Periodic Reporting for period 1 - ProDiME (Problem-based learning trajectories in discrete mathematics education)
Reporting period: 2022-09-01 to 2024-08-31
Problem Solving and Inquiry Based Mathematics Education are in the focus of numerous mathematics education research projects, educational policies and actions in various countries, since several decades. Nevertheless, the efficient integration of problem-solving activities with the learning of mathematical content in ordinary classrooms is not evident.
We proposed to confront these difficulties by developing a modelling tool in terms of problem-networks, combining our respective preliminary research on teaching strategies in the Hungarian Guided Discovery approach to mathematics education on one hand, and the modelling of problem-solving processes based on French theoretical frameworks on the other. The problem-networks modelling tool has two objectives: 1, serving for the analysis of problem-based teaching trajectories and of links between problems; and 2, supporting teachers’ planning work for developing problem-based teaching processes.
Discrete mathematics is a particularly relevant domain for the aforementioned questions. It is often considered as well adapted for developing problem solving skills, since problems of various difficulties can be posed without requiring complex preliminary theoretical knowledge from students. Recent research has also shown the various interests of teaching and learning discrete mathematics, including the potential role of the domain in the learning of mathematical abstraction and proof, and the various connections to other mathematical domains. Furthermore, the need for teaching discrete mathematics emerges recently on the behalf of computer science. Nevertheless, discrete mathematics is often marginal in curricula.
In the ProDiME project, we focused on the construction and analysis of problem-networks in discrete mathematics. Relying on our respective preliminary research on the didactics of discrete mathematics, and the analysis of Hungarian curricula which traditionally accords important place to discrete mathematics, we aimed to investigate the possible place of problem-based discrete mathematical activities in other curricula.
Although the understanding and the support of students’ learning processes remains an ultimate purpose of mathematics education research, the role of teachers in this process gains increasing attention in didactical research. Today it is clear that the implementation, dissemination of innovative teaching approaches is not possible in the form of a simple transmission process from researchers towards teachers: teachers have to be seen as crucial agents in innovation processes. This implies that collaboration between teachers and researchers plays an increasing role in mathematics education research as well as in teachers’ professional development and in the implementation of innovative projects. The ProDiME project implied collaboration with teachers for several purposes: furnishing a tool for teachers to support their work in planning their teaching processes; adapting specific problem-networks for the teachers’ own teaching contexts; analysing the teachers’ choices during the preparation and the implementation of their teaching trajectories and the impact of these choices on students’ work; developing innovative resources helping other teachers’ professional development.
We proposed to confront these difficulties by developing a modelling tool in terms of problem-networks, combining our respective preliminary research on teaching strategies in the Hungarian Guided Discovery approach to mathematics education on one hand, and the modelling of problem-solving processes based on French theoretical frameworks on the other. The problem-networks modelling tool has two objectives: 1, serving for the analysis of problem-based teaching trajectories and of links between problems; and 2, supporting teachers’ planning work for developing problem-based teaching processes.
Discrete mathematics is a particularly relevant domain for the aforementioned questions. It is often considered as well adapted for developing problem solving skills, since problems of various difficulties can be posed without requiring complex preliminary theoretical knowledge from students. Recent research has also shown the various interests of teaching and learning discrete mathematics, including the potential role of the domain in the learning of mathematical abstraction and proof, and the various connections to other mathematical domains. Furthermore, the need for teaching discrete mathematics emerges recently on the behalf of computer science. Nevertheless, discrete mathematics is often marginal in curricula.
In the ProDiME project, we focused on the construction and analysis of problem-networks in discrete mathematics. Relying on our respective preliminary research on the didactics of discrete mathematics, and the analysis of Hungarian curricula which traditionally accords important place to discrete mathematics, we aimed to investigate the possible place of problem-based discrete mathematical activities in other curricula.
Although the understanding and the support of students’ learning processes remains an ultimate purpose of mathematics education research, the role of teachers in this process gains increasing attention in didactical research. Today it is clear that the implementation, dissemination of innovative teaching approaches is not possible in the form of a simple transmission process from researchers towards teachers: teachers have to be seen as crucial agents in innovation processes. This implies that collaboration between teachers and researchers plays an increasing role in mathematics education research as well as in teachers’ professional development and in the implementation of innovative projects. The ProDiME project implied collaboration with teachers for several purposes: furnishing a tool for teachers to support their work in planning their teaching processes; adapting specific problem-networks for the teachers’ own teaching contexts; analysing the teachers’ choices during the preparation and the implementation of their teaching trajectories and the impact of these choices on students’ work; developing innovative resources helping other teachers’ professional development.
The theoretical part of the project implied the development of a modelling tool in terms of problem networks. Based on French theoretical frameworks of mathematics education, we developed representation and analysis methods of problem networks, including the characterisation of different types of links between problems as well as the detailed analysis of different problem-solving strategies of specific problems and the possible contribution of these different strategies to building links between problems. Our analyses can highlight logical and heuristic types of links as well as links related to specific mathematical content knowledge. Therefore, they can contribute to a better understanding of possible connections between problem-solving activities and curricular learning in mathematics education.
We analysed examples of discrete mathematical problem networks from the Hungarian and the French context. These examples were then used in collaboration with teachers. Ten teachers were implied in different phases of the project: a team in Montpellier (France), a team in Paris (France), and two teachers from Hungary. Six French teachers carried out experimentations in eleven classrooms, covering lower and upper secondary schools and professional high schools in France.
We focused particularly on French high schools’ final year, as a new combinatorics chapter has been introduced in this year’s curriculum shortly before the beginning of the project. In collaboration with the participating French teachers, we adapted a problem series issued from the Hungarian context (from the work of a participating Hungarian teacher) for high school combinatorics. This work helped the participating teachers to implement the new chapter. A resource based on this work is under development and will be available for French teachers online.
Furthermore, we organised two workshops. The first was a meeting between mixed teacher-researcher teams (called IREM groups) from different French cities, each of them working in connection with problem solving in mathematics education but adopting different approaches. The meeting allowed the comparison of these various approaches. The second workshop implied researchers from Europe and the US working on the didactics of discrete mathematics and served as a first step towards the establishment of a research network on discrete mathematics education.
We analysed examples of discrete mathematical problem networks from the Hungarian and the French context. These examples were then used in collaboration with teachers. Ten teachers were implied in different phases of the project: a team in Montpellier (France), a team in Paris (France), and two teachers from Hungary. Six French teachers carried out experimentations in eleven classrooms, covering lower and upper secondary schools and professional high schools in France.
We focused particularly on French high schools’ final year, as a new combinatorics chapter has been introduced in this year’s curriculum shortly before the beginning of the project. In collaboration with the participating French teachers, we adapted a problem series issued from the Hungarian context (from the work of a participating Hungarian teacher) for high school combinatorics. This work helped the participating teachers to implement the new chapter. A resource based on this work is under development and will be available for French teachers online.
Furthermore, we organised two workshops. The first was a meeting between mixed teacher-researcher teams (called IREM groups) from different French cities, each of them working in connection with problem solving in mathematics education but adopting different approaches. The meeting allowed the comparison of these various approaches. The second workshop implied researchers from Europe and the US working on the didactics of discrete mathematics and served as a first step towards the establishment of a research network on discrete mathematics education.
According to the first analyses carried out, our problem-network modelling tool allows detailed analysis of problem-based teaching processes and can support the conception of such processes. Nevertheless, the analysis of further examples is necessary, also beyond the domain of discrete mathematics, and will certainly lead to further development of the modelling tool.
The collaboration with teachers and preliminary analysis of this work shows that an approach based on problem-networks can efficiently support teachers’ planning work. A more detailed analysis of the collected data will be necessary to better understand teachers’ professional development, difficulties and needs related to this approach.
Concerning discrete mathematics, we could especially point out the importance of the use of different representations in the development students’ enumeration strategies in combinatorics, the difficulties related to the control of these strategies, and more generally to proving activities in discrete mathematics. The problem-networks studied in the project offered rich opportunities to work on these aspects in class. At the same time, it become clear that beyond the problem-networks themselves, teachers might need further work and support to treat these aspects efficiently in class.
The adaptation of a combinatorial problem series from the Hungarian to the French context showed high potentialities but highlighted also some difficulties of the implementation of the teaching progression in a new context. Several solutions were elaborated by the participating teachers to overcome these difficulties.
The collaboration with teachers and preliminary analysis of this work shows that an approach based on problem-networks can efficiently support teachers’ planning work. A more detailed analysis of the collected data will be necessary to better understand teachers’ professional development, difficulties and needs related to this approach.
Concerning discrete mathematics, we could especially point out the importance of the use of different representations in the development students’ enumeration strategies in combinatorics, the difficulties related to the control of these strategies, and more generally to proving activities in discrete mathematics. The problem-networks studied in the project offered rich opportunities to work on these aspects in class. At the same time, it become clear that beyond the problem-networks themselves, teachers might need further work and support to treat these aspects efficiently in class.
The adaptation of a combinatorial problem series from the Hungarian to the French context showed high potentialities but highlighted also some difficulties of the implementation of the teaching progression in a new context. Several solutions were elaborated by the participating teachers to overcome these difficulties.