2 | 2023

1. Relation between understandings of linear algebra concepts in the embodied world and in the symbolic world

Mitsuru Kawazoe.
For the use of embodied notions in teaching linear algebra, some studies indicate that it is helpful, whereas other studies indicate that it could be problematic or become an obstacle. Hence, additional research is needed. This study is focused on linear (in)dependence and basis, and investigates the relation between their understandings in the embodied and symbolic worlds. We also examine whether students' conceptions in the embodied world can be improved by the instruction emphasizing geometric images, as our previous studies identified some limitations of students' understanding in the embodied world. To address these issues, we designed four tasks aiming to assess students' conceptions of linear (in)dependence, basis, and dimension, and also designed linear algebra lessons emphasizing geometric images of these concepts. These tasks were conducted during the lessons and the data of 38 engineering students was collected. The analysis for the data showed that conceptions in the embodied world was positively associated with conceptions in the symbolic world; however, students' conceptions in the embodied world were not sufficiently improved by the geometric instruction implemented in this study.

2. Students using programming for pure and applied mathematics investigations

Chantal Buteau ; Laura Broley ; Kirstin Dreise ; Eric Muller.
In this paper, we recount our research on undergraduate mathematics students learning to use programming for mathematics investigation projects. More precisely, we focus on how a particular theoretical perspective (the Instrumental Approach) helps us better understand this student activity. Pulling data from students' and instructors' experiences in a sequence of courses (offered since 2001), our results expose, at the micro and macro levels, how the student activity is organized (through stable 'ways of doing'), and highlights the complexity of this activity (as an intertwined web of 'ways of doing' involving a combination of both mathematics and programming competencies). We end with concrete recommendations to instructors.

3. What is the place of examples in the teaching of abstract algebra?

Julie Stalder.
Teaching abstract algebra, seen as the study of structures and properties of structures, at a university level appears to be a challenge for both students and faculty. The professors in our study describe this passage through abstraction for the student body as a "killing game", an "impassable wall" or a "leap through abstraction". In this paper, based on a case study (Candy, 2020) we will investigate the choices of professors teaching abstract algebra at university. These professors were chosen because they teach abstract algebra at university in France, and they gave us access to their course corpus and agreed to be interviewed. In this article, we will choose to study in particular the teaching of the concept of ideal. An epistemological analysis will allow us to highlight its central role in the construction of abstract algebra. Then, using the anthropological theory of didactics (Chevallard, 1998), we will try to specify the place of examples and exercises in the praxis of the student body.

4. Analyzing today geometry on architectural heritage between mathematics and representation to define architect's background

Caterina Cumino ; Martino Pavignano ; Ursula Zich.
We present and analyze an example of activity aiming at introducing graphical language and standards to students of the first year bachelor's in architecture. Our example concerns the 2D representation of an architectural object, an essential competency in the architect's profession, and specifically deals with roofing systems made up of vaults generated by cylinders and their intersections. It is a first attempt to create activities that involve all students regardless of their different levels of geometric understanding and of familiarity with graphic language; in particular, this example exploits the introduction of physical or virtual models to support the mathematical thinking of students in completing the task and takes place during regular lesson times, without modifying or adding mathematical subject contents. Research literature on mathematical modeling, in particular on the so-called prescriptive one, provides us tools to frame and implement our study case; it also allows presenting mathematical modeling as a goal in an extra-mathematical educational context.

5. Conceptions des étudiants des classes préparatoires en Tunisie sur l'intégrale de Riemann

Inen Akrouti ; Slim Mrabet.
The integral is one of the most important topics in Calculus that is difficult to be understood by many students. When solving definite integral application problems, previous research emphasizes that students found the antiderivative procedure more useful and easier than the approximation process or area (Akrouti, 2020). This paper focuses on students' conceptions of the definite integral in the first year of preparatory class. Data were collected from students' written responses to questions that relate to their views of integration. The analysis shows that the majority of students choose the algebraic process to evaluate the proposed integrals. Participants were first-semester calculus students enrolled in a public university.