Long talks
Elise Lockwood, Oregon State University, USA
Valuing the Concrete while Striving for the Abstract: The Importance of Examples in Discrete Mathematics
The use of examples is a fundamental aspect of mathematical thinking and activity. In my work with undergraduate students in combinatorics, I have at times observed a tendency to resist work with concrete examples in favor of a focus on abstract reasoning and general formulas. In this talk, I consider this phenomenon, and I make a case for the importance of reasoning about and using examples within discrete mathematics. I frame some of this work within combinatorics, relating the idea of a set-oriented perspective to a focus on examples. I also offer some cases of mathematicians’ and students’ engagement with examples in other discrete contexts, including proof and number theory. I conclude by discussing ways in which engagement with examples might be unique in discrete (rather than continuous) contexts, and I suggest some pedagogical implications and potential directions for future research on this topic.
Cécile Ouvrier-Buffet, Université Paris-Cité and Université-Est-Créteil, France
Sylvain Gravier, Université Grenoble-Alpes and CNRS, France
Discrete mathematics for teaching "doing mathematics"? The example of the Maths à Modeler Research Federation
Since 1999, the French Maths à Modeler Research Federation has been promoting a collaborative work between mathematicians in discrete mathematics and mathematics education researchers. Using Brousseau’s Theory of Didactical Situations, we design innovative Research Situations for the Classroom based on real problems coming from ongoing mathematical research. The aim of such situations is to put students in the role of researchers and to engage them in "doing mathematics" i.e. to conjecture, to refute, to model, to extend but also to transform a questioning process, to be able to reason non-linearly, to define, and - of course - to prove.
All the research in mathematics education dealing specifically with the teaching of discrete mathematics at different school levels has pointed out the advantages of discrete mathematics for the teaching and learning of proof. The resolution of various kinds of problems (e.g. existence, characterization, recognition, optimization and extremal problems) mobilizes several rich proving processes (proofs by exhaustion, proofs by induction, algorithmic proofs, generic proofs, but also probabilistic methods, decomposition techniques, and structuration of objects (such as coloration in graph theory)). This talk will present the fruitful collaboration between discrete mathematicians and mathematics education researchers in Maths à Modeler. Examples of different Research Situations arising from this project will show the features of proving processes in discrete mathematics and their potential for education.
Short talks
Janka Medova, Department of Mathematics, Faculty of Natural Sciences and Informatics, Constantine the Philosopher University in Nitra, Slovakia
From Abstraction to Application: Varying Spectrum of Graph Complexity in Learning Graph Algorithms
Graph are useful tool for abstraction, used to understand the structure of various mathematical objects, scientific applications or real-life situations. Graph algorithms are used to solve various classes of the problems. Based on an abstract graph (AG), it would be possible to define a graph with unnecessary information (UI), a graph with redundant information (RI), a graph with unnecessary and redundant information (URI). Research shows that when introducing algorithm, abstract graph is easier to be understood. On the other hand, when students are already familiar with the algorithm, they are in some cases more successful when using model more similar to real situation, even though it is not abstract graph, but contains unnecessary and redundant information.
David Zenkl, Charles University in Prague, Czech Republic
The concept of experimental teaching of combinatorics at grammar school in the Czech Republic
My dissertation deals with experimental teaching of combinatorics in a constructivist style. Traditionally, combinatorics is taught in the Czech(oslovak) environment as a set of three to six formulas. Student's task is to learn to decide which one to use. This involves many misconceptions and errors. The innovative way of teaching tries to prevent these difficulties. It consists in the fact that the teacher does not emphasize combinatorial formulas and the related names like arrangements, permutations, combinations, but essential skills. These are, for example, the systematic listing of elements (systematic enumeration), the visualisation of the problem, the art of seeing the isomorphism of problems, the generalisation of problem solving, etc. At the seminar I will also discuss preliminary results of my design-based research.
Bernadett Szakácsné Györey, Eötvös Loránd University of Budapest, Hungary
The place and role of discrete mathematics in Hungarian curricula
Discrete mathematics plays an important role in the Hungarian curriculum. Several discrete mathematical topics appear in the curriculum as combinatorics, mathematical logic, sets, number theory and graph theory.
In the first half of my presentation, I will outline the curriculum structure and curricular requirements for topics of discrete mathematics. I will shortly discuss the appearance of autonomous discrete mathematics chapters in curricula and textbooks as sets and combinatorics in grades 9-10, mathematical logic in grade 10, and graph theory in grade 11.
I will also show that in the lower grades, elements of discrete mathematics topics appear, to a significant extent, embedded in algebra, geometry, probability, and statistics chapters. In the second half of my presentation, I will illustrate by an example how elements of discrete mathematics appear in the context of other topics.
Tamás Héger, Eötvös Loránd University of Budapest, Hungary
Discrete mathematics in teacher education at the Eötvös Loránd University in Hungary
Combinatorics, since the 1970s, is systematically present in all levels of Hungarian curricula. At the same time, our observation is that prospective teachers arrive at the university with formal but barely operational knowledge in combinatorics, and the experience they bring with them about teaching and learning combinatorics is focused on the learning of formulas for basic task types (permutations, variations, and combinations). Thus, at university discrete mathematical courses, we intend to fill this gap between principles and aims of the curriculum on one hand and the students’ actual knowledge in combinatorics (and, also, their conceptions of teaching combinatorics) on the other. In the talk I will present the structure of introductory discrete mathematical courses in Hungarian teacher education at Eötvös Loránd University with an emphasis on some practical details that were designed to achieve the aformentioned aim of these courses.
Luca Lamana, Free University of Bozen-Bolzano, Italy
Laura Branchetti, University of Milan, Italy
Secondary school teachers’ posture and interpretative processes in combinatorial problem solving
In combinatorial problem-solving, students spontaneously develop a wide set of heuristics to solve tasks, which often disappear after learning combinatorics at school. This presentation will focus on a work of research where the reasons behind this observed transformation were investigated, focusing on the figure of the secondary school teachers and their practice when teaching combinatorics. We investigate aspects of combinatorics teaching that influence the development of students' heuristics, such as teachers' processes of interpretation of students' solutions and the following interventions: What are the elements on which teachers focus when interpreting students' solutions? What kind of interpretations are made? What is the teachers' posture toward students' solutions and, in general, toward the teaching of combinatorics, and what are the aspects that influence their interventions? In our presentation, answers to these questions will be delineated, providing a first explanation for the process of polarization observed in students' heuristics before and after learning combinatorics.
Mickael Da Ronch, University of Teacher Education Valais, Switzerland
Epistemological analysis and design of research situations in discrete mathematics: method and examples
In France and more widely in French-speaking countries, the theory of didactic situations is one of the theoretical reference frameworks for the construction of teaching and learning situations, with a focus on didactic engineering as a research methodology. Epistemological or mathematical analysis is an essential phase in the design of such engineering to support research for mathematical situations. In particular, those that make it possible to mobilise the essential components of research activity in mathematics linked, for example, to experimentation, the formulation of conjectures or proof. Moreover, current research shows that the field of discrete mathematics is conducive to the design of situations, at all levels of teaching, which mobilise these components. Consequently, a 'generic' method will be proposed for carrying out this type of analysis, based on examples of problems from current research in discrete mathematics.
Eszter Bóra, Péter Juhász, Réka Szász, Gábor Szűcs
Alfréd Rényi Institute of Mathematics, and Eötvös Loránd University of Budapest, Hungary
Teaching enumeration strategies to secondary students with the Pósa method
The Pósa method is an approach within the Hungarian guided discovery tradition of teaching mathematics, where the aim is for students to discover concepts by themselves through scaffolding provided by the teacher. The method is mostly used in gifted education, but was also adapted to the secondary school curriculum. Discrete mathematics is an important topic area in the method because of its low threshold - high ceiling nature, and also because it lends itself to situations where students experience that problem-solving strategies are more effective than formulas. Specific techniques of the Pósa method include using problems that build on each other, setting engaging tasks, and exposing students to cognitive dissonance so that they can reassess their beliefs. In the talk we will introduce the basics of the Pósa method and its approach to discrete mathematics, which we will illustrate with a set of problems involving basic enumeration strategies.
Katalin Gosztonyi, Simon Modeste, University of Montpellier, France
Problem-networks in discrete mathematics education
In the frame of the MSCA ProDiME project, we are developing a modelling tool in terms of “problem-networks” to conceive, analyse and adapt problem-based teaching trajectories and to support teachers’ work. The modelling tool relies on French didactical theories, especially Vergnaud’s Theory of Conceptual Fields and Balacheff’s cKȼ model. We are focusing particularly on the domain of discrete mathematics. In our presentation, we will illustrate this work with examples of problem-networks issued from the Hungarian and French context of mathematics education. We will show how the identification of problem-networks with this modelling tool can help developing a priori analysis of problem-collections, and how it contributes to understanding and conceiving the development of mathematical knowledge through problem-solving. At the end of the presentation, we will shortly discuss our first observations concerning teachers’ appropriation of the problem-network approach.
