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From University Mathematics to Mathematics Education
Editors: Rolf Biehler ; Viviane Durand-Guerrier ; Nicolas Grenier-Boley ; Cécile Ouvrier-Buffet
Specificity of the knowledge in logic and its consequences on Klein's double discontinuity
In mathematics, logic is a field with a specific and transversal status. Indeed, it allows us to describe, control and validate mathematical activity. Therefore, it seems relevant to teach it in mathematics' classes.First, we present some research results from the DEMIPS’ « Logic and Proof » group. Based on interviews with university teachers, we will show different choices made for these kinds of courses and the different epistemologies relating to the proof that underlie them.Furthermore, since 2009, logic is faintly coming back in the french high school curriculum. Then, how can pre-service teachers rely on their university knowledges and transpose them to teach at high school level? We are going to answer this question of Klein's second discontinuity by presenting results from research in mathematics education about students’ difficulties and by explaining training situations we suggest.
From global to local aspects of Klein’s second discontinuity
The global question of how to identify, develop and assess mathematical knowledge that is relevant to future secondary school teachers, has been central in the emergence of mathematics education research from early on. We review parts of this history from the viewpoint of the anthropological theory of the didactic, and in particular the notion of relationships to mathematical praxeologies that are held by certain positions within school and university institutions. We also consider a modern case, where the questions arise in a very practical sense: how to bridge the gap between standard undergraduate mathematics courses and a school relevant model of real numbers and functions? We show how both theoretical and practical aspects of this more local question arise in a so-called capstone course for students with about two years of undergraduate mathematics experience.
Pre-service teachers' classroom stagings of proving-related activities and possible effects of Klein's discontinuity
Proof is a core element of mathematics. It therefore plays an essential role in university mathematics studies and thus also in the training of pre-service mathematics teachers. However, it has been shown that in-service teachers face problems in adequately implementing proof in their teaching. We consider this break between university studies and professional practice as part of what Klein called the "second discontinuity". Employing an activity-theoretical framework, we investigate in this paper the question how lesson stagings of preservice teachers at the end of their study program indicate systematic difficulties in staging proving-related activity for the classroom and how discontinuity experiences triggered by differences between proving-related practices in school and university could be considered as a possible cause for that. We present results from an observational study with pre-service teachers that reveal specific patterns in their proof-related behavior when planning and delivering lessons, and relate these back to sources for possible discontinuity experiences of pre-service teachers. As a practical implication, our results provide indications of how the topic of proof could be targeted in mathematics didactic teacher education.
Scriptwriting as a catalyst for linking undergraduate and school mathematics
Scripting tasks are a powerful tool for both mathematics education researchers and teacher educators, in part because the resultant dialogues provide insight into the scriptwriters' mathematical understanding and pedagogical inclinations. In this paper, we argue that scripting tasks used in mathematics education also provide an opportunity to deliver follow-up lessons that link undergraduate and school mathematics. These lessons facilitate mathematical connections by building directly on scriptwriters' experiences, as captured in their dialogues, and in turn, enrich their teaching practice.
Symmetry as a Topic for the University Education of Pre-service Teachers
This article provides a comprehensive mathematical-didactic analysis of how the highly relevant topic symmetry can be prepared for the university education of PSTs. Methodologically, the analysis is embedded in a design research cycle and serves as preparation for the actual design of learning activities. The procedure of "specifying and structuring" learning objects is used and adapted in such a way that, in addition to mathematical aspects, profession-oriented references to school mathematics are also considered. An essential result of the analysis is the formulation of so-called interface aspects to symmetry, which prove to be helpful in establishing such references.
Developing Kleinian Praxeologies: The Case of the Integral
In this paper, we pursue Winsløw's modelling of Klein's second discontinuity, within the Anthropological Theory of the Didactic (ATD), by introducing the notion of Kleinian praxeologies. These new praxeologies are built from praxeological blocks from existing praxeologies, from upper high school and university, to underline their links in mathematics teacher training. Then we present the results of an experiment, conducted according to the methodology of Didactic Engineering, which aims at the development of Kleinian praxeologies by teacher students. Our case study focuses on the integral of upper high school, in its links with Measure Theory taught at the university, in France. These links are described in terms of dominant praxeological models, enriched by Kleinian praxeologies. The data are analyzed using the different tools of the "questioning the world paradigm", in ATD (the questions-answers map and the Herbartian schema, for the study of chronogenesis and mesogenesis, respectively). The methodology allows a fine-grain analysis of the students' work and opens many perspectives for didactic research on Klein's second discontinuity, whether for the study of students' difficulties in linking elementary and advanced knowledge or for didactic engineering that aims to strengthen these links.
How to choose relevant mathematical contents to understand and overcome Klein’s second discontinuity with future teachers? The case of the scalar product
The issues related to Klein’s second discontinuity necessarily lead to the question of the choice of mathematical contents on which to focus and the way to consider the teaching of these contents in secondary education. In this article, we propose to explore this issue in the context of the French educational system for the case of the scalar product. We use examples that seem to us to highlight some important issues related to this second discontinuity, whether for researchers, future teachers or students.
Adding diversity to mathematical connections to counter Klein’s second discontinuity
For instructors that try to make university mathematics courses relevant to future secondary school teachers, doing so generally involves making connections between university mathematics content and school mathematics content–in attempts to counter what Felix Klein referred to as a “double discontinuity.” In this paper, I consider the nature of the mathematical connections that bridge these two domains, and common distinctions made in extant literature between them, such as directionality. Through this analysis, I point out another aspect of these connections that has been left implicit: university mathematics is primarily–and reasonably–framed as a superset of school mathematics content. In this paper I consider alternatives, in particular conceptualizing connections that invert this typical relational connection–i.e., a subset relational connection–and I exemplify these connections with concepts from university courses such as real analysis and abstract algebra. Then, I consider the rationale for doing so in terms of secondary teacher education, and the ways that diversifying our framework of connections in this way can be used to help counter Klein’s second discontinuity.