Pupils’ Conceptions about a historical open question: Goldbach’s conjecture. The improvement of mathematical education from a historical viewpoint.

Aldo Scimone

PhD Thesis, Bratislava University, 2003

The theoretical framework of this paper is the theory of didactical situations in mathematics by Guy Brousseau 1. It is common knowledge that this theory is based on the conception of the didactic situations, where a situation is defined like the set of circumstances into which a person is (a group, a collectivity, etc.), the relations linking him to the environment and the set of data characterizing an action or an evolution, i. e. an action at a certain moment.
In particular, this work concerns an a-didactic situation, namely that part of a didactic situation which teacher’s intention respect to pupils is not clear into. A didactic situation is really the moment of the didactic situation in which the teacher does not declare the task to be reached but he gets the pupil to think about the proposed task which is chosen in order to allow him to acquire a new knowledge and that it is to be looked for within the same logic of the problem. An a-didactic situation is a such one if it allows the pupils to appropriate and to manage the staking dinamycs, to get him to be a protagonist of the process, to get him to perceive the responsibility of it as a knowledge and not as a guilt of the saught result. The pupil must accept the suggested play (a-didactic situation) but he must put into action the best strategies allowing him to win.

All that is based on solving a problem, an open problem or a conjecture. So, the aim of this research is to analyze some conceptions of pupils while they are facing a conjecture, and in particular a famous historical conjecture like Goldbach’s one. Goldbach’s conjecture was
chosen because it has a long historical background allowing an efficient a-priori analyse, which is an important phase for the experimentation in order to foresee the possible pupils’ answers and behaviours in front of the conjecture.
Moreover, it has a fascinating formulation allowing pupils to mix many numerical examples , and to discuss fruitfully about its validity and some possible attempts of a demonstration.
So, the historical context is important because it suggestes an interplay between the history of mathematics and the mathematics education.
The content of the experimentation grows around the validation or the falsification of two hyphoteses of research: the first one concerning pupils’ inability to represent mentally any general method useful for a demonstration; the second one concerning their intuitive ability to recognize the validity of a conjecture. The validation or the falsification of these hypotheses are very useful in order to understand the metacognitive processes which are basic for the learning phase and the cultural growth of pupils.
Another important point for this experimentation is the fact that pupils did not know anything about the unsolvibility of Goldbach’s conjecture, so that the a-didactic situation could not be disturbed by any interference due to their knowledge of the failed attempts to solve the
conjecture.
Within Brousseau’s theory, such an experimentation was carried out by a quantitative analysis along with a qualitative analysis.
The statistical survey for the quantitative analysis was made by two phases: in the first experiment, which was realized with a sample of pupils attending the third and fourth year of study (16-17 years) of secondary school, the method of individual and matched activity was
used; the second experiment was carried out in three levels: pupils from the first school (6-10 years), pupils from primary school (11-15 years) and pupils from secondary school.

The quantitative analysis of the data drawn from pupils’ protocols was made by the software of inferential statistics 2 CHIC 2000 (Classification Hiérarchique Implicative et Cohésitive) and the factorial statistical survey S.P.S.S. (Statistical Package for Social
Sciences).
The research pointed out some important misconception by pupils and some knots in the passage from an argumentative phase to a demonstrative one of their activity which need to be deeped.