HAL Id: hal-02423544
https://hal.archives-ouvertes.fr/hal-02423544
Submitted on 24 Dec 2019
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
The potential of Problem Graphs as a representational tool with focus on the Hungarian Mathematics
Education tradition
Eszter Varga
To cite this version:
Eszter Varga. The potential of Problem Graphs as a representational tool with focus on the Hun- garian Mathematics Education tradition. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. �hal-02423544�
The potential of Problem Graphs as a representational tool with focus on the Hungarian Mathematics Education tradition
Eszter Varga
Eötvös Loránd University, Faculty of Science, Doctoral School of Mathematics, Budapest, Hungary; evarga@bpg.hu; vargaesztermail@gmail.com
Keywords: Series of Problems, Hungarian mathematics education, teacher design capacity
Introduction
Capturing teachers’ principles and working methods in instructional design can pose some challenges, as well as its representation and dissemination. In this work in progress, we suggest a new representation tool, the Problem Graph, that has the potential to reveal some of the Teacher Knowledge embedded in series of problems and task sequences. Problem Graphs are developed with dual purpose: as an analytical tool to explore the structure of existing resources through a priori analysis, and as a design tool that can help teachers plan their instructional sequences and long term learning trajectories. The potential of Problem Graphs in developing teachers’ design capacity were also investigated in two pilot experiments with the objective of introducing them in teacher education and PD programs.
Initially, this work is strongly rooted in the Hungarian Mathematics Education tradition, but both the approach in question and the representation tool is relevant to the international community for its special handling of Mathematical Knowledge for Teaching (Ball et al., 2008).
Background
My present and prospective work with Problem Graphs is a part of a large scale research project launched by the Hungarian Academy of Sciences.1 The project focuses on the Hungarian “Guided Discovery” approach, that was experimentally implemented nationwide by a reform movement led by Tamás Varga in the 1960s and 70s (see Gosztonyi et al., available in pre-Proceedings of ICMI 24; Varga, 1988). This approach is highly recognized among Hungarian experts. The research project is assigned to describe the approach and examine its background as reflected in the recent theoretical frameworks as well as revisiting the reform and communicating it to both national and international audiences.
In a preliminary study, Gosztonyi identified Series of Problems as a characteristic aspect of teachers’ work in the Guided Discovery approach. The research group started to analyze Series of Problems as a special type of resource, using the frames of the Documentational approach (Gosztonyi, 2018; Gueudet, Pepin, & Trouche, 2012). In this course of work, a need emerged for an efficient visual representation of the inner structure of Series of Problems, to which the graph
1The poster is made with the financial support of the MTA-ELTE Complex Mathematics Education Research Group, working in the frame of the Content Pedagogy Research Program of the Hungarian Academy of Sciences (ID number:
471028).
representation came up quite naturally as one of the possible answers. Problem Graphs were built to some of the examined Series of Problems and discussed in details with experts. In these structures, nodes represent the individual problems, while edges show the explored connections. The connections reflects Mathematical Knowledge for Teaching, one-way arrows show precedent- consequence relations. For the standards of further classification, the domains suggested by Ball et al. might be a good starting point, but this work is only at its first steps at the moment.
The poster will demonstrate that the name Series of Problems covers a notion that is more complex than a task sequence, and seems to be indigenous in the Hungarian tradition. Nevertheless, the graph representation can be an effective tool to support the analysis of other learning sequences independent from this cultural context.
Dealing with problem Graphs led to the idea that this type of work on set of problems could benefit teachers’ professional development greatly, especially if conducted in groupwork. Planning on investigating this opportunity, we conducted two pilot experiments with 10 and 18 participating teachers. In both experiments, teachers worked in groups on the same subset of a Problem Field which was examined previously by the research group. During the sessions, the teachers built a Problem Graph from the selected problems, followed by a whole group discussion and a questionnaire in the second experiment.
Poster Content
The poster will summarize some major principles of the Guided Discovery approach as a context. It will also give a working definition for Series of Problems. Limitations in space will not allow to provide full examples for Series of Problems, but some key problems will be highlighted. In the main part, Problems Graphs will be displayed, with one extensive example representing a Series of Problems from the field of Elementary Geometry in the center, with some specification of the connections. Graphs built by teachers from the same problems will also be presented, accompanied with some teacher responses from the questionnaires. This will allow us to draw some conclusion from the pilot experiment regarding the potential usage of the Problem Graph.
References
Ball, D.L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.
Gosztonyi, K. (2018). Teachers documentary work with series of problems. In V. Gitirana, T.
Miyakawa, M. Rafalska, S. Soury-Lavergne, & L. Trouche (Eds.). Proceedings of the Re(s)sources 2018 international conference (pp 195-198). ENS de Lyon
Gosztonyi, K., Vancsó, Ö., Pintér, K., Kosztolányi, J., Varga, E. (to be published). Varga’s
“Complex Mathematics Education” Reform: at the crossroad of the New Math and Hungarian Mathematical Traditions. Accepted to the ICMI Study 24: Curricular Reforms conference.
Gueudet, G., Pepin, B., & Trouche, L. (Eds.). (2012). From text to ‘lived’ resources: Mathematics curriculum materials and teacher development. Springer Netherlands.
Varga, T. (1988) Mathematics education in Hungary today. Educational Studies in Mathematics, 19, 291. https://doi.org/10.1007/BF00312449
