Operational Invariants in Two Conceptual Fields Transition: Studying Relative Time

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This thesis was born of a matter arising from the classroom, initially from mining concerns as a teacher and later as a researcher. It would be possible - and advisable - to teach the theory of relativity in high school? What are the conditions for such a school to succeed? How students would build representations of the physical quantities that have different statutes in everyday life in Classical Mechanics and Relativity?

The cut made by us, in making the concerns original research questions, was to investigate the construction of the concept of relative time. For this was a planned instructional sequence that based on an analysis of weaknesses of classical mechanics, relativity theory presented as a way to explain the points of tension and therefore be a more comprehensive theory. In this process, emphasis was placed on the study of the relativity of motion and how the appearance of the theory of relativity requires a change in the epistemological status of the time.

The research was applied for 16 teaching hours in a classroom of third grade high school of a federal public school in the state of Minas Gerais. Intervention activities were diversified teaching (lecturing by the teacher, reading texts, use of audio visual resources, discussion among students and small groups discussing me). All activities were recorded, analysed and, when necessary, transcribed.

The theoretical framework of this research is to plan activities, whether for data analysis was the Conceptual Fields Theory Gérard Vergnaud, especially formulations schema concept and operational invariants. In addition, we promote a link between the ideas and the formulation of Vergnaud scheme in Piaget, and use the research of the latter on the construction of the notion of time.

After the interpretation of the survey data, we were able to build cognitive trajectories for students with regard to the process of meaning relative time. For this, we analyse how the evolving operational invariants of students from each proposed activity throughout the research.

In this sense, we can highlight three main points:

following the formulation of Piaget, time is more than a concept, since it engenders organizing actions and the interrelation between concepts of speed, duration, frequency, etc.. So it is more productive to treat time as a scheme and analyse the necessary modifications to take account of a subject novelties brought by the Theory of Relativity.
in accordance with what was presented in the discussion (1), we conclude that the scheme requires time to absorb the movement that it is possible to operate the logic of relativity. Therefore, we observe that the operational invariants used by students to deal with situations related to motion analysis from different references were also present in cases of relative time.
Finally, as mentioned above assimilation is never only one copy but rather a reinterpretation, we find that the study of the Theory of Relativity provided a new context for the use of motion Scheme (alone or after the assimilation of the time) and allowed the occurrence of restatements in operational invariants for some students.

These points indicate the two main conclusions, one from the point of view of theoretical, the other practical standpoint or didactic. The first relates to the potential of Conceptual Fields Theory as a reference for students of meanings. At this point, the concepts of Scheme and invariant surgery are important in understanding how is the process of conceptualization in science. The second concerns the understanding by the students, the relative time. The central problem seems linked to a broader framework that encompasses the notions of movement and speed. In this sense, the teaching work should seek to build relationships between concepts and provide students with opportunities to apply their own formulations in various situations.


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VERGNAUD, G. Représentation et activité: deux concepts étroitement associés. Atas do 1º Congresso Internacional Lógico-Matemática en Educación Infantil. 2006. Disponível em <http://www.waece.org/cdlogicomatematicas>. Acesso em 10.dez.2007.

VERGNAUD, G. La conceptualisation, clef de voûte des rapports entre pratique et théorie. Disponível em <http://eduscol.education.fr/cid46598/la-conceptualisation-clef-de-voute-des-rapports-entre-pratique-et-theorie.html> Acesso em 10.fev.2012.


Gabriel Carvalho
IFMG - Campus Congonhas

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