Math has cost us dearly, and that’s to be expected. Many teachers have argued that you either have good math skills or you just don’t have them and we won’t get to know much about it.

However, this was not the opinion of many French intellectuals in the second half of the last century. They believed that mathematics, far from being learned by theory and that’s it, can be learned socially, by sharing possible ways of solving mathematical problems.

**The theory of didactic situations is the model derived from this philosophy**, Arguing that far from explaining mathematical theory and seeing if the students are doing well or not, it is better to make them discuss their possible solutions and make them see that it is perhaps themselves who come to discover the method for that. Let’s take a closer look.

Table of Contents

## What is the theory of didactic situations?

Guy Brousseau’s theory of didactic situations is a teaching theory that is part of the didactics of mathematics. It is based on the assumption that mathematical knowledge is not built spontaneously, but through **finding solutions on behalf of the learner, sharing with other students and understanding the path they followed to reach the solution** mathematical problems which are posed to him.

The point of view behind this theory is that teaching and learning mathematical knowledge, rather than something purely logical-mathematical, **it is a collaborative construction within an educational community**; it is a social process. By discussing and debating how a mathematical problem can be solved, he awakens in individual strategies to arrive at its resolution which, although they may be for some erroneous, are avenues which allow him to have a better understanding. understanding of mathematical theory given in class.

### Historical context

**The origins of the theory of didactic situations go back to the 1970s, when the didactics of mathematics began to appear in France.**, Having as intellectual orchestrators figures such as Guy Brousseau himself with Gérard Vergnaud and Yves Chevallard, among others.

It was a new scientific discipline that studied the communication of mathematical knowledge using experimental epistemology. He studied the relationship between the phenomena involved in the teaching of mathematics: the mathematical contents, the pedagogical agents and the pupils themselves.

Traditionally, the figure of the math teacher was not much different from that of other teachers, considered experts in their subjects. However, **the math teacher was considered a great master of this discipline, who was never wrong and always had a unique method to solve each problem**. This idea was based on the belief that mathematics is still an exact science and with only one way to solve each exercise, so any alternative not offered by the teacher is wrong.

However, entering the twentieth century and with the significant contributions of great psychologists such as Jean Piaget, Lev Vygotsky and David Ausubel begins to go beyond the idea that the teacher is the absolute expert and the learner the passive object of knowledge. Research in the field of learning and developmental psychology emphasizes that students can and should take an active role in building their knowledge, moving from a vision that must store all the data that is given to a vision. more united than he is the one who discovers, discusses with others and is not afraid of making mistakes.

This would lead us to the current situation and to the consideration of mathematics education as a science. This discipline has great respect for the contributions of the classical scene, focusing, as one would expect, on learning mathematics. It is up to the teacher to explain the mathematical theory, to wait for the pupils to do the exercises, to make mistakes and to show them what they have done wrong; now **it consists of students who consider different ways to arrive at 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 free situations. Guy Brousseau uses the term “didactic situations” to designate the way in which knowledge should be offered in the acquisition of mathematics, in addition to speaking of the way in which the students participate in it. It is here that we introduce the exact definition of the didactic situation and, on the other hand, the a-didactic situation of the model of the theory of didactic situations.

Brousseau evokes the “didactic situation” as **one that has been intentionally built by the educator, in order to help his students to acquire a certain 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. The resolution of these exercises establishes the mathematical knowledge offered in the classroom, since, as we have mentioned, this theory is mainly used in this field.

**The structure of didactic situations is the responsibility of the teacher**. It is he who must design them in such a way that the pupils can learn. However, this should not be misinterpreted, thinking that the teacher should give the solution directly. Yes, he teaches theory and gives time to practice it, but he does not teach all steps of problem-solving activities.

### A-didactic situations

During the didactic situation, there are “moments” called “a-didactic situations”. Such situations are **the moments when the pupil himself interacts with the proposed problem, not the moment when the educator explains the theory or gives the solution to the problem**.

These are the times when students take an active role in solving the problem by discussing with the rest of the classmates how to solve it or going through the steps to come up with the answer. The teacher should study the way in which the pupils “conceive” them.

The didactic situation must be posed in such a way as to invite the pupils to participate actively in the resolution of the problem. In other words, the didactic situations designed by the educator must help to create a-didactic situations and lead them to present cognitive conflicts and to ask questions.

At this point, the teacher should act as a guide, intervening or answering the questions but offering other questions or “clues” on the path to follow, he should never give them the solution directly.

This part is really difficult for the teacher, because he had to be careful and make sure not to give too revealing clues or directly ruin the process of finding the solution by giving everything to his students. **This is called the feedback process and the teacher should have thought through which questions should be answered and which should not.**, Make sure that this does not spoil the process of acquiring new content by the students.

## Types of situations

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

### 1. Situations of action

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

### 2. Formulation situations

In this part of the didactic situation **the information is worded verbally, that is, it explains how the problem could be solved**. In formulation situations, the ability of students 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 suggests, **the “paths” which have been proposed to reach the solution of the problem are validated**. The group members of the activity discuss how the problem proposed by the teacher could be solved, by testing the different experimental paths proposed by the students. The question is whether these alternatives give a single, many, none, and how likely they are to be true or false.

### 4. Status of institutionalization

The institutionalization situation would be **the “official” consideration that the object of the teaching has been acquired by the pupil and that the teacher takes it into account**. It is a very important social phenomenon and an essential phase of the didactic process. The teacher connects the knowledge freely constructed by the pupil in the a-didactic phase with cultural or scientific knowledge.

#### Bibliographical references:

- Brousseau G. (1998): Theory of didactic situations, La Pensée Sauvage, Grenoble, France.
- Chamorro, M. (2003): Didactics of mathematics. Pearson. Madrid, Spain.
- Chevallard, I, Bosch, M, Gascón, J. (1997): Studying Mathematics: The Lost Link between Teaching and Learning. Educational notebooks Nº 22.
- Horsori, University of Barcelona, Spain.
- Montoya, M. (2001). The teaching contract. Work document. Master in Mathematics Education. PUCV. Valparaiso, Chile.
- Panizza, M. (2003): Teaching mathematics at the initial level and at the first cycle of the EGB. Paidos. Buenos Aires, Argentina.