Revista Brasileira de Educação do Campo
The Brazilian Scientific Journal of Rural Education
ARTIGO/ARTICLE/ARTÍCULO
DOI: http://dx.doi.org/10.20873/uft.rbec.e7879
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
1
Este conteúdo utiliza a Licença Creative Commons Attribution 4.0 International License
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Realistic Mathematic Education: a theoretical
methodological approach to the teaching of mathematics in
countryside schools
Marcos Guilherme Moura-Silva
1
,
Rayza de Oliveira Souza
2
, Tadeu Oliver Gonçalves
3
, Ruy Guilherme Braga Borges
4
1, 2, 3, 4
Universidade Federal do Pará - UFPA. Instituto de Educação Matemática e Científica. Rua Augusto Corrêa, 1, Guamá.
Belém - PA. Brasil.
Author for correspondence: marcosgmouras@yahoo.com.br
ABSTRACT. The movement for a Rural Education still lacks
investigations of methodological theoretical assumptions for the
didactic field, based on the study of teaching practices that
consider the object of knowledge and, at the same time, value
the realistic/contextual aspect in which the student is inserted.
From this perspective, we investigate the methodological
theoretical implications of the theory of Realistic Mathematical
Education (EMR) for the teaching of mathematics in the
countryside school. Based on a qualitative methodological
approach, a hypothetical learning path was elaborated based on
the principles of EMR related to the teaching of analytical
geometry, from the practice of soil modeling in passion fruit
(passiflora edulis) cultivation. Our results point to the EMR as a
promising methodological theoretical approach of didactic
exploration to the countryside context capable of promoting
formal reasoning, concepts in realistic situations, appropriation
of mathematical language and potential for the development of
concepts in the field of Cartesian geometry.
Keywords: Realistic Mathematics Education, Emerging
Models, Analytical Geometry, Countryside School, Rural
Education.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
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2020
ISSN: 2525-4863
2
Educação matemática realística: uma abordagem teórico-
metodológica para o ensino de matemática nas escolas do
campo
RESUMO. O movimento por uma Educação do Campo ainda
carece de investigações de pressupostos teórico-metodológicos
para o campo didático, pautadas no estudo de práticas de ensino
que considerem o objeto de conhecimento e, ao mesmo tempo,
valorize o aspecto realístico/contextual onde o aluno está
inserido. Nessa perspectiva, investigamos as implicações
teórico-metodológicas da teoria da Educação Matemática
Realística (EMR) para o ensino de matemática na escola do
campo. Baseados em uma abordagem metodológica qualitativa,
elaborou-se uma trajetória hipotética de aprendizagem
fundamentada nos princípios da EMR relacionada ao ensino de
geometria analítica, a partir da prática de gabaritagem de terra
no cultivo do maracujá (passiflora edulis). Nossos resultados
apontam a EMR como uma via teórico-metodológica promissora
de exploração didática para o contexto do campo capaz de
promover raciocínios formais, conceitos em situações
realísticas, apropriação de linguagem matemática e potencial
para o desenvolvimento de conceitos no ramo da geometria
cartesiana.
Palavras-chave: Educação Matemática Realística, Modelos
Emergentes, Geometria Analítica, Escola Rural. Educação do
Campo.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
3
Educación Matemática realista: un enfoque teórico
metodológico para la enseñanza de las matemáticas en las
escuelas rurales
RESUMEN. El movimiento para una Educación del Campo aún
carece de investigaciones de supuestos teóricos metodológicos
para el campo didáctico, basados en el estudio de prácticas de
enseñanza que consideran el objeto del conocimiento y, al
mismo tiempo, valoran el aspecto realista / contextual en el que
se inserta el estudiante. Desde esta perspectiva, investigamos las
implicaciones teóricas metodológicas de la teoría de la
Educación Matemática Realista (EMR) para la enseñanza de las
matemáticas en la escuela del campo. Basado en un enfoque
metodológico cualitativo, se elaboró un camino de aprendizaje
hipotético basado en los principios de EMR relacionados con la
enseñanza de la geometría analítica, a partir de la práctica del
modelado del suelo en el cultivo de maracuyá (passiflora
edulis). Nuestros resultados apuntan a la RME como un enfoque
teórico metodológico prometedor de la exploración didáctica en
el contexto rural capaz de promover el razonamiento formal, los
conceptos en situaciones realistas, la apropiación del lenguaje
matemático y el potencial para el desarrollo de conceptos en el
campo de la geometría cartesiana.
Palabras clave: Educación Matemática Realista, Modelos
Emergentes, Geometría Analítica, Escuela Rural. Educación del
Campo.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
4
Introduction
The rural education movement lacks
investigations addressing
methodological/theoretical assumptions
based on the study of teaching practices
that consider the object of knowledge and
also value realistic/contextual aspects
familiar to the student. Few teaching
practices for mathematics are available to
assist the teacher in the classroom at rural
schools in order to implement the
principles of rural education.
From the standpoint of academic
performance, this scenario is critical and
requires interventions, as students at rural
schools in Brazil have a lower average
performance in mathematics than students
from urban schools. Data released by the
Instituto Nacional de Estudos e Pesquisas
Educacionais Anísio Teixeira (INEP [Anísio
Teixeira National Institute of Educational
Studies and Research]) show that only 6%
of students in the 5
th
and 9
th
grades of rural
schools perform adequately in
mathematics, which is half the rate
reported for urban schools (INEP, 2011).
More recent results from a large-scale
evaluation also show this disparity in the
level of proficiency between students at
urban and rural public schools, with an
average difference of 28.69 points (INEP,
2018).
To produce research aimed at
assisting in the teaching of mathematics at
rural schools in order to exert an impact on
the performance of the students, we bring
to the debate the assumptions of the theory
of Realistic Mathematics Education
(RME), highlighting it as a promising
means of teaching in the rural context. This
study presents the concepts and
foundations of RME, followed by
methodological procedures and a
discussion of the main findings. We aim to
establish RME as a
theoretical/methodological approach to the
teaching of mathematics at rural schools to
be explored by teachers in the classroom as
well as by researchers in the field of rural
education.
Realistic Mathematics Education
The instructional design theory of
RME had its origins in the Netherlands in
the 1970s during a universal effort to
improve mathematical thinking. It is based
on the interpretation of Hans Freudenthal,
who conceived mathematics as a human
activity (Freudenthal, 1983; Gravemeijer,
1994). In some ways, RME resembles
Decroly's “centers of interest”
(Gravemeijer 1994, 1999; Gravemeijer &
Terwel, 2000).
From Freudental's perspective,
students should learn mathematics through
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
5
a process of progressive mathematization
i
based on real or mathematically authentic
contextual problems. In this regard, RME
theory is primarily a knowledge-building
proposition; it does not focus on
motivating students in everyday life
contexts but on providing experientially
real contexts to be used in the progressive
mathematization process (Gravemeijer,
1999). According to Rasmussen &
Blumenfeld (2007), “RME is aimed at
enabling students to invent their own
reasoning methods and solution strategies,
leading to a stronger conceptual
understanding”. (p. 198).
The aim of contextual problems is a
process of reinvention on the part of
students to deal with formal mathematics.
It should be realistic from standpoint of
providing elements to imagine, turn into
ideas and become real in the minds of
students. This suggests that contextual
problems need not be authentically real but
must be imaginable, achievable and
conceivable (Van Den Heuvel-Panhuizen,
2005; Ferreira & Buriasco, 2016). Students
are able to extract information from a
contextual problem and use informal
strategies by trial-and-error to solve the
problem. This level in RME is
denominated horizontal mathematization.
The translation of this information into a
mathematical language using symbols and
progressing to the selection of algorithms,
such as an equation, is denominated
vertical mathematization (Figure 1). It is a
process involving the resolution of the
problem situation at different levels.
Figure 1. Horizontal and vertical mathematization.
Source: Gravemeijer (2004).
Contextual problems are considered
key elements in RME and must be able to
form concepts and models (Treffers &
Goffree, 1985). Based on De Lange
(1987), Ferreira and Buriasco (2015, p.
457) classify contextual problems in first,
second and third order according to the
objectives and mathematical potential, as
follows:
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
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Zero-Order Context: This is used to
make the problem look like a real-life
situation, denominated by De Lange
(1999) as a “false context” or
“camouflage context”. Problems with
this type of context should be
avoided. First-Order Context: This
presents “textually packaged”
mathematical operations, in which a
simple translation of the statement
into a mathematical language is
sufficient (DE LANGE, 1987). This
type of context is relevant and
necessary to solve the problem and
evaluate the response. Second-Order
Context: This is one with which the
student is faced with a realistic
situation and is expected to find
mathematical tools to organize,
structure and solve the task (De
Lange, 1987). According to De
Lange (1999), this type of context
involves mathematization, whereas
problems are already pre-
mathematized in first-order contexts.
Third Order Context: This enables
a “conceptual mathematization
process”. This type of context serves
to “introduce or develop a concept or
mathematical model”. (De Lange,
1987, p. 76, emphasis added).
Advancing the understanding of the
fundamentals of RME beyond contextual
problems and their classifications, Treffers
(1987) defined five principles for RME
(Table 1).
Table 1. Principles of Realistic Mathematics Education.
Phenomenological exploration
Mathematical activity is not initiated from the formal level but from a
situation that is experientially real for the student.
Use of models and symbols for
progressive mathematization
The second principle of RME is to move from the concrete level to the
more formal level using models and symbols.
Use of students' own construction
Students are free to use and find their own strategies for solving problems
as well as developing the next learning process.
Interactivity
The students' learning process is not only individual but also a social
process.
Interconnection
The development of an integrative view of mathematics, connecting
various domains of mathematics can be considered an advantage within
RME.
Source: Based on Treffers (1987).
A central heuristic of RME
encompassing all these principles is
denominated "emerging models", which
can promote ways of reasoning in students
for the development of formal mathematics
(Gravemeijer, 1999). Zandieh and
Rasmussen (2010) define models as ways
of organizing an activity, whether from
observable tools, such as graphs, diagrams
and objects, or mental tools, referring to
the ways in which students think and
reason while solving a problem (Treffers &
Goffree, 1985; Treffers 1987, 1991;
Gravemeijer, 1994). Models are
denominated emergent in the sense that the
various ways of creating and using tools,
graphics, analysis and expressions emerge
concomitantly with increasingly
sophisticated forms of reasoning.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
7
The emerging-models heuristic
involves four levels for the development of
this mathematical reasoning, starting with
the situational and moving toward the
formal, as shown in the figure below:
Figure 2. Levels of emerging models: from
situational to formal reasoning.
Source: Gravemeijer (2004).
The intention is that a student's
mathematical activity at each level changes
from a contextual solution (model of) to a
more general solution (model for)
(Gravemeijer, Bowers & Stephan, 2003).
The situational level is the basic level of
emerging models, where students work
towards mathematical goals within a
contextual problem. At this level, students
can use their own symbolism and related
models, regardless of the conceptual
mathematical rigors and the configuration
of the contextual problem. The referential
level involves building models based on
the initial configuration of the task. The
initially conceived models are adjusted
according to the contextual problem. At the
general level, the models built do not
depend on the configuration of the original
task. Finally, the formal level involves
reasoning with conventional symbolism
that reflects a new mathematical reality in
view of the contextual problem initially
posed.
In summary, the models appear in
specific contexts and refer to concrete,
experiential and real situations for students
associated with the RME principle of
“phenomenological exploration”. At this
level, the models must enable informal
strategies to solve the contextualized
problem. From then on, the model changes
its role and the students can establish
mathematical relationships and strategies
related to the principle of “using models
and symbols for progressive
mathematization”. Consequently, the
model becomes more objective and closer
to the level of formal mathematics. Thus,
RME argues that the modeling of informal
mathematical activities develops a more
formal mathematical reasoning model,
which can improve the level of
mathematical understanding and the
teaching-learning process. This issue
becomes critical when we consider rural
education, one of the pillars of which is
precisely the insertion of the context
experienced by the student in the learning
process.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
8
Methodological Path
The research takes a qualitative
approach following the
theoretical/methodological assumptions of
the instructional design theory of RME
based on the heuristic of emerging models.
The instructional design was developed for
teaching analytical geometry topics to
students in the 3
rd
year of rural high school.
The research involved the
exploration of the phenomenon of
preparing a plot of land
ii
for planting
passion fruit through discussions and video
footage. The rural property where the
investigation took place was located
around a settlement project in the
municipality of São João do Araguaia
(state of Paraíba, Brazil) and belonged to a
local resident who extracted fruit pulp as
an economic activity to acquire extra
income.
The actions were performed
considering the needs of the class, working
in the perspective of presenting a support
task to achieve the learning objectives and
thus develop concepts of analytical
geometry, as shown in the table below:
Table 2. Hypothetical learning trajectory for teaching analytical geometry according to assumptions of RME.
Learning objective
Concepts
Support Task
Understand elements of the
Cartesian plane
Point, plane and axes
Observe the process of preparing a
plot of land for planting passion
fruit and model the situation
mathematically.
Represent points on the Cartesian
plane
Coordinated pairs
Observe the process of preparing a
plot of land for planting passion
fruit and model the situation
mathematically.
Set distance and alignment
between points
Distance between two points and
alignment between three points
Second-order contextual problem
Source: research data (2019).
The support tasks were linked to the
principles of RME considering our
learning objectives. These tasks consisted
of the presentation of the practice of
preparing a plot of land for planting
passion fruit, from which the second-order
contextual problem was constructed,
according to De Lange (1987).
Intervention
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
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2020
ISSN: 2525-4863
9
The intervention carried out in the
classroom followed the seven steps defined
below:
1- Presentation of video footage
about passion fruit planting to the students,
configuring a farming practice with
emphasis placed on the phenomenon of
driving stakes at equidistant points. The
video was produced to compose the
teaching material to be used in the
classroom.
2- Moment of discussions for
students to express orally what they
understood about the phenomenon in order
to provide mental constructions of the first
mathematical relations.
3- Bring problematization to the
students' discussions on the geometric
knowledge in the preparing of the plot of
land aiming to develop at a more abstract
level the concepts of point, plane, axis,
Cartesian plane and coordinate pairs.
4- Creation of a model in groups
representing the phenomenon of preparing
a plot of land for planting passion fruit as a
way for students to put into practice the
geometric knowledge developed in the
previous steps.
5- Problematizing the geometric
concepts under study based on the
representative model of preparing the plot
of land, thus approaching a more general
level of school mathematics.
6- Demonstration in the framework
of the formulas necessary to calculate the
distance between two points and the
alignment of three points through
problematizations contextualized to the
phenomenon under study.
7- Development of questions about
distance between two points and the
alignment of three points from the
contextual problem.
Results and Discussion
The development of the concepts of
analytical geometry emerged through the
contextual problem of the measurement of
a plot in the practice of planting passion
fruit produced by the teacher, who took on
a mediating role in the process. The
subsequent problem is classified as a
second-order contextual problem in the
terms proposed by De Lange (1987), in
which the student is confronted with a
realistic situation and is expected to find
mathematical tools to organize, structure
and solve the task.
Contextual Problem
Before growing, passion fruit
seedlings need to be planted next to
wooden stakes with uniform spacing and
wire stretched between the tops of the
stakes following a single direction so that
the passion fruit grows easily. This method
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
10
of following a uniformity forms a grid,
with the equivalent distances from one
stake to the other.
Figure 3. Forming a grid: left, stakes are driven into soil at uniform distances; right, plants ready for harvesting.
Source: research data (2019).
During the grid-forming procedure,
the stakes are placed at two meters from
each other to facilitate the development of
plants until the harvesting of the fruit.
From the exploration of this specific
characteristic of passion fruit planting,
information was collected to serve as a
basis for the studies, fostering the first
emerging organizational models. Next, we
analyze the phases of the instructional
design from the levels of the emerging
models (situational, referential, general and
formal) as they were achieved by the
students in order to focus on the
contribution of the process to the transition
from intuitive, informal reasoning to more
sophisticated, formal modes of reasoning.
Situational Level
Considering the emerging models,
the situational level was obtained from the
students' engagement in the initial
observation of the educational video,
translating it into a contextual problem.
This was a phenomenological exploration
in terms of RME principles, in which the
situations are experientially real for
students and designed to support a
conceptual formalization process based on
informal reasoning. As a rich activity in
the generation of horizontal
mathematization, the students were asked
to watch and describe the environment of
the passion fruit plantation, seeking
general mathematical entities that would
later be discussed. In this instructional
design, conventional mathematics can be
reinvented, creating an opportunity for
progressive mathematization.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
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2020
ISSN: 2525-4863
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Reference Level
The students were asked to observe
the arrangement of the stakes in the soil
considering the visualization of the video
and to construct a representation of it. The
model was defined by the students and
each stake was represented by a “black
ball” numbered from 1 to 20, imagining an
aerial view of the passion fruit plantation.
The model was that of a small plantation
with 20 stakes placed equidistant from
each other at two-meter intervals, as shown
below:
Figure 4. Schematic model representing stakes for planting passion fruit.
Source: research data (2019).
The following figure shows the model developed by the students during the classroom
intervention:
Figure 5. Schematic model built by students representing positioning of stakes in grid format.
Source: research data (2019).
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
12
From this construction, the students
achieved the referential level with the
construction of a schematic model based
on the initial configurations of the
contextual problem: the positioning of
stakes for planting passion fruit.
General Level
Based on the previous schematic model,
the students were asked to imagine these
points on a plane with Cartesian axes,
expanding the idea of the situational model
to a more general model. Thus, the points
began to be understood in a Cartesian
perspective – a model that no longer
depended only on the initial contextual
problem, reaching the general level of
emerging models in the evolution of
mathematical reasoning.
Figure 6. Schematic model considering plane with Cartesian axes.
Source: research data (2019).
In this schematic model, the land
represents the plane and each stake
represents a point. With the help of a
system of axes associated with a plane, a
pair of coordinates was matched to each
point on the plane and vice versa.
Formal Level
The approach adopted to reach the
formal level, according to the emerging
models, started from problem situations
based on the initial contextual problem. At
the formal level, the students reasoned
about the concepts of analytical geometry
using conventional notations with a
reinvention guided by the teacher. For
such, the previously discussed two topics
of analytical geometry were addressed
during the classes to provide material to
support the investigative process.
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
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2020
ISSN: 2525-4863
13
The names of the subjects listed in
the following problems are fictitious, since
all activities in this stage were carried out
in the classroom and the students had
previous knowledge on the analytical
geometry topics studied during the
intervention.
Problem situation 1- Distance between
two points:
Supposing that we intend to
determine the distance between Ismael and
Tiago, two employees who work on the
passion fruit plantation harvesting the crop.
When you know that they are positioned
on stakes 16 and 8, respectively, as we can
see in the figure, what would this distance
be?
Figure 7. Schematic model of distance between two points.
Source: research data (2019).
When drawing two imaginary axes
corresponding to the ordinate and abscissa
axes in the representation of the passion
fruit plantation, the students were asked
about the distance between Ismael and
Tiago, as shown in the problem situation,
who were at points (0,6) and (4,2),
respectively.
Knowing that the distance between
two points is given by the equation:
,
The students developed the problem
solving as follows:
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
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2020
ISSN: 2525-4863
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Figure 8. Problem solving developed by students; student A on left and student B on right.
Source: research data (2019).
When carrying out the calculations,
the students obtained the answer 5.65; so,
the distance between Ismael and Tiago is 5
meters and 65 centimeters. The students'
resolution of this problem situation shows
that the sequence of actions promoted
formal reasoning and the appropriation of
mathematical language, as demonstrated
by the correct application of the distance
formula between two points by the
students.
Problem situation 2- Three-point
alignment
Imagine this situation: Ismael, Tiago
and Mateus are harvesting the passion fruit
and each is at a certain stake. Ishmael is at
stake 16, Tiago is at stake 12 and Matthew
is at stake 5. The employees have to work
in line for a better yield, thus reducing the
harvesting time and increasing
productivity. We will help Ismael, Tiago
and Mateus by checking if they are aligned
or not on the passion fruit plantation.
Figure 9. Schematic model of three-point alignment.
Source: research data (2019).
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
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We know that from three points A
( ), B( ) and C( ), they will
be aligned if, and only if, =
0. The students were able to identify the
three points that represented the stakes
where the employees were positioned, as
shown in Illustration 10: stake 16 (0,6),
stake 12 (2,4) and stake 5 (8,0). To verify
that these three points were aligned
considering the positioning on the
plantation, the students developed Sarrus'
Rule in the matrix formed from the points:
Figure 10. Problem solving developed by students; student A on left and student B on right.
Source: research data (2019).
As the answer, the students obtained
the value of the determinant equal to 4,
which does not satisfy the alignment
condition, as its value would have to be
equal to zero (det = 0). In this case, the
students managed to abstract key concepts
in the field of Cartesian geometry based on
the realistic situation and also managed to
advance in the understanding of resolution
rules, demonstrating that the actions have
the potential for the development of
concepts to teach mathematics at rural
schools.
Analyzing the two problem situations
described above, we can see that students
were able to use the appropriate
mathematical language to arrive at the
result of the problem, extracting
information and using informal strategies
(horizontal mathematization), followed by
the use of conventional mathematical
symbols and algorithms (vertical
mathematization).
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
16
Links with theoretical and
methodological assumptions of RME
During the intervention in the
classroom, the students were instructed to
build models in the terms proposed by
Zandieh and Rasmussen (2010), as the
concepts of the Cartesian plane, the
distance and alignment between points and
the relation between axis and pairs of
coordinates were developed in the
classroom. The flowchart below illustrates
the relations established in the instructional
design.
Figure 11. Flowchart demonstrating relations among RME principles, learning objectives and support tasks.
Source: research data (2019).
In analyzing the flowchart, we see
that the RME principles were achieved
during the classroom intervention, as
shown in the following table:
Table 3: Principles of RME and how each was achieved during classroom intervention.
Principles of RME
How it was achieved
Phenomenological exploration
In the exhibition of the video tutorial presenting the phenomenon under
study (measuring land in the practice of planting passion fruit), the students
had contact with an experientially real situation.
Use of models and symbols
for progressive
mathematization
From the moment that the students began to develop the geometric concepts
under study from the exploration of the contextual problem, especially
during discussions and problematizations about the theme, enabling a
natural evolution of knowledge.
Use of students' own
construction
In exploring the phenomenon under study during the construction of the
model, as the students could organize themselves in different ways; and in
discussions and problematizations where students' knowledge began to reach
more formal levels.
Interactivity
In the process of carrying out all actions planned for the classroom, with
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
17
greater emphasis during the construction of the representative model of
preparing the land, instigating students to apply the geometric knowledge in
development from a propitious situation to provide the approximation of
more general models of mathematics.
Interconnection
When students needed to mobilize other domains of mathematics to solve
the problems of calculating the distance between two points and alignment
between three points, using resources such as the formula to calculate the
distance in relation to the Pythagorean theorem and the determinant
calculation through 3 x 3 matrices (Sarrus' Rule) to determine the alignment
of the points. Thus, in the study of analytical geometry, interconnection is as
a tool present in all processes when combining algebra with geometry.
Source: research data (2019).
Linking our actions to the principles
of RME, the contribution was positive for
the teaching process of the topics of
analytical geometry, as it has its own
methodology that goes beyond traditional
teaching methods, bringing the
mobilization of mathematical knowledge
present in sociocultural practices into the
classroom.
Conclusion
Two specific mathematical practices
were promoted throughout the empirical
approach: i) the construction of a point
model on a plane, considering the stakes
used when planting passion fruit; ii)
reasoning about the Cartesian plane,
including the relation of axes with pairs of
coordinates. An interconnection between
several mathematical domains was
favored, such as the use of matrices and
determinants, and primitive concepts of
plane geometry, some naturally emerging
from the nature of the mathematical object
in question: analytical geometry as the
study of a plane and spatial geometry in an
algebraic perspective.
These mathematical practices arose
through contextual problems made
possible by the learning environment
designed by RME. This environment
enabled the students to produce their own
ideas in an examining and interactional
process in order to evolve from levels of
informal understanding to formal
reasoning. In these terms, this initial
contribution shows that RME can favor
mathematical practices and objectives and
develop conceptual understanding.
Specifically in addressing the
contextual problems developed, other
aspects stand out: (1) a textual model with
educational purposes conceptualized in
realistic situations; (2) the formulation of
mathematical schemes made possible
through statements; and (3) the promotion
of methodological teaching/learning to
explain and solve problems. Such aspects
indicate RME and the heuristic of
emerging models to be a significant
instructional design for teaching and
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
18
learning analytical geometry at rural
schools, surpassing traditional teaching
approaches in terms of mathematization.
As the methodological paths used by
the teacher in the classroom affect the
academic performance of the students, the
fundamental principles of RME in the
learning environment of emerging models
directly favor mathematical knowledge,
promoting more sophisticated and formal
reasoning. Therefore, teaching analytical
geometry from the context of socio-
cultural practices developed in the
students' communities is relevant, as
analytical geometry at most rural schools is
presented in a technical, abstract way.
The teaching of mathematics in rural
communities gives rise to a gamut of
realistic contextual problems, which makes
it possible to value the intrinsic
mathematical knowledge of the daily
activities of the different social groups and
merits greater theoretical and
methodological appropriation by
educators. Thus, Realistic Mathematics
Education is a promising way of
teaching/learning exploration.
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i
From a pedagogical perspective, mathematizing is
exploring contexts in order to understand / organize
/ mathematically.
ii
A land process template or process for defining or
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piles for implementing a crop.
Article Information
Received on November 07th, 2019
Accepted on January 20th, 2020
Published on May, 29th, 2020
Author Contributions: The author were 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 published.
Conflict of Interest: None reported.
Orcid
Marcos Guilherme Moura-Silva
http://orcid.org/0000-0003-3589-1897
Rayza de Oliveira Souza
http://orcid.org/0000-0002-8934-8614
Tadeu Oliver Gonçalves
http://orcid.org/0000-0002-2704-5853
Ruy Guilherme Braga Borges
http://orcid.org/0000-0001-7421-0105
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., & Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of mathematics in countryside schools…
RBEC
Tocantinópolis/Brasil
v. 5
e7879
10.20873/uft.rbec.e7879
2020
ISSN: 2525-4863
20
How to cite this article
APA
Moura-Silva, M. G., Souza, R. O., Gonçalves, T. O., &
Borges, R. G. B. (2020). Realistic Mathematic Education: a
theoretical methodological approach to the teaching of
mathematics in countryside schools. Rev. Bras. Educ.
Camp., 5, e7879. http://dx.doi.org/10.20873/uft.rbec.e7879
ABNT
MOURA-SILVA, M. G.; SOUZA, R. O.; GONÇALVES, T.
O.; BORGES, R. G. B. Realistic Mathematic Education: a
theoretical methodological approach to the teaching of
mathematics in countryside schools. Rev. Bras. Educ.
Camp., Tocantinópolis, v. 5, e7879, 2020.
http://dx.doi.org/10.20873/uft.rbec.e7879
