Since electrons determine most of the electronic, optical and thermal properties of matter, the challenge is to model them in order to explain and predict the properties of new materials. This modeling is highly complex, since it must take into account the infinitely small via quantum physics, and the infinitely large scale considering all the electrons in matter. Since the exact calculation is limited to just a few electrons, it is necessary to introduce strong approximations. In laboratories, this modeling is based on complex ab initio or semi-empirical methods. This raises the question of how to teach electron modeling. To address this issue, we have made several choices. Firstly, an epistemological choice by introducing a gradual approach starting from 2 atoms up to 10^22. We show that the formalism established for a few atoms becomes unusable in the face of very large numbers, and in fine to understand the need to introduce new concepts. Secondly, a pedagogical choice using Problem-Based Learning (PBL). Thirdly, a methodological choice by moving from one model to another that is more complex, but more realistic. This gradual approach "from one model to the next" allows us to reflect on the notion of models, their links with reality and their limits. This experience is an opportunity to think about more generally on model learning in physics, and to suggest ways of involving students in modeling activities.
This article develops a situation, coming from ongoing mathematical research, that can enable students to experience a mathematical activity involving the construction of definitions. An epistemological analysis “à la Lakatos” of this situation is developed: it highlights how the discrete situation allows a mathematical experience for undergraduate students and contains interesting and even challenging material for all levels, including teachers, lecturers and research mathematicians. The discrete situation opens perspectives for higher education, especially on defining processes and proofs, with concepts important across multiple areas of mathematics (generating set, minimality) in the background.
We are convinced of the usefulness of sketches and diagrams during mathematical work but the observation is made in our practices that they are not spontaneously used by students. In order to study the understanding and use of sketches by mathematics students, we designed and then proposed a test at different university levels. The test consists of five exercises.The first concerns different representation registers of a set of numbers, the second on a graphic proof of an implicative algebraic proposition and the last three on the graphic approch to the notions of injectivity, surjectivity, bijectivity in the context of the analysis. The sketches, proposed or requested in each exercise, are intended to be aids to changes of register and reasoning. We present what motivated the choices and developments of the exercises then we analyze the results of these tests. In each case, we see difficulties in understanding and the sketches proposed, which leads to think that the sketch must be the subject of specific work at the university level.
This article aims to further investigate how the teaching of mathematics in the classroom/lecture hall can both enhance students' thinking and enable them to master the analysis of complex mathematical knowledge through scientific debate in the classroom.
This article deals with peer-assessment in the context of higher education teaching in mathematics, and examines the nature of student activity when assessing work,more precisely proofs, produced by peers. After an overview of research on peer assessment, we propose an experiment with students in a specific post-secondary scientific class, and analyze the activity of the students involved. Our analyses are based on a priori analyses of the proposed tasks and tools from research on students personal mathematical work. We base on written notes of assessments, observations of pairs in assessment situations, and answers to a questionnaire on the perception of their activity. We have identified some specific student activities during the peer assessment process, in particular related to the analysis and correction of proofs, but also to the awareness of the issues associated with evaluation. This enables us to argue for the potential of peer assessment in higher mathematics education.