In the current digital age, (mathematics) teachers are provided with a profusion of digital resou... more In the current digital age, (mathematics) teachers are provided with a profusion of digital resources. Consequently, research in mathematics education has focused on teachers' interactions with resources as the heart of their professional activity. The term Pedagogical Design Capacity (PDC) was introduced to designate teacher's ability to perceive affordances of resources and to make decisions about how to use them efficiently. PDC has been further conceptualized to define teacher design capacity (TDC) as including a goal or points of reference for the design, a set of design principles, and reflection-in-action. This definition is illustrated with case studies involving teachers' design of digital resources, yet it does not seem to consider the specificities of the latter. We therefore aim to further refine the concept of TDC by considering crucial components that designing technology-based tasks and their enactment in the classroom entail. We ask: What are the unique components of TDC specific to digital technology and resources? We elaborate a framework to suggest dimensions and components of mathematics teacher Digital Resource Design Capacity (DRDC) to emphasize the focus on design of digital resources. This framework is built on previous studies and on the concept of instrumental orchestration, and is refined taking into account findings from two case studies in France and Israel.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 19, 2022
My contribution concerns the ways in which teachers' practices in supporting formative assessment... more My contribution concerns the ways in which teachers' practices in supporting formative assessment (FA) processes through digital technologies (DT) can be interpreted and analysed. After a reflection on the results of research studies on this issue, I present a model recently refined to characterize, at a macro level, the teachers' FA practices through DT and then the analysis of an example, developed at a micro level to highlight the roles that the teacher plays when interacting with students. I conclude with some reflections on the impact that the experience of distance teaching during the Covid-19 emergency could have on the future evolution of teachers' assessment practices through DT.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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.
Patterning activities, specifically those related to repeating patterns, may encourage young chil... more Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repeating patterns but had difficulties defining repeating patterns. In general, teachers had a high self-efficacy for all tasks. However, teachers’ had a significantly lower self-efficacy for defining repeating patterns than for drawing and continuing repeating patterns.
The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the... more The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the availability of different digital resources speci cally designed to help teaching and learning mathematics. We refer to platforms that were designed to allow for mathematical communication between multiple users. We seek to explore the affordances of such digital platforms to support mathematics teachers who wish to integrate technology as part of their practice. Speci cally, we ask: What are the affordances of the platforms that may support instrumentation and instrumentalization processes leading to the development of teacher's didactic instrument to plan and enact a mathematical activity in a digital environment? The four platforms we chose for analysis are STEP, DESMOS, Labomep and WIMS. Our analysis shows on the one hand that the platforms afford support to the teacher while enacting technology-based mathematics activities. On the other hand, we suggest several components of didactic instrumental genesis that mathematics teachers need to develop in order to take bene t from digital platform affordances. These components include, in particular, the ability to base decision making on data gathered and visualized in dashboard embedded in learning management systems.
Investigations in mathematics learning, Mar 1, 2012
The central role that tasks play in student learning has become clearer in the last decade. As a ... more The central role that tasks play in student learning has become clearer in the last decade. As a result of calls for a structured design process, task design has become one of the main fields of investigation in mathematics education. Here, we propose five design considerations (deciding on the mathematical content, choosing the context, deciding on the level of structural openness, choosing representations, and arranging a task sequence within an activity). We show how these considerations influenced the design of an activity, and compare students' actual work while solving this particular activity and the designers' considerations.
The 'Documentation Approach to Didactics' is an entry of the Encyclopedia of Mathematics Educatio... more The 'Documentation Approach to Didactics' is an entry of the Encyclopedia of Mathematics Education (Trouche, Gueudet & Pepin 2018). This entry has been updated in 2020 (Trouche, Gueudet & Pepin 2020). This article is a Portuguese adaptation of this updated version. It is part of a collection, gathering such adaptations in 14 languages (https://hal.archives-ouvertes.fr/DAD-MULTILINGUAL). The documentational approach to didactics is a theory in mathematics education. Its first aim is to understand teachers' professional development by studying their interactions with the resources they use and design in/for their teaching. In this text we briefly describe the emergence of the approach, its theoretical sources, its main concepts and the associated methodology. We illustrate these aspects with examples from different research projects. This synthetic presentation is written for researchers, but also for non-specialists (e.g. master students) interested in a first discovery of the documentational approach
HAL (Le Centre pour la Communication Scientifique Directe), Feb 4, 2015
Not all mathematical problems have a solution. This paper describes a decomposition problem, set ... more Not all mathematical problems have a solution. This paper describes a decomposition problem, set in a real-life context, for which no mathematical solution exists. It describes the strategies children use and how the realife context impacted on children's solutions. Results indicated that young children accept the possibility that a problem may not have a solution and that some turn to the context in order to find a practical solution instead.
Using examples and non-examples is a common technique to demonstrate concepts' characteristics an... more Using examples and non-examples is a common technique to demonstrate concepts' characteristics and boundaries. Based on their properties, certain objects are accepted as examples or non-examples intuitively, while others are accepted or neglected non-intuitively. This 2*2 classification is powerful when designing technology-enhanced learning experiences in which feedback could be provided in real-time. That is, feedback could be based not only on the correctness of student response, but also on the specifics of the objects with which they were engaged. Following this framework, we developed an interactive sorting task that aims at strengthening elementary school students' understanding of reflective symmetry. We studied learners' interaction with the objects presented to them, and their success. Our study included 29 elementary school students (ages 9 to 12) from both Israel and Germany. We used screen recording to code participants' shape-movements, and defined quantitative measures of these movements. Our findings support the need for designing feedback that takes into consideration object's properties and students' behavior.
The aim of this study is to investigate the geometrical knowledge as well as the geometrical self... more The aim of this study is to investigate the geometrical knowledge as well as the geometrical self-efficacy of kindergarten children, including abused and neglected kindergarten children. Individual interviews were conducted with 141 kindergarten children, ages 5–6 years old, of which 69 children were labeled as abused and neglected by the social welfare department of their municipality. Results indicated that both groups of kindergarten children had high self-efficacy beliefs related to identifying geometrical figures which were not significantly related to knowledge. In addition, significant differences in knowledge were found between the two groups.
Patterning activities, specifically those related to repeating patterns, may encourage young chil... more Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repeating patterns but had difficulties defining repeating patterns. In general, teachers had a high self-efficacy for all tasks. However, teachers’ had a significantly lower self-efficacy for defining repeating patterns than for drawing and continuing repeating patterns.
Qualitative research on classroom-based mathematics learning in inquiry-oriented classrooms is sc... more Qualitative research on classroom-based mathematics learning in inquiry-oriented classrooms is scarce. This paper presents a methodology aimed at developing a rich understanding of the interplay of mathematical progress in the different settings in which learning in such classrooms occurs - individuals, small groups, and the whole class. For this purpose, we enhance a theoretical-methodological approach of coordinating Documenting Collective Activity and the RBC-model of Abstraction in Context that has been developed in earlier studies. We do this using an intact lesson on the area and perimeter of the Sierpiński triangle in a mathematics education master’s level course on Chaos and Fractals. The enhancement of the methodology allowed integrating the Collective and Individual Mathematical Progress (CIMP) by Layering the Explanations (LE) provided by the two approaches and thus exhibiting the complexity of learning processes in inquiry-oriented classrooms.
This paper describes kindergarten children’s engagement with two patterning activities. The first... more This paper describes kindergarten children’s engagement with two patterning activities. The first activity includes two tasks in which children are asked to choose possible ways for extending two different repeating patterns and the second activity calls for comparing different pairs of repeating patterns. Children’s recognition of the unit of repeat and their recognition of the structure of the repeating patterns are investigated. Findings suggest differences between children’s responses to patterns that end with a complete unit of repeat and those that end with a partial unit. In addition, the issue of presenting repeating patterns using different media is discussed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
International Journal of Science and Mathematics Education, Apr 5, 2019
We focus on teachers’ ways of leading whole class discussions (WCDs) in mathematics, with the goa... more We focus on teachers’ ways of leading whole class discussions (WCDs) in mathematics, with the goal of uncovering their traces (if any) in their students’ responses (a) while participating in the WCDs and (b) in the written responses in a final test. For this purpose, two 8th-grade probability classes learning a 10-lesson unit with different teachers were observed. Our data sources include (1) video-recordings of the WCDs and (2) the responses of students to final test items. We analyzed the teachers’ talk-moves, students’ accountable participation, and students’ reasoning in the final test items. Interweaving the findings from all analyses we found differences between the classes in students’ ways of participation in WCDs and in their corresponding final test responses. The teachers’ ways of leading the WCDs contribute to the explanation of these differences.
In the current digital age, (mathematics) teachers are provided with a profusion of digital resou... more In the current digital age, (mathematics) teachers are provided with a profusion of digital resources. Consequently, research in mathematics education has focused on teachers' interactions with resources as the heart of their professional activity. The term Pedagogical Design Capacity (PDC) was introduced to designate teacher's ability to perceive affordances of resources and to make decisions about how to use them efficiently. PDC has been further conceptualized to define teacher design capacity (TDC) as including a goal or points of reference for the design, a set of design principles, and reflection-in-action. This definition is illustrated with case studies involving teachers' design of digital resources, yet it does not seem to consider the specificities of the latter. We therefore aim to further refine the concept of TDC by considering crucial components that designing technology-based tasks and their enactment in the classroom entail. We ask: What are the unique components of TDC specific to digital technology and resources? We elaborate a framework to suggest dimensions and components of mathematics teacher Digital Resource Design Capacity (DRDC) to emphasize the focus on design of digital resources. This framework is built on previous studies and on the concept of instrumental orchestration, and is refined taking into account findings from two case studies in France and Israel.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 19, 2022
My contribution concerns the ways in which teachers' practices in supporting formative assessment... more My contribution concerns the ways in which teachers' practices in supporting formative assessment (FA) processes through digital technologies (DT) can be interpreted and analysed. After a reflection on the results of research studies on this issue, I present a model recently refined to characterize, at a macro level, the teachers' FA practices through DT and then the analysis of an example, developed at a micro level to highlight the roles that the teacher plays when interacting with students. I conclude with some reflections on the impact that the experience of distance teaching during the Covid-19 emergency could have on the future evolution of teachers' assessment practices through DT.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 1, 2017
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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.
Patterning activities, specifically those related to repeating patterns, may encourage young chil... more Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repeating patterns but had difficulties defining repeating patterns. In general, teachers had a high self-efficacy for all tasks. However, teachers’ had a significantly lower self-efficacy for defining repeating patterns than for drawing and continuing repeating patterns.
The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the... more The slow uptake of technology by mathematics teachers is in contrast with the rapid growth in the availability of different digital resources speci cally designed to help teaching and learning mathematics. We refer to platforms that were designed to allow for mathematical communication between multiple users. We seek to explore the affordances of such digital platforms to support mathematics teachers who wish to integrate technology as part of their practice. Speci cally, we ask: What are the affordances of the platforms that may support instrumentation and instrumentalization processes leading to the development of teacher's didactic instrument to plan and enact a mathematical activity in a digital environment? The four platforms we chose for analysis are STEP, DESMOS, Labomep and WIMS. Our analysis shows on the one hand that the platforms afford support to the teacher while enacting technology-based mathematics activities. On the other hand, we suggest several components of didactic instrumental genesis that mathematics teachers need to develop in order to take bene t from digital platform affordances. These components include, in particular, the ability to base decision making on data gathered and visualized in dashboard embedded in learning management systems.
Investigations in mathematics learning, Mar 1, 2012
The central role that tasks play in student learning has become clearer in the last decade. As a ... more The central role that tasks play in student learning has become clearer in the last decade. As a result of calls for a structured design process, task design has become one of the main fields of investigation in mathematics education. Here, we propose five design considerations (deciding on the mathematical content, choosing the context, deciding on the level of structural openness, choosing representations, and arranging a task sequence within an activity). We show how these considerations influenced the design of an activity, and compare students' actual work while solving this particular activity and the designers' considerations.
The 'Documentation Approach to Didactics' is an entry of the Encyclopedia of Mathematics Educatio... more The 'Documentation Approach to Didactics' is an entry of the Encyclopedia of Mathematics Education (Trouche, Gueudet & Pepin 2018). This entry has been updated in 2020 (Trouche, Gueudet & Pepin 2020). This article is a Portuguese adaptation of this updated version. It is part of a collection, gathering such adaptations in 14 languages (https://hal.archives-ouvertes.fr/DAD-MULTILINGUAL). The documentational approach to didactics is a theory in mathematics education. Its first aim is to understand teachers' professional development by studying their interactions with the resources they use and design in/for their teaching. In this text we briefly describe the emergence of the approach, its theoretical sources, its main concepts and the associated methodology. We illustrate these aspects with examples from different research projects. This synthetic presentation is written for researchers, but also for non-specialists (e.g. master students) interested in a first discovery of the documentational approach
HAL (Le Centre pour la Communication Scientifique Directe), Feb 4, 2015
Not all mathematical problems have a solution. This paper describes a decomposition problem, set ... more Not all mathematical problems have a solution. This paper describes a decomposition problem, set in a real-life context, for which no mathematical solution exists. It describes the strategies children use and how the realife context impacted on children's solutions. Results indicated that young children accept the possibility that a problem may not have a solution and that some turn to the context in order to find a practical solution instead.
Using examples and non-examples is a common technique to demonstrate concepts' characteristics an... more Using examples and non-examples is a common technique to demonstrate concepts' characteristics and boundaries. Based on their properties, certain objects are accepted as examples or non-examples intuitively, while others are accepted or neglected non-intuitively. This 2*2 classification is powerful when designing technology-enhanced learning experiences in which feedback could be provided in real-time. That is, feedback could be based not only on the correctness of student response, but also on the specifics of the objects with which they were engaged. Following this framework, we developed an interactive sorting task that aims at strengthening elementary school students' understanding of reflective symmetry. We studied learners' interaction with the objects presented to them, and their success. Our study included 29 elementary school students (ages 9 to 12) from both Israel and Germany. We used screen recording to code participants' shape-movements, and defined quantitative measures of these movements. Our findings support the need for designing feedback that takes into consideration object's properties and students' behavior.
The aim of this study is to investigate the geometrical knowledge as well as the geometrical self... more The aim of this study is to investigate the geometrical knowledge as well as the geometrical self-efficacy of kindergarten children, including abused and neglected kindergarten children. Individual interviews were conducted with 141 kindergarten children, ages 5–6 years old, of which 69 children were labeled as abused and neglected by the social welfare department of their municipality. Results indicated that both groups of kindergarten children had high self-efficacy beliefs related to identifying geometrical figures which were not significantly related to knowledge. In addition, significant differences in knowledge were found between the two groups.
Patterning activities, specifically those related to repeating patterns, may encourage young chil... more Patterning activities, specifically those related to repeating patterns, may encourage young children’s appreciation for underlying structures. This paper investigates preschool teachers’ knowledge and self-efficacy for defining, drawing, and continuing repeating patterns. Results indicated that teachers were able to draw and continue various repeating patterns but had difficulties defining repeating patterns. In general, teachers had a high self-efficacy for all tasks. However, teachers’ had a significantly lower self-efficacy for defining repeating patterns than for drawing and continuing repeating patterns.
Qualitative research on classroom-based mathematics learning in inquiry-oriented classrooms is sc... more Qualitative research on classroom-based mathematics learning in inquiry-oriented classrooms is scarce. This paper presents a methodology aimed at developing a rich understanding of the interplay of mathematical progress in the different settings in which learning in such classrooms occurs - individuals, small groups, and the whole class. For this purpose, we enhance a theoretical-methodological approach of coordinating Documenting Collective Activity and the RBC-model of Abstraction in Context that has been developed in earlier studies. We do this using an intact lesson on the area and perimeter of the Sierpiński triangle in a mathematics education master’s level course on Chaos and Fractals. The enhancement of the methodology allowed integrating the Collective and Individual Mathematical Progress (CIMP) by Layering the Explanations (LE) provided by the two approaches and thus exhibiting the complexity of learning processes in inquiry-oriented classrooms.
This paper describes kindergarten children’s engagement with two patterning activities. The first... more This paper describes kindergarten children’s engagement with two patterning activities. The first activity includes two tasks in which children are asked to choose possible ways for extending two different repeating patterns and the second activity calls for comparing different pairs of repeating patterns. Children’s recognition of the unit of repeat and their recognition of the structure of the repeating patterns are investigated. Findings suggest differences between children’s responses to patterns that end with a complete unit of repeat and those that end with a partial unit. In addition, the issue of presenting repeating patterns using different media is discussed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
International Journal of Science and Mathematics Education, Apr 5, 2019
We focus on teachers’ ways of leading whole class discussions (WCDs) in mathematics, with the goa... more We focus on teachers’ ways of leading whole class discussions (WCDs) in mathematics, with the goal of uncovering their traces (if any) in their students’ responses (a) while participating in the WCDs and (b) in the written responses in a final test. For this purpose, two 8th-grade probability classes learning a 10-lesson unit with different teachers were observed. Our data sources include (1) video-recordings of the WCDs and (2) the responses of students to final test items. We analyzed the teachers’ talk-moves, students’ accountable participation, and students’ reasoning in the final test items. Interweaving the findings from all analyses we found differences between the classes in students’ ways of participation in WCDs and in their corresponding final test responses. The teachers’ ways of leading the WCDs contribute to the explanation of these differences.
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