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Delectus - Scientific Journal, Inicc-Perú - [ISSN: 2663-1148]




Vol. 4 No. 2 (2021): July-December [Edit closure: 01/07/2021]

RECEIVED: 07/06/2021 | ACCEPTED: 28/06/2021 | PUBLISHED: 01/07/2021

Suggested quote (APA, seventh edition)

Samaan Mardo, G. (2021). The geometrical radar: didactic resource for the education of the rotations of figures flat. Delectus4(2), 34-44.

The geometrical radar: didactic resource for the education of the rotations of figures flat

Graciela Samaan Mardo

Universidad Pedagógica Experimental Libertador (Upel), Venezuela

The present investigation allowed to the educational qualify to improve the process of education and learning do in of the educational context, therefore had by object, propose strategies and didactic tools for the education and the learning of the rotation of flat figures in the second year of half education general, that consented in the development of skills by part of the educational and students in the geometry, where, the student worked the forms and geometrical structures. With the end to strengthen the skill that the student has to have to guarantee the learning in the area of the Geometry, with didactic tools. For this, considered the theories: Model and Geometrical Reasoning of Go Hiele, Model of Hoffer and Theory of the didactic situations. The Investigation of campo, of descriptive design no experimental and of quantitative approach carried out in a to sample of twenty-two students of second year of the Or.And.N. Dr. José María Vargas to those who applied them a to didactic unit structured in presentation, purposes, competitions, activities and evaluation. Like instrument of recollection of data used observation direct and leaf of work. It concludes that the strategies and didactic tools used help to the educational in the process of the education of the geometry and allow to the student have understanding of the geometrical concepts and with this, can systematize mathematical procedures and execute procedures.

Keywords: Didactic Strategies; Education; Learning; Rotation of Flat figures.

The mathematical education, in the last decades, to world-wide level has been an area of study that has emphasized the development of critical approaches in the eagerness to improve the education and learning of his different branches. In the last years, in Venezuela the mathematical education has turned into scientific discipline, having narrow relation with several theoretical and methodological currents that comport to direct to the development of the investigation of the processes education and learning of the Mathematics, like this then, the educational live these changes in the mathematical contents, the strategies, material and didactic resources that can facilitate to the students for a better learning.

In this sense, the experience in the educational system Venezuelan highlights a restlessness by the mathematics and especially by the geometry, as seldom it arrives tackle all the geometrical contents in spite of being contemplated in all the programs of the half education general, what hampers that the mathematical knowledge’s are assimilated properly. about, Sánchez (2018) to severe that in the majority of the classrooms are not applied suitable didactic strategies to achieve the geometrical aims of significant way, by what is necessary to apply strategies and resources in the classroom so that the students can have practical and visual examples of the theory supplied by the educational.

The Geometry give of a pedagogical point of view is one of the matters that tends to have greater difficulties in the teaching and learning of the same, due to that the educational do not have the means and educational materials that allow him develop his contents in shape more interesting and effective. This problematic does notorious in the different levels and modalities of the educational system Venezuelan, but more highlights in the secondary education. Therefore, it is here where the educational has to do emphasis to teach this matter by means of didactic strategies that motivate the visualization of the contents and that, besides, allow to the student develop the mathematical skills.

For such end, the educational has to schedule didactic activities that motivate and wake up the interest of the students, so that they find sense and can visualize the content of the subject given, what stimulate to the experience to learn and participate actively in the sessions of classes.

Likewise, the didactic units contribute to that the students affirm the personality, develop the imagination and was able to solve problems. Of there that, this study, design a didactic unit for the education of the area of Geometry, having like theoretical and methodological base the model of geometrical reasoning of Go Hiele and the s geometrical skills of Hoffer, which indicate the form to learn and teach geometry.

The present study has considered work with the Model of Reasoning of Go Hiele that will back the design of the Strategies for the Education of the Rotation of Flat Figures in Second or Year of General Half Education. To continuation, mention the theories chosen pair to the development of this work.

Model of geometrical reasoning of Go Hiele

This model of reasoning geometric was designed by the husbands Go Hiele and Diana Go Hiele, where in his investigation reflected, carries out the process of education and learning of the geometry, besides relate the educational of mathematics and the students. This is, evaluate the way of how the educational of Mathematical by means of the utilization of didactic tools and following the levels of the model, can help to the students to obtain a greater degree of knowledge and skill of geometrical reasoning to the subject developed in classes.

In accordance with Jaime (1993) the model of Go Hiele comprises two basic appearances: Descriptive: by means of this identify different forms of geometrical reasoning of the individuals and can value his progress. Instructive: It marks guidelines to be followed by the professors to favor the advance of the students in the level of reasoning geometric in which they find.

The model structures in two parts: the first designated the levels of reasoning where the students are the main actors for sustain the theory based by Go them Hiele, each level identifies by a type of specific knowledge.

Level 1 (Recognition or Visualization): The student recognizes the geometrical figures by his form like an all, does not identify the parts neither components of the figure, is not able to recognize or explain the properties determinants of the figures, the descriptions are mainly visual and compares them with familiar elements of his surroundings. It lacks basic geometrical language to refer to geometrical figures by his name.

Level 2 (Analysis): The student can recognize and analyze the parts and properties of the geometrical figures, but is not he possible integrate relations between properties of a figure or between geometrical objects. Like this, the student discovers in shape experimental, relations between split them components of a figure, knows that change a figure of position to another does not affect to his notable attributes. It does not accept that a figure can belong to several general classes and that adopt several names.

Level 3 (Ordination or Classification): The student determines the figures by his properties. It establishes the necessary and sufficient conditions that have to fulfil the geometrical figures, by what the definitions purchase meant. It follows demonstrations, but is not able and understand them in his globalidad; it is not him possible organize a sequence of logical reasoning that justify his observations.

Level 4 (Deduction): the student in this level, is able to make deductions, logical and formal demonstrations, when recognizing his need to justify the propositions posed. It comprises and it handles the relations between properties and formalizes in axiomatic systems, by what already understands the axiomatic nature of the mathematical.

Level 5 (Rigor): The student in this level, sees in front of a study of the Geometry quite high comprises the meaning of axioms or postulates is qualified to analyze the degree of rigor of several axiomatic systems and compare them between himself.

It is important to highlight that the levels are consecutive to the moment to apply and evaluate the model. This model does not distinguish ages for the understanding of each one of the above-mentioned levels presented.

Now well, the second part of the model refers to the phases of learning where details the way how the educational of mathematical can undertake the activity so that the students attain to reach the level of upper reasoning to the that have at present.

Phase 1 (Information): The professor of Mathematical has to familiarize to the students the previous knowledge that can have on the geometrical work and his level of reasoning regarding east.

Phase 2 (Orientation guided): The professor of Mathematical guide to the students by means of activities and problems posed included exercises proposed by the students, with the end to make explorations with the geometrical objects that stimulates them to observe some notable elements related with the subject of geometry.

Phase 3 (Explicitness): The students have to try express in shape oral or written the results that have obtained, exchanging his experiences and arguing on them with the professor of Mathematical, with the purpose to reflect the results obtained through his geometrical explorations that had with the educational in the classroom of classes.

Phase 4 (Free Orientation): The educational of Mathematical poses to the students that make different general activities of the previous, with the purpose to reach a reasoning and language increasingly powerful.

Phase 5 (Integration): the educational resume the content tackled with the students where they establish a global vision of all the learnt on the subject, integrating these new knowledge, methods of work and forms of reasoning with which had previously.

This is not more than the guarantee that the education of the Mathematical applied, under the investigations that present along history, allow to solidify the base to reach new put.

Model of Hoffer

Development of the clear-cut strategies in the education of the geometry

For Hoffer (1981), signals that, the education of the geometry from his origin is important, in the development of the strategy to make the formal demonstrations, in which it requires some mathematical knowledge of the person, situating it in a high level in his intellectual progress. It is thus that, the author poses that, the education of the geometry promoted the process of new skills that can generate the expertise in the student in the geometrical field and to his mathematical time.

Considering the before exposed, will indicate the five skills that proposes Hoffer (1981):

  1. Visual strategies: primarily focused in the study of the geometrical subjects, east admits the mental representation by means of visual representations, reaching in the students the process even of the two hemispheres of the brain where the importance in the learning of the Mathematical is invigorate the resolution of a mathematical problem.
  2. Verbal strategies: it Pretends that the student handle by means of a geometrical language where are present the definitions, postulates and theorems, descriptions of figures geometrical, among others; making possible that it reveals concepts and terms by himself same, recognizing the lack that can have in the moment to describe some concept in Geometry.
  3. Strategy of drawing: adequate to the student to the observation of figures, charts, among others; giving place to that it experiences by own account, the comprise and analyze the geometrical concepts, flat figures, his properties, allowing him build an idea through the graphic representation, relating it with his surroundings.
  4. Logical strategies: Sand links with the students in the form of as they can develop his logical capacities of informal way, later to learn to write a formal demonstration inside the mathematical study. This method allows him expand even more, to develop the skills and skills has in the logical analysis-mathematical building it from the informal with graphic and verbal ideas until attaining systematize the demonstration formally.
  5. Strategies of application: Here the student has to procure visualize and solve the facts that poses them taking into account some models with geometrical applications in which they serve him like solution of the problem.

The author refers a series of didactic strategies which has to apply the educational of Mathematical during all the class, adapting to the moment of education, of way pleasant and dynamic, where the student generated his previous knowledge and the link with the new information, like this then, obtained a very significant learning that helped him later in his process to share knowledge and to manage in new environments of study.

Theory of the didactic situations

The didactics of the Mathematics of the French school, developed by Brousseau (1986) does quotation to a discipline called “Theory of Situations”. It refers to a study whose theory handles under the development of the process of education that goes in pro to reach the willing realities to promote the mathematical knowledge, under the conjecture that the same do not build of spontaneous way. Like this Brousseau base the following:

(...) The systematic description of the didactic situations is a half more direct to argue with the teachers about what do or could do, and to consider how these could take in account the results of the investigations in other fields. The theory of the situations appears then like a half privileged, not only to comprise what do the professors and the students, but also to produce problems or exercises adapted to the knowledge and to the students and to produce finally a media between the researchers and with the professors (Brousseau, 2000).

It is notable to mention then, that the professors and the students comprise what want to do of the didactic situations, this helps them to build exercises, problems and situations that comport to a better communication under the mathematical language. To his time, allows that the educational systematize all what the student employs in the moment to execute a mathematical procedure that promotes said resolution of the same.

It fits to stand out that the theory of situations is backed under the constructivist current that, in education, consists in giving to the study you the tools that access him to create his own procedures for the resolution of a problematic situation, which allows that his ideas transform and follow learning. The learning, Brousseau (1986), affirms that the student “learns adapting to a half that it is factor of contradictions, of difficulties, of disequilibrium’s, a bit as it does it the human society. This knowledge, fruit of the adaptation of the student, self-evident by new answers that they are the proof of the learning”. It means that the student give develops a level of understanding, adapting to the need for the resolution of mathematical problems having in consideration the importance of the method that uses the educational in said activities inside the classroom of classes.

Definition of rotation

Doing a review of texts of Mathematical, can notice that they define of alike ways the Rotation of flat figures. As it defines it Suárez & Durán (2006), pose the definition of the following way:

Sean Or a fixed point and ∝ an angle directed die. The rotation of centre Or and amplitude or angle ∝  is a transformation of the plane that assigns to each point P an only point P' so that 8 and the angle &9 is equal to ∝. This rotation can denote by10(Suárez & Durán, 2006, p. 96).

According to this definition, the necessary elements that knows to make a rotation are, his centre of rotation and his angle directed, like this can determine the image of any point P of the plane. For example: it was Or the point given and was ∝=45°

The student situated the centre zero (0) and later with the help of the educational attain trace the rotation that asks him in the example whose angle has to take into account that it is ∝=45° , the result would be what in the grafico 1. The rotation of a segment with centre 0 and p'.


It appears 1. Rotation given his centre and an angle directed

Following, presents a second definition. According to Mariño Et al (2012), where consider the following:

Symmetry of Rotation: a rotation, in geometry, is a movement of change in the orientation of a body or a figure; so that, given a point any one of the same, this remains to a constant distance of a fixed point, and has the following characteristic: a point designated centre of rotation, an angle 11 and a sense of rotation (p. 62).

The approach of the authors seems to address to use the didactic tools more common, establish it to continuation Mariño et al (2012), of the way that “and the process of rotation use the instruments of geometry, fundamentally the compass and the conveyor. The first, serves to trace the arches of circumferences that allow to turn around the centre of rotation. And second, to measure the angles that want to rotar” (p. 62).

Evidence then that, in both definitions, the authors pose to the participant the way of the technician and use of the didactic tools, to develop the subject of rotation. Besides, they mention the characteristics that wrap to the subject of study. They do quotation also, of how consider the angles directed (positive or negative), depending his sense. Mariño Et al (2012), taking like explanatory reference that “these transformations by rotation can be positive or negative depending of the sense of twist. For the first case has to be a twist in contrary sense to the hands of a clock and will be negative the twist when it was in sense of the hands (p. 62).

This involves, that the way of suggestion of the education keeps on being the traditional way that commonly is used by the educational of mathematical. It is important to clear, that, in this study, remembered him to the student that, to make the activity that the educational of mathematic develops in classes with the didactic tools, are necessary to obtain the end that comports the intentionality to improve what has proposed the educational to reach an effective education.

To finalize leaves clear that, and purpose of the investigation based in obtaining the greater scope regarding the results that obtained by part of the students, those who were guarantors so that said investigation carry his course taking in account that when applying the didactic unit with the didactic resource designated Geometrical Radar where applied him to the students to develop a subject linked with the mathematical which chose the rotations of flat figures, highlighting that it gave with fullness the development of the process of education in the classroom of classes.

The investigation answers to a quantitative approach (Hernández et al., 2018, p.4). It addressed under the modality of descriptive design no experimental, framing inside the modality of Investigation of Field. Therefore, this modality allowed to the researcher comprise and resolve some situation, need or problem in a context determined.

Consisted in the application of a didactic unit for the method of education and learning of the Rotation of Flat Figures in Second Year of Half Education General in the Or.And.N. “Dr. José María Vargas”, Maracay Edo. Aragua, Venezuela. Of this way, described the Didactic Unit In what consists?, to which level goes to give?, which are the resources to represent them?, which are the strategies of education to use?, and how can evaluate?; That is to say, they considered the technicians of appropriate evaluation to know the Rotation of Flat Figures so that all the students have the same opportunities to go out favored and attain the learning by means of the strategies applied.

The population was made up by all the students of the second year of the Or.And.N. Dr. José María Vargas, municipality Girardot, City of Maracay, State Aragua, Venezuela; working with a total of 44 students had to that they are two sections and each one consists of 22 students. Like sample they selected 22 students that cor answers to 50% of the total population.

The technical and instruments of recollection of data used were the observation direct: it is the one who allowed to the researcher be present in the place where presented the didactic unit so that each student worked without pressure no and be slope of the time and the leaf of work: it Allowed to the student develop the technician applied in the strategy inside the didactic unit elaborated in the classroom of class.

This activity carried to effect during four (4) sessions of classes, of the subject mathematical, specifically in the content of rotations of flat figures, of the second year of the III stage of General Half Education; the direct observation made with the purpose to describe the type of strategy of the method of education that explains in the didactic unit tackled in the chapter IV of this investigation. Besides, of formal way, conversed with the students of second year of said institution, in which it knew the opinion of these, in relation with the activities that made commonly during the classes of rotations of flat figures and the motivation that felt to the subject. The analysis of data based by means of is statistics descriptive (Palella & Martins, 2010), those who affirm that: “it consists especially in the presentation of data in shape of tables and graphics” (p. 175). This represented the information collected by means of a conceptual analysis that was reflected by graphic of bar and picture of analysis that illustrator the problematic situation.

It is important to mention that all the information generated of the interaction of the participants, where applied the contents of a didactic unit on rotation of flat figures that designated to the students, as also the interview to the students, where the researcher described all the situations observed and analyzed in the Or.And.N. “Dr. José María Vargas”, to obtain in quality of vigor, a fluent way and improve the process of learning fundamentally in the cognitive development of the student inside the geometrical analysis.

The experts elected for the present investigation determined the efficiency and internal consistency of the items of the instrument that applied in a determinate moment in the execution of the same.
This study carried out with the following activities:

  1. Investigation on studies made in the area; as well as the bibliographic review on inherent appearances to the study.
  2. Initial analysis of the problematic situation to study.
  3. Integration and synthesis of the theoretical appearances.
  4. Determination of the methodology to be followed, inside the modality of Investigation of Field.
  5. Definition of the population and sample in study.
  6. Preparation of the didactic unit to apply based in the contents of rotation of flat figures.
  7. Observation, by part of the researcher to the students of second year of the U.E.N. “Dr. José María Vargas” in several sessions of classes of the subject of mathematical.
  8. Informal conversation with the students of the second year on the development of the activities in the application of the didactic unit.
  9. Realization of activities by part of the students and his respective analysis of the researcher to these activities.
  10. Preparation of conclusions and recommendations.

Design of the didactic unit
This module, stands out some elements that compose and integrate the preparation and the development of a didactic unit employee by the educational assigned, where shows the labor execution inside the classroom, integrating like this the educational participation-student, with the purpose to obtain positive results for the aim that wants to reach

It appears 2. Representation of the Geometrical Radar

It fits to stand out that, to execute a didactic unit, is notable that the professor was consents that his planning is totally individual, since it has to be developed by the need that accredits to the educational to improve the process of education and learning concerning a subject explained in the area of mathematics. By his part apply a strategy based in the use of a resource designated Geometrical Radar where the intention is to explain by means of the resource used the rotations of flat figures, having the following representation of the resource used in the classroom of classes.

From the most particular perspective, in this section tackled the subject of rotations of flat figures whose level of study sees in the second year of half education general directed in the Or.And.N “Dr. José María Vargas”, requiring the context programmatic according to text that follows the aims programmed for mathematical of the official program of the system educational valid Venezuelan; whose authors are Suárez Bracho & Last Cepeda (2006), how will appreciate in the following contents.

It results timely, remember them to the students some previous knowledge that deserves to know of the study of the geometry as they are it:

  1. The concepts of point, straight and plane purchase intuitively. A point lacks dimensions and denotes by a capital or tiny letter, plants near of the point.
  2. A straight PQ extends indefinitely by both extremes. A point Or divides the straight in two equal half-lines and boundless.
  3. A segment of straight is the part of straight comprised between two points.

It appears 3. Segment of straight

  1. It gives you give two half-lines BA and BTo´of common origin B, gives the name of angle, to the twist that does to coincide one of the half-lines with the, keeping fixed the origin.

It appears 4. Angle of twist

Once done the explanation of the subject of rotations of flat figures, tackles the content with more depth, necessary mention to the students, that the rotation of centre Or and of amplitude 180° is a symmetry of centre Or, and in each case Or is the half point of the segment that joins each point with his image.

For this, asks him to the student build a triangle where will have to require, which type of triangle whose characteristic that represents it, will rotate in the leaf of work that the professor facilitates him, of the same way will choose the centre of rotation and will determine the amplitude that apropie his comfort to be able to develop of effective way showing the level of understanding that obtained in the process of education and learning.

Now well, inside the didactic unit scheduled some activities that the educational programmed in several classes with the students, due to the fact that once that it facilitated them the commonplace of rotations of flat figures pretended do an activity related with said subject. It fits to stand out that, in the first place, it initiated the activity with 22 students of second year of the Or.And.N “Dr. José María Vargas”, initiated the class asking to the students that defined of intuitive way what was a point, what determines a straight and in a third place what is constituted the plane cartesian, answer that obtained with very good soltura since, by the classes dictated by the educational, the students had a good reception of the subject. On the other hand, it delivered them to the students a didactic material (leaf squared in size letter) to be able to make the activities that the educational scheduled.

It initiated the activity asking them to the students plant in couple of such way that can to interact and exchange ideas based in the subject of rotations of flat figures, once organic the students formed 11 groups of two students, to each one delivered him the leaf of work (leaf squared) and initiates the activity giving the necessary indications for the execute the activity scheduled.

To continuation, shows the activities related with the rotation in the plane that the students with the leaf of work have to make to reach the aim that pretends show in the didactic unit.

Activity # 1
Given the points (To-2,5) and B(-4,-2)  and given the centre or axis of rotation Or(1,-3) and amplitude α=-90°  determine the rotation of the segment. It analyse his answer.

Activity #2
Build a triangle whose vertexes come given by(To 3,7) ; B(5,5) and C(2,3) , make the rotation of the polygon knowing that his axis of rotation is the point Or (0,0), whose amplitude α=180° 

We see the results that the students obtained in the development of the first activity pouted by the educational and made by each group.

Group #1
It appears 5. The image arose of the first activity and is an own expression of the group one

Group# 7
Graphic 5. The illustration shows the work developed by the students of the group #7 developing the activity #1 of the didactic unit developed in classes.

In the first activity of the didactic unit “Giving the Twist” where emphasizes in the rotation of flat figures, the students initiate with graphical the system of coordinates cartesian, situating some of the points that correspond to the straight numerical or in his effect to the axis of coordinate Cartesian, as evidence in each one of the works illustrated by the corresponding images to this activity, which are identified following the order of the groups from the first team until the eleventh group, observing total coherence in the application.

Afterwards of the explanation that the educational facilitated them in the previous classes, shows clearly the handle of the geometrical connotations as they reflect it in each development of the previously exposed illustrations in the present chapter, where indicate it in some of the cases of the following way: they situate the points in the plane cartesian and trace a segment that joins to the points To and B , situate the centre of rotation designating it with the letter Or, afterwards do use of the tool or material of work as it is it the conveyor and situate the centre of the conveyor in the centre of twist of rotation that this billed in the exercise proposed by the educational afterwards with previous knowledge of theories that relate the subject of rotation situate the centre of twist of angle, marking with a point the image of the point To rotated -90° time sense of the needles of the clock, obtaining to the image of To denoting it as To', of the same way does it for the point B, in this case achieving the image that results B´, also identify the segments under the geometrical nomenclatures and his measure from the centre of twist until the points like this denote them of the following way: g.

It asked them what observed to measure that was construing the rotation of the segment?, what occurred with the image of the polygon, (that in this case it treats of a triangle) once that it applied him the rotation whose amplitude of gyre is of a determinate degree?; Afterwards by means of his analysis geometric - analytical answer that the points were moved of a side to another, asked him what can conclude?, to which analysis can arrive?; Answering that the original figure is equal to the figure that moved .

Clearly it shows that the language Mathematic does him slope up handle it by what the educational him stresses that, the name that gives him to the movement of the figure denotes rotation and that the figure carries by name polygon, in this individual this polygon is represented with the three (3) points in the plane, besides the majority of the students take in account that, the measures of the segments are equal, to what answer that it does not change the measure of the straight. This so allows him to it to the student inquire and explore by means of his own criteria and his knowledge tied to the training that obtains in classes.

In a second I happen the students try to make and expose some examples of the daily life to refer to the angles of twists, take like base the twist that can give a cadet that belongs to the military school when a voice of control asks him give the turn to the right or in his effect to the left. When awake that, the subject does reference and relates with real situations, the students begin to formulate questions and begin to inquire between them same for effect of the relation that exists between the subject of rotations and flat figures and the reality.

It is important to highlight that the development of the first activity the student had the ease and the soltura to be able to rotar a divide, the situation complicated when in the second activity them asked rotar a polygon.

This investigation drove to the students to obtain a better fluidity, a greater development inside the conditions and needs that the students participated to have every time that it touched you a subject of mathematics, did not reach to imagine that the mathematical are part of an all in the life of the human being, where by means of his algorithms, equations, calculations, logical, analysis and, even more, his geometrical level; it can reach the limit of any resolution that can involve to the educational, to the student and to a simple social context.

Of all this process obtained some results that allowed to be evaluated by means of the development capacities and difficulties manifested during the application of the didactic unit, concerning the rotations of flat figures and that allowed to reach the following conclusions:

  1. The students confront some problems in the development of the didactic unit, because of the little training that in level of Secondary Education gives him in the area of geometry.
  2. The use of the leaf squared, made it easy to carry out rotations of figures like segments and triangles, which could appreciate the development of the didactic activity.
  3. The put in practice of the didactic unit allowed to boost in the students the interest by the education and learning of the geometry regarding the rotation of flat figures.
  4. In relation with the geometrical skills put in practices by the student’s participants, can affirm that they deployed the following skills:
    1. Recognition: Identification of the types of triangle, point of centre, felt of the rotation (time sense, sense anti schedule).
    2. Analysis: Had a better vision with concerning the path that leaves the figure from the position initial until the final position.
    3. Differentiate: they could deduce the type of angle of rotation and sense assigned to turn said figure.
    4. Classification: Separator the different figures to be rotates, giving like this shows of geometrical skills developed.

Conflict of interests
The author declares that there is no conflict of interest

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