Des travaux de recherche en ergonomie et en psychologie cognitive signalent une difficulté d’apprentissage du SIG, sans pour autant s’attarder sur le sujet (Medyckyj-Scott 1991, Nyerges 1991, Davies 1995 et, en France, Wurtz 1996). Notre recherche vient à la suite de ces travaux et tente d’apporter, par la compréhension des mécanismes de l’apprentissage du SIG dans les conditions offertes par la formation professionnelle continue, un éclairage sur les difficultés d’apprentissage du SIG et les manières possibles de les dépasser. Nous avons retenu – filmé et analysé – deux sessions de formation SIG d’une semaine (env. 30h chacune), ayant lieu à 3 années d’intervalle, dans le même centre de formation et étant co-animées par les mêmes deux formateurs. A la manière d’Aline Robert (2008), nous avons observé et enregistré l’activité en formation sans intervenir : une caméra fixe (image et son) a été placée à l’écart, de manière à couvrir le plus largement possible la partie occupée de la salle. Sur le plan théorique, la thèse centrale est celle de la possibilité et de la pertinence d’accéder aux processus et aux mécanismes d’apprentissage par les représentations que les professionnels adultes se construisent. Nous avons examiné cela avec la théorie de la représentation fonctionnelle (Leplat), également appelée image opérative (Ochanine) ou représentation pour l’action (Teiger, Weill-Fassina, Rabardel). Plusieurs autres cadres théoriques ont également été mobilisés : les théories de la formation pour adultes (Barbier, Bourgeois, Kaddouri), celles de la formation professionnelle (Mayen, Pastré, Vergnaud), la théorie des instruments et de l’action instrumentée (Rabardel), la théorie de la médiation et du guidage par autrui (Galperine, Savoyant). Nous avons rendu compte du processus d’apprentissage dans son déroulement, à travers l’évolution du système de représentation des personnes, auquel nous avons eu accès par l’analyse de l’activité verbale. En effet, dans l’interaction verbale entre les formateurs et les apprenants, ou entre des apprenants, les représentations des personnes ressortent comme des ressources (représentations construites avant la formation) ou comme des effets de la formation (représentations récentes, en cours de construction ou de transformation). L’évolution du système de représentations est systématiquement mise en relation avec les conditions qui le déterminent. Une organisation a priori structurée en séquences, mais qui autorise l’intervention de chacun, laisse place à l’expression des représentations des professionnels en formation. L’accent est cependant mis sur l’interaction verbale entre les formateurs et les apprenants et les effets produits. La fonction formative est assurée par deux formateurs, qui sont des professionnels expérimentés et qui agissent en co-animation simultanée. Cela offre un terrain intéressant d’analyse du partage de l’activité de transmission. Nous observons comment les formateurs organisent leur intervention commune : ils se complètent et se soutiennent, mais aussi se contredisent et s’empêchent…La thèse s’inscrit dans une nouvelle orientation du courant de la didactique professionnelle, orientation initiée par Mayen : il s’agit d’analyser la formation en vue du travail et non plus seulement le travail en vue de la formation. En analysant le processus d’apprentissage en formation, la thèse reste toutefois centrée sur le travail. D’abord, parce que l’objet de l’apprentissage, le SIG, est un instrument professionnel. Ensuite, parce que les sujets de la recherche sont des professionnels en activité, qui ne se détachent pas de leur travail en arrivant en formation (Pastré, Mayen et Vergnaud, 2007). Enfin, parce que les tâches et les situations de travail connues et vécues par chacun – et verbalisées devant le groupe – sont autant de ressources pour apprendre en situation de formation. / Research in ergonomics and cognitive psychology point out difficulties in learning GIS (Medyckyj-Scott 1991, Nyerges 1991, Davies 1995 et, en France, Wurtz 1996). Our research is a follow-up of this work. It attempts to highlight the difficulties of learning GIS and possible ways exceed them, by understanding the mechanisms of learning GIS in the conditions offered by the CVT (continuous vocational training). We filmed and analyzed two GIS training sessions (approx. 30 hours each). Taken place three years apart, in the same training center, the training sessions have been conducted by the same two trainers. As Aline Robert (2008), we observed and recorded training activity without interfering: a fixed camera was placed in the gap, to cover the widest possible the occupied part of the classroom.On a theoretical level, our central thesis is the possibility and relevance to access processes and mechanisms of learning through mental representations built by adult learners. We relied on the theory of functional representation (Leplat), also called operative image (Ochanine) or representation for the action (Teiger, Weill-Fassina, Rabardel). Several other theoretical frameworks have also been mobilized: theories of adults learning (Barber, Bourgeois, Kaddouri), theories of vocational training (Mayen, Pastré, Vergnaud), the theory of instruments and instrumented action (Rabardel), the theory of mediation and guidance by others (Galperin, Savoyant). We reported on the learning process in its development through the evolution of the system of representation of people, which we had access through the analysis of verbal activity. Indeed, in verbal interaction between instructors and learners or between learners, representations of people emerge as resources (representations built before training) or as effects of training (under construction or processing representations).The evolution of the system of representations is systematically related to the conditions that determine it. The organization of the training session allows everyone to express themselves and underscores its representations. Emphasis is however placed on verbal interaction between trainers and learners and the effects produced. The formative function is provided by two trainers who are experienced professionals and who act simultaneous. This offers an interesting field of analysis of sharing transmission activity. We observe how trainers organize their joint intervention: they complement and support each other, but also contradict and prevent.Our thesis is part of a new orientation of vocational didactics theoretical field. This orientation has initiated by Mayen and consists to analyze training to work and not only the work for training. By analyzing the learning process, the thesis remains focused on work process. First, because the subject of learning is a professional instrument (GIS). Second, because trainees are professionals who do not separate their work joining the training session (Pastré, Mayen and Vergnaud, 2007). Finally, because the work situations and tasks experienced and verbalized by everyone are so many resources to learn during the training session.

The study constituted an investigation for concept images and mathematical reasoning of Grade 11 learners on the concepts of reflection, translation and stretch of functions. The aim was to gain awareness of any conceptions that learners have about these transformations. The researcher’s experience in high school and university mathematics teaching had laid a basis to establish the research problem. The subjects of the study were 96 Grade 11 mathematics learners from three conveniently sampled South African high schools. The non-return of consent forms by some learners and absenteeism during the days of writing by other learners, resulted in the subsequent reduction of the amount of respondents below the anticipated 100. The preliminary investigation, which had 30 learners, was successful in validating instruments and projecting how the main results would be like. A mixed method exploratory design was employed for the study, for it was to give in-depth results after combining two data collection methods; a written diagnostic test and recorded follow-up interviews. All the 96 participants wrote the test and 14 of them were interviewed. It was found that learners’ reasoning was more based on their concept images than on formal definitions. The most interesting were verbal concept images, some of which were very accurate, others incomplete and yet others exhibited misconceptions. There were a lot of inconsistencies in the students’ constructed definitions and incompetency in using graphical and symbolical representations of reflection, translation and stretch of functions. For example, some learners were misled by negative sign on a horizontal translation to the right to think that it was a horizontal translation to the left. Others mistook stretch for enlargement both verbally and contextually. The research recommends that teachers should use more than one method when teaching transformations of functions, e.g., practically-oriented and process-oriented instructions, with practical examples, to improve the images of the concepts that learners develop. Within their methodologies, teachers should make concerted effort to be aware of the diversity of ways in which their learners think of the actions and processes of reflecting, translating and stretching, the terms they use to describe them, and how they compare the original objects to images after transformations. They should build upon incomplete definitions, misconceptions and other inconsistencies to facilitate development of accurate conceptions more schematically connected to the empirical world. There is also a need for accurate assessments of successes and shortcomings that learners display in the quest to define and master mathematical concepts but taking cognisance of their limitations of language proficiency in English, which is not their first language. Teachers need to draw a clear line between the properties of stretch and enlargement, and emphasize the need to include the invariant line in the definition of stretch. To remove confusion around the effect of “–” sign, more practice and spiral testing of this knowledge could be done to constantly remind learners of that property. Lastly, teachers should find out how to use smartphones, i-phones, i-pods, tablets and other technological devices for teaching and learning, and utilize them fully to their own and the learners’ advantage in learning these and other concepts and skills / Mathematics Education / D.Phil. (Mathematics, Science and Technology Education)