Habermas’ construct of rationality to bring out mathematics and physics disciplinary identities

The study is conceived within the European Erasmus+ project IDENTITIES, whose goal is to promote interdisciplinarity in prospective teachers' education. As a preliminary step in order to design teacher education activities, we investigate the disciplinary key aspects of mathematics and physics reasoning in disciplinary instructional materials (namely, physics textbooks) as well as interdisciplinary issues

“TEACHING REAL NUMBERS IN THE HIGH SCHOOL: AN ONTO-SEMIOTIC APPROACH TO THE INVESTIGATION AND EVALUATION OF THE TEACHERS' DECLARED CHOICES”

The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intuItive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic.The thesis addresses the topics of investigating teachers' declared choices of practices concerning real numbers and the continuum in the high school in Italy, evaluating their didactical suitability and the impact of a deep reflexion about some historical and didactical issues on the teachers' decision-making process. Our research hypothesis was that teachers' choices of teaching sequences concerning real numbers, with particular attention to the representations of real numbers, could be very relevant in order to interpret some of the well-known students' difficulties. After a pilot study in form of a teaching experiment and a literature review concerning students' and teachers' difficulties with real numbers and the continuum, we observed that some causes of potential difficulties could be situated indeed in the very beginning of the teaching-learning process, even before entering the classrooms: the phase in which a teacher choose the practices and objects by means of whom introducing and work with real numbers and the continuum. In particular the choice of the objects involved in the practice seemed to be relevant, since every object emerge from previous practices and its meaning is identified by the practices in which it emerged. Thus we got interested in investigating the personal factors that affect the process of selection of practices: personal meaning, goals and orientations, as it was stressed by Schoenfeold in his goal-oriented decision-making approach to the analysis of teachers choices. Furthermore we decided to explore the teachers' choices of sequences of practices and of representation of the mathematical objects and then to evaluate their suitability in relation to the literature review concerning students' difficulties with real numbers and to the complexity of the mathematical object as it emerge from an historical analysis. After having analysed the theoretical frameworks in mathematics education that could permit us to carry out our research, we decided to use the OSA, (onto-semiotioc approach) elaborated by Godino, Batanero & Font, described in their paper in 2007, and some evolutions like the CDM (conoscimiento didactico matematico) model proposed by Godino in 2009. We evaluated also other frameworks, in particular the ATD (Chevallard, 2014), but we found the OSA better for the analysis we would like to carry out. In particular the operationalization of the methodologies of analysis of the teachers' personal meaning of mathematical objects and the construct of didactical suitability were more effective for our porpouses. Our main results are the following: mny teachers' personal meanings of real numbers are far from the epistemic one; many of the teachers who studied real numbers at a formal level at school and at the University and perceived them as difficult and unuseful try to avoid to deepen the issues concerning real numbers with their stundent, thinking they would not understand; in general the experiences as students affect the teachers' choices; the teachers usually refer to real numbers also when the meaning is partial and doesn't coincide with one of the most general epistemic meanings of real numbers; very few teachers are aware of the complexity of the real numbers and are as aware of it to be able to control the relations between their many facets; also the teachers with a PhD in Mathematics operate choices that we can evaluate as unsuitable standing on the literature review and our framework; the teacher consider very hard to work with discrete and dense sets and prefer the intuitive approach to continuous sets rather then deepen the relation between dense and continuous sets, different degrees of infinity and so on; some teachers reasoning during the interviews changed their mind, getting aware of the complexity and admitting that simplifying too much can constitute a further cause of difficulty; the teachers refer to the students difficulties to justify their choice of simplifying, but when they face some crucial issues, often they admit to consider them unuseful or too difficult; nevertheless no teachers declare that would renounce to introduce the field of real numbers, at least intuitively; the most of the teachers declare that nothing more is introduced about real numbers in the last years and that the partial meanings introduced in the first years are used to face the Calculus problems (intutive approach to the Calculus); all the teachers consider necessary to introduce R or adequate subsets of R as domains of the functions expressed analytically because of their continuous graphic

High school students’ epistemological approaches to computer simulations of complex systems

The science of complex systems can provide not only scientist, but also professionals, policy-makers and citizens, with thinking resources to interpret and understand most of the modern global challenges. In this field, the widespread use of computational simulations, that are neither theoretical instruments nor laboratory experiments, has been contributing to the widening of the scientific skill gap between experts and citizens. The pilot study we present in this contribution aims at investigating high school students’ approaches towards simulations of complex systems, by searching for the criteria they use to evaluate their explanatory power and the reliability of their results. Preliminary analysis of the paired interviews has shown that (1) rarely students are able to elaborate explanations of the simulated complex phenomena, and (2) their critical attitude and trust towards simulations are strongly affected by their epistemological background. We argue that these findings deserve to be furtherly investigated, to understand in more details the sources of students’ difficulties in recognizing the epistemological and methodological value of simulations for scientific research and practice

Developing future-scaffolding skills through science education

Can science teaching contribute to developing skills for managing uncertainty towards the future and projecting imagination forwards? If so, how? In this paper, we outline an approach to 'teach the future' through science education. In the first part, we describe a framework that has been constructed to orient the design of teaching modules comprised of future-oriented educational activities. Then, a teaching module on climate change is described. The module was tested in a class of upper secondary school in Italy (grade12) and the main results are reported. They concern a change in perception of the future, as revealed by students: from far and unimaginable, the future became conceivable as a set of possibilities, addressable through concrete actions and within their reach, in the sense that they became able to view themselves as agents of their own future. The results lead us to argue that the approach appears promising in developing 'futurescaffolding skills', skills that enable people to construct visions of the future that support possible ways of acting in the present with one's eye on the horizon

Continuity of real functions in high school: a teaching sequence based on limits and topology

International audienceIt is well known that students have difficulties with the concept of continuity, specifically on points of discontinuity, and concepts like limits and infinity. In Italian textbooks, the continuity of functions is usually defined using limits, while an intuitive characterization of continuous functions is proposed without providing the students with formal tools to use it, like “the graphs of continuous functions can be drawn without lifting the pencil out of the paper”. Limits are one of the most complex subjects to learn and are usually introduced in an algorithmic way, without a true comprehension of the subject. We argue that introducing the definition of continuous functions using limits is problematic and we designed and tested a teaching sequence to investigate the potentiality of including a topological approach in high school

Fernandez Blanco, T,; Gonzales Roel, V.; Capone, R.; Branchetti, L.; Gaio, A. Inprasitha, M., Changsri, N. & Boonsena, N. (Eds). SYMMETRY-ART: A STEAM TRAINING WORKSHOP FOR PRIMARY SCHOOL TEACHERS (2020). Interim Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education. Khon Kaen, Thailand: PME.

The aim of this paper is to analyse a symmetry and art workshop from a STEAM perspective. The theoretical framework of the Meta-Didactical Transposition is taken as a reference. The sample consists of seven Primary School teachers. A qualitative methodology is followed that is developed in four phases: learning, planning, implementation and reflection. The results show that the teachers are not flexible in dealing with the different conceptions of symmetry and the creative aspect of the workshop. In general, there is a positive attitude towards the interdisciplinary character of the workshop, despite the fact that they were not able to connect both disciplines in a balanced way

Recognition and operationalization of Future-Scaffolding Skills : Results from an empirical study of a teaching-learning module on climate change and futures thinking

This article takes its point of departure from the younger generation's problematic relationship with time and the future. A general sense of changeability and directionlessness in society compromises young people's confidence in themselves to make a difference as individuals in important global issues affecting their futures, such as climate change. Given recent aims and commitments of science education to promote sustainable development and student agency, this study explores how science teaching can help students imagine and face possible future scenarios and develop agency in the present to influence them. This article presents a science education approach to equip secondary school students with skills of futures thinking and agency that we call "future-scaffolding skills." It also shows the process of building an operational definition for recognizing those skills in students' discourse and actions. For this purpose, an empirical study was carried out in the context of a teaching-learning module on climate change, consisting of activities inspired by the field of futures studies. Essays, individual and group interviews, questionnaires, and video recordings of students' final projects were collected from 24 students (16-19-years old) from three European countries. The results contribute to operationally defining "future-scaffolding skills," consisting of "structural skills" (the ability to recognize temporal, logical and causal relationships and build systemic views) and "dynamical skills" (the ability to navigate scenarios, relating local details to global views, past to present and future, and individual to collective actions).Peer reviewe

The I SEE project : An approach to futurize STEM education

In the world where young people feel that the future is no longer a promise but a threat, and science and technology are sources of fears and global problems, a challenging task for education is to support students in imagining a future for the world and for themselves. The aim of the EU-funded project “I SEE” is to create an approach in science education that addresses the problems posed by global unsustainability, the uncertainty of the future, social liquidity and the irrelevance of STEM education for young people. This way, we believe, STEM education can support young people in projecting themselves into the future as agents and active persons, citizens and professionals, and open their minds to future possibilities. In this paper we propose a teaching and learning approach for futurizing science education, and describe how that approach was used to develop the first I SEE module implemented in summer school in June 2017 with students from three countries. In sum, the I SEE teaching and learning approach consists of three stages and learning outcomes connected to each of them: encountering the focal issue; engaging with the interaction between science ideas and future dimensions, and synthesizing the ideas and putting them into practice. The middle stage of the model is the main part, involving future-oriented practices that turn knowledge into future- scaffolding skills. We describe four kinds of such future-oriented practices: a) activities to flesh out the future-oriented structure of scientific discourse, language and concepts; b) activities inspired by futures studies or by the working life and societal matters; c) exposure activities to enlarge the imagination about possible future STEM careers; and d) action competence activities. We conclude the paper by reflecting on our experiences of the implementation of the climate change module with upper secondary school students.In the world where young people feel that the future is no longer a promise but a threat, and science and technology are sources of fears and global problems, a challenging task for education is to support students in imagining a future for the world and for themselves. The aim of the EU-funded project “I SEE” is to create an approach in science education that addresses the problems posed by global unsustainability, the uncertainty of the future, social liquidity and the irrelevance of STEM education for young people. This way, we believe, STEM education can support young people in projecting themselves into the future as agents and active persons, citizens and professionals, and open their minds to future possibilities. In this paper we propose a teaching and learning approach for futurizing science education, and describe how that approach was used to develop the first I SEE module implemented in summer school in June 2017 with students from three countries. In sum, the I SEE teaching and learning approach consists of three stages and learning outcomes connected to each of them: encountering the focal issue; engaging with the interaction between science ideas and future dimensions, and synthesizing the ideas and putting them into practice. The middle stage of the model is the main part, involving future-oriented practices that turn knowledge into future- scaffolding skills. We describe four kinds of such future-oriented practices: a) activities to flesh out the future-oriented structure of scientific discourse, language and concepts; b) activities inspired by futures studies or by the working life and societal matters; c) exposure activities to enlarge the imagination about possible future STEM careers; and d) action competence activities. We conclude the paper by reflecting on our experiences of the implementation of the climate change module with upper secondary school students.In the world where young people feel that the future is no longer a promise but a threat, and science and technology are sources of fears and global problems, a challenging task for education is to support students in imagining a future for the world and for themselves. The aim of the EU-funded project “I SEE” is to create an approach in science education that addresses the problems posed by global unsustainability, the uncertainty of the future, social liquidity and the irrelevance of STEM education for young people. This way, we believe, STEM education can support young people in projecting themselves into the future as agents and active persons, citizens and professionals, and open their minds to future possibilities. In this paper we propose a teaching and learning approach for futurizing science education, and describe how that approach was used to develop the first I SEE module implemented in summer school in June 2017 with students from three countries. In sum, the I SEE teaching and learning approach consists of three stages and learning outcomes connected to each of them: encountering the focal issue; engaging with the interaction between science ideas and future dimensions, and synthesizing the ideas and putting them into practice. The middle stage of the model is the main part, involving future-oriented practices that turn knowledge into future- scaffolding skills. We describe four kinds of such future-oriented practices: a) activities to flesh out the future-oriented structure of scientific discourse, language and concepts; b) activities inspired by futures studies or by the working life and societal matters; c) exposure activities to enlarge the imagination about possible future STEM careers; and d) action competence activities. We conclude the paper by reflecting on our experiences of the implementation of the climate change module with upper secondary school students.Peer reviewe

Science of Complex Systems and Citizenship Skills: A Pilot Study with Adult Citizens

The issue of scientific citizenship in the context of STEM education has been under debate for over two decades. We present a preliminary study which aims to investigate if, how and why the development of hard scientific skills grounded in the discipline of complex systems (suitably simplified and adapted) may foster the development of citizenship skills that can impact on people's approaches to facing problems and making decisions. We carried out a pilot study with a group of 34 volunteer adult citizens. The data analysis showed that: i) in the beginning, only a few participants were comfortable dealing with scientific and epistemological concepts related to complex systems, favouring instead a "common sense" approach towards decision-making; ii) in some successful cases, there was an alignment between scientific competences and decision-making strategies, suggesting that it is possible to design activities based on authentic scientific concepts in order to develop citizenship skills