Las funciones didácticas en la enseñanza de la MatemáticaThe didactic functions in the teaching of MathematicsRisel Ruiz Cordovés Carlos Beltrán PazoUniversidad de Guantánamo. CubaORCID: http://orcid.org/0000-0001-6847-1878 http://orcid.org/0000-0003-3804-4159Correo electrónico:riselrc@cug.co.cucarlosbp@cug.co.cuRecibido: 25/04/2020Aceptado: 10/09/2020Resumen: La diversidad de criterios sobre cuáles son las funciones didácticas en la enseñanza y cómo abordarlas en las didácticas especiales, se ha convertido en un dilema para especialistas de estas disciplinas, de lo que no escapa la Didáctica de la Matemática. El empleo de métodos como el análisis-síntesis, el histórico lógico, el inductivo - deductivo y la observación, entre otros, propiciaron que en este artículo se lograra precisar qué entender por el término funciones didácticas y cómo se verifica su eficiencia en las clases de la disciplina Matemática.Palabras claves: Función didáctica; Proceso de enseñanza; Didáctica de la Matemática. Abstract: The diversity of criteria on what are the didactic functions in teaching and how to approach them in special didactics has become a dilemma for specialists in these disciplines, from which the Didactics of Mathematics does not escape. The use of methods such as analysis-synthesis, historical-logical, inductive-deductive and observation, among others, made it possible in this article to specify what to understand by the term didactic functions and how to verify their efficiency in mathematics classes.Key words: Didactic function; Teaching process; Didactics of mathematics. IntroductionIn the preparation of classes, it is impossible not to take into account the concept of didactic function and its transversality in the class. Teachers usually associate them to the "links" of the teaching process based on the criterion that they structure the class. However, difficulties also persist from the methodological order in the planning of classes and during the teaching of these in which the realization of the didactic functions and their relation with the parts of the class are not taken into account, which is evidenced in many of the controls to classes carried out at different levels of education, including the university. In the first instance, this is due to the pluralism that exists regarding two essential issues.The first of these questions refers to the didactic functions that cut across the classes as a basic teaching unit, while the second deals with the rather complex fact of how to give an outlet in mathematics classes to each of them.The authors of this article consider that there may be many more questions in this regard, but the result focuses on the answers to these two questions, although it is impossible to dissociate from others that are analyzed here.To these elements must be added the didactic fact that a class is not an isolated element within the teaching process, but that each class is closely related to the previous one and to the next one, and this in turn to the next one, thus forming the system of classes, which also have a logical structure.The objective of this paper is to explain some theoretical considerations related to didactic functions from their concept and definition, determining what they are and how they are performed in Mathematics classes.DevelopmentSome definitions of didactic functions can be found in the pedagogical literature. Some of them are listed below:According to the repository of articles Ecured (2021) didactic functions are concrete rules that are elaborated from the generalization of the Didactic Principles, which allow the teacher to apply them in a more specific, particular and sequential way. This concept does not specify whether the didactic functions structure the class and does not mention which didactic principles are derived from them. It also presents the limitation of not mentioning what these are. It is pointed out that the teacher can apply them in a specific and sequential way; however, it is not explained how, nor if it is part of the teaching process.According to M.A. Danilov and M.N. Skatkin (1975), the logic of the teaching process is given by constituent elements or links of the teaching process that have specific functions.These authors analyze the logic of the teaching process fundamentally organized in a given subject and at the level of a logical unit of learning (the class). They also clarify that although each link has specific functions, general teaching tasks are performed in any of them.They define the following as links in the teaching process:Problemical approach and awareness of cognitive tasks.Perception of objects and phenomena. Formation of concepts and development of the students' observation, imagination and reasoning abilities.Fixation and improvement of knowledge and development of skills and habits.Application of knowledge, skills and habits.Analysis of learners' achievements, testing and evaluation of their knowledge and revealing the level of intellectual development.For these authors, these links in the teaching process are the didactic functions. As can be seen, the first link or didactic function deals with motivation and orientation towards the objective; the second link deals with the formation of mental actions; the third link or didactic function deals with fixation; the fourth link deals with application; and the last link deals with evaluation.In this sense, it is the teacher who has the responsibility to determine on his own the answers to other questions. For example, there is no explicit didactic function that makes it possible to relate previous knowledge with the knowledge to be learned. On the other hand, this classification addresses fixation and application as two different links in the teaching process. Although it is true that they are two different concepts, from the didactic point of view they are closely related; therefore, the authors of this article do not consider it appropriate to approach them as separate didactic functions.According to Werner Jungk (1979), didactic functions are defined as the links in the teaching process.The author agrees with this criterion. Although it is true that didactic functions are links in the process, it is not made explicit in this definition that these links (didactic functions) guarantee the logic and effectiveness of the teaching process.Later this author reconceptualizes the links given by Danilov and Skatkin in the didactic functions as follows:1. Assurance of the starting level (ANP).2. Orientation towards the objective (OHO).3. Motivation.4. Elaboration of the new subject matter.5. Fixation.6. Control and evaluationAs can be seen, the first didactic function is included as the first didactic function, Assurance of the Starting Level, and Fixation and Application are merged into the didactic function Fixation.It also establishes that didactic functions 1, 2 and 3 are characteristic for the orientation phase both for the work with problems and for the formation of mental actions (definition of a concept, elaboration of a demonstration, realization of a calculation or geometric or construction exercise). Didactic functions 4 and 5 are characteristic of the phase of formation of an action or for work with problems and problem solving. The author himself considers that the didactic function Control and Evaluation in the sense that he uses it is not a component of the assimilation process because they are integral components of the teaching process as a whole. Control stimulates learning, and therefore contains certain elements of motivation.To these approaches, it is pointed out that the interrelation between the didactic functions in the logic of the teaching process in general and of the classes in particular is not made explicit.According to Machado (2016), didactic functions are those elements of the educational teaching process of the class or teaching that have a general and necessary character so that the established objectives are met.For the aforementioned author, these are didactic functions:1. Assurance of the starting level (ANP).2. Motivation and orientation towards the objective.3. Treatment of the new content.4. Fixing.5. Control or verification of learning.A novelty in this classification lies in the fact that, unlike other classifications of didactic functions, here an inseparable link is established between motivation and orientation towards the objective.This author frames didactic functions 1 and 2 within the activities at the beginning of the class or orienting phase. Those marked with the numbers 3 and 4 as part of the development activities of the class or executing phase. Number 5 is framed within the culmination activities or control phase.This classification is partly correct and, as a discordant element, they refer that in it the didactic functions are not combined or interrelated in the class. Each one is specific to a moment of the class and corresponds to the parts of the teaching activity, establishing a common thread of the process. For this, it is enough to think of something: if we motivate at the beginning of a class, will the students be motivated during the whole process? Certainly not. Motivation, orientation towards the objective, and the assurance of the preconditions are permanent elements or functions within any teaching process, especially in mathematics classes.According to a group of authors of the Universidad de Oriente (2019), the didactic functions characterize essential tasks (sometimes as stages, links or conductive threads) of the teaching-learning process derived from its regularities and that reflect and ensure step by step, in their integration and joint action, the assimilation of the content. These authors state that these functions make it possible to enforce the didactic principles in the teaching process by facilitating the didactic structure of the class.They classify them as follows: -Assurance of the starting level.-Orientation towards the objective.-Motivation. -Treatment of new content. -Fixation or consolidation of what has been learned. -Evaluation.According to this group of authors, these didactic functions (phases, steps or stages) constitute an element of special significance for the process of planning, structuring and carrying out classes.They also agree that in practice, didactic functions are developed in an integrated manner and complement each other. For this reason, many didacticians consider that the effectiveness of learning is also conditioned by their harmonious combination.The authors of this article observe that this conception of didactic functions, regardless of whether they are considered stages, steps or phases, is quite adequate to the teaching situations that arise within the teaching process, with particular importance in the classroom as the essential or primary link in the whole process.Taking into account the elements analyzed above and the specificities of the teaching-learning process of any subject, for the authors of this paper, the didactic functions are links to be taken into account in the structuring of the logic of the teaching process, specifically in the process of planning, structuring and conducting classes, contributing to the achievement of the objectives proposed in the program of the subject. They are fundamental in each phase of the class and are combined and interrelated throughout the teaching process.In this way, and as a consequence of the authors' work and research experience, for the purposes of this article we consider that the didactic functions are:1. Assurance of the starting level.2. Motivation.3. Orientation towards the objective.4. Treatment of the new content.5. Fixing the new content.6. EvaluationThese authors consider essential the understanding that each of these links crosses the whole teaching and learning activity in an indissoluble union, and whose permanent combination guarantees the positive result of the whole process. On the other hand, it should be taken into account that although they are separated for study, no part of the process escapes this combination. That is to say, for example, that the teacher constantly ensures the starting level or the preconditions that he considers the student needs to advance to the next step, the same happens with the motivation, which is permanent during the whole class, the orientation towards the objectives of each of the activities of the class, the constant evaluation of the whole process, and so on with the other functions. Thus, in the Orientation phase, the following didactic functions are fundamental: Assurance of the starting level (ANP), Motivation and orientation towards the objective (OHO), and Evaluation.In the phase of Elaboration of mental actions, the didactic function Treatment of the new content in combination and interrelation with ANP, Motivation and OHO intervene.In the consolidation phase, the didactic function Fixation of the new content in combination and interrelation with Evaluation intervenes.In practice, these functions penetrate each other and all work closely together. Therefore, it is necessary to carefully study the role that each activity will play in the class; whether it corresponds to the securing of preconditions, or the orientation towards the objective, the elaboration of new material, consolidation or control.Every teacher has to master each of these didactic functions. It is necessary to deepen the need to ensure the preconditions as a means to achieve affordability and systematization of teaching.The existing preconditions of the learners are conducive to the success of the teaching, since they form the starting level on which the teaching is developed. The importance of goal orientation must be mastered, for the more consciously the students learn and work, the more successful they will be in the process of assimilation. This important function must be present in every classroom activity.Goal orientation is a motivational process that must permeate every classroom activity. Consideration must be given to the activities and vocabulary through which students will understand what is expected of them in the class and in each of its activities.Knowing how to plan and manage the development of the new material is another essential aspect. It is necessary to take into account the particularities in directing the development of a concept, the beginning of the development of a skill or the formation of habits. The correct understanding of the new subject matter creates essential foundations for its subsequent fixation. After the students understand the new subject it becomes necessary that, in accordance with the objectives of the program, they memorize the essentials, are able to establish relationships, generalizations, and to apply them to new situations. As an example:In mathematics classes in which the teacher or professor deals with calculus with real numbers, it is not enough for the students to understand the way to solve the exercise; it is necessary that they develop skills so that they can solve it quickly and correctly. This will allow them to form new skills and will facilitate the assimilation of other concepts and relationships. This is achieved by means of a dosed, intense, varied and independent exercise.The systematic and planned control of the performance has to include all the stages of the class: it allows to know the progress of the teaching process, to discover the difficulties that appear and to take in time the measures directed to their eradication. It is also a guiding and educational element.In the analysis and determination of the structure of the class from the point of view of the didactic functions, it is essential to consider that these do not constitute a fixed set of formal steps. They guarantee the articulation of teaching and cover the whole process.An essential element to consider, once defined what is understood in this article by didactic function and its determination of which ones should be considered in the classes, is to specify some methodological aspects that are necessary to take into account to carry out the didactic functions in Mathematics classes.At this point, it is not the intention to expand on the theory but only to specify the essential methodological elements for the realization of the didactic functions, but with the specificities of the teaching of Mathematics. Didactic function Assurance of the starting level (ANP).In teaching practice, the new content is studied on the basis of consolidation and review of what has been studied before, which serves as a basis for learning the "new". In the teaching of mathematics in the Cuban school, the contents are organized by guidelines that organize the complex of subjects by grades according to the objectives to be achieved at the different levels of education. The contents of each grade constitute a starting level for learning the contents of subsequent grades.The starting level is constituted by the necessary preconditions (general and specific to Mathematics) that are the basis for learning the new content.The general preconditions refer to the student's qualities and the development of general and logical skills, as well as the mastery of general mathematical working techniques.The general preconditions are ensured before starting the corresponding class system through the comprehensive diagnosis of the student and its follow-up.Mathematics-specific preconditions refer to the mastery of concepts, procedures, propositions and the development of associated skills that serve as the basis for the corresponding subject complex.In order to ensure the specific preconditions of Mathematics, it is necessary for the teacher to follow the following steps:1. Determine the concepts, propositions, mathematical procedures and skills associated with the content to be studied and that serve as a basis.This will be done with the help of the program of the subject in the grade, the methodological guidelines and the Mathematics textbook.2.To make sure that the students possess these knowledge and skills. This can be done through written or oral questions in the class itself and through a diagnosis of knowledge at the beginning of the class system.3.Activation of knowledge and skills.In the classroom, the teacher can activate the previous knowledge and skills by reviewing the homework of the previous class at the beginning and also, at the moment it is needed during the development of the class. He can also activate knowledge and skills through review classes prior to learning the new content, considered in the dosage of the contents of the program of the subject.Didactic function Motivation.If the student is interested in what is new to learn, then learning will be effective and will have less chance of being forgotten.In mathematics classes, the teacher must motivate the student during the whole development of the class, thus motivating the occupation of the problem (learning a concept, a proposition, a procedure) as well as the way to solve the problem (why define in this way, how to find the proof of a theorem, how to find an algorithm, how to find an algorithm, etc.).The motivation of the occupation of the problem in mathematics classes can take two forms: practical or extramathematical motivation and intramathematical motivation.Practical or extramathematical motivation can be elaborated by:-Necessity and usefulness of the study of the new content. For this purpose, the student must be put in front of practical situations of life that are not solved with the knowledge he/she possesses up to that moment. These practical situations must be real or adapted didactically and are effective at the beginning of the class.-Historical situations that outline how mathematical discoveries took place and who were their protagonists. This form of motivation is effective at the beginning of a class system or content unit.-Intramathematical motivation can be elaborated on the basis of necessity, usefulness, ease, analogy, among other elements.-In the motivation of the solution of the problem, the teacher must encourage an active and conscious participation of the student during the class, starting with heuristic questions and using heuristic strategies and principles.Didactic function Target orientationThe learner must not only be motivated to learn the new content, but must also understand how he will learn it, under what conditions he will learn it, and what is expected of him during the course of the lesson and at the end of the lesson. In other words, goal orientation is information that the teacher gives the learner in advance about the outcome of his activity.The teacher should take into account the following indications in class:- Meditate on the objectives to be set for student learning in the next phase of teaching (sub-objectives).- Determine which path leads to this objective and how to make it evident through the sub-objectives.- Make the students recognize the objective of the class and, above all, make it their own.Didactic function Treatment of new knowledgeThe methodological orientations for the realization of this function are conformed in the phases of elaboration of concepts, obtaining a proposition or theorem and, consequently its demonstration, obtaining a succession of indications with algorithmic character and obtaining the solution way of a problem, according to the specificities of each one of these typical situations of the teaching of Mathematics. In this sense, the application of logical methods (reductive and deductive) acquires vital importance.Didactic function Fixation of new contentThe formation and development of skills is dependent on the acquisition of knowledge and is only possible through knowledge. In turn, with the formation and development of skills, premises are created to raise the quality of knowledge. From this derives the relevant role of fixation as a didactic function in the classroom.In this didactic function, some key concepts should be considered so that, when properly understood and developed, they facilitate the success of the process of formation and development of skills: fixation itself, exercise, review, deepening, systematization and application. They are separated only for their study, but it is recognized by these authors that they are closely related to each other and in many occasions, their combination in the mathematics class is the key to success.Something to be taken into account by Mathematics teachers is that in Mathematics classes, the fixation of new knowledge must be done according to the requirements of the fixation process of the typical situations of Mathematics teaching (Treatment of concepts, treatment of theorems and their proof, treatment of successions of indications with algorithmic character and treatment of exercises with text and application). Thus, when it comes to the fixation of a concept studied in a class, the teacher should begin with exercises that require developing the action of identification or realization of the concept, and in subsequent classes continue the fixation with application exercises.The mathematics teacher must take into consideration that another essential element in the formation and development of students' skills is in the exercise that, in the pedagogical sense, is the repeated performance of activities and actions that have the purpose of continuously perfecting the students' skills and habits. Both intellectual and practical skills have to be developed through practice. For its part, it should also be considered that review occupies in the framework of fixation a certain special position due to the way it is linked and integrated with the other forms of fixation. The conscious review that demands the activation of the students' knowledge and power, which is planned by the teacher, but may be necessary at any time (based on the individual differences of the students) for the achievement of the objectives of the class, constitutes an irreplaceable ally in the didactic battle in favor of solidity and durability and against the forgetting of mathematical knowledge.A partial conclusion that teachers should consider is that review is one of the ways to ensure the starting level for learning new knowledge, as discussed above.Deepening implies higher levels of understanding. It is deepened to the extent that constantly increasing demands are placed on the student's activity. It takes place when activities related to the content that broaden it are proposed. Likewise, in systematization, concepts are analyzed in detail, their collateral relations are studied, particular cases are studied and, consequently, a greater understanding and deeper ideas are achieved, and knowledge is organized in a system. Classification and analysis of objects, processes of objective reality, phenomena can be used. In this sense, different types of independent activity are used, according to which the student has to analyze, identify, classify and evaluate. Finally, the application is characterized by the confrontation of the students with exercises that present situations, conditions and contexts they are not accustomed to, problems, through which they are prepared for the independent solution of non-routine problems. These are problems that require: The application of various heuristic procedures, ways of working and thinking.Argumentation, substantiation and/or demonstration.The construction of geometric figures of non-algorithmic character.Interpretation and solution to extra-mathematical situations..Didactic function EvaluationDuring the class the teacher to perform this didactic function must:-Make detailed observations on the quality of answers, comments with emphasis on linguistic and mathematical representation, students' completion of tasks and at the end of the class inform the students of the result.-Try to get your students to recognize the source of their errors and to accept it. Write down common errors and exemplify them and show how to remedy them.Establish differentiated attention through exercises according to the performance ability of the students.-Pay attention to both the state of development of knowledge and mathematical skills, according to the objective of the evaluation.-Solve all exercises in the way you expect your students to solve them, in this way you will be able to determine the degree of difficulty and the time needed for the completion of the evaluation.-Reflect on the results of your students' performance with a view to improving your working methods.ConclusionsWhen analyzing the logic of the teaching process, the didactic functions must be considered separately and in their interrelation. The success of each didactic function leads to efficiency in student learning.It is necessary for teachers to investigate the best methods for performing the didactic functions in a way that activates student participation during the learning process and thus student assimilation of the content.Bibliographic referencesBallester Pedroso, S. (et-al). (1992). Metodología de la enseñanza de la Matemática. Tomo I. La Habana: Pueblo y Educación.Ballester Pedroso, S. (et-al). (2016). Didáctica de la Matemática. La Habana: Pueblo y Educación.Ferreiro Gravie, R. F. (2017).Teoría y modelo didáctico. Pedagogía piúDidattica. 3(2), 4-23.Guevara Machado, G.I (2016). Funciones didácticas .Recuperado de: http://es.scribd.com. Loyola Zorrilla, J.L. En web del maestro CMF. Recuperado de: (http//webdelmaestrocmf.com/portal/las-7-funciones-didácticas-la-sesión-aprendizaje/ón-aprendizaje/more-20072) Revista Pedagógica Nueva Escuela, 30 de junio 2017.M.A. Danilov y M. N. Skatkin. (1975). Didáctica de la escuela media. La Habana: Pueblo y Educación. W. Jungk. (1979). Conferencias de Metodología de la Enseñanza de la Matemática 2. La Habana: Pueblo y Educación.PAGE \* MERGEFORMAT1EduSol Vol.21 Núm.75 PAGE \* MERGEFORMAT2EduSol Vol.21 Núm.75 Risel Ruiz Cordovés; Carlos Beltrán Pazo: The didactic functions in the teaching of mathematics428053511049000EduSolVol. 21. Núm. 75ISSN: 1729-8091Publicada en línea: abril de 2021 (http://edusol.cug.co.cu)iVBORw0KGgoAAAANSUhEUgAAAboAAACOCAYAAABHa82KAAAAAXNSR0IArs4c6QAAAARnQU1BAACx
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