A theory developed by Guy Brousseau to understand the teaching of mathematics.
For many of us, mathematics has cost us a lot, and it is normal. Many teachers have defended the idea that you either have a good mathematical ability or you simply don’t have it and you will hardly be good at this subject.
However, this was not the opinion of various 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, putting in common the possible ways of solving mathematical problems.
The Theory of didactic situations is the model derived from this philosophy, holding that far from explaining mathematical theory and seeing if students are good at it or not, it is better to make them debate about their possible solutions and make them see that they can be the ones who come to discover the method for it. Let’s take a closer look at it.
What is the theory of didactic situations?
Guy Brousseau’s Theory of Didactic Situations is a teaching theory found within the didactics of mathematics. It is based on the hypothesis that mathematical knowledge is not built spontaneously, but through the search for solutions on the learner’s own account, sharing with the rest of the students and understanding the path that has been followed to reach the solution of the problems mathematicians who arise.
The vision behind this theory is that the teaching and learning of mathematical knowledge, more than something purely logical-mathematical, implies a collaborative construction within an educational community ; it is a social process. Through the discussion and debate of how a mathematical problem can be solved, strategies are awakened in the individual to reach its resolution that, although some of them may be wrong, are ways that allow them to have a better understanding of the mathematical theory given in class.
The origins of the Theory of didactic situations go back to the 1970s, a time when the didactics of mathematics began to appear in France, having as intellectual orchestrators figures such as 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. He studied the relationship between the phenomena involved in the teaching of mathematics: the mathematical content, 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 a great dominator of this discipline, who never made mistakes and always had a unique method to solve each problem. This idea started from the belief that mathematics is always an exact science and with only one way to solve each exercise, with which any alternative not proposed by the teacher is wrong.
However, entering the 20th 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 apprentice the passive object of knowledge is beginning to be overcome. Research in the field of learning and developmental psychology suggests that the student can and should take an active role in the construction of their knowledge, moving from a vision that they must store all data that is given to a more supportive that he is the one to discover, discuss with others and not be afraid of making mistakes.
This would lead us to the current situation and the consideration of the didactics of mathematics as a science. This discipline takes much into consideration the contributions of the classical stage, focusing, as might be expected, on learning mathematics. The teacher already explains the mathematical theory, waits for the students to do the exercises, make mistakes and makes them see what they have done wrong; now it consists of the students considering different ways to reach the solution of the problem, even if they deviate from the more classical path.
The 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 where we introduce the exact definition of the didactic situation and, as a counterpart, the a-didactic situation of the model of the theory of didactic situations.
Brousseau refers to a “didactic situation” as one that has been intentionally constructed by the educator, in order to help his students acquire a certain knowledge.
This didactic situation is planned based on problematizing activities, that is, activities in which there is a problem to be solved. Solving these exercises helps to establish the mathematical knowledge offered in class, since, as we have commented, this theory is used mostly in this area.
The structure of the didactic situations is the responsibility of the teacher. It is he who must design them in such a way that contributes to the students being able to learn. However, this should not be misinterpreted, thinking that the teacher must directly provide the solution. It does teach theory and offers the moment to put it into practice, but it does not teach each and every one of the steps to solve problem-solving activities.
The a-didactic situations
In the course of the didactic situation there appear some “moments” called “a-didactic situations”. These types 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, discussing with the rest of their classmates about what could be the way to solve it or trace the steps they should take to lead to the answer. The teacher must study how the students “manage”.
The didactic situation must be presented in such a way that it invites students to take an active part in solving the problem. That is, the didactic situations designed by the educator should contribute to the occurrence of a-didactic situations and cause them to 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” about what the way forward is like, he should never give them the solution directly.
This part is really difficult for the teacher, as he must have been careful and made sure not to give too revealing clues or, directly, ruin the process of finding the solution by giving his students everything. This is called the Return Process and it is necessary for the teacher to have thought about which questions to suggest their answer and which not, making sure that it does not spoil the process of acquisition of new content by the students.
Types of situations
Didactic situations are classified into three types: action, formulation, validation and institutionalization.
1. 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 medium that the teacher has proposed, 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 formulated verbally, that is, it is talked about how the problem could be solved. In formulation situations, the students’ ability to recognize, decompose and reconstruct the problem-solving activity is put into practice, trying to make others see through oral and written language how the problem can be solved.
3. Validation situations
In validation situations, as its name 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 ways proposed by the students. It is about finding out if these alternatives give a single result, several, none and how likely it is that they are right or wrong.
4. Institutionalization situation
The institutionalization situation would be the “official” consideration that the teaching object has been acquired by the student and the teacher takes it 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 cultural or scientific knowledge.
- 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): Studying Mathematics: the missing link between teaching and learning. Education Notebooks No. 22.
- Horsori, University of Barcelona, Spain.
- Montoya, M. (2001). The Didactic Contract. Work document. Master in Didactics of Mathematics. PUCV. Valparaíso, Chile.
- Panizza, M. (2003): Teaching Mathematics in the initial level and the first cycle of the EGB. Paidos. Buenos Aires, Argentina.