The study employed Ernest (2006) Theory of Semiotic Systems to investigate the use of and exposure to multiple representations in a 10th grade algebra II suburban high school class located in the southeastern region of the United States. The purpose of this exploratory case study (Yin, 2014) was to investigate the role of multiple representations in influencing and facilitating algebra II students’ conceptual understanding of piece-wise function, absolute-value functions, and quadratic functions. This study attempted to answer the following question: How does the use of and exposure to multiple representations influence algebra II students’ understanding and transfer of algebraic concepts? Furthermore, the following sub-questions assisted in developing a deeper understanding of the question: a) how does exposure to and use of multiple representations influence students’ identification of their pseudo-conceptual understanding of algebraic concepts?; b) how does exposure to and use of multiple representations influence students’ transition from pseudo-conceptual to conceptual understanding?; c) how does exposure to and use of multiple representations influence students’ transfer of their conceptual understanding to other related concepts? Understanding the notion of pseudo-conceptual understanding in algebra is significant in providing a tool for examining the veracity of algebra students’ conceptual understanding, where teachers have to consistently examine if students accurately understand the meanings of the mathematical signs that they are constantly using. The following data collection techniques were utilized: a) classroom observation, b) task based interviews, and c) study of documents. The unit of analysis was students’ verbal and written responses to task questions. Three themes emerged from the analysis of in this study: (a) re-imaging of conceptual understanding; (b) reflective approach to understanding and using mathematical signs; and (c) representational versatility in the use of mathematical signs. Findings from this study will contribute to the body of knowledge needed in research on understanding and assessing algebra students’ conceptual understanding of mathematics. In particular the findings from the study will contribute to the literature on understanding; the process of algebraic concepts knowledge acquisition, and the challenges that algebra students have with comprehension of algebraic concepts (Knuth, 2000: Zaslavsky et al., 2002).

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Notre travail de recherche porte sur l’analyse de l’action du professeur et ses interactions avec les élèves dans le domaine spécifique de l’étude du numérique en Cours Préparatoires. L’objectif de notre thèse est d’analyser l’action du professeur et ses pratiques didactiques en classes, et formatives dans une Cellule d’Animation Pédagogique. La première partie comporte une présentation du contexte socio économique et éducatif du Niger. Nous nous interrogeons ensuite sur les pratiques professorales permettant ou non aux élèves de prendre de réelles responsabilités par rapport aux savoirs, et sur la pertinence du travail de conception collective de séances. Dans la deuxième partie, nous exposons nos outils théoriques, les travaux antérieurs en lien avec nos objets de recherche, et la problématique générale de notre travail. La troisième partie est consacrée à la démarche méthodologique qui fonde notre étude sur l’analyse des manuels, des vidéos de classe et les observations in situ. La quatrième partie examine les ressources mathématiques utilisées à l’écoleprimaire au Niger, en interrogeant l’influence de ces ressources sur les pratiques habituelles des professeurs enquêtés. Les cinquième et sixième parties analysent les séances empiriques en termes de topogénèse, et de recours aux systèmes sémiotiques pour étudier le savoir en jeu dans les situations didactiques. Dans la septième partie, nous proposons un dispositif innovant de formation continue sur la conception collective de leçons dans une CAPED. En conclusion, dans une huitième partie nous présentons d’abord une synthèse de nos résultats, et nous exposons ensuite les perspectives que ce travail de recherche offre en termes de propositions en vue de l'élaboration d'ingénieries didactiques qui pourraient permettre la mise en place d’un travail coopératif entre professeurs et chercheurs / Our research focuses on the teacher's action and his interactions with students in the specific field of the study of digital in grade 2. The objective of this PhD dissertation is to analyze the teacher's action and his didactic practices in classes, and training practices in a CAPED in the Niger context. The first part of this PhD dissertation includes a presentation of the socio-economic and educational contexts in Niger. Then we examine the teaching practices in order to see if they allow students to take or not real responsibilities in relation to knowledge, and the appropriateness of collective design work sessions.In the second part, we present our theoretical tools, previous work related to our research objects, and the general problem tackled by our work. The third part is devoted to our methodological approach based on the study of textbook analyze, videos of classes and field observations. The fourth section discusses the mathematic resources used in the primary education in Niger and question the impact of these resources on the observed teachers' usual practices. The fifth and sixth sessions analyze empirical lessons in terms of topogenesis, and semiotic systems to study the knowledge involved in the didactic situations. In the seventh part, we propose an innovative in-service training based on the design of lessons in a collective CAPED. In conclusion, in an eighth part we first present a summary of our results, and then we present the perspectives that this research offers in terms of further developments of didactic engineering that could allow the implementation of a collaborative work between teachers and researchers