Authors: | Benrherbal, Abderrahmane |

Advisor: | DeBlois, Lucie |

Abstract: | This research focuses on the field of didactics of mathematics. It has two main objectives: the first is to understand how the use of the concepts of fraction and proportion in intra and interdisciplinary contexts can affect the learning and teaching of mathematics, science and technology as well as physics and chemistry. The second is to grasp whether the learning of geometry, probability, energy efficiency, concentration, stoichiometry , optical reflection, and uniformly accelerated rectilinear motion transforms the fraction and the proportion of the object to the tool (Douady, 1986). In order to achieve these goals, this research consists in identifying the nature of teacher-student interactions around the concepts of fraction and proportion taken in different contexts. The concepts of the fraction and the proportion play a crucial role in the training program of the Quebec school (MELS, 2001). They illustrate the intra and interdisciplinary character of their usages in mathematics and other disciplines. This diversity of use in view of intradisciplinary links (probability, statistics, homothetic, etc.) and interdisciplinary links (in science and technology, chemistry, biology, economics, etc.) makes their constructions fundamental. The conceptualization of fraction and proportion is based on various significances of the fraction (part of a whole, measure, ratio, quotient and operator) and on its assimilation (Proulx & Bednarz, 2009). However, developing the meaning of these two concepts represents a major challenge for students. This learning complexity is shared by researchers in didactics of mathematics (Brousseau, 1998; Kieren, 1988; G. Vergnaud, 1983, 1990) and by several teachers . This research studies the use of the status of fraction / proportion as per the dialectic tool / object (Douady, 1986) in intra and interdisciplinary contexts. Based on a qualitative / interpretative research, our analysis focuses mainly on the interactions between the teacher and the students as well as their productions. Our results on the nature of interactions between the teacher and the students and the task brought to light the didactic incidents (Roditi, 2005), the identification of breaks in the didactical contract (Brousseau, 1998) and the support given to students according to the types of proximities (Bridoux & al., 2015). Firstly, the analysis of interactions related to learning allowed us to identify the possible origin of the students’ errors and their characteristics which are grouped into three parts. The first part is linked to the data of the task statement when moving from a register of semiotic representation to another (Duval, 1993). When interpreting the data, the errors noted appear to be related to superfluous data and to certain terms used in the instructions. The second part is related to conceptual errors and generally affects proportional reasoning. When interpreting the ratio fraction, especially in the contexts of trigonometry and energy efficiency, the fraction is considered a quantity without establishing a relation between the numerator and the denominator. The third procedural aspect is related to the application of the cross-product procedure and the rules related to the various operations on fractions. In addition, this analysis allowed us to qualify the students' understanding of procedural according to the conceptual analysis of Bergeron and Herscovics (1989). In the mathematics class, the understanding of probability is interpreted according to the conceptual analysis performed by Savard (2008) and the understanding of trigonometry is examined according to the conceptual analysis performed by Sonja De kee, Dionne and Mura (1996). Secondly, the analysis of interactions linked to teaching allowed us to classify the types of help that teachers provide to students. We have categorized them according to three types of proximities (Bridoux et al., 2015): ascending proximities, descending proximities and horizontal proximities. We noted a predominance of the use of horizontal proximities among the four teachers. These horizontal proximities are very local in nature and their cognitive reach is limited (Bridoux et al., 2015, p. 22), thus contributing to the maintenance of the didactic contract. The effects of the didactic contract such as the Topaz effect, the effect of misunderstood expectation and the actor's paradox also influenced learning by maintaining the didactic contract. We have noted a frequent use of the Topaz effect, which in addition to maintaining the didactic contract, reduces the responsibility of the students and creates, in the student, expectations of solution from the teacher. Thus, this mode of intervention presents the cross-product procedure as the solution to the proposed tasks. The teacher/student’s relationship with knowledge also seems to influence learning and teaching in each discipline of our experimentation. The teaching seems to focus more on formal procedures than on understanding the underlying reasoning behind the concepts of fraction and proportion. Thus, students' understanding, and reasoning are abandoned in favor of procedures with rapid application. This relation to knowledge with regard to the concepts of fraction and proportion seems to be characterized by a desire to optimize the time devoted to their subject. This social dimension, although it responds to didactic time (Mercier, 1985, 1992), does not seem to contribute to the construction of the meaning of the concepts of fraction and proportion. By not being invited to use these concepts and develop their meaning, students could develop an instrumental relationship to knowledge. Finally, thanks to our results of student productions and verbatim, we were able to highlight the nature of the use of the fraction / proportion according to the dialectic as a tool or object (Douady, 1986). Analysis of the interventions of the four teachers, which are characterized by a predominance of horizontal proximities shows that these did not favor the transition from the fraction / the object proportion to the fraction / the tool proportion. This analysis revealed to us that the interpretations of the concepts of fraction, percentage, part-whole proportion and independent quantity proportion are still under construction and they are located at the "research" phase according to the operating cycle of the DOO. This analysis highlights many errors and confirms that the students' knowledge mainly relates to the procedures for carrying out the tasks. These errors show that their reasoning is based on the use of these concepts as "tools in development" in the resolution of tasks. Thus, these “tools in development” are more particularly observable in the “old” and “research” phases according to the operating cycle of the DOO. The notion of fraction and proportion play a significant role in learning and teaching in intra and interdisciplinary contexts and constitute a major challenge for students. This is how this study made explicit the fact that students use the concepts of fraction and proportion as a “tool in development” (Douady, 1986) when learning geometry, probability and energy efficiency, concentration, stoichiometry, optical reflection and uniformly accelerated rectilinear motion. As the fraction and the proportion are still in the state of a "tool in development" (Douady, 1986), their use in situations involving these two concepts influences the learning and teaching of these subjects. |

Document Type: | Thèse de doctorat |

Issue Date: | 2021 |

Open Access Date: | 9 August 2021 |

Permalink: | http://hdl.handle.net/20.500.11794/69908 |

Grantor: | Université Laval |

Collection: | Thèses et mémoires |

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