Teaching fractions through a Measurement Approach to prospective elementary teachers: A design experiment in a Math Methods course

In this study we give an account of a teaching experiment on fractions to prospective elementary teachers, which took place in winter 2014 in a Teaching Mathematics course in an Elementary Education undergraduate program at a North-American university. The experiment was an adaptation for teacher education of the “Measurement Approach” to teaching fractions developed by the psychologist V.V. Davydov for the elementary mathematics curriculum (Davydov & Tsvetkovich, 1991). The research had the characteristics of a design experiment, with a phase of reflection on the sources of meaning of fractions appropriate for the elementary school, as well as preliminary trials with one year before (winter 2013) preceding the implementation of the experiment in a “mature form.” We had two overarching goals in the design conception: fostering future teachers’ quantitative reasoning and cultivating a positioning relative to the course institution that is more conducive to accepting the approach – that of university students acquiring theoretical knowledge. In the description and the retrospective analysis of the teaching intervention we follow the realization of these goals at three levels: the overall organization of the material and tasks in the course by the instructor, the classroom interactions between the instructor and the students in lectures, and individual reasoning without mediation by the instructor. We found that the Measurement Approach encouraged a culture of systemic justification in the classroom with some students adopting flexibly and creatively the proposed models of reasoning within a given theory. However, the risk of students’ imitating only certain aspects of these models – such as words, sentence structures, or procedures – ran high, with many students using the theory only as “decoration”, without adequate understanding. Furthermore, although spontaneous engagement with quantitative reasoning for establishing validity of statements about fractions or for explaining realistic problems was rare, it was present in several students, in encouraging forms. Very few students adopted such reasoning, but those who did, exhibited sophisticated and varied strategies for solving problems, which demonstrated robust understanding of the fraction of quantity theory

Teaching fractions through a Measurement Approach to prospective elementary teachers: A design experiment in a Math Methods course

In this study we give an account of a teaching experiment on fractions to prospective elementary teachers, which took place in winter 2014 in a Teaching Mathematics course in an Elementary Education undergraduate program at a North-American university. The experiment was an adaptation for teacher education of the “Measurement Approach” to teaching fractions developed by the psychologist V.V. Davydov for the elementary mathematics curriculum (Davydov & Tsvetkovich, 1991). The research had the characteristics of a design experiment, with a phase of reflection on the sources of meaning of fractions appropriate for the elementary school, as well as preliminary trials with one year before (winter 2013) preceding the implementation of the experiment in a “mature form.” We had two overarching goals in the design conception: fostering future teachers’ quantitative reasoning and cultivating a positioning relative to the course institution that is more conducive to accepting the approach – that of university students acquiring theoretical knowledge. In the description and the retrospective analysis of the teaching intervention we follow the realization of these goals at three levels: the overall organization of the material and tasks in the course by the instructor, the classroom interactions between the instructor and the students in lectures, and individual reasoning without mediation by the instructor. We found that the Measurement Approach encouraged a culture of systemic justification in the classroom with some students adopting flexibly and creatively the proposed models of reasoning within a given theory. However, the risk of students’ imitating only certain aspects of these models – such as words, sentence structures, or procedures – ran high, with many students using the theory only as “decoration”, without adequate understanding. Furthermore, although spontaneous engagement with quantitative reasoning for establishing validity of statements about fractions or for explaining realistic problems was rare, it was present in several students, in encouraging forms. Very few students adopted such reasoning, but those who did, exhibited sophisticated and varied strategies for solving problems, which demonstrated robust understanding of the fraction of quantity theory

The development of the modern integration theory from Cauchy to Lebesgue : a historical and epistemological study with didactical implications

In this thesis, we describe the historical development of the modern integration theory by presenting the key ideas and insights that led to its shaping, from the first rigorous definition of the definite integral given by Cauchy in 1823, to Lebesgue's theory as it first appeared in his doctoral thesis of 1902. In the final part we also present recent approaches in integration theory. We show that various problems motivated the enrichment of the notion of integral, while one in particular constituted the most important trigger of this development from the beginnings well into the 20th century: the search for a better understanding of Fourier series. For our study, we principally look at original sources, and we provide detailed proofs for important results, often by elaborating on the authors' sketchy or heuristic arguments, malting thus the original results accessible to the modern reader. We then use this historical and epistemological analysis to raise some issues related to the teaching of integration at various levels at university, i.e., in calculus, analysis and measure theory courses. For this, we look at some typical textbooks and, inspired by the historical analysis, we give some suggestions for teaching. Our findings show that there might be some conceptual gaps between subsequent levels that are not necessarily insuperable, but require careful didactical analysis