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Most national curricula for both primary and secondary grades encourage the active involvement of learners through the manipulation
of materials (either concrete models or dynamic instruments). This trend is rooted in the emphasis given, at the dawn of ICMI,
to what might be called an experimental approach: the links between mathematics, natural sc...

## Context in source publication

**Context 1**

... Reuleaux incorporated mathematics into design and invention of machines in his work Kinematics of machinery. For mathematicians, he is best known for the Reuleaux triangle, which is one of the curves of constant width (see Fig. 4.) This curved triangle can be seen in some gothic windows; it also appears in some drawings of Leonardo da Vinci (1452-1519) and Leonhard Euler (1707-1783), but Reuleaux in his Kinematics gave the first applications and complete analysis of such triangles, and he also noticed that similar constant-width curves could be generated from ...

## Citations

... On the other hand, the European tradition of the so-called experimental approach, and the important recent contributions from these perspectives in research in mathematics education, have found a foothold in most national policies, which encourage teaching strategies in which students are actively involved through the use of manipulatives and tools (e.g. Bartolini Bussi et al., 2010). In Italian national policies, for instance, we could find it in the references to the Mathematical Laboratory, elaborated in the intended curriculum, both in the programmatic document Materiali UMI-CIIM Mathematica 2003, and in the institutional documents National Guidelines (MIUR, 2012) and National Guidelines and New Scenarios (MIUR, 2018). ...

Despite both the research in cognitive psychology and neuroscience, and in mathematics education has increasingly highlighted the relevance of active, bodily experience in mathematics learning, often school teaching seems to be far removed from these perspectives, still largely transmissive. The research presented in the paper is part of an ongoing study, carried out in two different cultural contexts (Italy and Australia), aims to explore the reasons for that distance, by investigating the perspective of primary and secondary school teachers on these teaching proposals and their implementation in the classroom. In addition to the direct involvement of teachers, the designed research includes interviews with experts in the field of mathematics education. After presenting the investigative perspective on the implementation of active, bodily experience learning activities proposed in the research project, an explanatory example will illustrate how the analysis of experts' interviews could contribute to identifying determinants of implementation.

... With the advent of the digital era, the new trends in modeling often moved from the physical world to the virtual space, and computers became a new sandbox to recreate either physical or mathematical objects. We can see this transition from physical models to digital ones across all technical and scientific disciplines, as well as in the educational context (see [11] for an account of the transition from analogue to digital devices in the classroom). ...

Our goal is to discuss the different issues that arise when attempting to visualize a joints-and-bars cube through GeoGebra, a widespread program that combines dynamic geometry (DGS) and computer algebra systems (CAS). As is standard in the DGS framework, the performance of the graphic model (i.e., the positions of the other vertices when dragging a given one) must correspond to a mathematically rigorous, symbolic computation-driven output. This requirement poses both computational algebraic geometry and dynamic geometry programming challenges that will be described, together with the corresponding proposed solutions. Among these, we include a complete determination of the dimension of the cubic linkage from an algebraic perspective, and introduce advanced 3D visualizations of this structure by using the GeoGebra software.

... We consequently proposed different tangible tools in the LA.M.PO tasks. Though the use of tangible models to introduce mathematical concepts may be natural in Engineering curricula (Bartolini Bussi et al. 2010), we realize it is rare, in Italy at least. In this sense, the work of Emma Castelnuovo (1972Castelnuovo ( , 2008 is an everlasting font of inspiration: updating her ideas to our context is a challenge for our lab. ...

... In our opinion, in both tasks presented, the most incisive part is the dynamic nature of the models: in Lab 3, composing and decomposing the vaults by means of the wedges and the acetate sheets and, in Lab 5, the construction and use of the parabola drawer. The 3D-printed models of vaults and conic sections are not so effective as objects themselves, but they become very interesting as far as Tinkercad (or possibly other programs) can be used in order to construct them (Baki et al. 2011;Bartolini Bussi et al. 2010): virtual tools are particularly effective as they have an inherent dynamic feature. ...

In the article, we present the tasks of the Maths lab at Politecnico di Torino that we tested for the first time in the academic year 2018–19, which were proposed for undergraduate Engineering students in their first year with the aim to integrate active learning into our traditional courses. The student activity in response to the tasks took place in classes of at most 25 individuals. The work was organized in small groups, mainly following a problem-solving approach and a central role was given to artefacts and technological applications. Here, we discuss the main motivations of our lab project, two specific tasks and some data collected by means of a questionnaire, in order to examine the interplay between tangible and virtual tools.

... Se espera que estos puedan formular, emplear e interpretar las matemáticas en distintos contextos, usarlas para describir, explicar y predecir fenómenos, y reconocer el papel que desempeñan en el mundo (OECD, 2016). A su vez, en las últimas décadas, diversas perspectivas teóricas han abordado la enseñanza de las matemáticas a través de problemas del mundo real; ejemplo de estas son la educación matemática realista (Bray y Tangney, 2016), la enseñanza mediante la resolución de problemas (Jurdak, 2016), la modelación en matemáticas (Stillman, Blum y Biembengut, 2015), la semiótica social (DeJarnette y González, 2016) y la construcción social e histórica del conocimiento matemático (Espinoza, Vergara y Valenzuela, 2017;Bartolini-Bussi, Taimina e Isoda, 2010;Cantoral, 2013;Montiel y Jácome, 2014). ...

... De modo similar, tanto Montiel y Jácome (2014), como DeJarnette y González (2016), plantean la necesidad de que en la escuela, más que incluir problemas matemáticos adaptados a situaciones reales, se trabaje con problemas del mundo real. Al respecto, tanto Cantoral (2013) como Bartolini-Bussi et al. (2010) articulan perspectivas histórico-culturales con la resolución de problemas del mundo real. En la presente investigación realizamos esta articulación para el caso específico del uso el teorema del ángulo inscrito en el estudio geométrico de la percepción visual (figura 1). ...

La contextualización en matemáticas adquiere relevancia debido al interés actual en que los estudiantes puedan usar lo que aprenden en la escuela para explicar fenómenos de la realidad. Por ello, el propósito de esta investigación es caracterizar las dificultades que surgen cuando estudiantes de secundaria abordan un problema del mundo real en el ámbito de la percepción visual. El método empleado corresponde a una ingeniería didáctica, mediante la cual se confrontan una indagación histórico-epistemológica de la Óptica de Euclides y las respuestas de los estudiantes, al abordar un problema diseñado sobre la base de dicha indagación. Los resultados revelan dificultades tanto en la tendencia que manifiestan los estudiantes a justificar sus respuestas desde el ámbito del fenómeno estudiado, como en las restricciones que genera el tratamiento escolar del teorema del ángulo inscrito.

... 9.1, 9.2 and 9.3. For example, Isoda (2008) developed e-textbooks 19 for Fig. 9.1 and engaged in lesson study to demonstrate their significance in mathematics education (see Bartolini Bussi et al. 2010). The objectives of lesson study with e-textbooks are firstly to demonstrate the close relationship between elementary geometry and mechanisms and recognize that elementary geometry provides the intuition for reasoning about mechanisms, and secondly, to recognize the difference between the mathematical systems for the solutions for the loci of the mechanism in Fig. 9.1 using elementary geometry, analytic geometry and vectors, and so on. ...

This open access book provides an overview of Felix Klein’s ideas, highlighting developments in university teaching and school mathematics related to Klein’s thoughts, stemming from the last century. It discusses the meaning, importance and the legacy of Klein’s ideas today and in the future, within an international, global context. Presenting extended versions of the talks at the Thematic Afternoon at ICME-13, the book shows that many of Klein’s ideas can be reinterpreted in the context of the current situation, and offers tips and advice for dealing with current problems in teacher education and teaching mathematics in secondary schools. It proves that old ideas are timeless, but that it takes competent, committed and assertive individuals to bring these ideas to life.
Throughout his professional life, Felix Klein emphasised the importance of reflecting upon mathematics teaching and learning from both a mathematical and a psychological or educational point of view. He also strongly promoted the modernisation of mathematics in the classroom, and developed ideas on university lectures for student teachers, which he later consolidated at the beginning of the last century in the three books on elementary mathematics from a higher standpoint.

... Примерно с 2003 года сотрудники Корнелльского университета и других универ-ситетов проводят обширную научно-исследовательскую работу по изучению, проек-тированию исторических механизмов, использованию исторических механических и математических коллекций в современном учебном процессе. Отметим работы [13]. Сотрудники СПбГУ также включились в подобную научно-иссле-довательскую работу [14][15][16]. ...

... Academics and educators have advocated for centuries for students to learn educational concepts while connecting to real-world concepts (Clairaut 1741(Clairaut /2006Bartolini-Bussi et al. 2010). Context-aware ulearning can be used to have students The learner's situation or the situation of the real-world environment in which the learner in location can be sensed, implying that the system is able to conduct the learning activities in the real world… context-aware ulearning can actively provide supports and hints to the learners in the right way, in the right place, and at the right time, based on the environmental contexts in the real world. ...

... Centuries ago, early scholars, (viz., Comenius 1657/1986) advocated for students to learn mathematics by connecting to real-world artifacts (environments and objects). Many since (viz., Bartolini-Bussi et al. 2010;Gainsburg 2008;Hiebert and Carpenter 1992;NCTM 2000;National Research Council 1990) have echoed that necessity to connect to the real world. Context-aware ulearning is a methodology to connect students with portable technological supports while connecting mathematical concepts to physical manifestations of those concepts in the real world. ...

The use of mobile learning has provided new pedagogical approaches to teaching geometry as a result of the technological affordances provided. One of the key affordances of mobile learning is the portability of the devices. This has untethered the learner from a particular environment to learn wherever and whenever the learner chooses. This chapter describes a subcategory of mobile learning called context-aware ubiquitous learning (context-aware ulearning) where learning happens in a real-world environment while using mobile devices to interact with that setting. This chapter delineates this subcategory and how this type of learning can be dichotomized into sensory and ambient context-aware ulearning. An argument is made that context-aware ulearning is a useful pedagogical approach for learning geometry.

... This epistemological problem was pointed out by Klein (Torretti, 1978). Klein invited students to create 3D models of geometrical problems (Bussi, Taimina & Isoda, 2010). The idea of combining the mathematical intuition and visualization is graphically presented in form of Klein's bottle. ...

... In several cases, geometry has been addressed to explore copies of historical artefacts such as instruments to draw parabolas (Bartolini Bussi 2010) at secondary level; instruments for perspective drawing at primary level (Maschietto & Bartolini Bussi 2009). These artefacts are part of the history of mathematics education in different countries (Bartolini Bussi, Taimina & Isoda 2010). ...

This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.

... The invitation to participate in that exhibition, extended to Italian mathematicians by the historian of mathematics Gino Loria in the pages of his Bollettino di 9 See the booklet by Del Re with the title Programma del corso e programma di esame per l'anno scolastico 1906-1907, which in Appendix includes the list of geometric models built by the students of the school of descriptive geometry of the University of Naples from 1901 to 1906. Only one of these models has survived; see F. Palladino 1992, tav bibliografia e storia delle scienze matematiche 15 , was ignored. This situation seems to be connected to the fact that models were mainly used for educational purposes, at least according to the data known at present, and it was thus more convenient to purchase ready-made collections from abroad. ...

Up to the present, the research on the collections of models in Italy has been limited to examining and cataloguing the collections existing at various universities, with a great deal of work being carried out mostly by Franco and Nicla Palladino, but an in-depth study of their use in university and pre-university teaching does not yet exist. In my paper I will deal with this problem, considering the period running from the mid-nineteenth century to the early decades of the twentieth century. In particular I will focus on the following points: the lack of interest in de-signing and making models for teaching at the university level at the turn of the twentieth century in Italy, and its causes; the one exception to this, i.e. , Beltrami’s cardboard model of the pseudospherical surface; models in pre-university schools in the 1800s; Corrado Segre and the use of models at the University of Turin; the role of models in the ”laboratory school” (Vailati, Marcolongo, Montessori, Emma Castelnuovo); the reconstruction of university collections of models in the 1950s. Finally I will try to draw some conclusions from this first historical analysis of the question.