correlate in the animal species. By
this term I understand that the world
is not seen simply in color and form,
but also as a mural with meaning and
meaning. (Vygotsky, 1995, p. 37)
v
.
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In this way, when the subject comes
into contact with the world of objects and
manipulates them, what he perceives is not
mere objects, but semiotic objects, that is,
things that have names, physical or
imaginary entities. Along with the visual
image of the object, the subject captures
the word that gives it meaning, even
though word and object seem to be
confused. In other words, the subject in the
process of literacy in the EJA is not
enough to draw the numeral 20, sometimes
related to the social use of money; more
importantly, the learner knows what 20 has
to do with 19 or 21, or what he can buy
with that grade, which is not always
considered in school and is meaningful to
him.
In the case of imitation, or in the case
of symbolic play, what makes semiotics
such activities, for Vygotsky (2009a), is
the meaning they have for the other, of the
social group, and not the similarity with
the represented object. This is a
fundamental conclusion, with which one
can define how obsolete are the didactic
procedures that prioritize only the
symbolic representation of numbers or
geometric entities.
Take, for example, the concept of
square, sometimes erroneously diffused in
school as a figure or polygon having four
equal sides. How or what, in this
definition, would a diamond be? Does not
a diamond also have four sides of equal
size? Pedagogically, it would be more
effective to conduct subjects to observe,
manipulate, overlap, and compare different
geometric figures in order to recognize
regularities present in them, so as to
establish that every square is, strictly
speaking, a rhombus, but not every
rhombus is a square. The same problem of
model association that does not actually
deal with conceptual elaboration can be
noted in relation to students'
incomprehension of the distinction
between circle and circle, or between
square and cube, or important conceptual
relations of social use between cm³ and
milliliter or between dm³ and liter, among
other difficulties.
Indeed,
The learning of Mathematics refers to
a set of concepts and procedures that
involve methods of investigation and
reasoning, forms of representation
and communication. As a science,
mathematics encompasses a broad
field of relationships, regularities and
coherences, arousing curiosity and
instigating the ability to generalize,
project, predict, and abstract. The
development of these procedures
expands the means to understand the
world around us, both in situations
nearer, present in daily life, and in
those of a more general character. On
the other hand, Mathematics is also
the basis for building knowledge
related to other areas of the
curriculum. It is present in the Exact
Sciences, in the Natural and Social
Sciences, in the varied forms of
communication and expression.
(Brasil, 2001, p. 99)
vi
.
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Without the necessary pedagogical
attitude of contextualization and
decontextualization there is no need to talk
about meaning production and negotiation
of learning meanings. It is about leading
EJA learners to understand that
mathematical ideas evolve. Thus, the
History of Mathematics reveals itself as an
important approach and motivation
strategy for the teaching of mathematical
concepts. For this reason, D'Ambrosio
(1996) asserts that the History of
Mathematics is central to the establishment
of a teaching process that presents
Mathematics as a cultural product, contrary
to the usual diffusion in the school system
that treats it as a ready, definitive truth and
finished, exact science that seems
unrelated to human vicissitudes.
Also, Miguel and Miorim (2004)
emphasize that the History of Mathematics
will help the student to realize that
Mathematics is not a science isolated from
other areas of knowledge, consolidating
itself as a human creation and, mainly,
indicating to the students the reasons by
which people do and practice mathematics,
that is, practical, economic and cultural
needs are the stimulus to the development
of mathematical ideas. It is, therefore, a
problem of development, being that in the
scope of the historical-cultural theory is the
learning that advances it.
Davidov (1988) relies on Vygotsky
(2009b) to cite the most recent edition, and
in Leontiev (1978) to establish that
education is the indispensable and general
internal source of development. That is,
teaching can only be effective, which,
through the content of knowledge to be
appropriate, anticipates and guides
development.
Therefore, according to Davidov
(1988), it is possible to develop in the
subject capacities and skills historically
formed and essential to contemporary
reality. It is not enough to teach the EJA
student the social function or use of a
particular concept, it is necessary for him
to develop (or reproduce) the human
abilities that are inherent in this object of
knowledge in order to use it properly.
The development of theoretical
thinking, particularly in relation to the
mathematical context, imposes the
students' need to establish relationships
between facts and things in order to
coordinate actions. It is remarkable the
pleasure of the students to sit on wheels to
talk, to relate life stories, to make pictures,
to participate in some game, to do folds, to
play and to listen to stories. All of these
situations can be exploited for the
formation of mathematical concepts,
considering the quantitative, logical,
topological and order aspects involved.
According to Melo & Cruz (2014), the
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conversation wheel has the characteristic
of allowing the participants to express their
impressions, concepts, opinions and
conceptions about the proposed theme, as
well as to reflect on the manifestations
presented by the group.
We note all these activities in the
daily life of the EJA. But these activities
that populate the universe of the young and
the adult need to be further explored in
order to know their interests, tastes,
difficulties, desires and insecurities.
Likewise, all these situations can be
quantified and explored mathematically
verbal or by the mediation of a scribe who
can be the teacher himself, when they still
do not master reading and writing. As an
example, the wheels of conversation and
life histories always present in the EJA
situations involving the functions of the
number that, in general, are rarely explored
in didactic work in Mathematics: codes
such as the ZIP code of the streets, the
telephone number, the identity folks;
ordinality when referring to an apartment
in a building or to a historical data or, still,
the measure as an extension of the idea of
number. But in general, the school treats
the number only with quantification.
On the other hand, pictorial
registration through drawing or coloring on
checkered paper allows the exploration of
graphics, tables, paths, maps and other
forms of representation absolutely possible
and allows the contextualized exploration
of notions of number and of fundamental
facts of elementary mathematical
operations. I mean that the talk wheel can
not be limited to the conversation, although
this in itself is already important, as
Bakhtin (2012) teaches us.
In fact, understanding how EJA
students think and how they organize their
thinking in different situations enables the
teacher to plan didactic referrals that favor
them to broaden their thinking strategies.
The complex issue of concept formation
and activity theory
Undoubtedly, mathematical learning
is conditioned by the internal structuring of
that science. The nature of the process of
its construction obliges us to return
periodically to the same contents with
increasing levels of complexity, abstraction
and formalization throughout the entire
schooling process. However, when the
student begins the construction of
mathematical notions, he does so by
making them cohesive with the concrete
situation in which they present themselves.
This reinforces the need for a formal
presentation from the environment itself
and the impossibility of arguing over
abstract situations without due discretion.
A concept is not something that is
formed a priori, as a ready, finished and
undeniable truth. To form a mathematical
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concept requires taking hypotheses,
drawing conclusions about them and
observing regularities, recording processes
and results and systematizing situations,
without losing sight of the playfulness and
pleasure of discovery inherent in
mathematical thinking.
Undoubtedly, by appropriating the
forms historically instituted for the proper
use of objects and knowledge, the subject
appropriates everything that is found in the
sphere of culture. It is through practical
activity, with objects of culture, that the
formation of what historical-cultural theory
calls "ideal," an internalized form of its
material, actual existence, whose
appropriation is made possible through
verbal communication between people,
that is, thanks to language.
In this way of conceiving the
relations between empirical thought and
theoretical thinking, Davidov (1988) thus
refers to the thesis of dialectical
materialistic logic:
... the original, starting and universal
form of existence of the logical
figure is the real, sensorial - practice
activity of man. Verbal thinking can
be understood scientifically as a
derived form of practical activity.
This thesis is, in our view,
unacceptable for traditional formal
logic and for the traditional
psychology of thought. On the
contrary, this thesis is completely
legitimate for the dialectical
materialist logic and for the
psychology which is consciously and
consequently based on its principles.
It is clear that logic and psychology
must start from a common
understanding of the activity that
tends to accomplish the goals of man
and his main types. (Davidov, 1988,
p. 20)
vii
.
It should be noted, then, that
dialectical materialist logic proposes a
differentiated understanding of human
activity, as responsible for the
appropriation of historically accumulated
culture, for the conditions of
objectification of the subject, and for the
development of the human psyche.
According to Vygotsky (1991), we
infer that everyday and scientific concepts
involve different experiences and attitudes
on the part of subjects of knowledge and
develop by different trajectories, too.
Sometimes the subject becomes aware of
his spontaneous concepts relatively late,
that is, the ability to operate with them at
will appears long after he has acquired the
concepts, that is, he has the concept, but he
is not aware of the his own act of thought.
In the case of the development of scientific
concepts, the process usually begins with
verbal definition and with application in
non-spontaneous operations. It is a process
that occurs at the level of interactions,
teacher-student and student-student, it
could be said.
In the context of historical-cultural
theory it is observed that the development
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of the subject occurs in a systemic way,
that is, with the development of all the
psychic functions taking place in an
integrated, joint way. Thus, in historical-
cultural theory there is no specific
approach to the development of logical
reasoning in students, and this type of
reasoning is considered, together with the
other higher psychic functions, a product
of the type of activity that is provided to
the subject since the beginning of its
development. Logical reasoning would
therefore be a specific form of organization
of broad thinking, theoretical thinking.
Leontiev (1988) calls activity not to
any student's doing, but that it is
meaningful and especially, that it has a
purpose. According to him,
By activity, we designate the
processes psychologically
characterized by that to which the
process as a whole is directed (its
object), always coinciding with the
objective that stimulates the subject
to perform this activity, that is, the
motive. (Leontiev, 1988, p. 68).
viii
There will only be activity, therefore,
when the motive and the goal coincide. It
is important, then, that mediator teachers
consider the concept of activity and its
implications for the teaching process,
which brings us back to the problem of
training.
The study activity then reports to a
specific form of activity directed towards
the assimilation of theoretical knowledge,
with a view to the formation of theoretical
thinking.
For Davidov (1988), psychic activity
develops as the subject plans and selects
the objects (instruments) and the way
(strategy) to use them, according to the
purpose of the moment. The ability of
planning, in turn, depends on the
development of needs, which are always
social.
Every activity is triggered by a need.
Activity always seeks, as purpose, that
which is not yet real, but that there is
possibility of being real - this is the main
characteristic of vital activity. Thus
Davidov (1988, p. 33, my translation of the
original in Spanish) refers to the subject:
To seek what does not yet exist, but
which is possible and which is given
to the subject only as a purpose, but
not yet realized: this is the main
characteristic of the vital activity of
any sentient and thinking being, that
is, of the subject. The paradoxical
character of the quest consists in the
combination in itself of the possible
with the real. ... The subject
organizes his actions in dependence
of what can happen in the future and
a future that does not exist yet! Here
the purpose, as an image of the
future, as the image of what is to be
determines the present, defines the
real action and state of the subject.
It is commonplace that we find in
basic education mistaken assertions by
students about geometric forms. A circle is
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sometimes referred to as a ball, a sphere
may also be a ball or circle, and a cube is
sometimes taken as a square, among other
vicissitudes of that nature. Among other
reasons, this is due to the fragmented way
in which the geometric entities are
presented to the students, starting from the
flat forms to reach the spatial forms. It is a
consequence of the multifaceted
presentation of mathematical fact in school
and a contradiction, since knowledge is
constituted from what is general to what is
specific, from what is broad to what is
particular.
In this perspective, the formation of
geometric concepts is inversely related to
the way in which the school usually
explores the geometric concepts, besides
disregarding that the knowledge process is
established by the force of social and
interpersonal relations.
Thus, Davidov (1982) considers that
the objective of schooling should be the
parsimonious pursuit of the development
of theoretical thinking, beyond empirical
thought. In its formulation, the study
activity, through specific tasks, has as a
goal to lead the student to the appropriation
of more general laws that involve a
mathematical concept, so that they are
directed towards the concrete relations
being, through the appropriation, the
relation is transforms into abstraction with
content.
By this way of conceiving the
appropriation of scientific knowledge,
mathematical knowledge and geometry, in
particular, is taken as human, historical,
and social production. Being in study
activity requires that teaching material
should prioritize the solution of cognitive
tasks in a context of investigative and
exploratory action in order to ensure the
creative experience.
The EJA student seeks to satisfy his /
her cognitive interests by communicating
with others and by observing their
surroundings. It is with the process of
formal schooling, according to Davidov
(1988) that there is a new stage of
development whose main activity becomes
the activity of study.
Under the guidance of the teacher,
the learner systematically assimilates the
theoretical content in the form of scientific
concepts, moral values and artistic images,
the developed forms of social
consciousness observed in science, art, and
morals and capacities to act in accordance
with the demands of these organized forms
of thought.
Therefore, Davidov (1988) considers
that the organization of the study activity
requires the introduction of new forms for
full realization, and the general cultural
habits of reading, writing and calculation
are not enough. They must constitute a
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general psychic development in the form
of capacity to study.
It is a question of seeking an
organization of teaching in the EJA that
contemplates appropriations of scientific
concepts, not just everyday ones. This is an
essential condition for the development of
theoretical thinking. It is the coexistence
with representations that allows the
students to elaborate thinking capable of
articulating the various conceptual
meanings, be they arithmetic, algebraic or
geometric. This requires the insertion of
the student in investigative activity,
assuming the development of his ability to
elaborate questions, starting from the
mediation of the teacher who is responsible
for elaborating and organizing particular
tasks aimed at achieving this goal.
Once these considerations have been
made it is necessary to establish that,
according to Duval (2003), the
understanding of information or
mathematical activity lies in the
simultaneous mobilization of at least two
registers of representation, or in the
possibility of changing at any moment of
registration of representation. The
coordination of at least two registers of
representation is manifested by the speed
and spontaneity of the cognitive activity of
conversion. This, in theory, explains a
great part of the students' difficulties with
the mathematical activity because a
process of teaching by association of
models is still manifest in school,
practically devoid of the foundations of the
dialogical relation necessary for the
production of meanings and for the
negotiation of meanings in mathematical
education as we defend throughout this
article.
As learning can be understood as the
possibility of making connections and
associations between different meanings of
each new idea, it depends, then, on the
multiplicity of relations that the student
establishes among these different
meanings. Hence communication is a
resource that helps the learner to establish
the connections between his spontaneous
conceptions and what he is learning again
with a view to establishing meaningful
learning in Mathematics.
Incoherently, sometimes the school
still explores in the schooling process the
work with the reproduction, without
understanding, of ideas and concepts, of
texts, of copies, in short. On the one hand,
it loses the possibility of exploring in the
process of mathematical literacy the rich
body of symbols that the pupil lives even
when he leaves for the streets or to
organize the physical space destined to the
construction of his own dwelling; On the
other hand, when the school inserts the
EJA student in the mathematical world it
seems that it does it starting from nothing
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and it usually sins by the repetition of
symbols devoid of meaning, losing the
possibility of thinking the negotiation of
mathematical meanings.
Researches such as those developed
by Bruner (1997) or Lins & Gimenez
(1994), for example, emphasize the
processes of signification in the historical-
cultural perspective, emphasizing that the
production of meanings and negotiation of
meanings are concepts that encompass
both meanings already consolidated as the
meanings that things, words, events,
gestures, actions, etc., can assume for
people.
In this way, all signification is a
social production. From this point of view,
producing meaning for mathematical
concepts implies linking them to other
factors internal or external to the
Mathematical discipline. It is a didactic-
pedagogical action that involves activities
of investigation, contextualized action,
comparison, and observation of
mathematical facts of reality, relations
between the constituent elements of objects
created by humanity and, in particular, the
relation between informal knowledge and
systematized mathematical knowledge.
Therefore, the teaching of
Mathematics for the subjects of the EJA
can be initiated by the coordination of
quantitative relations of the cultural
universe of students such as those
involving quantitative data of reality, play
activities and sensorial exploration of the
physical environment, spatial perception,
from the moment the subject identifies and
perceives their location in space, even if
this space is as close to it as the classroom,
home, street where they live, among other
things.
From the exploration of the sensory
space it is possible to lead the students to
observe the objects that also occupy this
space, to make relation between one and
another object, the size that they have;
identify and relate similarities and
differences; recognize the geometric forms,
dates or times; and other mathematical
ideas present in the daily routine, in
particular, those that go back to the
numbering.
However, the study activity aims at
the appropriation of socially elaborated
experience, knowledge and skills, which
presupposes the formation by the students
of the abstractions and generalizations that
form the basis of theoretical thinking. Yet
according to Davidov (1988), content is the
basis of teaching that promotes
development.
In this way, subjects of the EJA need
to establish the connection of the universal
or the general with the particular or
singular, that is, operate with the concept
in the transition from general to specific. In
the words of the author,
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By its content, the theoretical concept
appears as a reflection of the
processes of development, of the
relation between the universal and
the singular, the essence and the
phenomena; by its form, appears as a
procedure of ascension from the
abstract to the concrete. (Davidov,
1988, p. 152)
ix
.
Thus, to teach geometric concepts in
the EJA we can organize ourselves from
relations between Space and Form and
Greatness and Measures, without the
preoccupation with formalization too early
as it happens, but with the development of
geometric ideas that can throughout the
development sustain the structuring of
mathematical thinking. Activities of
recognition of physical space, indications
from objects, locality, near, far, right, left,
above, below, here, there, etc., are revealed
as necessary for the development of what
can be called fundamental vocabulary of
Mathematics.
To direct the conclusion of the text,
we emphasize that in the daily activity the
student develops the capacity of
imagination, being that when getting
involved in the activity of study,
appropriates the capacity to think
theoretically. It should also be noted,
however, that such capacities are not
innate, they are developed in a process in
which the individual reproduces, by his
own activity, the human capacities
historically formed.
It seems to us fundamental to
consider in due proportion theoretical
elements that point to the game as a phase
of transition to more developed stages of
thought, that is,
... in the later stages of game
development, the object already
manifests itself as a sign of the thing
through the word that dominates it,
and the action with gestures
abbreviated and synthesized
concomitantly with speech. Thus,
ludic actions have an intermediate
character and gradually acquire that
of mental acts with meanings of
objects that take place in the plane of
speech aloud and still rely on
external actions that nevertheless
have already acquired the character
of gesture- synthetic indication.
(Elkonin, 2009, p. 415)
x
.
Again as an example, from activities
aimed at the exploration of the
environment, which for students is a game,
they do not only learn about space, but also
exercise and learn the vocabulary
necessary for such communication.
Constructions with different materials,
setting up models, routes and labyrinths
and the exploration of larger spaces, which
can be done from the explanation of the
path they make from home to school, from
classroom to bathroom, from where they
live for work, the path they have taken on a
tour they have done, and several other
paths, can enable EJA subjects to relate
reality and mathematical ideas. Likewise,
measuring shapes and paths with the most
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different nonstandard quantities such as the
span, the foot, a piece of string or wood
may prove salutary so that they gradually
understand the need to use a tape measure,
instruments of standardized measures.
Likewise, the exploration and
quantification of work reality data can
relate the mathematical concepts that are
explored daily and the school mathematics.
When students begin to recognize
geometric forms, it is common to use
names created by them, names that are not
specific, and it is then up to the teacher to
know how to respect the nomenclature
created by the subjects, but as soon as the
opportunity arises to speak the name of the
form that the subject he has to explore the
regularities of the figures in order to lead
the students to recognize the specific name
of a geometric form in recognition of their
particularities.
Elkonin (1987) warns, however, that
the child does not live his work as the
adult; in the same way, I would say that the
adult cannot be infantilized, that is, he has
his own form of experiences in the context
of his cultural activity, his work and even
in the act of playing or playing. Therefore,
based on the author's thinking, it is
important to approach the game as
fundamental to psychic development,
without limiting it to the merely didactic
question. I emphasize that an analysis
based only on the skills, abilities and
notions that the game can contribute to
form in the students restricts its
possibilities, putting in second place its
specificity by the observation and
representation of the social relations of
adults in the game.
In light of the above, we need to
develop actions in order to mobilize the
groups constituted in the school of EJA,
directing them to reflection, so that the
contradictions between thought and action,
between the lived and the conceived ,
become explicit, driving them to change.
This leads us to think that:
a) the disinterest and low achievement of
students in Mathematics, historically due
to the traditional way of transmitting
mathematical knowledge, contrasts with
the playful content and formal beauty of
Mathematics;
b) the exaggerated emphasis on the
logical-formal symbolism of Mathematics
reinforces the pedagogical tendency to
"pass content" to the detriment of a
process of formation of mathematical
concepts;
c) the preoccupation with routine
operations and memorization impairs the
cognitive development of the student
determining, in association with other
factors, the failure of teaching;
d) the lack of integration between the
themes according to the linear
organization of the curriculum ("ladder
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curriculum") contrasts with the idea of
"spiral organization" or with the
contemporary conception of curriculum as
a "network of meanings" and reinforces
the fragmentation of Mathematics teaching
programs.
To summarize, the discussion about
the problem of the formation of
mathematical concepts must consider as
central theses of the action in the teaching
situation a broad process of production of
meanings of learning and negotiation of
meanings based on the pedagogical
implications of the historical-cultural
theory for education mathematics of
young people and adults:
a) Contextualized problematization:
consideration in the pedagogical work
with Mathematics of the socio-cultural
contributions of the student to be
considered in the school situations
experienced by students outside it, what
could be called cultural mathematics, that
is, the various forms of mathematization
developed by the various social groups, in
order to allow the interaction between
these two forms of mathematical thinking.
b) Historicization: to show students how
mathematical ideas evolve and
complement each other in an organic and
flexible whole, is a basic assumption to
understand mathematics as a construction
process.
c) Transdisciplinary entanglement:
organization of mathematical ideas in
articulation with the various areas of
knowledge since they do not arise from
nothing; on the contrary, many
mathematical ideas did not even arise in
exclusively mathematical contexts.
Mathematical learning is
conditioned by the internal structuring of
this science. The nature of the process of
its construction forces us to lead the
student to return periodically on the same
contents with levels of complexity,
abstraction and formalization increasing.
When the student begins the construction
of mathematical notions, he / she makes
them cohesive with the concrete situation
in which they present themselves. This
reinforces the need for a formal
presentation from the environment itself
and the impossibility of arguing over
abstract situations without due discretion.
In such a way, to form a
mathematical concept demands to raise
hypotheses, to draw conclusions about
them and to observe regularities,
registering processes and results and
systematizing situations, without losing
sight of the playfulness and pleasure of
discovery, inherent in mathematical
thinking.
Possibly, when observing the
speech, the experiences and the usual
mathematical knowledge of the students of
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the EJA, the teacher will see that even
when registering incorrectly, the adult may
be understanding the content presented to
him, and manifesting this learning in his
own way. To what extent has this been
considered in the daily pedagogical
practice of the initial years of schooling, in
particular, in what concerns the teaching
of mathematical concepts in the EJA?
Conclusion considerations
In recent theoretical formulations,
the attempts to renew Mathematics
teaching programs in the Brazilian
context, especially in the last thirty years,
seem evident in recent theoretical
formulations. In general, the teaching
programs planned in this period address
issues that can be considered as current in
this curricular context such as the notion
of mathematical literacy, the quest for
overcoming linearity of the curriculum,
the perspective of meaningful learning,
relations of mutual impregnation between
the mother tongue and mathematical
language, and in particular the
understanding of problem solving as the
matrix that generates a process of
formation of concepts in Mathematics.
In this sense, we highlight
throughout the article some contradictions
that stand out in the daily life of
mathematics education and the
implications for the initial schooling of the
youth and adults of the EJA. On the one
hand, one loses the possibility of exploring
in the educational process the symbolic
richness that the subject experiences even
when going out into the streets; On the
other hand, when the school inserts the
adult in the world of numbers, he thinks
that he does it out of nothing and sins
again by the repetition of symbols devoid
of meaning and does not consider the
possibility of thinking about the
production of meanings and the
negotiation of meanings in Mathematics.
Not infrequently, we observe in some
classes of the EJA the eagerness for the
introduction of the symbolic language of
way too precocious.
We start from the idea that, given
these findings about the recent curricular
movement in mathematics education, EJA
and academic debate in a national context,
the analysis of teachers' representations
about these attempts to renew the teaching
process still turns out to be somewhat
contradictory, to be judged by the chosen
pedagogical behaviors and the
mathematical learning evaluation
indicators.
Analyzing some invariants of this
situation we establish that the development
of theoretical thinking, in the historical-
cultural perspective, consolidates parallel
to the sensorial development and it is the
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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task of the teacher of the EJA to contribute
to the development of both through a
pedagogical action that puts as a
perspective the problem of the sense and
meaning of teaching and learning of
Mathematics.
We consider that the predominant
preoccupation with the development of a
symbolic language that is excessively
abstract and patterned by repetition and
memorization still prevails, practically at
all levels of mathematics teaching. Thus,
topics that should be dealt with in an
integrated way with other areas of
knowledge, involving practical activities,
have been approached in isolation from
each other, making it difficult for students
to learn and synthesize.
Thus, the contemporary tendency of
curricular organization sees the formation
of concepts as a vast field of decisive
formulations for the development of
logical reasoning, in the resolution of
problems that require visualization and
manipulation of mathematical facts as well
as for the establishment of relations
between facts of other knowledge areas.
The work with mathematical notions
significantly from the first steps in the
schooling process also contributes to the
expansion of number and measure ideas,
since it stimulates the student to observe,
perceive similarities and differences, and
identify certain regularities inherent in
mathematical thinking.
Thus, without the pretension of
exhausting the discussion on the subject,
we establish throughout this article that it
is from the exploration and manipulation
of ideas and data of the physical world that
the student will be allowed to establish
connections between Mathematics and
other areas of knowledge. Basically, a
process of teaching and learning
Mathematics significantly in the EJA, as
discussed, cannot do without the broad
process of negotiation of meanings and
production of senses of learning; of
historicization that allows to consider the
process of evolution of the mathematical
ideas and, also, of transdisciplinary
treatment of the themes developed in the
mathematical education that can make
possible for the students the perception that
the Mathematics is, primordially, a science
that deals with the search of solution of
problems faced by humanity in the course
of historical development.
The research in Mathematics
Education reports extensively to these
particularities of the teaching of this
science and denounces the neglect with
which the learning of this language in the
school has been treated. Among other
actions, it is salutary the discussion with
teachers and those interested in the subject
about the importance of the presence of the
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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transmission of these mathematical ideas in
the different levels of education in order to
meet the needs of the subjects of the EJA
to build knowledge articulated with the
various domains of thought and the social
imposition of instrumentalizing them better
to live in a world that progressively
requires the most different knowledge and
skills.
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i
My translation of the original in Portuguese as
referenced at the end of the article.
ii
My translation of the original in Portuguese as
referenced at the end of the article.
iii
My translation of the original in Portuguese as
referenced at the end of the article.
iv
My translation of the original in Portuguese as
referenced at the end of the article.
v
My translation of the original in Portuguese as
referenced at the end of the article.
vi
My translation of the original in Portuguese as
referenced at the end of the article.
vii
My translation of the original in Spanish as
referenced at the end of the article.
viii
My translation of the original in Portuguese as
referenced at the end of the article.
ix
My translation of the original in Spanish as
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x
My translation of the original in Spanish as
referenced at the end of the article.
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
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v. 3
n. 2
p. 519-548
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2018
ISSN: 2525-4863
547
Article Information
Received on April 16th, 2018
Accepted on May 17th, 2018
Published on June 23th, 2018
Author Contributions: The author was responsible for the
designing, delineating, analyzing and interpreting the data,
production of the manuscript, critical revision of the content
and approval of the final version to be published.
Conflict of Interest: None reported.
Orcid
José Carlos Miguel
http://orcid.org/0000-0001-9660-3612
How to cite this article
APA
Miguel, J. C. (2018). Mathematical education of young and
adults: pedagogical implications of historical-cultural
theory. Rev. Bras. Educ. Camp., 3(2), 519-548. DOI:
http://dx.doi.org/10.20873/uft.2525-4863.2018v3n2p519-2
ABNT
MIGUEL, J. C. Mathematical education of young and
adults: pedagogical implications of historical-cultural
theory. Rev. Bras. Educ. Camp., Tocantinópolis, v. 3, n.
2, mai./ago., p. 519-548, 2018. DOI:
http://dx.doi.org/10.20873/uft.2525-4863.2018v3n2p519-2
Miguel, J. C. (2018). Mathematical education of young and adults: pedagogical implications of historical-cultural theory...
Tocantinópolis
v. 3
n. 2
p. 519-548
may/aug.
2018
ISSN: 2525-4863
548