## Seminar's aim

Research on the transition between secondary education and university education (in the sense of postsecondary education) have been developed for long in various mathematical fields (Gueudet 2008). More recently, work has been carried out on what is referred to in the literature as "Klein's double discontinuity" (Winsløw & Grønbæk, 2013). These research are consistent with the view developed by Felix Klein who pointed out that university mathematics and school mathematics seem to have few connections (Kilpatrick 2019).

This question is still relevant today (eg Winsløw & Grønbæk, 2013, Gueudet & al. 2016), especially since many future mathematics teachers share the idea that university mathematics does not meet the needs of their future profession (Gueudet & al. 2016). There are therefore two challenges ; on the one hand, to highlight for students the links between university mathematics and school mathematics; on the other hand, provide future teachers with access to effective tools for their didactic work.

The seminar is intended to present recent researches on these questions, in particular those conducted by young researchers, including PhD students. The presentation will focus on well-identified mathematical fields (Analysis, Geometry, Arithmetic, Probability, Discrete Mathematics, Logic, Topology etc ..). Avenues for the training of university teachers training could be considered.

#### References

Gueudet, G. (2008). Investigating the secondary-tertiary transition. *Educational Studies in Mathematics , 67* (3), 237-254.

Kilpatrick, J. (2019). A double Discontinuity and a Triple Approach : Felix Klein’s perspective on Mathematics Teacher Education. In H.G. Weigand., W. McCallum, M. Menghini, M. Neubrad, Schubring, G. (eds) The Legacy of Felix Klein. ICME 13 Monographs. Springer, 215-225.

Gueudet, G., Bosch, M., diSessa, A., Kwon, O.N. & Verschaffel, L. (2016). Transitions in Mathematics Education. Springer, 2016, ICME 13 Topical survey, Gabriele Kaiser.

Winsløw, C. & Grønbæk, N. (2014). Klein’s double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus. *Recherches en Didactique des Mathématiques**, 34/1*, 59-86.

## Seminar leaders

The seminar is jointly organized by the Laboratoire de Didactique André Revuz (LDAR, EA 44E4) and the Institut Montpelliérain Alexander Grothendieck (IMAG, UMR 5149).

Seminar leader: Nicolas Genier-Boley, University of Rouen Normandie, France.

Seminar co-leader: Viviane Durand-Guerrier, University of Montpellier, France.

## Practical organization of the seminar

The seminar will take place online between September 2021 and December 2022 at the rate of one session per month. Except in exceptional cases, this session will be held on the second Monday of each month from 2:00 pm to 4:00 pm, Paris time (UTC+2 between March and October, UTC+1 between October and March).

Each session will consist of an hour of presentation by the speaker, followed by an hour of discussion.) The language of the seminar will be English.

#### Contact

The people interested in this seminar are invited to write to Nicolas Grenier-Boley (nicolas.grenier-boley@univ-rouen.fr), in order to be inscribed on the seminar mailing list and to receive the videoconference link at each session.

## Seminar sessions

### Monday, September 13, 2021, 2-4pm (UTC+2)

## Carl Winsløw (University of Copenhagen) : "The real numbers as a key challenge in upper secondary teacher education"

Abstract: Upper secondary mathematics teacher education was the context in which Felix Klein formulated his famous second discontinuity problem (beginning with his 1872 inaugural lecture at Erlangen). This problem exists today in much the same form, and consists in the gap between the mathematical education delivered in universities, and the task of teaching in (high) schools. The problem has probably even widened today. Here, we shall focus on one central mathematical object - the real number system - and on what can be done at University to develop students’ relationship to this object so that it is both grounded in present-day mathematics, and adapted to the needs of teaching in present-day high schools. In particular we will attend to the role of computer tools and programming in both contexts.

### Monday, October 11, 2021, 2-4pm (UTC+2)

## Nicolas Grenier-Boley (University of Rouen Normandie): "Choosing relevant content for Klein's second transition? The case of bilinear or multilinear algebra"

Abstract: The design and implementation of dedicated research to Klein's second transition necessarily brings the question of the choice of relevant mathematical content on which to focus. In this presentation, we will propose to explore this problem within the framework of the French educational system with regard to the case of bilinear or multilinear algebra. The presentation will be based on examples that will highlight some challenges for such a program, whether for researchers, future teachers or students.

### Monday, November 8, 2021, 2-4pm (UTC+1)

## Max Hoffmann (University of Paderborn): "Geometry for Student Teachers: A holistic course concept as a contribution to overcoming the second discontinuity"

Abstract: At Paderborn University, student teachers (for the German "Gymnasium") attend some of their mathematics courses not together with Bachelor of Science students. This provides the opportunity to design and research course concepts which implement profession orientation in a holistic way. One of these courses is the "Geometry for Student Teachers". First, I will present theoretical considerations on the functions of academic mathematics competencies for professional teacher action and derive design principles for the implementation of profession orientation in mathematics courses for student teachers. Based on this, I will introduce the design of the course. Finally, I will describe in detail the development of selected interface learning opportunities, discuss real student work and present results of the accompanying research.

oduce the design of the course. Finally, I will describe in detail the development of selected interface learning opportunities, discuss real student work and present results of the accompanying research.

### Monday, December 13, 2021, 2-4pm (UTC+1)

## Katalin Gosztonyi (University Eötvös Loránd) : "Discrete mathematics in the secondary curriculum and in teacher education in Hungary: issues and perspectives for the second transition"

Abstract: In Hungary, discrete mathematics has important traditions in research as well as in mathematics education. In mathematics education, several domains of discrete mathematics appear in, the curricula, and especially combinatorics is systematically present from primary school to the baccalaureate. By consequence, discrete mathematics constitutes also an autonomous domain in teacher education.

In the first part of the presentation, I will briefly present the place and role of discrete mathematics in school curricula. In the second part, I will talk about discrete mathematics courses in teacher education, explaining aspects of continuity and rupture with secondary school education. In the third part, I will discuss questions related to our teacher education course on the teaching of discrete mathematics, our current developments and endeavours to prepare teachers for the second transition.

### Monday, January 10, 2022, 2-4pm (UTC+1)

## Laura Branchetti (University of Milan): "The interplay between Mathematics and Physics in secondary teacher-student education: activities and research developed within an interdisciplinary research team"

Abstract: Since 2014, in collaboration with a small group of researchers in Physics and Mathematics education and teachers-researchers (led by Prof. Olivia Levrini), I have been investigating the issues arisen by interdisciplinarity in secondary teaching, with particular attention to high school. The main motivations were the well-known difficulties emerged in research in physics education due to mathematics and the feeling that the research in didactics of these two disciplines had much more in common than it was actually represented in research and institutions in Italy. In the Italian context these connections are even more important to explore since teachers with a background in mathematics or physics can teach both disciplines at secondary school and the mathematical curriculum includes explicitly interactions with physics. Discussing and exploring the existing literature we realized that the main limitation of the existing studies was the “deformation” of the main aims and values of the other discipline once the issue of interdisciplinarity was formulated in a disciplinary way (Physics in Mathematics or Mathematics in Physics). This phenomenon affects also the teacher-students in the Klein's second transition, since they are taught in institutional contexts where epistemologies and identities of discipline are conceived as clearly separated in teaching (in particular at the university). Indeed, often the interdisciplinarity that characterizes the discipline at the research level is not presented to Bachelor or Master students. Teacher-students education is thus a key action to reach innovation. We designed activities for teacher-students impemlented in courses attended by students with a background in both disciplines and carried out research starting from our implementations. In 2019 we developed an international research project with the Universities of Montpellier, Crete and Barcelona, that was funded by the Erasmus project, enlarging the perspective to an international context.

In this contribution I will resume the main issues, the theoretical framework and some examples of design, implementation and analysis of modules for teacher-students, facing in particular the issue of Klein's second transition.

### Monday, March 7, 2022, 2-4pm (UTC+1)

## Thomas Bauer (University of Marburg) & Eva Müller-Hill (University of Rostock) : "Approaches in pre-service teacher education to counter Klein's double discontinuity

Abstract: Felix Klein's observation of double discontinuity poses several challenges at several stages of pre-service teacher education: 1) At the transition from school to university, how can students be provided with the experience that school mathematics and university mathematics are interrelated and useful for each other? 2) In mathematics content courses, how can students experience mathematics in such a way that they are both willing and able to make good use of academic knowledge later in their job? 3) And finally, how can students learn to make full use of their knowledge (math content and math education) when planning classroom implementations?

In the talk we present approaches to these challenges from our teaching practice: With respect to the first discontinuity, we show how the concept of "interface tasks" in real analysis can help to strengthen the relationship between school and university mathematics. We present design principles and empirical results from the use of these tasks. Concerning the second discontinuity, we present math education course designs that relate proof and argumentation as core mathematical activities to implementations in the classroom. We use an activity theoretic framework to uncover and explain phenomena observed in students' classroom implementations.

### Monday, March 28, 2022, 2-4pm (UTC+2)

## Avenilde Romo (Center for Research and Advanced Studies of the National Polytechnic Institute, Mexico): "Klein's second discontinuity as seen through the lens of a Mexican program of professional development for mathematics teachers in service"

Abstract: There is no specific training for future high school and university mathematics teachers in Mexico. Most mathematics teachers at this level are mathematicians, physicists, engineers or professionals with a five-year undergraduate degree in an area with a specific mathematical, scientific orientation who have a vocation and interest in teaching. Given the need to have specialised teachers, different professional training programmes have emerged to address the didactic needs of these in-service teachers. In particular, in 2000, was created the programme of professional development for mathematics teachers in service at the Instituto Politécnico Nacional, where I have worked for ten years. This seminar session will reflect, partially, how this programme addresses Klein's second discontinuity. To do so, I will consider some examples of the courses offered in a master's program and also the master's thesis developed by the Mexican and Latin American teachers of mathematics in service.

### Monday, June 13, 2022, 2-4pm (UTC+2)

## Virginie Deloustal-Jorrand (University Lyon 1) & Zoé Mesnil (University Paris Cité): "Specificity of knowledge in logic and its consequences on Klein's double discontinuity"

Abstract. In mathematics, logic is a field with a specific and transverse status. Indeed, it allows to describe, control and validate mathematical activity. Therefore, it seems relevant to teach it in mathematics' classes in order to help understand and use mathematical language, necessary to mathematical activity, and to produce proofs.

In French university (as elsewhere) proof is taught in different courses. Those are linked at different degrees to mathematical logic. First, we will present some research results from the DEMIPS’ « Logic » group (Didactique et Épistémologie des Mathématiques, lien avec l'Informatique et la Physique, dans le supérieur, https://demips.math.cnrs.fr). Based on interviews with university teachers, we will show different choices made for these kind 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 future teachers rely on their university knowledges and transpose them to teach at high school level? We will try to answer this question of Klein's second discontinuity by presenting results from research in didactics about student’s difficulties and by showing training situations we suggest.

### Monday, July 11, 2022, 11am-1pm (UTC+2)

## Oh Nam Kwon (Seoul National University): "Didactic Transposition in College and School Mathematics: Novice Teachers’ Understanding of Factorization"

Abstract. The factorization of polynomials is a basic concept for solving problems in various domains of school mathematics. It is relevant to the greatest common divisors and least common multiples of polynomials, and is needed to solve equations and inequalities. However, although the concept of factorization comprises important content, it is recognized only as procedural knowledge required to solve equations. Thus, students memorize the formula as the inverse process of expansion without necessarily understanding its importance. From an advanced standpoint, polynomial factorization in school mathematics could be related to the unique factorization domain (UFD). Thus, I examine the understanding of polynomial factorization of novice teachers with relatively vivid advanced mathematical experience, in order to compare the relationship between school and college mathematics, and to investigate what influence these teachers’ advanced mathematical experience has on their teaching of polynomial factorization.

In the first part of the presentation, I will present the curricula for mathematics teacher education and teacher employment tests focusing on teacher knowledge in Korean context. In the second part of the presentation, I will examine how the content of school mathematics is described from an advanced standpoint and how teachers utilize their college mathematics knowledge in the context of school mathematics. Focusing on the UFD in college mathematics and polynomial factorization in school mathematics, I will analyze school and algebra textbooks, and explored teacher knowledge of the UFD and how teachers’ factorization concepts occur in the teaching context in particular the issue of Klein's second transition.

### Monday, September 12, 2022, 2-4pm (UTC+2)

## Nick Wasserman (Columbia University): " Upgrading Learning for Teachers in Real Analysis: A Look at Diversifying Mathematical Connections to Counter Klein's Second Discontinuity"

Abstract: Secondary mathematics teachers in the United States often have to take a course in Real Analysis as part of their degree program. Many see little value in these advanced courses. This talk explores ideas and results from one project, Upgrading Learning for Teachers in Real Analysis (ULTRA), in which modules for a real analysis course were designed to make more explicit the connection to school mathematics teaching. The talk will explore the overarching instructional model for module design and, in particular, discuss how diversifying the kinds of mathematical connections between school and university mathematics was used to help counter Klein's second discontinuity.

### Monday, October 10, 2022, 2-4pm (UTC+2)

## Lisa Hefendehl-Hebeker (Duisburg-Essen University) : "Epistemological demands of the second discontinuity"

Abstract: "Mathematics is an organ of knowledge and an infinit refinement of language. It rises out of ordinary language and imagination like a plant out of the soil, and its roots are numbers and simple spatial ideas" (Kähler, 1955). In comparison with their experiences at school, university students of mathematics must cope with a faster pace, a broader scope of material and a higher degree of abstraction and formalization. In addition, they must attain professional habits with accompanying attitudes and norms. When they return to the school as teachers, they must go through a reverse process and go back to the roots. But they should do this in a professional way that is suitable for further developing the naïve view of the learners. The presentation will address the epistemological requirements in this regard.

### Monday, November 7, 2022, 2-4pm (UTC+1)

## Gaëtan Planchon (University of Montpellier): "A didactic engineering on integral to address Klein's second discontinuity"

Abstract:

Before entering the mathematics teacher training cycle, students, future teachers of mathematics, are confronted with different theories of integration (Riemann, Lebesgue, measure theory). In doing so, they often have difficulties to perceive the links between these theories and the integral taught in high school, which illustrates the problem of Klein's second discontinuity.

In this talk, we will present some elements of an engineering experiment in Montpellier which aims at reducing this discontinuity by making visible the links between the theory of measurement and the intuitive notion of area which is the basis of the integral in high school. The design of this engineering is based on the modeling of Klein's second discontinuity proposed by Winslow in the framework of the Anthropological Theory of Didactics and the description of knowledge in terms of dominant praxeological models, in the different institutions (high school and university). We also use the tools of ATD related to the paradigm of questioning the world to carry out analyses (a priori and a posteriori) of the tasks proposed to the students. This leads us to formalize the process of linking high school and university knowledge in terms of the development of new praxeologies, called Kleinian praxeologies, based on the examples present in our device.

### Monday, November 21, 2022, 2-4pm (UTC+1)

#### Rina Zazkis (Simon Fraser University): "Accessing and enhancing mathematical knowledge via scripting tasks"

Abstract: How can a teacher get a “scan” of a group’s knowledge and understanding of a mathematical topic in order to plan for, or adjust, subsequent instruction? To address this question, I introduce the notion of “lesson play” and its evolution to “script writing” or “scripting”. I short, scripting involves teachers in writing a dialogue between a teacher and students based on a particular mathematical issue. The teacher-character in such a dialogue demonstrates how the script-writer envisions a particular mathematical situation. I will exemplify how the instruction that follows students scripts can be adjusted to connect school teaching to ideas of undergraduate mathematics.