Realistic Mathematic Education: a theoretical methodological approach to the teaching of mathematics in countryside schools

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.


Introduction
The rural education movement lacks

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(Gravemeijer , 1999Gravemeijer & Terwel, 2000).

From
Freudental's perspective, students should learn mathematics through 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,  Contextual problems are considered key elements in RME and must be able to form concepts and models (Treffers & Goffree, 1985 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 premathematized 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).

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 1987Treffers , 1991Gravemeijer, 1994). 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).

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 support tasks were linked to the principles of RME considering our

Results and Discussion
The 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

Formal Level
The approach adopted to reach the   In analyzing the flowchart, we see that the RME principles were achieved during the classroom intervention, as shown in the following table: 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 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 Education is a promising way of teaching/learning exploration.