Download the module’s lesson plan (pdf, editable format)
The module concerns the curricular theme of modelling, which is transversal to mathematics, physics and other sciences, with specificities in the different disciplines but also deeply interdisciplinary in the case of mathematical modelling in other disciplines and in real-life problems. The themes have been chosen since modelling appears in secondary school curricula in all countries, but its interpretation can be affected by habits of the disciplinary community that can vary a lot among different disciplines. In particular, it can be flattened or trivialized when dealing with mathematical modelling in physics, or it can be turned into an object (“the model”) rather than considering the process of modelling. The module has been designed to emphasize the contribution of different disciplines to an authentic modelling process suitable for secondary school students and exploiting only disciplinary contents and methods that can be considered curricular in all countries.
The contamination between different methods and issues is alternated to the disciplinary analysis of the questions posed of their relevance to the discipline, in order to make the epistemological role of models and modelling within different disciplines visible and create a space where the student-teachers were encouraged to make deep interdisciplinary experiences.
The IDENTITIES approach for the design and implementation of the module is mainly evident in the choice of the following design principles:
- Keep disciplinary technicalities as much simpler as possible so as to emphasize the interdisciplinary potential of the topic that emerges by modelling real problems and disciplinary problems and focus on the issues emerging in the process of modelling
- Unpack the epistemic core of disciplinary knowledge through the Family Resemblance Approach (Erduran & Dagher, 2014)
- Establish the boundary zone by introducing boundary objects (“model” and “approximation”), and activating a variety of boundary-crossing mechanisms (Akkerman & Bakker, 2011)
- Create a safe learning environment, where student-teachers feel free to play with their ideas and are supported to recognize the influence of disciplines in the formulation of questions to investigate a phenomenon. In particular, they will move from a paradigm of “modelling the real world”, typical of some approaches to mathematical modelling in education, to the paradigm of modelling as the elaboration of data collected and organized according to goals. Such goals could be related to issues posed by different agents (companies, societal agencies, research institutes, ..) or to the curiosity of the students themselves while exploring a situation.
Aims: The first submodule aims to involve students in a mathematical modelling experience and to make them deal with two directions of mathematical modelling: vertical and horizontal modelization (Yvain-Prebiski and Modeste, 2019). The situations concern life sciences and industrial sciences. We use the results of this contemporary epistemological study to analyze and support the relevance of some problems designed to foster the devolution of the mathematical modelling process to the students. Moreover, in this submodule, students are asked to reflect on the novelties, choices and difficulties that emerged when dealing with a problem formulated by members of other fields.
Activities: First, an a-didactical situation is proposed to students concerning life science and industry; they are involved into a milieu where their exploration, guided by a question and supported by suitable media, should make emerge modelling strategies that the trainer could institutionalize as vertical and horizontal modelization in the “Becoming students” phase. We will use the strategy of “realistic fiction” in order to reduce the complexity of horizontal mathematization. The second activity guides the students to get acquainted with the vocabulary of Akkerman and Bakker by making explicit the assumptions that make this approach to modelling a mathematical one (identification) and what new aspects of mathematics they met in this process. Moreover, they are asked to reflect on eventual difficulties they have met during the activity due to the fact that the problem was formulated within other fields.
Introduction to the module
Ice-breaking and boundary activities (pdf, editable format)
Introduction to the IDENTITIES project: disciplines, interdisciplinarity, and key questions (pdf, editable format)
Curricular and S-T-E-M advanced interdisciplinarity (video)
Images of interdisciplinarity (pdf, editable format)
Taxonomy of interdisciplinarity (video)
Modelling the growth of trees
Preliminary ideas about modelling in mathematics and physics (pdf, editable format)
A-didactical situation about the growth of trees (pdf, editable format)
Aims: The second submodule aims to introduce the main epistemological features of mathematical modelling and modelling in sciences. After making explicit the notions of vertical and horizontal mathematization and their intertwining, students are introduced to the epistemological role of ‘models and the game of modelling’ in science and particularly when dealing with the issue of radiation and absorbance in the greenhouse effect. The purpose of the submodule is to discuss with the students the epistemological dimension of modelling in different disciplines, and to show that modelling is a boundary object that can trigger interesting identification and reflection learning processes at the boundary between mathematics and sciences at secondary school.
Activities: At the beginning, through interactive lectures, the student-teachers, in the role of “students”, are guided to recognize what are the invariant practices of horizontal mathematization they carried out in the explorer phase, and how they interplay with context-specific practices. In the second activity, the student-teachers are presented with a lecture about epistemological issues that can represent a demanding challenge for students, and more generally for citizens, related to climate science. While usually the attention is posed to the difficulties due to the use of non-linear models whose features are very different from the classic and mechanistic view of modelling (Pasini, 2003), puzzling epistemological issues also arise when very basic models are used in a curricular introduction to the greenhouse effect, which could easily leave students sceptical and detached (Tasquier, 2015a; Tasquier et al., 2014). After being presented with two different disciplinary epistemological aspects of modelling, students are asked to deal with mathematization in the organization of empirical data concerning the greenhouse effect into a mathematical model and come back to the scientific issues.
Simulation of a teaching situation and solution of the re-launched realistic fiction (pdf, editable format)
Vertical and horizontal mathematization and their relationships (pdf)
Reflection about implementations in secondary school (pdf)
Production of a home report
Modelling in science: the case of the greenhouse effect
Modelling in science: questionnaires about the Cartographer and Palomar and about their own conceptions about modelling in science (pdf, editable format)
The interaction between matter and radiation: heat and temperature (pdf, editable format)
Experiments and empirical data about the interaction between matter and radiation (pdf, editable format)
Aims: The third submodule concerns the analysis of the mathematization in scientific modelling and interdisciplinary issues arising when different epistemologies of modelling are combined.
Activities: To the participants, epistemological and linguistic grids are provided as a summary or enrichment of the second submodule. Through this activity, they are guided to apply the disciplinary and interdisciplinary knowledge and tools they have learned as lenses to break down stereotypes about modelling procedures in sciences and mathematics, highlighting their similarities, differences and interplays.
Modelling and interpretation of empirical data related to the interaction between matter and radiation (pdf, editable format)
Modelling in science education: data from students’ answers (pdf)
Boundary objects and boundary crossing
Introducing boundary objects and boundary-crossing mechanisms (video)
Discussing experiences as boundary people
Identification of mathematical and physical processes of modelling
Interdisciplinary analysis of modelling processes between mathematics and physics: the case of the interaction between matter and radiation
Analysis of coordination in interdisciplinary modelling: boundary crossing mechanisms in the activity about modelling of the interaction matter-radiation and horizontal mathematization in the activity about modelling of the interaction matter-radiation (pdf, editable format)
The contribution of modelling to the development of interdisciplinarity at school: final discussion
Download the module’s lesson plan (pdf, editable format)