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Abstract

Considering the general problem of integrating mathematical modelling into current educational systems, this paper focuses on the ecological dimension of this problem—the institutional constraints that hinder the development of mathematical modelling as a normalised teaching activity—and the inevitable step of the professional development of teachers. Within the framework of the Anthropological Theory of the Didactic, this step is approached using the study and research paths for teacher education (SRP-TE), an inquiry-based process combining practical and theoretical questioning of school mathematical activities. We present a research study focusing on the design and analysis of an online and distance-learning course for in-service mathematics teachers based on the SRP-TE methodology. This course starts from the initial question of how to analyse, adapt and integrate a learning process related to mathematical modelling and how to sustain its long-term development. Our analysis is based on a case study consisting in four successive editions of a course for Latin American in-service mathematics teachers held at the Centre for Applied Research in Advanced Science and Technology in Mexico. The starting point is a modelling activity about forecasting the number of Facebook users, which includes functional modelling and regression. The results show how the course represents a valuable instrument to help teachers progress in the critical issue of identifying institutional constraints—most of them beyond the scope of action of teachers and students and not approached by previous research—hindering the integration of mathematical modelling in current secondary schools.

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... Throughout the years, several examples of in-service teacher courses on modelling have been presented with the purpose of supporting teachers to design modelling activities, promoting different theoretical approaches to mathematical modelling (Kjeldsen & Blomhøj, 2006;Doerr 2007;Blum & Borromeo Ferri, 2009;Barquero, Bosch, & Romo, 2018). These approaches have ways to engage students in modelling activities, where students sometimes choose strategies different from those foreseen by researchers or teachers. ...
... This was also the purpose of the SRPs, developed by the participants of our PD. However, it has been reported that Study and Research Paths for Teacher Education, SRP-TE (Barquero et al., 2018) proved to be difficult for teachers to implement and they turned to transmission of knowledge when back in their own classrooms with designs developed in a PD. In our study, we draw on the suggestion of creating supporting structures for in-service teachers as part of courses on SRP (Muñoz, García, & Fernández, 2018). ...
... In our PD, the participants were encouraged to develop these maps based on their knowledge on the students' prerequisites, knowledge to be taught, preferred textbooks, possible google hits etc. This is one way to engage the teachers in studying the MO of their teaching, which is the first challenge for teachers engaging with SRP in PD (Barquero et al., 2018). In our PD, the participants were suggested to use the tree-diagrams, named 'knowledge-maps', as navigation tools when teaching (see Jessen & Rasmussen, 2018). ...
Conference Paper
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In this paper, we present a model for upper secondary in-service teacher courses based on the Anthropological Theory of the Didactics and explore how we can teach teachers to design and implement mathematical modelling in their classrooms. The course evolves around Study and Research Path based teaching and strives to create para-didactic infrastructures as a framework for teachers' development of teaching practice. The novelty in this study is the sequence of shared preparation, observation and evaluation of teaching in the course. We describe the structures and their functioning through an example of a group of teachers' work. Based on the activity we discuss the potentials of creating such structures and the needs for further research in this field.
... While it is a common approach for science, so far, K-12 mathematics curricula have paid limited attention to helping students develop modelling skills [7,12]. Even when modelling is included in the curriculum, teachers often do not have the necessary background to incorporate it into their teaching [45] and rarely see it as relevant to mathematics they are teaching [46]. Yet, modelling could help struggling students to learn how to communicate mathematics, use multiple representations, and connect abstract concepts to real-life [12,47]. ...
... Modelling is becoming more prominent in both mathematics and science school curricula [4,27,29] and consequently in teacher education [45]. There is mounting evidence that the emphasis on reallife connections in modelling leads to remarkable learning gains, especially in underserved student populations and students at risk, or as Lesh, Young, and Fennewald [59] write, "modeling is virtually unparalleled in the successes that it has produced" (p. ...
... At the same time, the diagram focusses on what the students should be doing during the modelling activity, but it completely ignores the teacher. As a result, teachers may not see it as relevant or misinterpret it [45,46]. ...
Article
The goal of this paper is to examine the place of modelling in STEM education and teacher education. First, we introduce modelling as a cyclical process of generating, testing, and applying knowledge while highlighting the epistemological commonalities and differences between the STEM disciplines. Second, we build on the four well-known frameworks, to propose an Educational Framework for Modelling in STEM, which describes both teacher and student roles in the modelling cycle. Third, we use this framework to analyze how modelling is presented in the new mathematics and science school curricula in two Canadian provinces (Ontario and British Columbia), and how it could be implemented in teacher education. Fourth, we emphasize the epistemological aspects of the Educational Framework for Modelling in STEM, as disciplinary epistemological foundations may seem too abstract to both teacher educators and teachers of STEM school subjects. Yet, epistemologies are the driving forces within each discipline and must be considered while teaching STEM as a unified field. To nurture critical thinkers and innovators, it is critical to pay attention to what knowledge is and how it is created and tested. The Educational Framework for Modelling in STEM may be helpful in introducing students and future teachers to the process of modelling, regardless of if they teach it in a single- or a multi-discipline course, such as STEM. This paper will be of interest to teacher educators, teachers, researchers, and policy makers working within and between the STEM fields and interested in promoting STEM education and its epistemological foundations.
... Across research studies of mathematical modeling there is also agreement that mathematical modeling is mathematically demanding and instructionally complex requiring new ways of reasoning and ways of participating in classrooms (Cai et al., 2014;Blum, 2015;Jung & Newton, 2018). Less well articulated are the instructional dilemmas faced in teaching while supporting the mathematical and instructional demands of modeling (Barquero, Bosch, Romo, 2018). How teachers orchestrate the interpretive and strategic space of modeling is the aim of the study reported on in this paper. ...
... Researchers investigating teachers' mathematical modeling instruction highlight cognitive, social, and contextual challenges that impact teachers' knowledge, practice, and the educational environment of modeling instruction (Barquero, et al., 2018;Blum, 2015;Cai, et al., 2014). With most research informed by a cognitive perspective, fewer studies take up an ecological view of teaching of modeling to understand the dilemmas or tensions that may exist in the educational system when shifting classroom instruction to incorporate modeling (Barquero, et al., 2018). ...
... Researchers investigating teachers' mathematical modeling instruction highlight cognitive, social, and contextual challenges that impact teachers' knowledge, practice, and the educational environment of modeling instruction (Barquero, et al., 2018;Blum, 2015;Cai, et al., 2014). With most research informed by a cognitive perspective, fewer studies take up an ecological view of teaching of modeling to understand the dilemmas or tensions that may exist in the educational system when shifting classroom instruction to incorporate modeling (Barquero, et al., 2018). One agreement across these reviews is that teachers matter and teaching modeling is a complex process drawing upon highly demanding instructional competence or knowing-toact and action. ...
Conference Paper
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Research on two teacher-designed and enacted modeling routines for secondary classrooms documents the tensions that emerge as activity systems of school mathematics and modeling instruction were coordinated. Analysis revealed that teachers modified routines in productive and less productive ways to attend to students’ tendency toward traditional school mathematics. Productive modifications show that tensions across activity systems were opportunities for expansive teacher and student learning.
... In the present special issue we identified contributions on fundamental research that aimed at developing a measurement instrument (Vorhölter 2018), investigated students' reasoning (Degrande et al. 2018), clarified the relation between modelling, motivation and other factors (Krawitz and Schukajlow 2018;Plath and Leiss 2018), described students' modelling processes and developed comprehensive theoretical frameworks (Almeida 2018;Barquero et al. 2018;Brady 2018;Cantoral et al. 2018;English and Watson 2018;Galleguillos and Borba 2018) and give an overview on modelling projects (Burkhardt 2018). A valuable part of contributions are focused on applied and development research. ...
... An important problem of integrating mathematical modelling into educational systems, the institutional constraints that hinder the development of mathematical modelling as an integral part of everyday practice in mathematics classes was investigated by Barquero et al. (2018) using the Anthropological Theory of the Didactic in the context of an online and distance learning course for inservice Latin American mathematics teachers. Results of an ecological analysis brought to light three forms of constraints: the need for a mathematical discourse concerning modelling, applicationism as the dominant pedagogy and strong local constraints (such as using only short tasks). ...
Article
The teaching and learning of mathematical modelling is an important research field all over the world. In this paper we present a survey of the state-of-the-art on empirical studies in this field. We analyse the development of studies focusing on cognitive aspects of the promotion of modelling, i.e. the promotion of modelling abilities resp. skills, or in newer terminology, modelling competencies. Furthermore, we provide a literature search on the role of empirical research in important mathematics education journals and point out that this topic is only seldom treated in these journals. In addition, Proceedings of the conference series on the teaching and learning of mathematical modelling and applications were analysed in order to identify the role of empirical research in this important series and the kind of topics which are examined. The literature research points out the dominance of case study approaches and cognitively oriented studies compared to studies which used quantitative research methods or focused on affect-related issues. Finally, the papers in this special issue are described and developments and future prospects are identified.
... The logic that developing EdTech initiatives in the most challenging contexts and extending them to more favourable conditions is compelling but rarely implemented. 12 ...
... Yet, school cultures can also often pose significant constraints on a teacher's creativity, inhibiting their inherent urges to "go off script" in order to really engage students [45]. As such, providing teachers with new analytic and design tools to overcome institutional constraints by taking on diverse and changing roles during TPD activities can shed light on the embedded institutional (often 'naturalised', or deep-rooted) behaviours that may constrain teachers in the pedagogical process ( [12], p. 41). Technology can play a significant role in drawing in remote, external peers' perspectives. ...
Article
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Pre-service education and in-service teacher professional development (collectively termed teacher professional development or TPD here) can play a pivotal role in raising teaching quality and, therefore, learning outcomes for children and young people in low- and middle-income countries (LMICs). However, TPD opportunities in LMICs are limited, unsustained, and often not informed by recent research evidence, and outcomes are mixed. Educational technologies offer potential to enhance formally provided programmes and informal peer-learning forms of TPD. We present the first systematic review of the literature pertaining to technology-mediated TPD for educators of school-aged learners in LMICs, aiming to characterise appropriate and effective uses of technology along with specific constraints operating in those contexts. An in-depth synthesis of 170 studies was undertaken, considering macro-, meso- and micro-level factors during TPD design and implementation in the 40 LMICs represented. Volume of publications increased dramatically over the review period (2008–2020), indicating that the field is rapidly developing. Results largely showed benefits for teachers, but evidence for sustainability, cost-effectiveness or tangible impacts on classroom practice and student outcomes was thin. Promising, locally-contextualised forms of technology-mediated TPD included virtual coaching, social messaging, blended learning, video-stimulated reflection, and use of subject-specific software/applications. We report on the variable effectiveness of programmes and limited attention to marginalised groups. To maximise effectiveness of technology-enhanced TPD, the role of facilitators or expert peers is paramount – yet often glossed over – and the interpersonal dimension of teacher learning must be maintained. Recommendations are made for researchers, policymakers, teachers and teacher educators.
... According to these assumptions, Ruiz-Olarría (2015) and Barquero, Bosch and Romo (2018) present the proposal of study and research paths for teacher education (SRPs-TE) as an educational format to provide teachers with pertinent (theoretical and practical) tools to nourish and sustain this type of questioning, making it also operative for the design and management of didactic processes. During the last ten years, diverse experiences of design, implementation and analysis of SRPs-TE have been developed (see Table 12.1) at different school levels and in different conditions (pre-service and in-service teacher education, online or in distance training, etc.). ...
... Even if prospective teachers did not implement an SRP during the course, in the first case (primary education) and the third one (university), some teachers have started implementing SRPs in their daily practice since the course finished. In contrast, Barquero, Bosch and Romo (2018) present different experiences of an SRP-TE where this module is fully implemented and this kind of discussions takes place. This research study also shows that the initial modules are crucial to creating the appropriate milieu to analyse and discuss the teachers' implementations, a result that is shared with the three case studies considered in this chapter. ...
... These formats are called study and research paths for teacher education (SRP-TE) (Ruiz-Olarría 2015) and consist of a five-phase process aimed at helping teachers generate answers to the initial professional question. Even though the SRP-TE structure is well defined and has been implemented in different institutional settings (Barquero et al. 2018;Ruiz-Olarría 2015), the conclusions of these specific studies, in line with other teacher education experiences, claim for new tools originated in research in mathematics education to be incorporated into teacher educational courses. These research works describe the lack of tools that would enable teachers to characterise, analyse, question and reorganise the mathematical content to be taught. ...
... This course is presented in detail in Barquero et al. (2015Barquero et al. ( , 2018. The SRP proposed in M1 was initiated by this generating question: Q 0 : How to make a short and long-term sales forecast for each variable (sold t-shirts, revenue) for a clothing company considering the weekly sales in several shops (see Table 1) and the evolution of the benefits? ...
Article
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Research in teacher education is still a field dealing with open questions. One of these open fields is the incorporation of new pedagogical principles such as inquiry-based processes. In this transition encouraged by most national curricula, knowledge involved in study processes changes heavily: it moves from well-established and static knowledge to more dynamic and provisional structures. This transition causes the description, management and analysis of knowledge to be a challenge for teachers. Researchers within the Anthropological Theory of the Didactic have used question–answer maps to describe knowledge involved in the implementation of inquiry teaching formats. We propose to use these question–answer maps as a research tool to be incorporated in teacher education to empower teachers to deal with this new nature of knowledge to be taught. We present in this paper an exploratory case study of an online course for secondary teachers using question–answer maps as a tool to describe knowledge.
... For example, institutional constraints have been identified that make it difficult to fit modeling activities into the normal functioning of educational centers (Barquero, Bosch, and Romo [16]). It has also been observed that teachers' attitudes and their training are key to the regular use of modeling activities (Blum and Leiß [17]; Schmidt [18]). ...
Article
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This paper presents a case study carried out at an elementary school that led to a characterization of mathematical modeling projects aimed at generating social impact. It shows their potential as generators of mathematical learning opportunities. In the school project, upper-grade students (sixth grade, 11-year-olds) studied the way in which the rest of the students at the institution traveled from their homes to school. Its purpose was to identify risk points from the standpoint of road safety and to develop a set of recommendations so that all the children could walk safely to school. In our study, we identified, on the one hand, the mathematical learning opportunities that emerged during the development of the project and, on the other, the mathematical models created by the students. We discuss the impact of the project on the different groups in the school community (other students, parents, and teachers). We conclude with a characterization of the mathematical modeling projects oriented towards social impact and affirm that they can be generators of mathematical learning opportunities.
... The generating question (Q 0 ) gives rise to a rooted tree sequence of derived questions and answers that can be modelled using question--answer (Q-A) maps and the media-milieu dialectics (Kidron et al. 2014;Winsløw, Matheron, and Mercier 2013). Examples of experimentations of SRP at different educational levels and in different fields like Mathematics, Epidemiology, Management or Sports exist (Serrano, Bosch and Gascón 2010;Fonseca, Gascón, and Lucas 2014;Barquero, Bosch and Romo 2018;García and Ruiz-Higueras 2013;Jessen 2014;Thrane 2009). ...
Article
Study and research path (SRP) is an inquiry-based teaching format grounded in the Anthropological Theory of the Didactic. One of the specific characteristics of SRP is its task-design basis, which includes explicitly questioning the knowledge at stake in study processes. In this work, we propose general guidelines for the design of SRPs for Engineering courses. We have explored the validity of the SRP methodology for learning Strength of Materials, a key subject in Mechanical Engineering. The use of question–answer maps and the media–milieu dialectics throughout the development of the SRP are discussed, together with the student’s perceptions on the new learning process.
... We have already mentioned that the online courses offer a means of supporting the interaction between experience and competence. This is made possible by developing collaborative knowledge mediated by the activities performed in each LU, as the analysis of various LU illustrates (Covián and Romo, 2017;Barquero et al., 2018;Romo and Covián, 2018;Romo-Vásquez and Hache (in press). As an example, we take the analysis elaborated for the LU entitled 'Processes for the Instit utionalization of Scholastic Mathematics', presented in Romo-Vázquez and Gómez-Blancarte (2018). ...
Article
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Drawing on Communities of Practice theory, we use the notions of experience and competence to analyze two of the main training processes that the Online Program for Mathematics Education offers to foster the professional development of mathematics teachers: online courses and dissertation projects. We examine how these two processes promote coordination and alignment between teachers' experience in mathematics teaching and competence, defined by the results of research in Mathematics Education. The analysis provides insights that contribute to the current need to investigate how an educational community can be sustained and how to best determine its role in facilitating the professional development of teachers.
... Los datos que se recopilen de la experiencia serán analizados bajo la mirada de las macro-herramientas del EOS, como parte de un proceso de investigación que da continuidad al trabajo de Ledezma (2017) sobre el tema de la modelización, e inspirados por otros estudios que también han profundizado en las competencias en modelización matemática, tanto durante la formación académica como la práctica profesional de los docentes (por ejemplo, Anhalt, Cortez y Bennett, 2018; Barquero, Bosch y Romo, 2018;Blomhøj y Kjeldsen, 2006;Doerr, 2006, entre otros). ...
Conference Paper
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Este taller va dirigido a profesores de matemática de nivel medio (13 a 18 años), cuyo objetivo es introducir a los participantes en la modelización desde un en-foque didáctico-cognitivo. Para ello, nos basaremos en la propuesta de Borro-meo Ferri (2018) para la constitución de un curso de modelización matemática, considerando macro-herramientas propuestas en el Enfoque Ontosemiótico, así como el ciclo de modelización propuesto en los estudios de Blomhøj y Højgaard. Con estos elementos, se pretende que los participantes resuelvan problemas, identifiquen las fases del proceso y reflexionen sobre su aplicación en el aula.
... Aunque todavía no podemos decir gran cosa de este módulo, puesto que son pocas las experimentaciones realizadas (Barquero, Bosch y Romo, 2015, 2018Florensa, Bosch, y Gascón, 2016), consideramos que es clave por diversos motivos. En primer lugar, porque constituye un ámbito de validación experimental final del REI-FP llevado a cabo y de las nuevas posibles formas de organizar 139 tanto los contenidos matemáticos curriculares como su enseñanza y aprendizaje. ...
Article
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We present a methodology for the reconstruction of a mathematical praxeology for teaching based on study and research paths for teacher education. This methodology is illustrated through the construction of a praxeology for teaching that emerges as an answer to the generating question: “How to organize the teaching of about elementary functional modeling at lower secondary school (12-16 years old) and what role can be assigned to proportionality in this organization?” The experimentation took place in the Master for secondary school teacher education.
... El dispositivo didáctico denominado REI ya comentado previamente se ha utilizado también en investigaciones sobre la formación continua de profesores. En Barquero, Bosch y Romo (2018) se propone y describe el diseño y la implementación de los denominados REI para la formación del profesorado (REI-FP) en un curso en línea y a distancia dirigido a futuros profesores de Educación Secundaria y Universitaria de México, en un curso denominado "Procesos de Institucionalización de las Matemáticas Escolares". Se pretende conectar los resultados de investigaciones con las dificultades de la práctica educativa para la enseñanza de la modelización matemática. ...
Chapter
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En este trabajo se aborda una revisión de los avances de investigación desarrollados en España sobre el uso de la modelización en el proceso de enseñanza y aprendizaje de las matemáticas. Se pretende obtener una recopilación organizada de los trabajos realizados en los últimos años. Una de las características de los estudios descritos es la variedad de enfoques teóricos que se utilizan y que, sin duda, da cuenta de la diversidad de investigaciones realizadas en nuestro país centradas en la modelización; un área de investigación que ya cuenta con una larga tradición en el ámbito internacional y que, en los últimos años, se ha enriquecido de las contribuciones internacionales de distintos grupos de investigación españoles.
... Actualmente la investigación respecto al cómo la modelización puede favorecer el proceso de enseñanza-aprendizaje de las matemáticas ha cobrado gran protagonismo (Trelles y Alsina, 2017). En este sentido, son varias las investigaciones en los últimos años que abordan el uso de la modelización matemática en entornos educativos, por ejemplo: Lu y Kaiser (2022) Barquero et al. (2018). Este hecho ha repercutido de manera directa en el diseño curricular de varios países, pues, cada vez son más los países como Alemania, Colombia, Chile, Ecuador, España, Estados Unidos, Perú, Singapur, Suecia, entre otros, que han incorporado explícitamente ...
Article
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En la actualidad la modelización matemática dentro de la investigación en ambientes educativos ha adquirido notable presencia, por ello, muchos países han decidido incorporarla tanto en sus documentos curriculares como en su práctica de enseñanza; Ecuador no es la excepción y en su última propuesta curricular la incorpora explícitamente. Los libros de texto forman parte de los recursos más utilizados por el profesorado, por tanto, el objetivo de esta investigación es determinar la presencia de la modelización matemática en el bloque de estadística y probabilidad mediante un estudio comparativo de los libros de texto entregados por el Ministerio de Educación en los años 2016-2019 frente a los libros utilizados actualmente y entregados al estudiantado desde el año 2020. La metodología utilizada responde a un enfoque cuantitativo, con un alcance descriptivo, 497 actividades fueron analizadas mediante un análisis de contenido. Los resultados muestran la poca presencia de la modelización matemática, si bien los libros actuales presentan en conjunto problemas más contextualizados, las actividades de modelización matemática han desaparecido completamente; se concluye que es necesaria una reformulación urgente de los libros de texto en lo que a estadística y probabilidad se refiere, en la que se incorpore en mayor medida actividades de modelización. Los hallazgos sirven para que el estado ecuatoriano y los autores de libros de texto tomen decisiones acerca de cómo mantener una posición coherente entre lo manifestado en los lineamientos curriculares y lo que se propone al estudiantado en su proceso de enseñanza aprendizaje a través de los libros de texto.
... Indeed, the didactic divide can be overcome-consciously or not-in special cases. For example, Barquero, Bosch, & Romo (2018) reports their teacher-educational experimentation, which includes the teaching of ATD, especially the notion of study and research path. However, its reproducibility in other contexts seems to be quite low under the abovementioned existing conditions. ...
... Scholars have noted, however, that day-to-day mathematics teaching includes few mathematical modeling examples (e.g., Blum, 2015;Tan & Ang, 2016). Teachers' professional mathematical modeling competencies have been widely accepted as the key factor that determines the successful implementation of mathematical modeling in mathematics teaching (Barquero et al., 2018;Greefrath & Vorhölter, 2016). ...
Article
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Although mathematical modeling plays an important role in many curricula worldwide, significant discrepancies persist in the importance of mathematical modeling in ordinary mathematics classrooms and teacher education. This paper compares pre-service mathematics teachers’ professional mathematical modeling competencies in three different regions—Germany, Mainland China, and Hong Kong—where educational and cultural traditions differ, including the role of mathematical modeling. In total, 232 pre-service mathematics teachers from the three regions completed a modeling task covering mathematics content knowledge (MCK) of modeling and mathematical pedagogical content knowledge (MPCK) of modeling. The results show that pre-service teachers from Germany demonstrated the strongest MCK and MPCK of mathematical modeling; by contrast, pre-service mathematics teachers from Mainland China and Hong Kong demonstrated relatively weaker MCK and MPCK of mathematical modeling. MCK and MPCK of mathematical modeling were also found to be unevenly developed at different competence levels for the three regions. These differences may be attributed to the history of mathematical modeling in mathematics curricula, teacher education, and teaching culture in these three regions.
... Ferri (2014) indicated that future instructors must have a solid understanding of many aspects of modelling as well as effective strategies for teaching modelling. Several research indicate also in the same direction in terms to the importance of the teacher's role in the modelling cycle (Barquero et al., 2018;Stillman et al., 2013). To engage in modeling-based activities in the classroom, instructors must possess a broader range of abilities. ...
Article
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This work sets out to examine the validity and reliability of the Mathematical Modeling Attitude Scale (MMAS), a instrument measuring teachers’ attitude towards mathematical modeling. A cross-sectional survey research was utilised to describe the validity and reliability of the MMAS. The population of the present study focused on Malaysian mathematics teacher in primary and secondary schools (N = 171) and this was achieved using convenience sampling. Exploratory factor analysis (EFA), confirmatory factor analysis (CFA) and Rasch analysis were utilised in analysing the data in the present work. EFA revealed that the data from the teachers had a four-factor structure: constructivism, relevance and real-life, understanding and motivation and interest. The CFA confirmed that the model fit indices established the four-factor structure of the first and second-order model. Although the Rasch analysis generally supported the finding of EFA and CFA, there was still room for improvement in terms of the rating scale and DIF criterion.
... However, mathematical modelling still does not see a prominent role in classroom practice worldwide (Barquero, Bosch, & Romo, 2018). The main reason is that mathematical modelling is difficult both for students and teachers. ...
Thesis
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This research explored the beliefs, knowledge and skills of Bhutanese pre-service mathematics teachers and their preparedness to teach a reformed mathematics curriculum. Data were collected when the participants joined a series of inquiry-oriented, technology-enabled calculus workshops. These data analyses revealed many inconsistencies between the participants’ espoused and enacted beliefs and is alignments between their mathematical knowledge and skills when compared to the intentions of the newly reformed curriculum. From this, implications have been drawn about the preparedness of the soon-to-graduate pre-service mathematics teachers, and recommendations are made regarding ways future pre-service teachers mathematical beliefs, knowledge and skills may be enhanced.
... Se asume la Teoría Antropológica de lo Didáctico (TAD), la cual predice que los cambios que experimentan las instituciones forzosamente producen cambios epistemológicos en la enseñanza de los saberes, estos procesos describen y caracterizan el caso específico de la matemática (Chevallard,1985(Chevallard, , 1994(Chevallard, , 2015(Chevallard, , 2019. La investigación didáctica desde la TAD requiere describir aspectos relacionados con las tres dimensiones que fundamentan el problema didáctico, a saber: la epistemológica, la económica y la ecológica (Chevallard & Bosch, 2020;Gascón, 1999;Gascón, 2011;Licera, 2017;Lucas & Gascón, 2019 Barquero, Bosch & Romo, 2018, 2019Costa, Arlego & Otero, 2015;Florensa, Bosch & Gascón, 2016, 2019Florensa, Bosch, Gascón & Winslow, 2018;Gazzola, Otero & Llanos, 2020. Este artículo recoge estas ideas y pretende generar un marco conceptual inicial para una investigación científica profunda dentro del DCNS que se relaciona con el CDM de la LMOES buscando impactar la educación secundaria de la República Dominicana. ...
Article
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Se presenta un proceso constructivo del problema didáctico para abordar el tema de la enseñanza en el componente de la especialidad Matemática que se imparte a los estudiantes de la “Licenciatura en Matemática Orientada a la Educación Superior”, carrera que se dicta en las universidades de la República Dominicana. El problema está relacionado con la propuesta de un modelo didáctico que busque transformar la enseñanza de los saberes inherentes a la formación de los futuros maestros que se desenvolverán profesionalmente en la educación secundaria dominicana teniendo en cuenta el nuevo currículo presentado por el Ministerio de Educación de República Dominicana. El proceso se plantea utilizando la Teoría Antropológica de lo Didáctico, teniendo como sustento la heurística de un patrón inscrito en las tres dimensiones que definen un problema didáctico constituidas por: la epistemología, la economía y la ecología. Este artículo toma en cuenta la metodología propuesta por Ruiz-Olarría A., Bosch M. & Gascón J. (2019) para la construcción de una praxeología para la enseñanza en la institución de formación del profesorado utilizando los procedimientos de los recorridos de estudio e investigación.
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El desarrollo profesional online del profesorado permite articular la difusión de nuevas propuestas basadas en la investigación didáctica con la experiencia del docente y sus dificultades para integrarlas en el aula. Presentamos una investigación sobre el diseño e implementación de Unidades de Aprendizaje online y a distancia para profesores de matemáticas en activo basada en una adaptación de la metodología de los Recorridos de Estudio e Investigación para la Formación del Profesorado (REI-FP) propuesta por Ruiz-Olarría (2015). De acuerdo con esta metodología, la Unidad de Aprendizaje (en adelante, UA) pretende abordar, junto con los profesores participantes, los obstáculos institucionales que plantea la enseñanza de la modelización matemática en relación con la enseñanza por transmisión de conocimientos preestablecidos: rigidez del currículo, gestión del tiempo, escasez de dispositivos de evaluación adaptados, control del uso de las TIC, multidisciplinariedad, entre otros.La investigación se basa en un estudio de caso formado por cuatro ediciones sucesivas de una UA para profesores de matemáticas latinoamericanos en activo que se realizó en el Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada de México. La UA se centra en una actividad de modelización sobre previsiones de ventas que adapta la metodología de los REI-FP al caso online y a distancia. Después de describir el contenido de la UA, se realiza un análisis clínico exploratorio del proceso de formación implementado, usando como material empírico las producciones de un grupo de profesores. Los resultados muestran la viabilidad de la adaptación metodológica, poniéndose de manifiesto que los dispositivos de formación implementados facilitan la emergencia e identificación de restricciones institucionales generalmente ocultas a los docentes en su quehacer cotidiano. También muestran limitaciones del proceso de formación que requieren mayor explotación de las herramientas multimedia, así como la necesidad de extender el proceso de formación impartido más allá de la estricta temporalidad de la UA.
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Resumo Neste trabalho apresentamos um estudo das condições e restrições que incidem sobre um percurso de estudo e pesquisa (PEP) ao utilizá-lo para o ensino de cônicas (parábola, elipse e hipérbole) a um grupo de estudantes do terceiro ano do Ensino Médio de Itabaiana/Sergipe/Brasil. A pesquisa seguiu a metodologia da teoria antropológica do didático que é baseada em princípios da Engenharia Didática e que possui quatro fases. Na primeira foram desenvolvidos os estudos epistemológico e econômico, na segunda realizamos a análise a priori e o planejamento do PEP, que foi experimentado na terceira fase e, por fim, na quarta, realizamos a análise a posteriori do experimento vivido. O planejamento, a construção e a gestão do PEP em sala de aula tiveram como base um modelo epistemológico de referência (MER) a respeito de cônicas, que mostra o estudo desse objeto em três modelos de geometria: sintética, analítica e linear. Os resultados apontam para um conjunto de condições e restrições, originadas do modelo de ensino atual, em que a instituição cobra do professor o cumprimento do conteúdo em determinado prazo; em que os alunos, não habituados com um dispositivo didático baseado em uma investigação, apresentam dificuldades no processo de elaboração de questões, no uso de fontes confiáveis de pesquisa na Internet, na falta de critérios para validar os conteúdos encontrados e no trabalho com elementos da geometria sintética.
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https://www.routledge.com/Working-with-the-Anthropological-Theory-of-the-Didactic-in-Mathematics/Bosch-Chevallard-Garcia-Monaghan/p/book/9780367187705 . . . . . . . This book presents the main research veins developed within the framework of the Anthropological Theory of the Didactic (ATD), a paradigm that originated in French didactics of mathematics. While a great number of publications on ATD are available in French and Spanish, Working with the Anthropological Theory of the Didactic in Mathematics Education is the first directed at English-speaking international audiences.
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Nuestra investigación se centra en la implementación de propuestas de enseñanza a nivel universitario que promuevan un cambio del paradigma pedagógico. Partimos del paradigma dominante en estas instituciones, principalmente basado en la visita de los saberes como si fueran monumentos, para avanzar hacia un paradigma más centrado en el estudio de cuestiones. En el marco de teoría antropológica de lo didáctico, hemos trabajado en el diseño de los recorridos de estudio e investigación (REI) como dispositivos de enseñanza en la transición entre paradigmas, los cuales se han desarrollado en distintas modalidades. En este trabajo, nos centramos en la formación universitaria del profesorado como agente crucial en este cambio, y nos centramos en la adaptación de los REI al caso de la formación del profesorado (REI-FP). Veremos en qué sentido los REI-FP permiten avanzar en el estudio de las condiciones y restricciones que emergen en la difusión de los REI.
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This paper focuses on how to link and combine activities of posing and answering questions with the ones of studying pre-existing answers at university level. Our research, developed within the framework of the Anthropological Theory of the Didactic, presents the design and use of an instructional device – study and research paths (SRP) – aimed at encouraging the interaction between inquiry and study activities. In this paper, we focus on an SRP about the evolution of the number of Facebook users, which was designed and implemented with first-year university students. We discuss the SRP analysis paying special attention to: (a) how to enhance the internal dynamics of the SRP to engage students in self-sustained study and research activities, and (b) how the integration of the SRP in the regular mathematics course helped to combine inquiry with more ‘transmissive’ instructional devices and why this interaction was essential to enrich the students’ milieu.
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We believe there are three fundamental design criteria that help specify essential characteristics of a world learning system capable of addressing the scope and scale of the global challenges we face today. Problems such as overpopulation, world hunger, poverty, illiteracy, armed conflict, inequity, disease, and environmental degradation are inextricably interconnected. Moreover, they are complex, dynamic, and globally distributed. To address such challenges, we must increase our global intelligence along several dimensions: cognitive, behavioral, and moral. We must increase, by orders of magnitude, our societal capacity for inquiry; our ability to continuously create, adapt, and transfer solutions. A world learning system that can match the challenges we face must meet three basic specifications: • Action-learning capacity to address problems while continuously reflecting on what approaches are working and why—and then using these insights to guide future actions. • Cross-boundary representation that includes participants from all sectors—private, public, and nonprofit—and from a sufficient range of demographic constituencies and professional disciplines to match the complexity of factors and stakeholders driving the problem. • Cross-level linkages that connect learning-system activities at local, national, and global levels—wherever civic problems and opportunities arise. Civic development is essentially a social process of action learning, in which practitioners from diverse sectors, disciplines, and organizations work together to share ideas and best practices, create new approaches, and build new capabilities. The full potential of this learning process is only realized when it connects all the players at various levels who can contribute to it.
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This survey provides an overview of the German discussion on modelling and applications in schools. It considers the development from the beginning of the 20th century to the present, and discusses the term “mathematical model” as well as different representations of the modelling process as modelling cycles. Different trends in the historical and current debate on applications and modelling can be differentiated as perspectives of modelling. Modelling is now one of the six general mathematical competencies defined in the educational standards for mathematics introduced in Germany in 2003, and there have been several initiatives to implement modelling in schools, as well as a whole range of empirical research projects focusing on teachers or students in modelling processes. As a special kind for implementing modelling into school, modelling weeks and days carried out by various German universities have been established.
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The German discussion about modelling and applications in school has a long tradition. In this topical survey, we consider the development from the beginning of the 20th century until today and discuss the term mathematical model as well as different representations of the modelling process. The various tendencies of the historical and current debate on applications and modelling can be differentiated as perspectives on modelling. Nowadays modelling is one of the six general mathematical competencies of the educational standards for mathematics that were introduced in Germany beginning in 2003, and there have been many initiatives to implement modelling into school and carry out a whole range of empirical research projects with the focus on teachers or students in modelling processes. As a special way of implementing modelling into schools, modelling days or weeks carried out by various German universities have been established.
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Considering the general problem of integrating mathematical modelling into current educational systems, this paper focuses on the ineluctable step of the professional development of teachers. Within the framework of the Anthropological Theory of the Didactic, the use of study and research paths for teacher education (SRP-TE) was recently proposed as a means to combine a constant practical and theoretical questioning of mathematical modelling school activities. After presenting the rationale of our proposal, we will illustrate the phases of the SRP-TE design and some preliminary results with the implementation of an on-line course for in-service secondary school teachers in CICATA-IPN (Mexico).
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The topic of this paper is mathematical modelling or—as it is often, more broadly, called—applications and modelling. This has been an important topic in mathematics education during the last few decades, beginning with Pollak’s survey lecture (New Trends in Mathematics Teaching IV, Paris, pp. 232–248, 1979) at ICME-3, Karlsruhe 1976. By using the term “applications and modelling”, both the products and the processes in the interplay between the real world and mathematics are addressed. In this paper, I will try to summarize some important aspects, in particular, concerning the teaching of applications and modelling.
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Recent research on the teaching and learning of mathematical modelling highlights the existence of strong institutional constraints impinging on the large-scale dissemination of mathematics as a modelling activity in current educational systems at all school levels. The study of these constraints and the way new teaching proposals can overcome these, what we call the ecology of modelling, appears as a necessary step for mathematics education research. Within the framework of the anthropological theory of the didactic, the hierarchy of levels of didactic codetermination is used to identify and analyse this ecology, not only to deal with the variety of constraints appearing in classroom activities, but mainly to know at what level action is necessary. A teaching proposal in terms of study and research paths in tertiary education shows new possibilities to overcome some of the constraints that hinder the development of mathematics as a modelling activity.
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We assume that task design-based research refers to any research that is experimentally based on an educational context and pays special attention to the description, analysis, and organization of the content to be taught (the “task” in the expression “task design”). This chapter focuses on the field of didactics of mathematics as it emerged in the middle of the 1970s with the works of the French researcher Guy Brousseau (1997). The notion of didactic engineering was used early in this field to define the relationships between the theoretical developments of didactics and the empirical reality of the classrooms. It can be presented as a research methodology structured in distinct intertwined phases with the double aim of studying didactic phenomena and developing new educational proposals (Artigue, 1990, 2008). We are illustrating this methodology by means of two examples: the measure of quantities at primary school and the modeling of a population growth at university level. The first one is taken from the theory of didactic situations which is at the origin of didactics as a scientific discipline; the second one corresponds to an evolved conception of didactic engineering within the anthropological theory of the didactic.
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This paper presents a mathematical modelling activity experienced with students of first year university level centred on a problem of forecasting sales using one-variable functions. It then focuses on the back and forth movements between the initial system – a time-series of the term sales of a firm – and the different models proposed to make the forecasting. The analysis of these movements, that are at the core of the 'mathematising step' of the modelling cycle, shows how the initial empirical system is being enlarged and progressively enriched with new variables and mathematical objects. Thus the development of a modelling activity initiated with a real-situation may soon lead to a process where the mathematising affects both the system and the model.
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How do practicing mathematics teachers continue to develop the knowledge and habits of mind that enable them to teach well and to improve their teaching over time? The question of how (and what) teachers learn lies at the crux of any effort to provide high-quality mathematics teaching for all students. This article reviews 106 articles written between 1985 and 2008 related to the professional learning of practicing teachers of mathematics. We offer a synthesis of this research, guided by Clarke and Hollingsworth’s (Teach Teach Educ 18(8):947–967, 2002) dynamic model of teacher growth. Their model emphasizes the recursive nature of teachers’ learning and suggests that growth in one aspect of teachers’ knowledge and practice may promote subsequent growth in other areas. We report the results in six major areas of teacher learning, identify several crosscutting themes in the literature, and make recommendations for future research aimed at understanding teachers’ professional learning.
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The aim of this paper is explain how the notion of “study and research course”, a recent construct in the anthropological theory of didactics, provides a general tool to model mathematical knowledge from a didactical perspective. We do this from two points of view. First, the notion itself arose as a tool for didactic design, particularly in mathematics teaching involving modelling type subjects, and we introduce a new diagrammatic representation in order to present and analyse a selection of such designs. Secondly, we demonstrate how the same representational tool can be used to analyse other didactic processes as study and research courses. In both cases, our examination of cases reveals that designers and teachers are often more explicit about their intention to enable students to do “research” (e.g. exploring a challenging problem) than to enable students to engage in autonomous “study”; moreover, the role for students is mostly confined to search for answers to questions raised, essentially, by the teacher. These observations point to directions which deserve more attention in design research related to the notion of study and research course.
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We live in a small world, where a rural Chinese butcher who contracts a new type of deadly flu virus can infect a visiting international traveller, who later infects attendees at a conference in a Hong Kong hotel, who within weeks spread the disease to Vietnam, Singapore, Canada, and Ireland. Fortunately, the virulence of the Severe Acute Respiratory Syndrome (SARS) was matched by the passion and skill of a worldwide community of scientists, health care workers, and institutional leaders who stewarded a highly successful campaign to quarantine and treat those who were infected while identifying the causes of the disease and ways to prevent its spread. In such a world, we depend on expert practitioners to connect and collaborate on a global scale to solve problems like this one – and to prevent future ones.
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In this paper, qualitative results are presented for a case study about future math teachers’ professional knowledge restricted to the area of modeling competencies. Based on interviews and the results of open questionnaires, the future teachers’ competencies were evaluated in the areas of mathematical knowledge (so-called mathematical content knowledge), knowledge of mathematics pedagogy (so-called pedagogical content knowledge in mathematics) and knowledge of educational psychology (so-called general pedagogical knowledge). The study shows that in order to develop a comprehensive understanding of modeling and its pedagogical value, future teachers need appropriate knowledge and competencies in mathematics, mathematics pedagogy and general pedagogy. The study underlines the central role pedagogical content knowledge plays in the development of professional knowledge of teachers. Key WordsAssessment–Beliefs about mathematics–Competency/competencies–Correlation–Mathematical modeling–°competency/competencies–°cycle–Motivation–Network–Word problems
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This paper describes the development of mathematical modelling as an element in school mathematics curricula and assessments. After an account of what has been achieved over the last forty years, illustrated by the experiences of two mathematician-modellers who were involved, I discuss the implications for the future—for what remains to be done to enable modelling to make its essential contribution to the «functional mathematics», the mathematical literacy, of future citizens and professionals. What changes in curriculum are likely to be needed? What do we know about achieving these changes, and what more do we need to know? What resources will be needed? How far have they already been developed? How can mathematics teachers be enabled to handle this challenge which, scandalously, is new to most of them? These are the overall questions addressed. The lessons from past experience on the challenges of large-scale of implementation of profound changes, such as teaching modelling in school mathematics, are discussed. Though there are major obstacles still to overcome, the situation is encouraging.
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The paper presents and analyses experiences from developing and running an inservice course in project work and mathematical modelling for mathematics teachers in the Danish gymnasium, e.g. upper secondary level, grade 10~12. The course objective is to support the teachers to develop, try out in their own classes, evaluate and report a project based problem oriented course in mathematical modelling. The in-service course runs over one semester and includes three seminars of 3, 1 and 2 days. Experiences show that the course objectives in general are fulfilled and that the course projects are reported in manners suitable for internet publication for colleagues. The reports and the related discussions reveal interesting dilemmas concerning the teaching of mathematical modelling and how to cope with these through «setting the scene» for the students modelling projects and through dialogues supporting and challenging the students during their work. This is illustrated and analysed on the basis of two course projects.
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The paper deals with the gap between the relevance of applications and modelling in didactical discussions and its minor importance in everyday mathematics teaching. Results of our own empirical studies that describe mathematical beliefs of teachers and students as being central obstacles are presented. Further, the studies demonstrate the possibility to change these beliefs as well as ways to promote modelling competencies.
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This chapter argues that while there have been some notable achievements incorporating aspects of applications and modelling into educational programmes, this is no cause for complacency. In addition to making current knowledge and initiatives available to a wider spectrum of the educational community, future advancement requires that new questions be posed, existing conceptualisations deepened, and dilemmas identified and addressed. The chapter samples a selection of issues, some ongoing, some emerging that are illustrative of these challenges.
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In this chapter, we present part results of an empirical study on tensions in discourses manifested by teachers when they implemented mathematical modelling in the pedagogic practices. The focus is on analysing the tension of students’ mathematical performance. Using Bernstein’s theoretical frame, we followed three teachers from the lower secondary school level from Brazilian public schools. These teachers were videotaped during their modelling-based lessons. The nature of the research analysis is qualitative. The procedures used for collecting data were observations accomplished through recordings of lessons, interviews after each lesson and teachers’ narratives on their lessons. The results have shown that the tension of students’ mathematical performance is related to what and how to teach mathematical content in the modelling environment, when students do not have a mathematical performance to solve problems from daily life situations.
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The historical analysis of mathematics teaching at secondary level shows the succession in time of different school paradigms. The present paper describes and tries to analyse a new didactic paradigm, still at an early age, the paradigm “of questioning the world”, which relies heavily on four interrelated concepts, that of inquiry and of being “Herbartian”, “procognitive”, and “exoteric”. It is the author’s ambition to show, however succinctly, how the present crisis in mathematics education could hopefully be solved along these lines, which preclude recurring to strategies seeking only to patch up the old, still dominant paradigm “of visiting works”.
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Mathematics is becoming ever more important to the engineer. Rather than being given a `toolkit¿ of techniques, students should be taught the basic concepts of mathematics and given confidence in their use
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Online mathematics teacher education is characterized as an emergent area of research in mathematics education. We identify some key topics that require further research: communities and networks of teachers in online environments; sustainability of these communities and kinds of organizational structures; knowledge-building practices in technology-mediated work group interactions; and online interactions among teachers. The emergence of new research issues also gives rise to new theoretical approaches or the adaptation of existing theoretical perspectives that are presented in this special issue. We summarize some of these theoretical perspectives and attempt to show how online environments have changed them, as well as some theoretical problems that remain to be solved.
Article
This paper describes an attempt to design, analyse and refine professional development (PD) resources that encourage the implementation of inquiry-based learning (IBL). We describe the iterative development of the resources in England with over 100 mathematics teachers from secondary, tertiary and adult education and then analyse the impact these resources had on teachers’ beliefs and practices and the issues arising. This evaluation revealed that teachers had moved away from transmission-based orientations, encouraged by the use of less structured tasks and sample lesson plans, but some found it difficult to adopt IBL pedagogies. The most significant issues for teachers may be summarised as: confusing IBL with ‘discovery’ learning; developing and managing collaborative cultures within the classroom; and planning lessons that adapt to the emerging needs of learners.
Article
In this paper, we analyse the conditions and constraints which might favour, or on the contrary hinder, a large-scale implementation of inquiry-based mathematics and science education, on the basis of our work within the PRIMAS project in 12 European countries. As a complement to the approach through the analysis of teachers’ beliefs and practices (see Engeln et al. in ZDM Int J Math Educ 45(6), this issue, 2013), we tackle this issue from a systemic institutional perspective. Indeed, in our approach, we consider teachers as actors of institutions, representing some disciplines, embedded in a school system, sharing some common pedagogical issues, in relation to society. Our sources of information are easily accessible public documents. With a theoretical background from Chevallard’s anthropological theory of didactics, we organized our analysis according to four levels of institutional organization that co-determine both content and didactical aspects in the teaching of mathematics and sciences: society, school, pedagogy and disciplinary. Our approach is systemic in the sense that we do not focus on teachers as individuals, nor on the curricula, the organization of teachers’ training or the textbooks themselves. Rather, we trace the way the conditions and constraints are operative, provide the main results of our analysis and draw out a few perspectives according to our four levels of didactical determination. Finally, in the conclusion, we reflect on the limits and potential of our analysis.
Article
In this article a review of the literature on the theme of mathematics teachers’ work and interactions with resources is provided, taking a particular perspective, the so-called ‘collective perspective’ on resources, their use and transformation. The review is presented under three headings: (1) theoretical frameworks commonly used in this area of research; (2) teachers' interactions with resources in terms of their design and use; and (3) teachers' interactions with resources in terms of teacher learning and professional development. From the literature, and the collection of articles in this issue, we argue that the collective dimensions play an important role in mathematics teachers’ work with resources and in their professional learning/development. How teachers may work in collectives and with resources, and in which ways ‘productive’ collectives may form and work together; which roles particular resources can play in these delicate constellations and how particular resources may support teachers in their work and learning; and which kinds of resources offer opportunities for community building; is likely to need further empirical investigations.
Article
LEMA was an international project which aimed at designing a professional development course for modelling. Materials for professional development which were to be used in different national contexts were designed, piloted and evaluated. In this chapter, we present the overall framework of the project and its evaluation by outlining the theoretical background, the design of the professional development course, the design of the evaluation, and summative results. In brief, the summative results of the evaluation showed that the professional development course had no effect on teachers’ beliefs but a strong positive effect on their pedagogical content knowledge and self-efficacy in terms of modelling, as well as a high degree of satisfaction among participants regarding their professional development.
Article
Modelling is not only written into educational standards throughout Germany; other European countries also demand the integration of reality-based, problem-solving tasks into school mathematics. In reality, however, things look quite different: in many places mathematics lessons are still dominated by exercises in simple calculation. So why? What is stopping teachers from introducing modelling? In order to explore this issue in depth, an empirical study was conducted. A 55-item questionnaire to investigate teachers’ perspectives on mathematical modelling in classrooms was developed and refined. A sample of 101 teachers, 52 of whom had undergone a training programme in modelling, completed the questionnaire. Major factors perceived by teachers as hindrances to using modelling were lack of time, assessment of performance and lack of materials.
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Research based on models and modelling perspectives (MMP) has shown that, in order for mathematical concepts and abilities to be useful beyond school, new levels and types of understandings are needed beyond those that have been emphasized in even the most innovative and future-oriented statements of curriculum standards. Similarly, teacher-level knowledge and abilities consists of a great deal more than the kind of beliefs, dispositions, and pedagogical content knowledge that have been emphasized in most past research on teacher development.
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This chapter deals with empirical findings on the teaching and learning of mathematical modelling, with a focus on grades 8–10, that is, 14–16-year-old students. The emphasis lies on the actual behaviour of students and teachers in learning environments with modelling tasks. Most examples in this chapter are taken from our own empirical investigations in the context of the project DISUM. In the first section, the terms used in this chapter are recollected from a cognitive point of view by means of examples, and reasons are summarised why modelling is an important and also demanding activity for students and teachers. In the second section, examples are given of students’ difficulties when solving modelling tasks, and some important findings concerning students dealing with modelling tasks are presented. The third section concentrates on teachers; examples of successful interventions are given, as well as some findings concerning teachers treating modelling examples in the classroom. In the fourth section, some implications for teaching modelling are summarised, and some encouraging (though not yet fully satisfying) results on the advancement of modelling competency are presented.
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This paper begins by describing teachers’ knowledge as the creation and development of increasingly sophisticated models or ways of interpreting the tasks of teaching. One study illuminates several ways that pre-service teachers perceive the processes of modelling and the limits of their experiences with stochastic models. Results from a second study indicate that teachers need to have a broad and deep understanding of the diversity of approaches that students might take with modeling tasks. The second study also suggests a reversal in the usual roles of teachers and students by engaging students as evaluators of models.
Mathematical modelling in teacher education-experiences from a modelling seminar
  • R Borromeo Ferri
  • W Blum
Borromeo Ferri, R., & Blum, W. (2010). Mathematical modelling in teacher education-experiences from a modelling seminar. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 2046-2055). Lyon: Institut National de Recherche Pédagogique.
Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers
  • M Blomhøj
  • T Kjeldsen
Blomhøj, M., & Kjeldsen, T. (2006). Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers. ZDM, 38(1), 163-177.
Anwendungsbezüge im Mathematikunterricht-Trends und Perspektiven
  • W Blum
Blum, W. (1996). Anwendungsbezüge im Mathematikunterricht-Trends und Perspektiven. In G. Kadunz, H. Kautschitsch, G. Ossimitz & E. Schneider (Eds.), Trends und Perspektiven (pp. 15-38). Wien: Hölder-Pichler-Tempsky.
Mathematical modeling in school education: Mathematical, cognitive, curricular, instructional and teacher education perspectives
  • J Cai
  • M Cirillo
  • J A Pelesko
  • B Ferri
  • R Borba
  • M Geiger
  • V Stillman
  • G English
  • L D Wake
  • G Kaiser
  • G Kwon
Cai, J., Cirillo, M., Pelesko, J. A., Ferri, B., Borba, R., Geiger, M., Stillman, V., English, G., Wake, L. D., Kaiser, G., G., & Kwon, O. N. (2014). Mathematical modeling in school education: Mathematical, cognitive, curricular, instructional and teacher education perspectives. In P. Liljedahl, C. Nicol, S. Oesterle & D. Allan (Eds.), Proceedings of the joint meeting of PME 38 and PME-NA 36, PME-NA, Vancouver (pp. 145-172).
Modeling students' mathematical modeling competencies
  • G Kaiser
  • B Schwarz
  • S Tiedemann
Kaiser, G., Schwarz, B., & Tiedemann, S. (2010). Future teachers' professional knowledge on modeling. In R. Lesh, P. L. Galbraith, C. R. Haines & A. Hurford (Eds.), Modeling students' mathematical modeling competencies, ICTMA 13 (pp. 433-444). New York: Springer.
Re-sourcing teacher work and interaction: New perspectives on resource design, use and teacher collaboration
  • B Pepin
  • G Gueudet
  • L Trouche
Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teacher work and interaction: New perspectives on resource design, use and teacher collaboration. ZDM, 45(7), 929-943.
Fitting models to data. The mathematising step in the modelling process
  • L Serrano
  • M Bosch
  • J Gascón
Serrano, L., Bosch, M., & Gascón, J. (2010). Fitting models to data. The mathematising step in the modelling process. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME 6 (pp. 2185-2196). Lyon: INRP.