Katalin Gosztonyi: Series of problems in the Hungarian „guided discovery” approach (MTA-ELTE project)

Series of problems play crucial role in Varga’s conception on mathematics education (the starting point of the MTA-ELTE research project), and more generally in the Hungarian „guided discovery” teaching tradition (existing only in a successful but very limited circle of teachers). Teachers following the guided discovery approach recognize the selection and the ordering of problems as a key activity in the planning of their teaching. Furthermore, numerous Hungarian teaching resources offer ordered series of problems, but commentaries, explications of the organizing principles are quite rare. In the frame of the MTA-ELTE Complex Mathematics Education project, we collect series of problems from Hungarian resources and practice, analyse their structure and their role in the teaching process. We also examine possibilities of diffusion, not only of the specific series, but also of the organizing principles, amongst teachers not familiar with the approach. I will show some examples of our analysis and present our current dilemmas concerning resource construction and diffusion.

Alain Bernard: The whys and hows of the historical research project on „series of problems”: the cases of Diophantus, Alcuin, and Clairaut.

Within the historical and anthropological research project “series of problems” developed in the HASTEC framework (see the participating projects for details) we paid attention, among other collections of problems or questions, to treatises of mathematical content. For this purpose, we developed various methodologies to analyse these texts, focused on their organization, intentionality (when present or provable) and purpose. Given the particularity of texts and the variety of their cultural contexts, there is no unique methodology for them: each production usually deserves its own approach – even though there are issues common to all. We shall illustrate this by briefly discussing three kinds of texts of highly different supports and periods: Diophantus’s Arithmetica, stemming from Greek antiquity, the “propositions to sharpen the young” attributed to Alcuin and belonging to the High Middle Ages, and Alexis Clairaut’s Eléments de Géométrie, an introduction to geometry first published in 1741 and a typical product of the French Enlightenment.

Marie-Line Gardes: DREAM : Research and resource to spread mathematics research problems in the classroom

DREAM is a research team which is composed by tearchers in secondary school, teacher’s trainers and researchers in math education. The aims of the group are :

• Elaborate and analyze situations of research problems
• Elaborate a numerical resource the aim of which is to give aid to mathematics teachers to use research problems in their teaching
• Create a progression of mathematics teaching based on research problems

In the communication, I will present the research group, its aims and methodology of work. Then I will show the numerical resource - the site http://dreamaths.univ-lyon1.fr/. Finally I will explain a progression of mathematics teaching based on research problems for 14-15 years students.

Dániel Katona: Web of problem threads in the Pósa method

The Pósa method for discovery learning mathematics has been being developed for 3 decades now, in out-of-school talent-care settings. In the frame of the MTA-Rényi Reseach Group on Discovery Learning in Mathematics, current experiments are are conducted for broadening the set of target students, e.g. the Flying School - study circles all around the country, workshops for indigent students and constructing the modification of the method to be applied in public education. During the problem-based sessions of the method, students solve and pose problems, forming a P set of problems. Within some P1, P1, ...Pn subsets of P, the problems are connected, sometimes in a multiple way, by some common factors, such a common heuristic method or shared content. We call these common features the kernels of the so called threads, which are subsets themselves. The subsets could also be series, though the order of the problems may be changed. Important is, that these hreads regularly cross each other, that is, they have common elements (problems), which creates P to be a web of problem set (WPT). What is role of this structure, the WPT in triggering the discovery process? Our research hopes to find some answers.

Erika Jakucs: Experimentation in the mathematics classroom

During this session, participants will be put into the situation of middle-school students and discover a lesson as I do it with my own students. I will show how I lead the construction of mathematical notions and intuitions by experimentation, laying some traps for the students' thinking and making some psychological digressions. At the end, we will discuss the role of series of problems in the construction of the experienced teaching process.

Klára Pintér: Different representations of word problems in primary school

Most of word problems in primary school are arithmetical problems where an operation gives the solution. The problems we deal with are more complex then arithmetical problems, they need problem solving strategies. We consider problems which can be solved by representing quantities with segments. We show series of problems and activity with cuisenaire rods to introduce this representation.

Stéphane Herrero, Laure Theoden, Alain Bernard: Meeting Diophantus' problems at highschool level, between French and Mathematics classes.

The workshop should enable the participants to discover, and eventually participate in, the conception of a booklet on the possible dialogue between Mathematics, French and Classical culture around key humanists figures having some relation to science and mathematics, and belonging to classical literature. These are for example Diophantus and Hero of Alexandria, Pythagoras, Thales, Vitruvius. The project takes its source in an interdisciplinary teaching project developed in 2016-17 in a French highschool by two of us (Stéphane, Laure), the project being partly inspired by a teacher training session organized in 2014 on series of problems in Diophantus (by Alain). The booklet in preparation is conceived as a resource for teachers, by letting them discover both the spirit and genesis of this pedagogical project, as well as some of the historical texts and interpretations that were exploited for it. The basic aim is to trigger reflections and similar projects. We take as starting point a document created for the pupils, enabling them to discover and study excerpts of Diophantus’ Arithmetica.

Emmanuelle Rocher, Alain Bernard: Mathematical and pedagogical variations on Alcuin of York's "Problems to Sharpen the Young".

The workshop should enable the participants to discover, and eventually participate in, the conception of a booklet around the socalled “Problems to sharpen the Young”. These are a collection of some fifty mathematical riddles stemming from the Early Middle Ages (8^th-9^th cent. CE?) and usually attributed to the famous clergyman and scholar Alcuin of York. One aim of the booklet, as well as of the workshop, is to get the readers (and participants) acquainted with some typical problems belonging to this collection,. Some of them are to the present day very well known –even when their origin is forgotten- because they still circulate in textbooks or children's literature. The other purpose is to study and trigger variations on theses problems: mathematical variations (what kinds of methods could we think of, to solve these enigmatas in a modern way?), pedagogical variations (what pedagogical use could we imagine for them?).

Eszter Varga: Distance under the magnifying glass. A possible utilization of Series of Problems as a design tool for teachers’ long-term planning

On this workshop, we will analyze in detail a subset of a certain Series of Problems, identifying the underlying concepts and possible connections while building the Problem Graph. Following the presumed interest of the participants, we will focus on the preparatory course (middle school), but we will also make an attempt to explore some intriguing ramifications issued from my own high-school practice.

Our Series of Problems is organized around a core problem about the distance of sets of points, and concerns some key concepts within the field of (non-analytic) geometry, but also has some combinatoric vibe in it.