2 | 2023

Research Work from INDRUM Network – contributions to thinking differently about teaching mathematics. Issue Editors: Nicolas Grenier-Boley, Hussein Sabra, Marianna Bosch, & Carl Winsløw.

Editorial note

The present Volume 2 of EpiDÉMES “Epijournal de Didactique et d'épistémologie des mathématiques du supérieur” bring to you works by the International Network of Research in Didactics of University Mathematics (INDRUM ). This special issue builds upon the topics presented in Volume 1, offering complementary perspectives and insights into the possible contributions of research to improve the teaching and learning of mathematics at the university level. Supported by the European Association for Research in Mathematics Education (ERME), INDRUM unites researchers from diverse countries or even continents, to explore and advance Mathematics Education at all the levels of tertiary education. The research work presented in this issue perfectly illustrates this widening of the network. We would like to mention that one of the objectives of the INDRUM network is to reinforce the visibility of research in University Mathematics Education and with respect to the Mathematics community, which is concomitant with the editorial line of EpiDÉMES. The authors of the articles presented in this special issue have all positioned their research in terms of its specific relevance to higher education, while highlighting the possible contributions of this research for practitioners in higher education. In the first paper by Inen Akrouti and Slim Mrabet, students’ conceptions of the definite integral in the first year of preparatory class are investigated. The study reveals that many students tend to rely on algebraic procedures when evaluating integrals, highlighting the need to explore alternative approaches. By analyzing students’ productions, this research provides valuable insights into students’ perspectives on this crucial topic in teaching calculus. It lays the groundwork for refining instructional strategies and enhancing students’ understanding of the definite integral. Julie Stalder’s paper delves into the choices made by teachers teaching abstract algebra. With a focus on the concept of “Ideal”, the study employs an epistemological analysis to highlight its central role in abstract algebra. By examining the role of examples and exercises in students’ practices, the research sheds light on effective teaching approaches in this field. Caterina Cumino, Martino Pavignano, and Ursula Zich present in the third paper an example of integrating graphic language and standards into the education of architecture students. The study focuses on the 2D representation of architectural objects, such as roofing systems generated by cylinders and their intersections. By emphasizing the use of physical or virtual models to support students’ mathematical thinking, irrespective of their varying levels of geometric understanding, this research enriches the learning experiences of architecture students. The connection between mathematical modeling and architectural contexts adds depth to their education. In the fourth paper, Chantal Buteau, Laura Broley, Kirstin Dreise, and Eric Muller explore undergraduate mathematics students’ experiences in using programming for mathematics investigation projects. The study reveals the organization and complexity of students’ activities, which intertwine mathematics and programming competencies. Through a analysis of student and instructor data, this research provides practical recommendations for instructors engaged in similar projects. The final paper by Mitsuru Kawazoé focuses on the use of embodied notions in teaching linear algebra, particularly examining linear (in)dependence and basis concepts. The research investigates the relationship between students’ understandings in both the embodied and symbolic worlds. It also evaluates the impact of instruction that emphasizes geometric images. The findings shed light on the association between conceptions in different worlds and the challenges of improving students’ understanding through geometric instruction. The INDRUM special issue of the “Epijournal de Didactique et d'épistémologie des mathématiques du supérieur” offers valuable insights and recommendations for teachers and researchers alike. We extend our appreciation to the authors for their contributions. May this special issue inspire further research, foster collaborative discussions, and lead to the continual improvement of mathematics teaching and learning in higher education.

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.