Entering the field of university research: a mathematics-physics adventure
I entered research in university mathematics education at the turn of the eighties, when a group of mathematicians and physicists in my university, including researchers in mathematics and physics education, decided to create an experimental course for students entering the mathematics and science programme. Our main goal was to challenge the compartmentalization of the two disciplines. The design of this course was very innovative, with regular lectures on topics of common interest jointly prepared and given by one mathematician and one physicist, along with common tests and interdisciplinary projects (Artigue, 1981). The whole team met once a week and everyone attended the common lectures.
Everything worked fine with the exception of the planned common lecture on the notion of differential for which the mathematicians and physicists could not reach an agreement. This âdifferential clashâ became a research issue that I first addressed jointly with the two didacticians of physics in the team, Laurence Viennot and Edith Saltiel, then with Marc Legrand and his colleagues in Grenoble, who were working on the teaching of the Riemann integral.
If I consider retrospectively the research we developed (Artigue, Menigaux & Viennot, 1990; Alibert et al., 1988), it is rather representative of some predominant characteristics of university research in mathematics and physics education at that time. The research has a clear cognitive orientation. It aims at understanding the conceptions of the differential developed by our students, the difficulties they meet with this notion and the associated processes in mathematics and in physics, and from a methodological perspective, it mainly uses a series of questionnaires and interviews.
This research also demonstrates a strong epistemological sensitivity, perhaps more characteristic of the French didactic community (Artigue et al., 2019). With the help of historians, indeed, we made specific efforts to understand the source of the âdifferential clashâ observed, why mathematicians seemed so rigidly attached to their vision of the differential as a function, in fact a differential form, and why physicists seemed so rigidly attached to their pragmatic vision of differentials as very small, if not infinitesimal, increments. For that purpose, we worked on primary and secondary historical sources, and also systematically studied the traces of the differential/derivative educational debate.
This research also shows the strong desire of researchers to translate their research findings into educational action. The results of both the cognitive and epistemological analyses were used to make visible the negative effects of certain usual teaching practices to mathematicians and physicists. Students, for instance, declared in the interviews that it was better for them not to try to understand what a differential was, and to work mechanically, both in mathematics and in physics; they were not able to distinguish between situations requiring or not the use of differential processes, and they only succeeded in solving the tasks proposed to them because they had learned to detect the linguistic hints calling for the use of such processes in the texts, and to mobilize the respective rituals of each discipline. These results were also used to develop a compromise acceptable by the two teams. Moreover, math-physics tasks in line with this compromise were designed and implemented in specific workshops in the following years (Artigue, Menigaux & Viennot, 1988).
However, this research, and more globally the work carried out in this experimental course, shows that issues of ecology (Artaud, 1997) and sustainability were not really part of our agenda. In fact, when some years later the team moved to other projects and teaching activities, this experimental course disappeared.
Experiencing the âschizophreniaâ of undergraduate university teachers: the teaching of differential equations
The second research I will evoke is the research I developed regarding the teaching of differential equations some years later. At that time, I was working on issues related to dynamical systems with some colleagues, with the support of a specialist of this domain, the mathematician Adrien Douady. I experienced the type of âschizophreniaâ which is rather common to those who teach undergraduate courses: the complete disconnection between what is their lived experience as mathematicians and their lived experience as university teachers. The programme of the course on ordinary differential equations for second year students I had to teach was focused on solving algebraically by integration in finite terms some specific forms of equations, which led students to think that the goal of research in this domain was to progressively complete the book of recipes they were introduced to. Some exact solutions in terms of power series were also looked for but only advanced students could have access to other views.
With some mathematician colleagues, I began to reflect on the possibility of developing elementary ODE courses more in line with the epistemology of this domain, thanks to the use of computers (Artigue, Gautheron & Sentenac, 1988). Once again, the epistemological work was an essential dimension of the research, leading to the identification of three main historical strands, each one of them having its own problĂŠmatique and development: the algebraic, the numerical and the geometrical-topological strand initiated much later than the two first ones by Henri PoincarĂŠ at the end of the 19th century. Only the first strand, in its most elementary forms, was part of traditional courses (the theory of exact solving as initiated by Condorcet and Liouville was not considered).
In the mid-eighties, the creation of an experimental section for first year students at the University of Lille 1 offered new research possibilities. To address the issue at stake (the viability of a first ODE course reflecting the current epistemology of the field), we used a methodology of didactic engineering (Artigue, 2014), and relied on didactic constructs familiar to French didacticians â such as the notions of setting and tool-object dialectics 1 due to Regine Douady (1986) â together with fundamental constructs of the Theory of Didactical Situations (TDS) (Brousseau, 1997). This methodology was of course adapted to the specific context of university education. The didactic engineering was collaboratively designed with Marc Rogalski and his colleagues from the University of Lille 1, revised after its first experimentation, and then successfully implemented there for nearly a decade (Artigue & Rogalski, 1990). In fact, the first experimentation showed that the viability of such a course required a different institutional status for graphical representations than the limited heuristic status given to them in university courses; graphical representations had to be credited as a legitimate tool for reasoning and proof, of course in appropriate forms such as those developed in the didactic engineering (Artigue, 1992). Moreover, we discovered that such a change could not be limited to the teaching of differential equations; for evident reasons of coherence, this change impacted the didactic contract (Brousseau, 1997) of the whole Analysis course the teaching of ODE belonged to. This certainly helps to understand why, despite its repeated successful implementation, only the first situations of this design were more widely used. Today, conceptual tools such as the hierarchy of levels of didactic co-determinacy proposed by the Anthropological Theory of the Didactic (ATD) (Chevallard, 2002; 2019) help us to systematically consider the different conditions and constraints governing the possible ecology of didactic designs, beyond those situated at the level of the mathematical theme or sector directly addressed2 â and to better anticipate their possible effects. We were less equipped to address these ecological issues thirty years ago. It must also be mentioned that, since then, research on the teaching and learning of differential equations has developed internationally, studying more systematically the accessibility and learning potential of teaching designs combining algebraic, numeric and geometrical-topological approaches as shown in the synthesis (Kwon, 2020).
These are just two examples among many others. They reflect the cognitive and epistemological focus of the research carried out at that time, and also the form that this focus was likely to take in the French didactic culture where TDS, with its underlying systemic perspective was the predominant theoretical approach. They also reflect the engagement of researchers in action, but without the conceptual tools that would have allowed them to seriously address dissemination and sustainability issues.
The international scene: the AMT Working Group of PME and the ICMI Study on the teaching and learning of mathematics at university level
On the international scene, the state of research at that time is well reflected by the work of the Advanced Mathematic Thinking (AMT) working group of the International Group for the Psychology of Mathematics Education (PME) which I entered in the late eighties, and the AMT book coordinated by David Tall (1991) resulting from this work. This book confirms the cognitive orientation, already mentioned, of research towards the study of studentsâ learning processes, thinking modes, conceptions and difficulties, and also the strong influence of constructivist perspectives. One of the main aims of the working group was to elucidate the specific nature, if any, of what its participants called âadvanced mathematical thinkingâ, and thanks to this elucidation, to better understand what differentiated learning processes at university from those experienced before by students. Even if a definitive answer is not provided in the book, some criteria are proposed in terms of relationship to abstraction, symbolism and generalization, role of definitions, formal reasoning and proof. One can also observe the important role played in research by constructions that take the form of...
