The theory of didactic situations: what it is and what it explains about teaching.

A theory developed by Guy Brousseau to understand the teaching of mathematics.

Many of us have had a hard time with mathematics, and this is normal. Many teachers have defended the idea that either you have a good mathematical ability or you simply do not have it and you will hardly be good at this subject.

However, this was not the opinion of several French intellectuals in the second half of the last century. They considered that mathematics, far from being learned through theory and that's it, can be acquired in a social way, by sharing possible ways of solving mathematical problems.

The Theory of Didactic Situations is the model derived from this philosophy.The author argues that far from explaining mathematical theory and seeing whether students are good at it or not, it is better to have them discuss possible solutions and make them see that they can discover the method themselves. Let's take a closer look.

What is the theory of didactic situations?

Guy Brousseau's theory of didactic situations is a theory of teaching that is found within the didactics of mathematics. It is based on the hypothesis that mathematical knowledge is not constructed spontaneously, but by means of the search for solutions on the learner's own account, sharing it with the rest of the students and understanding the path followed to reach the solution of the mathematical problems he/she is given. of the mathematical problems posed to him/her.

The vision behind this theory is that the teaching and learning of mathematical knowledge, rather than something purely logical-mathematical, involves a collaborative construction within an educational community, involves a collaborative construction within an educational community; it is a social process.It is a social process. Through discussion and debate on how a mathematical problem can be solved, strategies are awakened in the individual to arrive at its resolution which, although some of them may be erroneous, are ways that allow him/her to have a better understanding of the mathematical theory given in class.

Historical background

The origins of the theory of didactic situations date back to the 1970s, a time when the didactics of mathematics began to appear in France.The intellectual orchestrators were Guy Brousseau himself, together with Gérard Vergnaud and Yves Chevallard, among others.

It was a new scientific discipline which studied the communication of mathematical knowledge using an experimental epistemology. It studied the relationship between the phenomena involved in the teaching of mathematics: the mathematical contents, the educational agents and the students themselves.

Traditionally, the figure of the mathematics teacher was not very different from that of other teachers, seen as experts in their subjects. However, the mathematics teacher was seen as an expert in his or her subject, the mathematics teacher was seen as a great master of this discipline, who was never wrong and always had a unique method for solving each problem.. This idea was based on the belief that mathematics is always an exact science with only one way to solve each exercise, making any alternative not proposed by the teacher erroneous.

However, in the twentieth century and with the significant contributions of great psychologists such as Jean Piaget, Lev Vigotsky and David Ausubel, the idea that the teacher is the absolute expert and the learner the passive object of knowledge began to be overcome. Research in the field of the psychology of learning and development suggests that the learner can and should take an active role in the construction of his knowledge, moving from a view that he should store all the information he is given to one that is more in favor of him being the one who discovers, debates with others and is not afraid of making mistakes.

This would lead us to the current situation and the consideration of didactics of mathematics as a science. This discipline takes much into consideration the contributions of the classical stage, focusing, as one would expect, on the learning of mathematics. It is no longer for the teacher to explain the mathematical theory, wait for the students to do the exercises, make mistakes and make them see what they have done wrong; now it is for the students to come up with different ways to solve the problem, even if they deviate from the more classical path. is for the students to come up with different ways to reach the solution of the problem, even if they deviate from the more classical path..

Didactic situations

The name of this theory does not use the word situations for free. Guy Brousseau uses the expression "didactic situations" to refer to how knowledge should be offered in the acquisition of mathematics, in addition to talking about how students participate in it. It is here that we introduce the exact definition of didactic situation and, as a counterpart, the a-didactic situation of the didactic situation theory model.

Brousseau refers to the "didactic situation" as that which has been intentionally constructed by the educator, with the purpose of helping the students to acquire a specific knowledge..

This didactic situation is planned on the basis of problem-solving activities, that is, activities in which a problem to be solved is presented. Solving these exercises contributes to consolidate the mathematical knowledge offered in class, since, as we have mentioned, this theory is mostly used in this area.

The structure of the didactic situations is the teacher's responsibility.. It is the teacher who must design them in such a way that they contribute to the students' ability to learn. However, this should not be misinterpreted as meaning that the teacher must directly provide the solution. He/she does teach the theory and offers the moment to put it into practice, but does not teach each and every one of the steps to solve the problem-solving activities.

The a-didactic situations

In the course of the didactic situation there are some "moments" called "a-didactic situations". This type of situations are the moments in which the student himself interacts with the proposed problem, not the moment in which the educator explains the theory or gives the solution to the problem..

These are the moments in which the students take an active role in solving the problem by discussing with the rest of the classmates about what could be the way to solve it or to trace the steps that should be taken to reach the answer. The teacher must study how the students "manage" to solve the problem.

The didactic situation should be designed in such a way that it invites the students to take an active part in solving the problem. That is to say, the didactic situation designed by the educator must contribute to a-didactic situations and make them present cognitive conflicts and ask questions.

At this point the teacher must act as a guide, intervening or answering the questions but offering other questions or "clues" as to how to proceed, never giving them the solution directly.

This part is really difficult for the teacher, as he must have been careful and made sure not to give clues that are too revealing or, directly, ruin the process of finding the solution by giving his students everything. This is called the Process of Return and it is necessary for the teacher to have thought through which questions to suggest his or her answer and which not toThe teacher must make sure that he/she does not spoil the process of acquisition of new content by the students.

Types of situations

Teaching situations are classified into three types: action, formulation, validation and institutionalization.

Action situations

In action situations there is an exchange of non-verbalized information, represented in the form of actions and decisions. The student must act on the environment presented by the teacher, putting into practice the implicit knowledge acquired in the explanation of the theory. acquired in the explanation of the theory.

2. Formulation situations

In this part of the didactic situation the information is formulated verbally, i.e., the student talks about how the problem could be solved.. In the formulation situations, the students' ability to recognize, decompose and reconstruct the problem-solving activity is put into practice, trying to show others through oral and written language how the problem can be solved.

3. Validation situations

In validation situations, as the name itself indicates, the "paths" that have been proposed to reach the solution of the problem are validated.. The members of the activity group discuss how the problem proposed by the teacher could be solved, testing the different experimental paths proposed by the students. The aim is to find out whether these alternatives give a single result, several, none, and how likely they are to be right or wrong.

4. Institutionalization situation

The institutionalization situation would be the "official" consideration that the object of teaching has been acquired by the learner and the teacher takes this into account.. It is a very important social phenomenon and an essential phase during the didactic process. The teacher relates the knowledge freely constructed by the student in the a-didactic phase with the cultural or scientific knowledge.

Bibliographical references:

• Brousseau G. (1998): Théorie des Situations Didactiques, La Pensée Sauvage, Grenoble, France.
• Chamorro, M. (2003): Didactics of Mathematics. Pearson. Madrid, Spain.
• Chevallard, Y, Bosch, M, Gascón, J. (1997): Estudiar Matemáticas: el eslabón perdido entre enseñanza y aprendizaje. Cuadernos de Educación Nº 22.
• Horsori, Universitat de Barcelona, Spain.
• Montoya, M. (2001). El Contrato Didáctico. Working paper. Magíster en Didáctica de la Matemática. PUCV. Valparaíso, Chile.
• Panizza, M. (2003): Enseñar Matemáticas en el nivel inicial y el primer ciclo de la EGB. Paidos. Buenos Aires, Argentina.